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List of Theorems
RefDescription
a1ii 1 (_Note_: This inference r...
idi 2 This inference, which requ...
mp2 9 A double modus ponens infe...
mp2b 10 A double modus ponens infe...
a1i 11 Inference introducing an a...
2a1i 12 Inference introducing two ...
mp1i 13 Inference detaching an ant...
a2i 14 Inference distributing an ...
mpd 15 A modus ponens deduction. ...
imim2i 16 Inference adding common an...
syl 17 An inference version of th...
3syl 18 Inference chaining two syl...
4syl 19 Inference chaining three s...
mpi 20 A nested modus ponens infe...
mpisyl 21 A syllogism combined with ...
id 22 Principle of identity. Th...
idALT 23 Alternate proof of ~ id . ...
idd 24 Principle of identity ~ id...
a1d 25 Deduction introducing an e...
2a1d 26 Deduction introducing two ...
a1i13 27 Add two antecedents to a w...
2a1 28 A double form of ~ ax-1 . ...
a2d 29 Deduction distributing an ...
sylcom 30 Syllogism inference with c...
syl5com 31 Syllogism inference with c...
com12 32 Inference that swaps (comm...
syl5 33 A syllogism rule of infere...
syl6 34 A syllogism rule of infere...
syl56 35 Combine ~ syl5 and ~ syl6 ...
syl6com 36 Syllogism inference with c...
mpcom 37 Modus ponens inference wit...
syli 38 Syllogism inference with c...
syl2im 39 Replace two antecedents. ...
syl2imc 40 A commuted version of ~ sy...
pm2.27 41 This theorem, called "Asse...
mpdd 42 A nested modus ponens dedu...
mpid 43 A nested modus ponens dedu...
mpdi 44 A nested modus ponens dedu...
mpii 45 A doubly nested modus pone...
syld 46 Syllogism deduction. Dedu...
mp2d 47 A double modus ponens dedu...
a1dd 48 Double deduction introduci...
2a1dd 49 Double deduction introduci...
pm2.43i 50 Inference absorbing redund...
pm2.43d 51 Deduction absorbing redund...
pm2.43a 52 Inference absorbing redund...
pm2.43b 53 Inference absorbing redund...
pm2.43 54 Absorption of redundant an...
imim2d 55 Deduction adding nested an...
imim2 56 A closed form of syllogism...
embantd 57 Deduction embedding an ant...
3syld 58 Triple syllogism deduction...
sylsyld 59 A double syllogism inferen...
imim12i 60 Inference joining two impl...
imim1i 61 Inference adding common co...
imim3i 62 Inference adding three nes...
sylc 63 A syllogism inference comb...
syl3c 64 A syllogism inference comb...
syl6mpi 65 A syllogism inference. (C...
mpsyl 66 Modus ponens combined with...
mpsylsyld 67 Modus ponens combined with...
syl6c 68 Inference combining ~ syl6...
syl6ci 69 A syllogism inference comb...
syldd 70 Nested syllogism deduction...
syl5d 71 A nested syllogism deducti...
syl7 72 A syllogism rule of infere...
syl6d 73 A nested syllogism deducti...
syl8 74 A syllogism rule of infere...
syl9 75 A nested syllogism inferen...
syl9r 76 A nested syllogism inferen...
syl10 77 A nested syllogism inferen...
a1ddd 78 Triple deduction introduci...
imim12d 79 Deduction combining antece...
imim1d 80 Deduction adding nested co...
imim1 81 A closed form of syllogism...
pm2.83 82 Theorem *2.83 of [Whitehea...
peirceroll 83 Over minimal implicational...
com23 84 Commutation of antecedents...
com3r 85 Commutation of antecedents...
com13 86 Commutation of antecedents...
com3l 87 Commutation of antecedents...
pm2.04 88 Swap antecedents. Theorem...
com34 89 Commutation of antecedents...
com4l 90 Commutation of antecedents...
com4t 91 Commutation of antecedents...
com4r 92 Commutation of antecedents...
com24 93 Commutation of antecedents...
com14 94 Commutation of antecedents...
com45 95 Commutation of antecedents...
com35 96 Commutation of antecedents...
com25 97 Commutation of antecedents...
com5l 98 Commutation of antecedents...
com15 99 Commutation of antecedents...
com52l 100 Commutation of antecedents...
com52r 101 Commutation of antecedents...
com5r 102 Commutation of antecedents...
imim12 103 Closed form of ~ imim12i a...
jarr 104 Elimination of a nested an...
pm2.86d 105 Deduction associated with ...
pm2.86 106 Converse of axiom ~ ax-2 ....
pm2.86i 107 Inference associated with ...
pm2.86iALT 108 Alternate proof of ~ pm2.8...
loolin 109 The Linearity Axiom of the...
loowoz 110 An alternate for the Linea...
con4 111 Alias for ~ ax-3 to be use...
con4i 112 Inference associated with ...
con4d 113 Deduction associated with ...
mt4 114 The rule of modus tollens....
pm2.21i 115 A contradiction implies an...
pm2.24ii 116 A contradiction implies an...
pm2.21d 117 A contradiction implies an...
pm2.21ddALT 118 Alternate proof of ~ pm2.2...
pm2.21 119 From a wff and its negatio...
pm2.24 120 Theorem *2.24 of [Whitehea...
pm2.18 121 Proof by contradiction. T...
pm2.18i 122 Inference associated with ...
pm2.18d 123 Deduction based on reducti...
notnotr 124 Double negation eliminatio...
notnotri 125 Inference associated with ...
notnotriOLD 126 Obsolete proof of ~ notnot...
notnotrd 127 Deduction associated with ...
con2d 128 A contraposition deduction...
con2 129 Contraposition. Theorem *...
mt2d 130 Modus tollens deduction. ...
mt2i 131 Modus tollens inference. ...
nsyl3 132 A negated syllogism infere...
con2i 133 A contraposition inference...
nsyl 134 A negated syllogism infere...
notnot 135 Double negation introducti...
notnoti 136 Inference associated with ...
notnotd 137 Deduction associated with ...
con1d 138 A contraposition deduction...
mt3d 139 Modus tollens deduction. ...
mt3i 140 Modus tollens inference. ...
nsyl2 141 A negated syllogism infere...
con1 142 Contraposition. Theorem *...
con1i 143 A contraposition inference...
con4iOLD 144 Obsolete proof of ~ con4i ...
pm2.24i 145 Inference associated with ...
pm2.24d 146 Deduction form of ~ pm2.24...
con3d 147 A contraposition deduction...
con3 148 Contraposition. Theorem *...
con3i 149 A contraposition inference...
con3rr3 150 Rotate through consequent ...
mt4d 151 Modus tollens deduction. ...
mt4i 152 Modus tollens inference. ...
nsyld 153 A negated syllogism deduct...
nsyli 154 A negated syllogism infere...
nsyl4 155 A negated syllogism infere...
pm3.2im 156 Theorem *3.2 of [Whitehead...
mth8 157 Theorem 8 of [Margaris] p....
jc 158 Deduction joining the cons...
impi 159 An importation inference. ...
expi 160 An exportation inference. ...
simprim 161 Simplification. Similar t...
simplim 162 Simplification. Similar t...
pm2.5 163 Theorem *2.5 of [Whitehead...
pm2.51 164 Theorem *2.51 of [Whitehea...
pm2.521 165 Theorem *2.521 of [Whitehe...
pm2.52 166 Theorem *2.52 of [Whitehea...
expt 167 Exportation theorem expres...
impt 168 Importation theorem expres...
pm2.61d 169 Deduction eliminating an a...
pm2.61d1 170 Inference eliminating an a...
pm2.61d2 171 Inference eliminating an a...
ja 172 Inference joining the ante...
jad 173 Deduction form of ~ ja . ...
jarl 174 Elimination of a nested an...
pm2.61i 175 Inference eliminating an a...
pm2.61ii 176 Inference eliminating two ...
pm2.61nii 177 Inference eliminating two ...
pm2.61iii 178 Inference eliminating thre...
pm2.01 179 Reductio ad absurdum. The...
pm2.01d 180 Deduction based on reducti...
pm2.6 181 Theorem *2.6 of [Whitehead...
pm2.61 182 Theorem *2.61 of [Whitehea...
pm2.65 183 Theorem *2.65 of [Whitehea...
pm2.65i 184 Inference rule for proof b...
pm2.21dd 185 A contradiction implies an...
pm2.65d 186 Deduction rule for proof b...
mto 187 The rule of modus tollens....
mtod 188 Modus tollens deduction. ...
mtoi 189 Modus tollens inference. ...
mt2 190 A rule similar to modus to...
mt3 191 A rule similar to modus to...
peirce 192 Peirce's axiom. This odd-...
looinv 193 The Inversion Axiom of the...
bijust 194 Theorem used to justify de...
impbi 197 Property of the biconditio...
impbii 198 Infer an equivalence from ...
impbidd 199 Deduce an equivalence from...
impbid21d 200 Deduce an equivalence from...
impbid 201 Deduce an equivalence from...
dfbi1 202 Relate the biconditional c...
dfbi1ALT 203 Alternate proof of ~ dfbi1...
biimp 204 Property of the biconditio...
biimpi 205 Infer an implication from ...
sylbi 206 A mixed syllogism inferenc...
sylib 207 A mixed syllogism inferenc...
sylbb 208 A mixed syllogism inferenc...
biimpr 209 Property of the biconditio...
bicom1 210 Commutative law for the bi...
bicom 211 Commutative law for the bi...
bicomd 212 Commute two sides of a bic...
bicomi 213 Inference from commutative...
impbid1 214 Infer an equivalence from ...
impbid2 215 Infer an equivalence from ...
impcon4bid 216 A variation on ~ impbid wi...
biimpri 217 Infer a converse implicati...
biimpd 218 Deduce an implication from...
mpbi 219 An inference from a bicond...
mpbir 220 An inference from a bicond...
mpbid 221 A deduction from a bicondi...
mpbii 222 An inference from a nested...
sylibr 223 A mixed syllogism inferenc...
sylbir 224 A mixed syllogism inferenc...
sylbbr 225 A mixed syllogism inferenc...
sylbb1 226 A mixed syllogism inferenc...
sylbb2 227 A mixed syllogism inferenc...
sylibd 228 A syllogism deduction. (C...
sylbid 229 A syllogism deduction. (C...
mpbidi 230 A deduction from a bicondi...
syl5bi 231 A mixed syllogism inferenc...
syl5bir 232 A mixed syllogism inferenc...
syl5ib 233 A mixed syllogism inferenc...
syl5ibcom 234 A mixed syllogism inferenc...
syl5ibr 235 A mixed syllogism inferenc...
syl5ibrcom 236 A mixed syllogism inferenc...
biimprd 237 Deduce a converse implicat...
biimpcd 238 Deduce a commuted implicat...
biimprcd 239 Deduce a converse commuted...
syl6ib 240 A mixed syllogism inferenc...
syl6ibr 241 A mixed syllogism inferenc...
syl6bi 242 A mixed syllogism inferenc...
syl6bir 243 A mixed syllogism inferenc...
syl7bi 244 A mixed syllogism inferenc...
syl8ib 245 A syllogism rule of infere...
mpbird 246 A deduction from a bicondi...
mpbiri 247 An inference from a nested...
sylibrd 248 A syllogism deduction. (C...
sylbird 249 A syllogism deduction. (C...
biid 250 Principle of identity for ...
biidd 251 Principle of identity with...
pm5.1im 252 Two propositions are equiv...
2th 253 Two truths are equivalent....
2thd 254 Two truths are equivalent ...
ibi 255 Inference that converts a ...
ibir 256 Inference that converts a ...
ibd 257 Deduction that converts a ...
pm5.74 258 Distribution of implicatio...
pm5.74i 259 Distribution of implicatio...
pm5.74ri 260 Distribution of implicatio...
pm5.74d 261 Distribution of implicatio...
pm5.74rd 262 Distribution of implicatio...
bitri 263 An inference from transiti...
bitr2i 264 An inference from transiti...
bitr3i 265 An inference from transiti...
bitr4i 266 An inference from transiti...
bitrd 267 Deduction form of ~ bitri ...
bitr2d 268 Deduction form of ~ bitr2i...
bitr3d 269 Deduction form of ~ bitr3i...
bitr4d 270 Deduction form of ~ bitr4i...
syl5bb 271 A syllogism inference from...
syl5rbb 272 A syllogism inference from...
syl5bbr 273 A syllogism inference from...
syl5rbbr 274 A syllogism inference from...
syl6bb 275 A syllogism inference from...
syl6rbb 276 A syllogism inference from...
syl6bbr 277 A syllogism inference from...
syl6rbbr 278 A syllogism inference from...
3imtr3i 279 A mixed syllogism inferenc...
3imtr4i 280 A mixed syllogism inferenc...
3imtr3d 281 More general version of ~ ...
3imtr4d 282 More general version of ~ ...
3imtr3g 283 More general version of ~ ...
3imtr4g 284 More general version of ~ ...
3bitri 285 A chained inference from t...
3bitrri 286 A chained inference from t...
3bitr2i 287 A chained inference from t...
3bitr2ri 288 A chained inference from t...
3bitr3i 289 A chained inference from t...
3bitr3ri 290 A chained inference from t...
3bitr4i 291 A chained inference from t...
3bitr4ri 292 A chained inference from t...
3bitrd 293 Deduction from transitivit...
3bitrrd 294 Deduction from transitivit...
3bitr2d 295 Deduction from transitivit...
3bitr2rd 296 Deduction from transitivit...
3bitr3d 297 Deduction from transitivit...
3bitr3rd 298 Deduction from transitivit...
3bitr4d 299 Deduction from transitivit...
3bitr4rd 300 Deduction from transitivit...
3bitr3g 301 More general version of ~ ...
3bitr4g 302 More general version of ~ ...
notnotb 303 Double negation. Theorem ...
notnotdOLD 304 Obsolete proof of ~ notnot...
con34b 305 A biconditional form of co...
con4bid 306 A contraposition deduction...
notbid 307 Deduction negating both si...
notbi 308 Contraposition. Theorem *...
notbii 309 Negate both sides of a log...
con4bii 310 A contraposition inference...
mtbi 311 An inference from a bicond...
mtbir 312 An inference from a bicond...
mtbid 313 A deduction from a bicondi...
mtbird 314 A deduction from a bicondi...
mtbii 315 An inference from a bicond...
mtbiri 316 An inference from a bicond...
sylnib 317 A mixed syllogism inferenc...
sylnibr 318 A mixed syllogism inferenc...
sylnbi 319 A mixed syllogism inferenc...
sylnbir 320 A mixed syllogism inferenc...
xchnxbi 321 Replacement of a subexpres...
xchnxbir 322 Replacement of a subexpres...
xchbinx 323 Replacement of a subexpres...
xchbinxr 324 Replacement of a subexpres...
imbi2i 325 Introduce an antecedent to...
bibi2i 326 Inference adding a bicondi...
bibi1i 327 Inference adding a bicondi...
bibi12i 328 The equivalence of two equ...
imbi2d 329 Deduction adding an antece...
imbi1d 330 Deduction adding a consequ...
bibi2d 331 Deduction adding a bicondi...
bibi1d 332 Deduction adding a bicondi...
imbi12d 333 Deduction joining two equi...
bibi12d 334 Deduction joining two equi...
imbi12 335 Closed form of ~ imbi12i ....
imbi1 336 Theorem *4.84 of [Whitehea...
imbi2 337 Theorem *4.85 of [Whitehea...
imbi1i 338 Introduce a consequent to ...
imbi12i 339 Join two logical equivalen...
bibi1 340 Theorem *4.86 of [Whitehea...
bitr3 341 Closed nested implication ...
con2bi 342 Contraposition. Theorem *...
con2bid 343 A contraposition deduction...
con1bid 344 A contraposition deduction...
con1bii 345 A contraposition inference...
con2bii 346 A contraposition inference...
con1b 347 Contraposition. Bidirecti...
con2b 348 Contraposition. Bidirecti...
biimt 349 A wff is equivalent to its...
pm5.5 350 Theorem *5.5 of [Whitehead...
a1bi 351 Inference rule introducing...
mt2bi 352 A false consequent falsifi...
mtt 353 Modus-tollens-like theorem...
imnot 354 If a proposition is false,...
pm5.501 355 Theorem *5.501 of [Whitehe...
ibib 356 Implication in terms of im...
ibibr 357 Implication in terms of im...
tbt 358 A wff is equivalent to its...
nbn2 359 The negation of a wff is e...
bibif 360 Transfer negation via an e...
nbn 361 The negation of a wff is e...
nbn3 362 Transfer falsehood via equ...
pm5.21im 363 Two propositions are equiv...
2false 364 Two falsehoods are equival...
2falsed 365 Two falsehoods are equival...
pm5.21ni 366 Two propositions implying ...
pm5.21nii 367 Eliminate an antecedent im...
pm5.21ndd 368 Eliminate an antecedent im...
bija 369 Combine antecedents into a...
pm5.18 370 Theorem *5.18 of [Whitehea...
xor3 371 Two ways to express "exclu...
nbbn 372 Move negation outside of b...
biass 373 Associative law for the bi...
pm5.19 374 Theorem *5.19 of [Whitehea...
bi2.04 375 Logical equivalence of com...
pm5.4 376 Antecedent absorption impl...
imdi 377 Distributive law for impli...
pm5.41 378 Theorem *5.41 of [Whitehea...
pm4.8 379 Theorem *4.8 of [Whitehead...
pm4.81 380 Theorem *4.81 of [Whitehea...
imim21b 381 Simplify an implication be...
pm4.64 386 Theorem *4.64 of [Whitehea...
pm2.53 387 Theorem *2.53 of [Whitehea...
pm2.54 388 Theorem *2.54 of [Whitehea...
ori 389 Infer implication from dis...
orri 390 Infer disjunction from imp...
ord 391 Deduce implication from di...
orrd 392 Deduce disjunction from im...
jaoi 393 Inference disjoining the a...
jaod 394 Deduction disjoining the a...
mpjaod 395 Eliminate a disjunction in...
orel1 396 Elimination of disjunction...
orel2 397 Elimination of disjunction...
olc 398 Introduction of a disjunct...
orc 399 Introduction of a disjunct...
pm1.4 400 Axiom *1.4 of [WhiteheadRu...
orcom 401 Commutative law for disjun...
orcomd 402 Commutation of disjuncts i...
orcoms 403 Commutation of disjuncts i...
orci 404 Deduction introducing a di...
olci 405 Deduction introducing a di...
orcd 406 Deduction introducing a di...
olcd 407 Deduction introducing a di...
orcs 408 Deduction eliminating disj...
olcs 409 Deduction eliminating disj...
pm2.07 410 Theorem *2.07 of [Whitehea...
pm2.45 411 Theorem *2.45 of [Whitehea...
pm2.46 412 Theorem *2.46 of [Whitehea...
pm2.47 413 Theorem *2.47 of [Whitehea...
pm2.48 414 Theorem *2.48 of [Whitehea...
pm2.49 415 Theorem *2.49 of [Whitehea...
pm2.67-2 416 Slight generalization of T...
pm2.67 417 Theorem *2.67 of [Whitehea...
pm2.25 418 Theorem *2.25 of [Whitehea...
biorf 419 A wff is equivalent to its...
biortn 420 A wff is equivalent to its...
biorfi 421 A wff is equivalent to its...
biorfiOLD 422 Obsolete proof of ~ biorfi...
pm2.621 423 Theorem *2.621 of [Whitehe...
pm2.62 424 Theorem *2.62 of [Whitehea...
pm2.68 425 Theorem *2.68 of [Whitehea...
dfor2 426 Logical 'or' expressed in ...
imor 427 Implication in terms of di...
imori 428 Infer disjunction from imp...
imorri 429 Infer implication from dis...
exmid 430 Law of excluded middle, al...
exmidd 431 Law of excluded middle in ...
pm2.1 432 Theorem *2.1 of [Whitehead...
pm2.13 433 Theorem *2.13 of [Whitehea...
pm4.62 434 Theorem *4.62 of [Whitehea...
pm4.66 435 Theorem *4.66 of [Whitehea...
pm4.63 436 Theorem *4.63 of [Whitehea...
imnan 437 Express implication in ter...
imnani 438 Infer implication from neg...
iman 439 Express implication in ter...
annim 440 Express conjunction in ter...
pm4.61 441 Theorem *4.61 of [Whitehea...
pm4.65 442 Theorem *4.65 of [Whitehea...
pm4.67 443 Theorem *4.67 of [Whitehea...
imp 444 Importation inference. (C...
impcom 445 Importation inference with...
impd 446 Importation deduction. (C...
imp31 447 An importation inference. ...
imp32 448 An importation inference. ...
ex 449 Exportation inference. (T...
expcom 450 Exportation inference with...
expd 451 Exportation deduction. (C...
expdimp 452 A deduction version of exp...
expcomd 453 Deduction form of ~ expcom...
expdcom 454 Commuted form of ~ expd . ...
impancom 455 Mixed importation/commutat...
con3dimp 456 Variant of ~ con3d with im...
pm2.01da 457 Deduction based on reducti...
pm2.18da 458 Deduction based on reducti...
pm3.3 459 Theorem *3.3 (Exp) of [Whi...
pm3.31 460 Theorem *3.31 (Imp) of [Wh...
impexp 461 Import-export theorem. Pa...
pm3.2 462 Join antecedents with conj...
pm3.21 463 Join antecedents with conj...
pm3.22 464 Theorem *3.22 of [Whitehea...
ancom 465 Commutative law for conjun...
ancomd 466 Commutation of conjuncts i...
ancomst 467 Closed form of ~ ancoms . ...
ancoms 468 Inference commuting conjun...
ancomsd 469 Deduction commuting conjun...
pm3.2i 470 Infer conjunction of premi...
pm3.43i 471 Nested conjunction of ante...
simpl 472 Elimination of a conjunct....
simpli 473 Inference eliminating a co...
simpld 474 Deduction eliminating a co...
simplbi 475 Deduction eliminating a co...
simpr 476 Elimination of a conjunct....
simpri 477 Inference eliminating a co...
simprd 478 Deduction eliminating a co...
simprbi 479 Deduction eliminating a co...
adantr 480 Inference adding a conjunc...
adantl 481 Inference adding a conjunc...
adantld 482 Deduction adding a conjunc...
adantrd 483 Deduction adding a conjunc...
impel 484 An inference for implicati...
mpan9 485 Modus ponens conjoining di...
syldan 486 A syllogism deduction with...
sylan 487 A syllogism inference. (C...
sylanb 488 A syllogism inference. (C...
sylanbr 489 A syllogism inference. (C...
sylan2 490 A syllogism inference. (C...
sylan2b 491 A syllogism inference. (C...
sylan2br 492 A syllogism inference. (C...
syl2an 493 A double syllogism inferen...
syl2anr 494 A double syllogism inferen...
syl2anb 495 A double syllogism inferen...
syl2anbr 496 A double syllogism inferen...
syland 497 A syllogism deduction. (C...
sylan2d 498 A syllogism deduction. (C...
syl2and 499 A syllogism deduction. (C...
biimpa 500 Importation inference from...
biimpar 501 Importation inference from...
biimpac 502 Importation inference from...
biimparc 503 Importation inference from...
animorl 504 Conjunction implies disjun...
animorr 505 Conjunction implies disjun...
animorlr 506 Conjunction implies disjun...
animorrl 507 Conjunction implies disjun...
ianor 508 Negated conjunction in ter...
anor 509 Conjunction in terms of di...
ioran 510 Negated disjunction in ter...
pm4.52 511 Theorem *4.52 of [Whitehea...
pm4.53 512 Theorem *4.53 of [Whitehea...
pm4.54 513 Theorem *4.54 of [Whitehea...
pm4.55 514 Theorem *4.55 of [Whitehea...
pm4.56 515 Theorem *4.56 of [Whitehea...
oran 516 Disjunction in terms of co...
pm4.57 517 Theorem *4.57 of [Whitehea...
pm3.1 518 Theorem *3.1 of [Whitehead...
pm3.11 519 Theorem *3.11 of [Whitehea...
pm3.12 520 Theorem *3.12 of [Whitehea...
pm3.13 521 Theorem *3.13 of [Whitehea...
pm3.14 522 Theorem *3.14 of [Whitehea...
iba 523 Introduction of antecedent...
ibar 524 Introduction of antecedent...
biantru 525 A wff is equivalent to its...
biantrur 526 A wff is equivalent to its...
biantrud 527 A wff is equivalent to its...
biantrurd 528 A wff is equivalent to its...
mpbirand 529 Detach truth from conjunct...
jaao 530 Inference conjoining and d...
jaoa 531 Inference disjoining and c...
pm3.44 532 Theorem *3.44 of [Whitehea...
jao 533 Disjunction of antecedents...
pm1.2 534 Axiom *1.2 of [WhiteheadRu...
oridm 535 Idempotent law for disjunc...
pm4.25 536 Theorem *4.25 of [Whitehea...
orim12i 537 Disjoin antecedents and co...
orim1i 538 Introduce disjunct to both...
orim2i 539 Introduce disjunct to both...
orbi2i 540 Inference adding a left di...
orbi1i 541 Inference adding a right d...
orbi12i 542 Infer the disjunction of t...
pm1.5 543 Axiom *1.5 (Assoc) of [Whi...
or12 544 Swap two disjuncts. (Cont...
orass 545 Associative law for disjun...
pm2.31 546 Theorem *2.31 of [Whitehea...
pm2.32 547 Theorem *2.32 of [Whitehea...
or32 548 A rearrangement of disjunc...
or4 549 Rearrangement of 4 disjunc...
or42 550 Rearrangement of 4 disjunc...
orordi 551 Distribution of disjunctio...
orordir 552 Distribution of disjunctio...
jca 553 Deduce conjunction of the ...
jcad 554 Deduction conjoining the c...
jca31 555 Join three consequents. (...
jca32 556 Join three consequents. (...
jcai 557 Deduction replacing implic...
jctil 558 Inference conjoining a the...
jctir 559 Inference conjoining a the...
jccir 560 Inference conjoining a con...
jccil 561 Inference conjoining a con...
jctl 562 Inference conjoining a the...
jctr 563 Inference conjoining a the...
jctild 564 Deduction conjoining a the...
jctird 565 Deduction conjoining a the...
syl6an 566 A syllogism deduction comb...
ancl 567 Conjoin antecedent to left...
anclb 568 Conjoin antecedent to left...
pm5.42 569 Theorem *5.42 of [Whitehea...
ancr 570 Conjoin antecedent to righ...
ancrb 571 Conjoin antecedent to righ...
ancli 572 Deduction conjoining antec...
ancri 573 Deduction conjoining antec...
ancld 574 Deduction conjoining antec...
ancrd 575 Deduction conjoining antec...
anc2l 576 Conjoin antecedent to left...
anc2r 577 Conjoin antecedent to righ...
anc2li 578 Deduction conjoining antec...
anc2ri 579 Deduction conjoining antec...
pm3.41 580 Theorem *3.41 of [Whitehea...
pm3.42 581 Theorem *3.42 of [Whitehea...
pm3.4 582 Conjunction implies implic...
pm4.45im 583 Conjunction with implicati...
anim12d 584 Conjoin antecedents and co...
anim12d1 585 Variant of ~ anim12d where...
anim1d 586 Add a conjunct to right of...
anim2d 587 Add a conjunct to left of ...
anim12i 588 Conjoin antecedents and co...
anim12ci 589 Variant of ~ anim12i with ...
anim1i 590 Introduce conjunct to both...
anim2i 591 Introduce conjunct to both...
anim12ii 592 Conjoin antecedents and co...
prth 593 Conjoin antecedents and co...
pm2.3 594 Theorem *2.3 of [Whitehead...
pm2.41 595 Theorem *2.41 of [Whitehea...
pm2.42 596 Theorem *2.42 of [Whitehea...
pm2.4 597 Theorem *2.4 of [Whitehead...
pm2.65da 598 Deduction rule for proof b...
pm4.44 599 Theorem *4.44 of [Whitehea...
pm4.14 600 Theorem *4.14 of [Whitehea...
pm3.37 601 Theorem *3.37 (Transp) of ...
nan 602 Theorem to move a conjunct...
pm4.15 603 Theorem *4.15 of [Whitehea...
pm4.78 604 Implication distributes ov...
pm4.79 605 Theorem *4.79 of [Whitehea...
pm4.87 606 Theorem *4.87 of [Whitehea...
pm3.33 607 Theorem *3.33 (Syll) of [W...
pm3.34 608 Theorem *3.34 (Syll) of [W...
pm3.35 609 Conjunctive detachment. T...
pm5.31 610 Theorem *5.31 of [Whitehea...
imp4b 611 An importation inference. ...
imp4a 612 An importation inference. ...
imp4aOLD 613 Obsolete proof of ~ imp4a ...
imp4bOLD 614 Obsolete proof of ~ imp4b ...
imp4c 615 An importation inference. ...
imp4d 616 An importation inference. ...
imp41 617 An importation inference. ...
imp42 618 An importation inference. ...
imp43 619 An importation inference. ...
imp44 620 An importation inference. ...
imp45 621 An importation inference. ...
imp5a 622 An importation inference. ...
imp5d 623 An importation inference. ...
imp5g 624 An importation inference. ...
imp55 625 An importation inference. ...
imp511 626 An importation inference. ...
expimpd 627 Exportation followed by a ...
exp31 628 An exportation inference. ...
exp32 629 An exportation inference. ...
exp4b 630 An exportation inference. ...
exp4a 631 An exportation inference. ...
exp4aOLD 632 Obsolete proof of ~ exp4a ...
exp4bOLD 633 Obsolete proof of ~ exp4b ...
exp4c 634 An exportation inference. ...
exp4d 635 An exportation inference. ...
exp41 636 An exportation inference. ...
exp42 637 An exportation inference. ...
exp43 638 An exportation inference. ...
exp44 639 An exportation inference. ...
exp45 640 An exportation inference. ...
expr 641 Export a wff from a right ...
exp5c 642 An exportation inference. ...
exp5j 643 An exportation inference. ...
exp5l 644 An exportation inference. ...
exp53 645 An exportation inference. ...
expl 646 Export a wff from a left c...
impr 647 Import a wff into a right ...
impl 648 Export a wff from a left c...
impac 649 Importation with conjuncti...
exbiri 650 Inference form of ~ exbir ...
simprbda 651 Deduction eliminating a co...
simplbda 652 Deduction eliminating a co...
simplbi2 653 Deduction eliminating a co...
simplbi2comt 654 Closed form of ~ simplbi2c...
simplbi2com 655 A deduction eliminating a ...
simpl2im 656 Implication from an elimin...
simplbiim 657 Implication from an elimin...
dfbi2 658 A theorem similar to the s...
dfbi 659 Definition ~ df-bi rewritt...
pm4.71 660 Implication in terms of bi...
pm4.71r 661 Implication in terms of bi...
pm4.71i 662 Inference converting an im...
pm4.71ri 663 Inference converting an im...
pm4.71d 664 Deduction converting an im...
pm4.71rd 665 Deduction converting an im...
pm5.32 666 Distribution of implicatio...
pm5.32i 667 Distribution of implicatio...
pm5.32ri 668 Distribution of implicatio...
pm5.32d 669 Distribution of implicatio...
pm5.32rd 670 Distribution of implicatio...
pm5.32da 671 Distribution of implicatio...
biadan2 672 Add a conjunction to an eq...
pm4.24 673 Theorem *4.24 of [Whitehea...
anidm 674 Idempotent law for conjunc...
anidms 675 Inference from idempotent ...
anidmdbi 676 Conjunction idempotence wi...
anasss 677 Associative law for conjun...
anassrs 678 Associative law for conjun...
anass 679 Associative law for conjun...
sylanl1 680 A syllogism inference. (C...
sylanl2 681 A syllogism inference. (C...
sylanr1 682 A syllogism inference. (C...
sylanr2 683 A syllogism inference. (C...
sylani 684 A syllogism inference. (C...
sylan2i 685 A syllogism inference. (C...
syl2ani 686 A syllogism inference. (C...
sylan9 687 Nested syllogism inference...
sylan9r 688 Nested syllogism inference...
mtand 689 A modus tollens deduction....
mtord 690 A modus tollens deduction ...
syl2anc 691 Syllogism inference combin...
hypstkdOLD 692 Obsolete proof of ~ mpidan...
sylancl 693 Syllogism inference combin...
sylancr 694 Syllogism inference combin...
sylanbrc 695 Syllogism inference. (Con...
sylancb 696 A syllogism inference comb...
sylancbr 697 A syllogism inference comb...
sylancom 698 Syllogism inference with c...
mpdan 699 An inference based on modu...
mpancom 700 An inference based on modu...
mpidan 701 A deduction which "stacks"...
mpan 702 An inference based on modu...
mpan2 703 An inference based on modu...
mp2an 704 An inference based on modu...
mp4an 705 An inference based on modu...
mpan2d 706 A deduction based on modus...
mpand 707 A deduction based on modus...
mpani 708 An inference based on modu...
mpan2i 709 An inference based on modu...
mp2ani 710 An inference based on modu...
mp2and 711 A deduction based on modus...
mpanl1 712 An inference based on modu...
mpanl2 713 An inference based on modu...
mpanl12 714 An inference based on modu...
mpanr1 715 An inference based on modu...
mpanr2 716 An inference based on modu...
mpanr12 717 An inference based on modu...
mpanlr1 718 An inference based on modu...
pm5.74da 719 Distribution of implicatio...
pm4.45 720 Theorem *4.45 of [Whitehea...
imdistan 721 Distribution of implicatio...
imdistani 722 Distribution of implicatio...
imdistanri 723 Distribution of implicatio...
imdistand 724 Distribution of implicatio...
imdistanda 725 Distribution of implicatio...
anbi2i 726 Introduce a left conjunct ...
anbi1i 727 Introduce a right conjunct...
anbi2ci 728 Variant of ~ anbi2i with c...
anbi12i 729 Conjoin both sides of two ...
anbi12ci 730 Variant of ~ anbi12i with ...
syldanl 731 A syllogism deduction with...
sylan9bb 732 Nested syllogism inference...
sylan9bbr 733 Nested syllogism inference...
orbi2d 734 Deduction adding a left di...
orbi1d 735 Deduction adding a right d...
anbi2d 736 Deduction adding a left co...
anbi1d 737 Deduction adding a right c...
orbi1 738 Theorem *4.37 of [Whitehea...
anbi1 739 Introduce a right conjunct...
anbi2 740 Introduce a left conjunct ...
bitr 741 Theorem *4.22 of [Whitehea...
orbi12d 742 Deduction joining two equi...
anbi12d 743 Deduction joining two equi...
pm5.3 744 Theorem *5.3 of [Whitehead...
pm5.61 745 Theorem *5.61 of [Whitehea...
adantll 746 Deduction adding a conjunc...
adantlr 747 Deduction adding a conjunc...
adantrl 748 Deduction adding a conjunc...
adantrr 749 Deduction adding a conjunc...
adantlll 750 Deduction adding a conjunc...
adantllr 751 Deduction adding a conjunc...
adantlrl 752 Deduction adding a conjunc...
adantlrr 753 Deduction adding a conjunc...
adantrll 754 Deduction adding a conjunc...
adantrlr 755 Deduction adding a conjunc...
adantrrl 756 Deduction adding a conjunc...
adantrrr 757 Deduction adding a conjunc...
ad2antrr 758 Deduction adding two conju...
ad2antlr 759 Deduction adding two conju...
ad2antrl 760 Deduction adding two conju...
ad2antll 761 Deduction adding conjuncts...
ad3antrrr 762 Deduction adding three con...
ad3antlr 763 Deduction adding three con...
ad4antr 764 Deduction adding 4 conjunc...
ad4antlr 765 Deduction adding 4 conjunc...
ad5antr 766 Deduction adding 5 conjunc...
ad5antlr 767 Deduction adding 5 conjunc...
ad6antr 768 Deduction adding 6 conjunc...
ad6antlr 769 Deduction adding 6 conjunc...
ad7antr 770 Deduction adding 7 conjunc...
ad7antlr 771 Deduction adding 7 conjunc...
ad8antr 772 Deduction adding 8 conjunc...
ad8antlr 773 Deduction adding 8 conjunc...
ad9antr 774 Deduction adding 9 conjunc...
ad9antlr 775 Deduction adding 9 conjunc...
ad10antr 776 Deduction adding 10 conjun...
ad10antlr 777 Deduction adding 10 conjun...
ad2ant2l 778 Deduction adding two conju...
ad2ant2r 779 Deduction adding two conju...
ad2ant2lr 780 Deduction adding two conju...
ad2ant2rl 781 Deduction adding two conju...
adantl3r 782 Deduction adding 1 conjunc...
adantl4r 783 Deduction adding 1 conjunc...
adantl5r 784 Deduction adding 1 conjunc...
adantl6r 785 Deduction adding 1 conjunc...
simpll 786 Simplification of a conjun...
simplld 787 Deduction form of ~ simpll...
simplr 788 Simplification of a conjun...
simplrd 789 Deduction eliminating a do...
simprl 790 Simplification of a conjun...
simprld 791 Deduction eliminating a do...
simprr 792 Simplification of a conjun...
simprrd 793 Deduction form of ~ simprr...
simplll 794 Simplification of a conjun...
simpllr 795 Simplification of a conjun...
simplrl 796 Simplification of a conjun...
simplrr 797 Simplification of a conjun...
simprll 798 Simplification of a conjun...
simprlr 799 Simplification of a conjun...
simprrl 800 Simplification of a conjun...
simprrr 801 Simplification of a conjun...
simp-4l 802 Simplification of a conjun...
simp-4r 803 Simplification of a conjun...
simp-5l 804 Simplification of a conjun...
simp-5r 805 Simplification of a conjun...
simp-6l 806 Simplification of a conjun...
simp-6r 807 Simplification of a conjun...
simp-7l 808 Simplification of a conjun...
simp-7r 809 Simplification of a conjun...
simp-8l 810 Simplification of a conjun...
simp-8r 811 Simplification of a conjun...
simp-9l 812 Simplification of a conjun...
simp-9r 813 Simplification of a conjun...
simp-10l 814 Simplification of a conjun...
simp-10r 815 Simplification of a conjun...
simp-11l 816 Simplification of a conjun...
simp-11r 817 Simplification of a conjun...
jaob 818 Disjunction of antecedents...
adant423OLD 819 Obsolete as of 2-Oct-2021....
jaoian 820 Inference disjoining the a...
jao1i 821 Add a disjunct in the ante...
jaodan 822 Deduction disjoining the a...
mpjaodan 823 Eliminate a disjunction in...
pm4.77 824 Theorem *4.77 of [Whitehea...
pm2.63 825 Theorem *2.63 of [Whitehea...
pm2.64 826 Theorem *2.64 of [Whitehea...
pm2.61ian 827 Elimination of an antecede...
pm2.61dan 828 Elimination of an antecede...
pm2.61ddan 829 Elimination of two anteced...
pm2.61dda 830 Elimination of two anteced...
condan 831 Proof by contradiction. (...
abai 832 Introduce one conjunct as ...
pm5.53 833 Theorem *5.53 of [Whitehea...
an12 834 Swap two conjuncts. Note ...
an32 835 A rearrangement of conjunc...
an13 836 A rearrangement of conjunc...
an31 837 A rearrangement of conjunc...
bianass 838 An inference to merge two ...
an12s 839 Swap two conjuncts in ante...
ancom2s 840 Inference commuting a nest...
an13s 841 Swap two conjuncts in ante...
an32s 842 Swap two conjuncts in ante...
ancom1s 843 Inference commuting a nest...
an31s 844 Swap two conjuncts in ante...
anass1rs 845 Commutative-associative la...
anabs1 846 Absorption into embedded c...
anabs5 847 Absorption into embedded c...
anabs7 848 Absorption into embedded c...
a2and 849 Deduction distributing a c...
anabsan 850 Absorption of antecedent w...
anabss1 851 Absorption of antecedent i...
anabss4 852 Absorption of antecedent i...
anabss5 853 Absorption of antecedent i...
anabsi5 854 Absorption of antecedent i...
anabsi6 855 Absorption of antecedent i...
anabsi7 856 Absorption of antecedent i...
anabsi8 857 Absorption of antecedent i...
anabss7 858 Absorption of antecedent i...
anabsan2 859 Absorption of antecedent w...
anabss3 860 Absorption of antecedent i...
an4 861 Rearrangement of 4 conjunc...
an42 862 Rearrangement of 4 conjunc...
an43 863 Rearrangement of 4 conjunc...
an3 864 A rearrangement of conjunc...
an4s 865 Inference rearranging 4 co...
an42s 866 Inference rearranging 4 co...
anandi 867 Distribution of conjunctio...
anandir 868 Distribution of conjunctio...
anandis 869 Inference that undistribut...
anandirs 870 Inference that undistribut...
syl2an2 871 ~ syl2an with antecedents ...
syl2an2r 872 ~ syl2anr with antecedents...
impbida 873 Deduce an equivalence from...
pm3.48 874 Theorem *3.48 of [Whitehea...
pm3.45 875 Theorem *3.45 (Fact) of [W...
im2anan9 876 Deduction joining nested i...
im2anan9r 877 Deduction joining nested i...
anim12dan 878 Conjoin antecedents and co...
orim12d 879 Disjoin antecedents and co...
orim1d 880 Disjoin antecedents and co...
orim2d 881 Disjoin antecedents and co...
orim2 882 Axiom *1.6 (Sum) of [White...
pm2.38 883 Theorem *2.38 of [Whitehea...
pm2.36 884 Theorem *2.36 of [Whitehea...
pm2.37 885 Theorem *2.37 of [Whitehea...
pm2.73 886 Theorem *2.73 of [Whitehea...
pm2.74 887 Theorem *2.74 of [Whitehea...
orimdi 888 Disjunction distributes ov...
pm2.76 889 Theorem *2.76 of [Whitehea...
pm2.75 890 Theorem *2.75 of [Whitehea...
pm2.8 891 Theorem *2.8 of [Whitehead...
pm2.81 892 Theorem *2.81 of [Whitehea...
pm2.82 893 Theorem *2.82 of [Whitehea...
pm2.85 894 Theorem *2.85 of [Whitehea...
pm3.2ni 895 Infer negated disjunction ...
orabs 896 Absorption of redundant in...
oranabs 897 Absorb a disjunct into a c...
pm5.1 898 Two propositions are equiv...
pm5.21 899 Two propositions are equiv...
norbi 900 If neither of two proposit...
nbior 901 If two propositions are no...
pm3.43 902 Theorem *3.43 (Comp) of [W...
jcab 903 Distributive law for impli...
ordi 904 Distributive law for disju...
ordir 905 Distributive law for disju...
pm4.76 906 Theorem *4.76 of [Whitehea...
andi 907 Distributive law for conju...
andir 908 Distributive law for conju...
orddi 909 Double distributive law fo...
anddi 910 Double distributive law fo...
pm4.39 911 Theorem *4.39 of [Whitehea...
pm4.38 912 Theorem *4.38 of [Whitehea...
bi2anan9 913 Deduction joining two equi...
bi2anan9r 914 Deduction joining two equi...
bi2bian9 915 Deduction joining two bico...
pm4.72 916 Implication in terms of bi...
imimorb 917 Simplify an implication be...
pm5.33 918 Theorem *5.33 of [Whitehea...
pm5.36 919 Theorem *5.36 of [Whitehea...
bianabs 920 Absorb a hypothesis into t...
oibabs 921 Absorption of disjunction ...
pm3.24 922 Law of noncontradiction. ...
pm2.26 923 Theorem *2.26 of [Whitehea...
pm5.11 924 Theorem *5.11 of [Whitehea...
pm5.12 925 Theorem *5.12 of [Whitehea...
pm5.14 926 Theorem *5.14 of [Whitehea...
pm5.13 927 Theorem *5.13 of [Whitehea...
pm5.17 928 Theorem *5.17 of [Whitehea...
pm5.15 929 Theorem *5.15 of [Whitehea...
pm5.16 930 Theorem *5.16 of [Whitehea...
xor 931 Two ways to express "exclu...
nbi2 932 Two ways to express "exclu...
dfbi3 933 An alternate definition of...
pm5.24 934 Theorem *5.24 of [Whitehea...
xordi 935 Conjunction distributes ov...
biort 936 A wff disjoined with truth...
pm5.55 937 Theorem *5.55 of [Whitehea...
ornld 938 Selecting one statement fr...
pm5.21nd 939 Eliminate an antecedent im...
pm5.35 940 Theorem *5.35 of [Whitehea...
pm5.54 941 Theorem *5.54 of [Whitehea...
baib 942 Move conjunction outside o...
baibr 943 Move conjunction outside o...
rbaibr 944 Move conjunction outside o...
rbaib 945 Move conjunction outside o...
baibd 946 Move conjunction outside o...
rbaibd 947 Move conjunction outside o...
pm5.44 948 Theorem *5.44 of [Whitehea...
pm5.6 949 Conjunction in antecedent ...
orcanai 950 Change disjunction in cons...
intnan 951 Introduction of conjunct i...
intnanr 952 Introduction of conjunct i...
intnand 953 Introduction of conjunct i...
intnanrd 954 Introduction of conjunct i...
mpbiran 955 Detach truth from conjunct...
mpbiran2 956 Detach truth from conjunct...
mpbir2an 957 Detach a conjunction of tr...
mpbi2and 958 Detach a conjunction of tr...
mpbir2and 959 Detach a conjunction of tr...
pm5.62 960 Theorem *5.62 of [Whitehea...
pm5.63 961 Theorem *5.63 of [Whitehea...
bianfi 962 A wff conjoined with false...
bianfd 963 A wff conjoined with false...
pm4.43 964 Theorem *4.43 of [Whitehea...
pm4.82 965 Theorem *4.82 of [Whitehea...
pm4.83 966 Theorem *4.83 of [Whitehea...
pclem6 967 Negation inferred from emb...
biantr 968 A transitive law of equiva...
orbidi 969 Disjunction distributes ov...
biluk 970 Lukasiewicz's shortest axi...
pm5.7 971 Disjunction distributes ov...
bigolden 972 Dijkstra-Scholten's Golden...
pm5.71 973 Theorem *5.71 of [Whitehea...
pm5.75 974 Theorem *5.75 of [Whitehea...
pm5.75OLD 975 Obsolete proof of ~ pm5.75...
bimsc1 976 Removal of conjunct from o...
4exmid 977 The disjunction of the fou...
ecase2d 978 Deduction for elimination ...
ecase3 979 Inference for elimination ...
ecase 980 Inference for elimination ...
ecase3d 981 Deduction for elimination ...
ecased 982 Deduction for elimination ...
ecase3ad 983 Deduction for elimination ...
ccase 984 Inference for combining ca...
ccased 985 Deduction for combining ca...
ccase2 986 Inference for combining ca...
4cases 987 Inference eliminating two ...
4casesdan 988 Deduction eliminating two ...
niabn 989 Miscellaneous inference re...
consensus 990 The consensus theorem. Th...
dedlem0a 991 Lemma for an alternate ver...
dedlem0b 992 Lemma for an alternate ver...
dedlema 993 Lemma for weak deduction t...
dedlemb 994 Lemma for weak deduction t...
pm4.42 995 Theorem *4.42 of [Whitehea...
ninba 996 Miscellaneous inference re...
prlem1 997 A specialized lemma for se...
prlem2 998 A specialized lemma for se...
oplem1 999 A specialized lemma for se...
dn1 1000 A single axiom for Boolean...
bianir 1001 If a wff is equivalent to ...
jaoi2 1002 Inference removing a negat...
jaoi3 1003 Inference separating a dis...
cases 1004 Case disjunction according...
cases2 1005 Case disjunction according...
dfifp2 1008 Alternate definition of th...
dfifp3 1009 Alternate definition of th...
dfifp4 1010 Alternate definition of th...
dfifp5 1011 Alternate definition of th...
dfifp6 1012 Alternate definition of th...
dfifp7 1013 Alternate definition of th...
anifp 1014 The conditional operator i...
ifpor 1015 The conditional operator i...
ifpn 1016 Conditional operator for t...
ifptru 1017 Value of the conditional o...
ifpfal 1018 Value of the conditional o...
ifpid 1019 Value of the conditional o...
casesifp 1020 Version of ~ cases express...
ifpbi123d 1021 Equality deduction for con...
ifpimpda 1022 Separation of the values o...
1fpid3 1023 The value of the condition...
elimh 1024 Hypothesis builder for the...
dedt 1025 The weak deduction theorem...
con3ALT 1026 Proof of ~ con3 from its a...
elimhOLD 1027 Old version of ~ elimh . ...
dedtOLD 1028 Old version of ~ dedt . O...
con3OLD 1029 Old version of ~ con3ALT ....
3orass 1034 Associative law for triple...
3anass 1035 Associative law for triple...
3anrot 1036 Rotation law for triple co...
3orrot 1037 Rotation law for triple di...
3ancoma 1038 Commutation law for triple...
3orcoma 1039 Commutation law for triple...
3ancomb 1040 Commutation law for triple...
3orcomb 1041 Commutation law for triple...
3anrev 1042 Reversal law for triple co...
3anan32 1043 Convert triple conjunction...
3anan12 1044 Convert triple conjunction...
anandi3 1045 Distribution of triple con...
anandi3r 1046 Distribution of triple con...
3anor 1047 Triple conjunction express...
3ianor 1048 Negated triple conjunction...
3ioran 1049 Negated triple disjunction...
3oran 1050 Triple disjunction in term...
3simpa 1051 Simplification of triple c...
3simpb 1052 Simplification of triple c...
3simpc 1053 Simplification of triple c...
simp1 1054 Simplification of triple c...
simp2 1055 Simplification of triple c...
simp3 1056 Simplification of triple c...
simpl1 1057 Simplification rule. (Con...
simpl2 1058 Simplification rule. (Con...
simpl3 1059 Simplification rule. (Con...
simpr1 1060 Simplification rule. (Con...
simpr2 1061 Simplification rule. (Con...
simpr3 1062 Simplification rule. (Con...
simp1i 1063 Infer a conjunct from a tr...
simp2i 1064 Infer a conjunct from a tr...
simp3i 1065 Infer a conjunct from a tr...
simp1d 1066 Deduce a conjunct from a t...
simp2d 1067 Deduce a conjunct from a t...
simp3d 1068 Deduce a conjunct from a t...
simp1bi 1069 Deduce a conjunct from a t...
simp2bi 1070 Deduce a conjunct from a t...
simp3bi 1071 Deduce a conjunct from a t...
3adant1 1072 Deduction adding a conjunc...
3adant2 1073 Deduction adding a conjunc...
3adant3 1074 Deduction adding a conjunc...
3ad2ant1 1075 Deduction adding conjuncts...
3ad2ant2 1076 Deduction adding conjuncts...
3ad2ant3 1077 Deduction adding conjuncts...
simp1l 1078 Simplification of triple c...
simp1r 1079 Simplification of triple c...
simp2l 1080 Simplification of triple c...
simp2r 1081 Simplification of triple c...
simp3l 1082 Simplification of triple c...
simp3r 1083 Simplification of triple c...
simp11 1084 Simplification of doubly t...
simp12 1085 Simplification of doubly t...
simp13 1086 Simplification of doubly t...
simp21 1087 Simplification of doubly t...
simp22 1088 Simplification of doubly t...
simp23 1089 Simplification of doubly t...
simp31 1090 Simplification of doubly t...
simp32 1091 Simplification of doubly t...
simp33 1092 Simplification of doubly t...
simpll1 1093 Simplification of conjunct...
simpll2 1094 Simplification of conjunct...
simpll3 1095 Simplification of conjunct...
simplr1 1096 Simplification of conjunct...
simplr2 1097 Simplification of conjunct...
simplr3 1098 Simplification of conjunct...
simprl1 1099 Simplification of conjunct...
simprl2 1100 Simplification of conjunct...
simprl3 1101 Simplification of conjunct...
simprr1 1102 Simplification of conjunct...
simprr2 1103 Simplification of conjunct...
simprr3 1104 Simplification of conjunct...
simpl1l 1105 Simplification of conjunct...
simpl1r 1106 Simplification of conjunct...
simpl2l 1107 Simplification of conjunct...
simpl2r 1108 Simplification of conjunct...
simpl3l 1109 Simplification of conjunct...
simpl3r 1110 Simplification of conjunct...
simpr1l 1111 Simplification of conjunct...
simpr1r 1112 Simplification of conjunct...
simpr2l 1113 Simplification of conjunct...
simpr2r 1114 Simplification of conjunct...
simpr3l 1115 Simplification of conjunct...
simpr3r 1116 Simplification of conjunct...
simp1ll 1117 Simplification of conjunct...
simp1lr 1118 Simplification of conjunct...
simp1rl 1119 Simplification of conjunct...
simp1rr 1120 Simplification of conjunct...
simp2ll 1121 Simplification of conjunct...
simp2lr 1122 Simplification of conjunct...
simp2rl 1123 Simplification of conjunct...
simp2rr 1124 Simplification of conjunct...
simp3ll 1125 Simplification of conjunct...
simp3lr 1126 Simplification of conjunct...
simp3rl 1127 Simplification of conjunct...
simp3rr 1128 Simplification of conjunct...
simpl11 1129 Simplification of conjunct...
simpl12 1130 Simplification of conjunct...
simpl13 1131 Simplification of conjunct...
simpl21 1132 Simplification of conjunct...
simpl22 1133 Simplification of conjunct...
simpl23 1134 Simplification of conjunct...
simpl31 1135 Simplification of conjunct...
simpl32 1136 Simplification of conjunct...
simpl33 1137 Simplification of conjunct...
simpr11 1138 Simplification of conjunct...
simpr12 1139 Simplification of conjunct...
simpr13 1140 Simplification of conjunct...
simpr21 1141 Simplification of conjunct...
simpr22 1142 Simplification of conjunct...
simpr23 1143 Simplification of conjunct...
simpr31 1144 Simplification of conjunct...
simpr32 1145 Simplification of conjunct...
simpr33 1146 Simplification of conjunct...
simp1l1 1147 Simplification of conjunct...
simp1l2 1148 Simplification of conjunct...
simp1l3 1149 Simplification of conjunct...
simp1r1 1150 Simplification of conjunct...
simp1r2 1151 Simplification of conjunct...
simp1r3 1152 Simplification of conjunct...
simp2l1 1153 Simplification of conjunct...
simp2l2 1154 Simplification of conjunct...
simp2l3 1155 Simplification of conjunct...
simp2r1 1156 Simplification of conjunct...
simp2r2 1157 Simplification of conjunct...
simp2r3 1158 Simplification of conjunct...
simp3l1 1159 Simplification of conjunct...
simp3l2 1160 Simplification of conjunct...
simp3l3 1161 Simplification of conjunct...
simp3r1 1162 Simplification of conjunct...
simp3r2 1163 Simplification of conjunct...
simp3r3 1164 Simplification of conjunct...
simp11l 1165 Simplification of conjunct...
simp11r 1166 Simplification of conjunct...
simp12l 1167 Simplification of conjunct...
simp12r 1168 Simplification of conjunct...
simp13l 1169 Simplification of conjunct...
simp13r 1170 Simplification of conjunct...
simp21l 1171 Simplification of conjunct...
simp21r 1172 Simplification of conjunct...
simp22l 1173 Simplification of conjunct...
simp22r 1174 Simplification of conjunct...
simp23l 1175 Simplification of conjunct...
simp23r 1176 Simplification of conjunct...
simp31l 1177 Simplification of conjunct...
simp31r 1178 Simplification of conjunct...
simp32l 1179 Simplification of conjunct...
simp32r 1180 Simplification of conjunct...
simp33l 1181 Simplification of conjunct...
simp33r 1182 Simplification of conjunct...
simp111 1183 Simplification of conjunct...
simp112 1184 Simplification of conjunct...
simp113 1185 Simplification of conjunct...
simp121 1186 Simplification of conjunct...
simp122 1187 Simplification of conjunct...
simp123 1188 Simplification of conjunct...
simp131 1189 Simplification of conjunct...
simp132 1190 Simplification of conjunct...
simp133 1191 Simplification of conjunct...
simp211 1192 Simplification of conjunct...
simp212 1193 Simplification of conjunct...
simp213 1194 Simplification of conjunct...
simp221 1195 Simplification of conjunct...
simp222 1196 Simplification of conjunct...
simp223 1197 Simplification of conjunct...
simp231 1198 Simplification of conjunct...
simp232 1199 Simplification of conjunct...
simp233 1200 Simplification of conjunct...
simp311 1201 Simplification of conjunct...
simp312 1202 Simplification of conjunct...
simp313 1203 Simplification of conjunct...
simp321 1204 Simplification of conjunct...
simp322 1205 Simplification of conjunct...
simp323 1206 Simplification of conjunct...
simp331 1207 Simplification of conjunct...
simp332 1208 Simplification of conjunct...
simp333 1209 Simplification of conjunct...
3adantl1 1210 Deduction adding a conjunc...
3adantl2 1211 Deduction adding a conjunc...
3adantl3 1212 Deduction adding a conjunc...
3adantr1 1213 Deduction adding a conjunc...
3adantr2 1214 Deduction adding a conjunc...
3adantr3 1215 Deduction adding a conjunc...
3ad2antl1 1216 Deduction adding conjuncts...
3ad2antl2 1217 Deduction adding conjuncts...
3ad2antl3 1218 Deduction adding conjuncts...
3ad2antr1 1219 Deduction adding conjuncts...
3ad2antr2 1220 Deduction adding conjuncts...
3ad2antr3 1221 Deduction adding conjuncts...
3anibar 1222 Remove a hypothesis from t...
3mix1 1223 Introduction in triple dis...
3mix2 1224 Introduction in triple dis...
3mix3 1225 Introduction in triple dis...
3mix1i 1226 Introduction in triple dis...
3mix2i 1227 Introduction in triple dis...
3mix3i 1228 Introduction in triple dis...
3mix1d 1229 Deduction introducing trip...
3mix2d 1230 Deduction introducing trip...
3mix3d 1231 Deduction introducing trip...
3pm3.2i 1232 Infer conjunction of premi...
pm3.2an3 1233 Version of ~ pm3.2 for a t...
pm3.2an3OLD 1234 Obsolete proof of ~ pm3.2a...
3jca 1235 Join consequents with conj...
3jcad 1236 Deduction conjoining the c...
mpbir3an 1237 Detach a conjunction of tr...
mpbir3and 1238 Detach a conjunction of tr...
syl3anbrc 1239 Syllogism inference. (Con...
3anim123i 1240 Join antecedents and conse...
3anim1i 1241 Add two conjuncts to antec...
3anim2i 1242 Add two conjuncts to antec...
3anim3i 1243 Add two conjuncts to antec...
3anbi123i 1244 Join 3 biconditionals with...
3orbi123i 1245 Join 3 biconditionals with...
3anbi1i 1246 Inference adding two conju...
3anbi2i 1247 Inference adding two conju...
3anbi3i 1248 Inference adding two conju...
3imp 1249 Importation inference. (C...
3imp31 1250 The importation inference ...
3impa 1251 Importation from double to...
3impb 1252 Importation from double to...
3impia 1253 Importation to triple conj...
3impib 1254 Importation to triple conj...
ex3 1255 Apply ~ ex to a hypothesis...
3exp 1256 Exportation inference. (C...
3expa 1257 Exportation from triple to...
3expb 1258 Exportation from triple to...
3expia 1259 Exportation from triple co...
3expib 1260 Exportation from triple co...
3com12 1261 Commutation in antecedent....
3com13 1262 Commutation in antecedent....
3com23 1263 Commutation in antecedent....
3coml 1264 Commutation in antecedent....
3comr 1265 Commutation in antecedent....
3adant3r1 1266 Deduction adding a conjunc...
3adant3r2 1267 Deduction adding a conjunc...
3adant3r3 1268 Deduction adding a conjunc...
3imp21 1269 The importation inference ...
3imp3i2an 1270 An elimination deduction. ...
3an1rs 1271 Swap conjuncts. (Contribu...
3imp1 1272 Importation to left triple...
3impd 1273 Importation deduction for ...
3imp2 1274 Importation to right tripl...
3exp1 1275 Exportation from left trip...
3expd 1276 Exportation deduction for ...
3exp2 1277 Exportation from right tri...
exp5o 1278 A triple exportation infer...
exp516 1279 A triple exportation infer...
exp520 1280 A triple exportation infer...
3impexp 1281 Version of ~ impexp for a ...
3anassrs 1282 Associative law for conjun...
3an4anass 1283 Associative law for four c...
ad4ant13 1284 Deduction adding conjuncts...
ad4ant14 1285 Deduction adding conjuncts...
ad4ant123 1286 Deduction adding conjuncts...
ad4ant124 1287 Deduction adding conjuncts...
ad4ant134 1288 Deduction adding conjuncts...
ad4ant23 1289 Deduction adding conjuncts...
ad4ant24 1290 Deduction adding conjuncts...
ad4ant234 1291 Deduction adding conjuncts...
ad5ant12 1292 Deduction adding conjuncts...
ad5ant13 1293 Deduction adding conjuncts...
ad5ant14 1294 Deduction adding conjuncts...
ad5ant15 1295 Deduction adding conjuncts...
ad5ant23 1296 Deduction adding conjuncts...
ad5ant24 1297 Deduction adding conjuncts...
ad5ant25 1298 Deduction adding conjuncts...
ad5ant245 1299 Deduction adding conjuncts...
ad5ant234 1300 Deduction adding conjuncts...
ad5ant235 1301 Deduction adding conjuncts...
ad5ant123 1302 Deduction adding conjuncts...
ad5ant124 1303 Deduction adding conjuncts...
ad5ant125 1304 Deduction adding conjuncts...
ad5ant134 1305 Deduction adding conjuncts...
ad5ant135 1306 Deduction adding conjuncts...
ad5ant145 1307 Deduction adding conjuncts...
ad5ant1345 1308 Deduction adding conjuncts...
ad5ant2345 1309 Deduction adding conjuncts...
3adant1l 1310 Deduction adding a conjunc...
3adant1r 1311 Deduction adding a conjunc...
3adant2l 1312 Deduction adding a conjunc...
3adant2r 1313 Deduction adding a conjunc...
3adant3l 1314 Deduction adding a conjunc...
3adant3r 1315 Deduction adding a conjunc...
syl12anc 1316 Syllogism combined with co...
syl21anc 1317 Syllogism combined with co...
syl3anc 1318 Syllogism combined with co...
syl22anc 1319 Syllogism combined with co...
syl13anc 1320 Syllogism combined with co...
syl31anc 1321 Syllogism combined with co...
syl112anc 1322 Syllogism combined with co...
syl121anc 1323 Syllogism combined with co...
syl211anc 1324 Syllogism combined with co...
syl23anc 1325 Syllogism combined with co...
syl32anc 1326 Syllogism combined with co...
syl122anc 1327 Syllogism combined with co...
syl212anc 1328 Syllogism combined with co...
syl221anc 1329 Syllogism combined with co...
syl113anc 1330 Syllogism combined with co...
syl131anc 1331 Syllogism combined with co...
syl311anc 1332 Syllogism combined with co...
syl33anc 1333 Syllogism combined with co...
syl222anc 1334 Syllogism combined with co...
syl123anc 1335 Syllogism combined with co...
syl132anc 1336 Syllogism combined with co...
syl213anc 1337 Syllogism combined with co...
syl231anc 1338 Syllogism combined with co...
syl312anc 1339 Syllogism combined with co...
syl321anc 1340 Syllogism combined with co...
syl133anc 1341 Syllogism combined with co...
syl313anc 1342 Syllogism combined with co...
syl331anc 1343 Syllogism combined with co...
syl223anc 1344 Syllogism combined with co...
syl232anc 1345 Syllogism combined with co...
syl322anc 1346 Syllogism combined with co...
syl233anc 1347 Syllogism combined with co...
syl323anc 1348 Syllogism combined with co...
syl332anc 1349 Syllogism combined with co...
syl333anc 1350 A syllogism inference comb...
syl3an1 1351 A syllogism inference. (C...
syl3an2 1352 A syllogism inference. (C...
syl3an3 1353 A syllogism inference. (C...
syl3an1b 1354 A syllogism inference. (C...
syl3an2b 1355 A syllogism inference. (C...
syl3an3b 1356 A syllogism inference. (C...
syl3an1br 1357 A syllogism inference. (C...
syl3an2br 1358 A syllogism inference. (C...
syl3an3br 1359 A syllogism inference. (C...
syl3an 1360 A triple syllogism inferen...
syl3anb 1361 A triple syllogism inferen...
syl3anbr 1362 A triple syllogism inferen...
syld3an3 1363 A syllogism inference. (C...
syld3an1 1364 A syllogism inference. (C...
syld3an2 1365 A syllogism inference. (C...
syl3anl1 1366 A syllogism inference. (C...
syl3anl2 1367 A syllogism inference. (C...
syl3anl3 1368 A syllogism inference. (C...
syl3anl 1369 A triple syllogism inferen...
syl3anr1 1370 A syllogism inference. (C...
syl3anr2 1371 A syllogism inference. (C...
syl3anr3 1372 A syllogism inference. (C...
3impdi 1373 Importation inference (und...
3impdir 1374 Importation inference (und...
3anidm12 1375 Inference from idempotent ...
3anidm13 1376 Inference from idempotent ...
3anidm23 1377 Inference from idempotent ...
syl2an3an 1378 ~ syl3an with antecedents ...
syl2an23an 1379 Deduction related to ~ syl...
3ori 1380 Infer implication from tri...
3jao 1381 Disjunction of three antec...
3jaob 1382 Disjunction of three antec...
3jaoi 1383 Disjunction of three antec...
3jaod 1384 Disjunction of three antec...
3jaoian 1385 Disjunction of three antec...
3jaodan 1386 Disjunction of three antec...
mpjao3dan 1387 Eliminate a three-way disj...
3jaao 1388 Inference conjoining and d...
syl3an9b 1389 Nested syllogism inference...
3orbi123d 1390 Deduction joining 3 equiva...
3anbi123d 1391 Deduction joining 3 equiva...
3anbi12d 1392 Deduction conjoining and a...
3anbi13d 1393 Deduction conjoining and a...
3anbi23d 1394 Deduction conjoining and a...
3anbi1d 1395 Deduction adding conjuncts...
3anbi2d 1396 Deduction adding conjuncts...
3anbi3d 1397 Deduction adding conjuncts...
3anim123d 1398 Deduction joining 3 implic...
3orim123d 1399 Deduction joining 3 implic...
an6 1400 Rearrangement of 6 conjunc...
3an6 1401 Analogue of ~ an4 for trip...
3or6 1402 Analogue of ~ or4 for trip...
mp3an1 1403 An inference based on modu...
mp3an2 1404 An inference based on modu...
mp3an3 1405 An inference based on modu...
mp3an12 1406 An inference based on modu...
mp3an13 1407 An inference based on modu...
mp3an23 1408 An inference based on modu...
mp3an1i 1409 An inference based on modu...
mp3anl1 1410 An inference based on modu...
mp3anl2 1411 An inference based on modu...
mp3anl3 1412 An inference based on modu...
mp3anr1 1413 An inference based on modu...
mp3anr2 1414 An inference based on modu...
mp3anr3 1415 An inference based on modu...
mp3an 1416 An inference based on modu...
mpd3an3 1417 An inference based on modu...
mpd3an23 1418 An inference based on modu...
mp3and 1419 A deduction based on modus...
mp3an12i 1420 ~ mp3an with antecedents i...
mp3an2i 1421 ~ mp3an with antecedents i...
mp3an3an 1422 ~ mp3an with antecedents i...
mp3an2ani 1423 An elimination deduction. ...
biimp3a 1424 Infer implication from a l...
biimp3ar 1425 Infer implication from a l...
3anandis 1426 Inference that undistribut...
3anandirs 1427 Inference that undistribut...
ecase23d 1428 Deduction for elimination ...
3ecase 1429 Inference for elimination ...
3bior1fd 1430 A disjunction is equivalen...
3bior1fand 1431 A disjunction is equivalen...
3bior2fd 1432 A wff is equivalent to its...
3biant1d 1433 A conjunction is equivalen...
intn3an1d 1434 Introduction of a triple c...
intn3an2d 1435 Introduction of a triple c...
intn3an3d 1436 Introduction of a triple c...
an3andi 1437 Distribution of conjunctio...
an33rean 1438 Rearrange a 9-fold conjunc...
nanan 1441 Write 'and' in terms of 'n...
nancom 1442 The 'nand' operator commut...
nannan 1443 Lemma for handling nested ...
nanim 1444 Show equivalence between i...
nannot 1445 Show equivalence between n...
nanbi 1446 Show equivalence between t...
nanbi1 1447 Introduce a right anti-con...
nanbi2 1448 Introduce a left anti-conj...
nanbi12 1449 Join two logical equivalen...
nanbi1i 1450 Introduce a right anti-con...
nanbi2i 1451 Introduce a left anti-conj...
nanbi12i 1452 Join two logical equivalen...
nanbi1d 1453 Introduce a right anti-con...
nanbi2d 1454 Introduce a left anti-conj...
nanbi12d 1455 Join two logical equivalen...
xnor 1458 Two ways to write XNOR. (C...
xorcom 1459 The connector ` \/_ ` is c...
xorass 1460 The connector ` \/_ ` is a...
excxor 1461 This tautology shows that ...
xor2 1462 Two ways to express "exclu...
xoror 1463 XOR implies OR. (Contribut...
xornan 1464 XOR implies NAND. (Contrib...
xornan2 1465 XOR implies NAND (written ...
xorneg2 1466 The connector ` \/_ ` is n...
xorneg1 1467 The connector ` \/_ ` is n...
xorneg 1468 The connector ` \/_ ` is u...
xorbi12i 1469 Equality property for XOR....
xorbi12d 1470 Equality property for XOR....
anxordi 1471 Conjunction distributes ov...
xorexmid 1472 Exclusive-or variant of th...
trujust 1477 Soundness justification th...
tru 1479 The truth value ` T. ` is ...
fal 1482 The truth value ` F. ` is ...
dftru2 1483 An alternate definition of...
trud 1484 Eliminate ` T. ` as an ant...
tbtru 1485 A proposition is equivalen...
nbfal 1486 The negation of a proposit...
bitru 1487 A theorem is equivalent to...
bifal 1488 A contradiction is equival...
falim 1489 The truth value ` F. ` imp...
falimd 1490 The truth value ` F. ` imp...
a1tru 1491 Anything implies ` T. ` . ...
truan 1492 True can be removed from a...
dfnot 1493 Given falsum ` F. ` , we c...
inegd 1494 Negation introduction rule...
efald 1495 Deduction based on reducti...
pm2.21fal 1496 If a wff and its negation ...
truantru 1497 A ` /\ ` identity. (Contr...
truanfal 1498 A ` /\ ` identity. (Contr...
falantru 1499 A ` /\ ` identity. (Contr...
falanfal 1500 A ` /\ ` identity. (Contr...
truortru 1501 A ` \/ ` identity. (Contr...
truorfal 1502 A ` \/ ` identity. (Contr...
falortru 1503 A ` \/ ` identity. (Contr...
falorfal 1504 A ` \/ ` identity. (Contr...
truimtru 1505 A ` -> ` identity. (Contr...
truimfal 1506 A ` -> ` identity. (Contr...
falimtru 1507 A ` -> ` identity. (Contr...
falimfal 1508 A ` -> ` identity. (Contr...
nottru 1509 A ` -. ` identity. (Contr...
notfal 1510 A ` -. ` identity. (Contr...
trubitru 1511 A ` <-> ` identity. (Cont...
falbitru 1512 A ` <-> ` identity. (Cont...
trubifal 1513 A ` <-> ` identity. (Cont...
falbifal 1514 A ` <-> ` identity. (Cont...
trunantru 1515 A ` -/\ ` identity. (Cont...
trunanfal 1516 A ` -/\ ` identity. (Cont...
falnantru 1517 A ` -/\ ` identity. (Cont...
falnanfal 1518 A ` -/\ ` identity. (Cont...
truxortru 1519 A ` \/_ ` identity. (Cont...
truxorfal 1520 A ` \/_ ` identity. (Cont...
falxortru 1521 A ` \/_ ` identity. (Cont...
falxorfal 1522 A ` \/_ ` identity. (Cont...
hadbi123d 1525 Equality theorem for the a...
hadbi123i 1526 Equality theorem for the a...
hadass 1527 Associative law for the ad...
hadbi 1528 The adder sum is the same ...
hadcoma 1529 Commutative law for the ad...
hadcomb 1530 Commutative law for the ad...
hadrot 1531 Rotation law for the adder...
hadnot 1532 The adder sum distributes ...
had1 1533 If the first input is true...
had0 1534 If the first input is fals...
hadifp 1535 The value of the adder sum...
cador 1538 The adder carry in disjunc...
cadan 1539 The adder carry in conjunc...
cadbi123d 1540 Equality theorem for the a...
cadbi123i 1541 Equality theorem for the a...
cadcoma 1542 Commutative law for the ad...
cadcomb 1543 Commutative law for the ad...
cadrot 1544 Rotation law for the adder...
cadnot 1545 The adder carry distribute...
cad1 1546 If one input is true, then...
cad0 1547 If one input is false, the...
cadifp 1548 The value of the carry is,...
cad11 1549 If (at least) two inputs a...
cadtru 1550 The adder carry is true as...
minimp 1551 A single axiom for minimal...
minimp-sylsimp 1552 Derivation of sylsimp ( ~ ...
minimp-ax1 1553 Derivation of ~ ax-1 from ...
minimp-ax2c 1554 Derivation of a commuted f...
minimp-ax2 1555 Derivation of ~ ax-2 from ...
minimp-pm2.43 1556 Derivation of ~ pm2.43 (al...
meredith 1557 Carew Meredith's sole axio...
merlem1 1558 Step 3 of Meredith's proof...
merlem2 1559 Step 4 of Meredith's proof...
merlem3 1560 Step 7 of Meredith's proof...
merlem4 1561 Step 8 of Meredith's proof...
merlem5 1562 Step 11 of Meredith's proo...
merlem6 1563 Step 12 of Meredith's proo...
merlem7 1564 Between steps 14 and 15 of...
merlem8 1565 Step 15 of Meredith's proo...
merlem9 1566 Step 18 of Meredith's proo...
merlem10 1567 Step 19 of Meredith's proo...
merlem11 1568 Step 20 of Meredith's proo...
merlem12 1569 Step 28 of Meredith's proo...
merlem13 1570 Step 35 of Meredith's proo...
luk-1 1571 1 of 3 axioms for proposit...
luk-2 1572 2 of 3 axioms for proposit...
luk-3 1573 3 of 3 axioms for proposit...
luklem1 1574 Used to rederive standard ...
luklem2 1575 Used to rederive standard ...
luklem3 1576 Used to rederive standard ...
luklem4 1577 Used to rederive standard ...
luklem5 1578 Used to rederive standard ...
luklem6 1579 Used to rederive standard ...
luklem7 1580 Used to rederive standard ...
luklem8 1581 Used to rederive standard ...
ax1 1582 Standard propositional axi...
ax2 1583 Standard propositional axi...
ax3 1584 Standard propositional axi...
nic-dfim 1585 Define implication in term...
nic-dfneg 1586 Define negation in terms o...
nic-mp 1587 Derive Nicod's rule of mod...
nic-mpALT 1588 A direct proof of ~ nic-mp...
nic-ax 1589 Nicod's axiom derived from...
nic-axALT 1590 A direct proof of ~ nic-ax...
nic-imp 1591 Inference for ~ nic-mp usi...
nic-idlem1 1592 Lemma for ~ nic-id . (Con...
nic-idlem2 1593 Lemma for ~ nic-id . Infe...
nic-id 1594 Theorem ~ id expressed wit...
nic-swap 1595 The connector ` -/\ ` is s...
nic-isw1 1596 Inference version of ~ nic...
nic-isw2 1597 Inference for swapping nes...
nic-iimp1 1598 Inference version of ~ nic...
nic-iimp2 1599 Inference version of ~ nic...
nic-idel 1600 Inference to remove the tr...
nic-ich 1601 Chained inference. (Contr...
nic-idbl 1602 Double the terms. Since d...
nic-bijust 1603 Biconditional justificatio...
nic-bi1 1604 Inference to extract one s...
nic-bi2 1605 Inference to extract the o...
nic-stdmp 1606 Derive the standard modus ...
nic-luk1 1607 Proof of ~ luk-1 from ~ ni...
nic-luk2 1608 Proof of ~ luk-2 from ~ ni...
nic-luk3 1609 Proof of ~ luk-3 from ~ ni...
lukshef-ax1 1610 This alternative axiom for...
lukshefth1 1611 Lemma for ~ renicax . (Co...
lukshefth2 1612 Lemma for ~ renicax . (Co...
renicax 1613 A rederivation of ~ nic-ax...
tbw-bijust 1614 Justification for ~ tbw-ne...
tbw-negdf 1615 The definition of negation...
tbw-ax1 1616 The first of four axioms i...
tbw-ax2 1617 The second of four axioms ...
tbw-ax3 1618 The third of four axioms i...
tbw-ax4 1619 The fourth of four axioms ...
tbwsyl 1620 Used to rederive the Lukas...
tbwlem1 1621 Used to rederive the Lukas...
tbwlem2 1622 Used to rederive the Lukas...
tbwlem3 1623 Used to rederive the Lukas...
tbwlem4 1624 Used to rederive the Lukas...
tbwlem5 1625 Used to rederive the Lukas...
re1luk1 1626 ~ luk-1 derived from the T...
re1luk2 1627 ~ luk-2 derived from the T...
re1luk3 1628 ~ luk-3 derived from the T...
merco1 1629 A single axiom for proposi...
merco1lem1 1630 Used to rederive the Tarsk...
retbwax4 1631 ~ tbw-ax4 rederived from ~...
retbwax2 1632 ~ tbw-ax2 rederived from ~...
merco1lem2 1633 Used to rederive the Tarsk...
merco1lem3 1634 Used to rederive the Tarsk...
merco1lem4 1635 Used to rederive the Tarsk...
merco1lem5 1636 Used to rederive the Tarsk...
merco1lem6 1637 Used to rederive the Tarsk...
merco1lem7 1638 Used to rederive the Tarsk...
retbwax3 1639 ~ tbw-ax3 rederived from ~...
merco1lem8 1640 Used to rederive the Tarsk...
merco1lem9 1641 Used to rederive the Tarsk...
merco1lem10 1642 Used to rederive the Tarsk...
merco1lem11 1643 Used to rederive the Tarsk...
merco1lem12 1644 Used to rederive the Tarsk...
merco1lem13 1645 Used to rederive the Tarsk...
merco1lem14 1646 Used to rederive the Tarsk...
merco1lem15 1647 Used to rederive the Tarsk...
merco1lem16 1648 Used to rederive the Tarsk...
merco1lem17 1649 Used to rederive the Tarsk...
merco1lem18 1650 Used to rederive the Tarsk...
retbwax1 1651 ~ tbw-ax1 rederived from ~...
merco2 1652 A single axiom for proposi...
mercolem1 1653 Used to rederive the Tarsk...
mercolem2 1654 Used to rederive the Tarsk...
mercolem3 1655 Used to rederive the Tarsk...
mercolem4 1656 Used to rederive the Tarsk...
mercolem5 1657 Used to rederive the Tarsk...
mercolem6 1658 Used to rederive the Tarsk...
mercolem7 1659 Used to rederive the Tarsk...
mercolem8 1660 Used to rederive the Tarsk...
re1tbw1 1661 ~ tbw-ax1 rederived from ~...
re1tbw2 1662 ~ tbw-ax2 rederived from ~...
re1tbw3 1663 ~ tbw-ax3 rederived from ~...
re1tbw4 1664 ~ tbw-ax4 rederived from ~...
rb-bijust 1665 Justification for ~ rb-imd...
rb-imdf 1666 The definition of implicat...
anmp 1667 Modus ponens for ` \/ ` ` ...
rb-ax1 1668 The first of four axioms i...
rb-ax2 1669 The second of four axioms ...
rb-ax3 1670 The third of four axioms i...
rb-ax4 1671 The fourth of four axioms ...
rbsyl 1672 Used to rederive the Lukas...
rblem1 1673 Used to rederive the Lukas...
rblem2 1674 Used to rederive the Lukas...
rblem3 1675 Used to rederive the Lukas...
rblem4 1676 Used to rederive the Lukas...
rblem5 1677 Used to rederive the Lukas...
rblem6 1678 Used to rederive the Lukas...
rblem7 1679 Used to rederive the Lukas...
re1axmp 1680 ~ ax-mp derived from Russe...
re2luk1 1681 ~ luk-1 derived from Russe...
re2luk2 1682 ~ luk-2 derived from Russe...
re2luk3 1683 ~ luk-3 derived from Russe...
mptnan 1684 Modus ponendo tollens 1, o...
mptxor 1685 Modus ponendo tollens 2, o...
mtpor 1686 Modus tollendo ponens (inc...
mtpxor 1687 Modus tollendo ponens (ori...
stoic1a 1688 Stoic logic Thema 1 (part ...
stoic1b 1689 Stoic logic Thema 1 (part ...
stoic2a 1690 Stoic logic Thema 2 versio...
stoic2b 1691 Stoic logic Thema 2 versio...
stoic3 1692 Stoic logic Thema 3. Stat...
stoic4a 1693 Stoic logic Thema 4 versio...
stoic4b 1694 Stoic logic Thema 4 versio...
alnex 1697 Theorem 19.7 of [Margaris]...
eximal 1698 A utility theorem. An int...
nf2 1702 Alternate definition of no...
nf3 1703 Alternate definition of no...
nf4 1704 Alternate definition of no...
nfi 1705 Deduce that ` x ` is not f...
nfri 1706 Consequence of the definit...
nfd 1707 Deduce that ` x ` is not f...
nfrd 1708 Consequence of the definit...
nftht0 1709 Closed form of ~ nfth . (...
nfntht 1710 Closed form of ~ nfnth . ...
nfntht2 1711 Closed form of ~ nfnth . ...
gen2 1714 Generalization applied twi...
mpg 1715 Modus ponens combined with...
mpgbi 1716 Modus ponens on biconditio...
mpgbir 1717 Modus ponens on biconditio...
nfth 1718 No variable is (effectivel...
nfnth 1719 No variable is (effectivel...
hbth 1720 No variable is (effectivel...
nftru 1721 The true constant has no f...
nex 1722 Generalization rule for ne...
nffal 1723 The false constant has no ...
sptruw 1724 Version of ~ sp when ` ph ...
nfiOLD 1725 Obsolete proof of ~ nf5i a...
nfthOLD 1726 Obsolete proof of ~ nfth a...
nfnthOLD 1727 Obsolete proof of ~ nfnth ...
alim 1729 Restatement of Axiom ~ ax-...
alimi 1730 Inference quantifying both...
2alimi 1731 Inference doubly quantifyi...
al2im 1732 Closed form of ~ al2imi . ...
al2imi 1733 Inference quantifying ante...
alanimi 1734 Variant of ~ al2imi with c...
alimdh 1735 Deduction form of Theorem ...
albi 1736 Theorem 19.15 of [Margaris...
albii 1737 Inference adding universal...
2albii 1738 Inference adding two unive...
sylgt 1739 Closed form of ~ sylg . (...
sylg 1740 A syllogism combined with ...
alrimih 1741 Inference form of Theorem ...
hbxfrbi 1742 A utility lemma to transfe...
alex 1743 Theorem 19.6 of [Margaris]...
exnal 1744 Theorem 19.14 of [Margaris...
2nalexn 1745 Part of theorem *11.5 in [...
2exnaln 1746 Theorem *11.22 in [Whitehe...
2nexaln 1747 Theorem *11.25 in [Whitehe...
alimex 1748 A utility theorem. An int...
aleximi 1749 A variant of ~ al2imi : in...
alexbii 1750 Biconditional form of ~ al...
exim 1751 Theorem 19.22 of [Margaris...
eximi 1752 Inference adding existenti...
2eximi 1753 Inference adding two exist...
eximii 1754 Inference associated with ...
ala1 1755 Add an antecedent in a uni...
exa1 1756 Add an antecedent in an ex...
19.38 1757 Theorem 19.38 of [Margaris...
imnang 1758 Quantified implication in ...
alinexa 1759 A transformation of quanti...
alexn 1760 A relationship between two...
2exnexn 1761 Theorem *11.51 in [Whitehe...
exbi 1762 Theorem 19.18 of [Margaris...
exbiOLD 1763 Obsolete proof of ~ exbi a...
exbii 1764 Inference adding existenti...
2exbii 1765 Inference adding two exist...
3exbii 1766 Inference adding three exi...
nfnt 1767 If ` x ` is not free in ` ...
nfn 1768 Inference associated with ...
nfnd 1769 Deduction associated with ...
nfbii 1770 Equality theorem for not-f...
nfxfr 1771 A utility lemma to transfe...
nfxfrd 1772 A utility lemma to transfe...
exanali 1773 A transformation of quanti...
exancom 1774 Commutation of conjunction...
exan 1775 Place a conjunct in the sc...
exanOLD 1776 Obsolete proof of ~ exan a...
alrimdh 1777 Deduction form of Theorem ...
eximdh 1778 Deduction from Theorem 19....
nexdh 1779 Deduction for generalizati...
albidh 1780 Formula-building rule for ...
exbidh 1781 Formula-building rule for ...
exbidhOLD 1782 Obsolete proof of ~ exbidh...
exsimpl 1783 Simplification of an exist...
exsimpr 1784 Simplification of an exist...
19.40 1785 Theorem 19.40 of [Margaris...
19.26 1786 Theorem 19.26 of [Margaris...
19.26-2 1787 Theorem ~ 19.26 with two q...
19.26-3an 1788 Theorem ~ 19.26 with tripl...
19.29 1789 Theorem 19.29 of [Margaris...
19.29r 1790 Variation of ~ 19.29 . (C...
19.29rOLD 1791 Obsolete proof of ~ 19.29r...
19.29r2 1792 Variation of ~ 19.29r with...
19.29x 1793 Variation of ~ 19.29 with ...
19.35 1794 Theorem 19.35 of [Margaris...
19.35i 1795 Inference associated with ...
19.35ri 1796 Inference associated with ...
19.25 1797 Theorem 19.25 of [Margaris...
19.30 1798 Theorem 19.30 of [Margaris...
19.43 1799 Theorem 19.43 of [Margaris...
19.43OLD 1800 Obsolete proof of ~ 19.43 ...
19.33 1801 Theorem 19.33 of [Margaris...
19.33b 1802 The antecedent provides a ...
19.40-2 1803 Theorem *11.42 in [Whitehe...
19.40b 1804 The antecedent provides a ...
19.40bOLD 1805 Obsolete proof of ~ 19.40b...
albiim 1806 Split a biconditional and ...
2albiim 1807 Split a biconditional and ...
exintrbi 1808 Add/remove a conjunct in t...
exintrbiOLD 1809 Obsolete proof of ~ exintr...
exintr 1810 Introduce a conjunct in th...
alsyl 1811 Theorem *10.3 in [Whitehea...
nfimd 1812 If in a context ` x ` is n...
nfim 1813 If ` x ` is not free in ` ...
nfand 1814 If in a context ` x ` is n...
nf3and 1815 Deduction form of bound-va...
nfan 1816 If ` x ` is not free in ` ...
nfanOLD 1817 Obsolete proof of ~ nfan a...
nfnan 1818 If ` x ` is not free in ` ...
nf3an 1819 If ` x ` is not free in ` ...
nfbid 1820 If in a context ` x ` is n...
nfbi 1821 If ` x ` is not free in ` ...
nfor 1822 If ` x ` is not free in ` ...
nf3or 1823 If ` x ` is not free in ` ...
nfbiiOLD 1824 Obsolete proof of ~ nfbii ...
nfxfrOLD 1825 Obsolete proof of ~ nfxfr ...
nfxfrdOLD 1826 Obsolete proof of ~ nfxfrd...
ax5d 1828 ~ ax-5 with antecedent. U...
ax5e 1829 A rephrasing of ~ ax-5 usi...
nfv 1830 If ` x ` is not present in...
nfvd 1831 ~ nfv with antecedent. Us...
alimdv 1832 Deduction form of Theorem ...
eximdv 1833 Deduction form of Theorem ...
2alimdv 1834 Deduction form of Theorem ...
2eximdv 1835 Deduction form of Theorem ...
albidv 1836 Formula-building rule for ...
exbidv 1837 Formula-building rule for ...
2albidv 1838 Formula-building rule for ...
2exbidv 1839 Formula-building rule for ...
3exbidv 1840 Formula-building rule for ...
4exbidv 1841 Formula-building rule for ...
alrimiv 1842 Inference form of Theorem ...
alrimivv 1843 Inference form of Theorem ...
alrimdv 1844 Deduction form of Theorem ...
exlimiv 1845 Inference form of Theorem ...
exlimiiv 1846 Inference associated with ...
exlimivv 1847 Inference form of Theorem ...
exlimdv 1848 Deduction form of Theorem ...
exlimdvv 1849 Deduction form of Theorem ...
exlimddv 1850 Existential elimination ru...
nexdv 1851 Deduction for generalizati...
nexdvOLD 1852 Obsolete proof of ~ nexdv ...
2ax5 1853 Quantification of two vari...
stdpc5v 1854 Version of ~ stdpc5 with a...
19.21v 1855 Version of ~ 19.21 with a ...
19.32v 1856 Version of ~ 19.32 with a ...
19.31v 1857 Version of ~ 19.31 with a ...
nfvOLD 1858 Obsolete proof of ~ nfv as...
nfvdOLD 1859 Obsolete proof of ~ nfvd a...
nfdvOLD 1860 Obsolete proof of ~ nf5dv ...
weq 1861 Extend wff definition to i...
equs3 1862 Lemma used in proofs of su...
speimfw 1863 Specialization, with addit...
speimfwALT 1864 Alternate proof of ~ speim...
spimfw 1865 Specialization, with addit...
ax12i 1866 Inference that has ~ ax-12...
sbequ2 1869 An equality theorem for su...
sb1 1870 One direction of a simplif...
spsbe 1871 A specialization theorem. ...
sbequ8 1872 Elimination of equality fr...
sbimi 1873 Infer substitution into an...
sbbii 1874 Infer substitution into bo...
ax6v 1876 Axiom B7 of [Tarski] p. 75...
ax6ev 1877 At least one individual ex...
exiftru 1878 Rule of existential genera...
19.2 1879 Theorem 19.2 of [Margaris]...
19.2d 1880 Deduction associated with ...
19.8w 1881 Weak version of ~ 19.8a an...
19.8v 1882 Version of ~ 19.8a with a ...
19.9v 1883 Version of ~ 19.9 with a d...
19.3v 1884 Version of ~ 19.3 with a d...
spvw 1885 Version of ~ sp when ` x `...
19.39 1886 Theorem 19.39 of [Margaris...
19.24 1887 Theorem 19.24 of [Margaris...
19.34 1888 Theorem 19.34 of [Margaris...
19.23v 1889 Version of ~ 19.23 with a ...
19.23vv 1890 Theorem ~ 19.23v extended ...
19.36v 1891 Version of ~ 19.36 with a ...
19.36iv 1892 Inference associated with ...
pm11.53v 1893 Version of ~ pm11.53 with ...
19.12vvv 1894 Version of ~ 19.12vv with ...
19.27v 1895 Version of ~ 19.27 with a ...
19.28v 1896 Version of ~ 19.28 with a ...
19.37v 1897 Version of ~ 19.37 with a ...
19.37iv 1898 Inference associated with ...
19.44v 1899 Version of ~ 19.44 with a ...
19.45v 1900 Version of ~ 19.45 with a ...
19.41v 1901 Version of ~ 19.41 with a ...
19.41vv 1902 Version of ~ 19.41 with tw...
19.41vvv 1903 Version of ~ 19.41 with th...
19.41vvvv 1904 Version of ~ 19.41 with fo...
19.42v 1905 Version of ~ 19.42 with a ...
exdistr 1906 Distribution of existentia...
19.42vv 1907 Version of ~ 19.42 with tw...
19.42vvv 1908 Version of ~ 19.42 with th...
exdistr2 1909 Distribution of existentia...
3exdistr 1910 Distribution of existentia...
4exdistr 1911 Distribution of existentia...
spimeh 1912 Existential introduction, ...
spimw 1913 Specialization. Lemma 8 o...
spimvw 1914 Specialization. Lemma 8 o...
spnfw 1915 Weak version of ~ sp . Us...
spfalw 1916 Version of ~ sp when ` ph ...
equs4v 1917 Version of ~ equs4 with a ...
equsalvw 1918 Version of ~ equsal with t...
equsexvw 1919 Version of ~ equsexv with ...
cbvaliw 1920 Change bound variable. Us...
cbvalivw 1921 Change bound variable. Us...
ax7v 1923 Weakened version of ~ ax-7...
ax7v1 1924 First of two weakened vers...
ax7v2 1925 Second of two weakened ver...
equid 1926 Identity law for equality....
nfequid 1927 Bound-variable hypothesis ...
equcomiv 1928 Weaker form of ~ equcomi w...
ax6evr 1929 A commuted form of ~ ax6ev...
ax7 1930 Proof of ~ ax-7 from ~ ax7...
equcomi 1931 Commutative law for equali...
equcom 1932 Commutative law for equali...
equcomd 1933 Deduction form of ~ equcom...
equcoms 1934 An inference commuting equ...
equtr 1935 A transitive law for equal...
equtrr 1936 A transitive law for equal...
equeuclr 1937 Commuted version of ~ eque...
equeucl 1938 Equality is a left-Euclide...
equequ1 1939 An equivalence law for equ...
equequ2 1940 An equivalence law for equ...
equtr2 1941 Equality is a left-Euclide...
equequ2OLD 1942 Obsolete proof of ~ equequ...
equtr2OLD 1943 Obsolete proof of ~ equtr2...
stdpc6 1944 One of the two equality ax...
stdpc7 1945 One of the two equality ax...
equvinv 1946 A variable introduction la...
equviniva 1947 A modified version of the ...
equvinivOLD 1948 The forward implication of...
equvinvOLD 1949 Obsolete version of ~ equv...
equvelv 1950 A specialized version of ~...
ax13b 1951 An equivalence between two...
spfw 1952 Weak version of ~ sp . Us...
spfwOLD 1953 Obsolete proof of ~ spfw a...
spw 1954 Weak version of the specia...
cbvalw 1955 Change bound variable. Us...
cbvalvw 1956 Change bound variable. Us...
cbvexvw 1957 Change bound variable. Us...
alcomiw 1958 Weak version of ~ alcom . ...
hbn1fw 1959 Weak version of ~ ax-10 fr...
hbn1w 1960 Weak version of ~ hbn1 . ...
hba1w 1961 Weak version of ~ hba1 . ...
hba1wOLD 1962 Obsolete proof of ~ hba1w ...
hbe1w 1963 Weak version of ~ hbe1 . ...
hbalw 1964 Weak version of ~ hbal . ...
spaev 1965 A special instance of ~ sp...
cbvaev 1966 Change bound variable in a...
aevlem0 1967 Lemma for ~ aevlem . Inst...
aevlem 1968 Lemma for ~ aev and ~ axc1...
aeveq 1969 The antecedent ` A. x x = ...
aev 1970 A "distinctor elimination"...
hbaevg 1971 Generalization of ~ hbaev ...
hbaev 1972 Version of ~ hbae with a D...
aev2 1973 A version of ~ aev with tw...
aev2ALT 1974 Alternate proof of ~ aev2 ...
axc11nlemOLD2 1975 Lemma for ~ axc11n . Chan...
aevlemOLD 1976 Old proof of ~ aevlem . O...
wel 1978 Extend wff definition to i...
ax8v 1980 Weakened version of ~ ax-8...
ax8v1 1981 First of two weakened vers...
ax8v2 1982 Second of two weakened ver...
ax8 1983 Proof of ~ ax-8 from ~ ax8...
elequ1 1984 An identity law for the no...
cleljust 1985 When the class variables i...
ax9v 1987 Weakened version of ~ ax-9...
ax9v1 1988 First of two weakened vers...
ax9v2 1989 Second of two weakened ver...
ax9 1990 Proof of ~ ax-9 from ~ ax9...
elequ2 1991 An identity law for the no...
ax6dgen 1992 Tarski's system uses the w...
ax10w 1993 Weak version of ~ ax-10 fr...
ax11w 1994 Weak version of ~ ax-11 fr...
ax11dgen 1995 Degenerate instance of ~ a...
ax12wlem 1996 Lemma for weak version of ...
ax12w 1997 Weak version of ~ ax-12 fr...
ax12dgen 1998 Degenerate instance of ~ a...
ax12wdemo 1999 Example of an application ...
ax13w 2000 Weak version (principal in...
ax13dgen1 2001 Degenerate instance of ~ a...
ax13dgen2 2002 Degenerate instance of ~ a...
ax13dgen3 2003 Degenerate instance of ~ a...
ax13dgen4 2004 Degenerate instance of ~ a...
ax13dgen4OLD 2005 Obsolete proof of ~ ax13dg...
hbn1 2007 Alias for ~ ax-10 to be us...
hbe1 2008 The setvar ` x ` is not fr...
hbe1a 2009 Dual statement of ~ hbe1 ....
nf5-1 2010 One direction of ~ nf5 can...
nf5i 2011 Deduce that ` x ` is not f...
nf5dv 2012 Apply the definition of no...
nf5dh 2013 Deduce that ` x ` is not f...
nfe1 2014 The setvar ` x ` is not fr...
nfa1 2015 The setvar ` x ` is not fr...
nfna1 2016 A convenience theorem part...
nfia1 2017 Lemma 23 of [Monk2] p. 114...
nfnf1 2018 The setvar ` x ` is not fr...
modal-5 2019 The analogue in our predic...
nfe1OLD 2020 Obsolete proof of ~ nfe1 a...
alcoms 2022 Swap quantifiers in an ant...
hbal 2023 If ` x ` is not free in ` ...
alcom 2024 Theorem 19.5 of [Margaris]...
alrot3 2025 Theorem *11.21 in [Whitehe...
alrot4 2026 Rotate four universal quan...
nfa2 2027 Lemma 24 of [Monk2] p. 114...
hbald 2028 Deduction form of bound-va...
excom 2029 Theorem 19.11 of [Margaris...
excomim 2030 One direction of Theorem 1...
excom13 2031 Swap 1st and 3rd existenti...
exrot3 2032 Rotate existential quantif...
exrot4 2033 Rotate existential quantif...
ax12v 2035 This is essentially axiom ...
ax12v2 2036 It is possible to remove a...
ax12vOLD 2037 Obsolete proof of ~ ax12v2...
ax12vOLDOLD 2038 Obsolete proof of ~ ax12v ...
19.8a 2039 If a wff is true, it is tr...
19.8aOLD 2040 Obsolete proof of ~ 19.8a ...
sp 2041 Specialization. A univers...
spi 2042 Inference rule reversing g...
sps 2043 Generalization of antecede...
2sp 2044 A double specialization (s...
spsd 2045 Deduction generalizing ant...
19.2g 2046 Theorem 19.2 of [Margaris]...
19.21bi 2047 Inference form of ~ 19.21 ...
19.21bbi 2048 Inference removing double ...
19.23bi 2049 Inference form of Theorem ...
nexr 2050 Inference associated with ...
qexmid 2051 Quantified excluded middle...
nf5r 2052 Consequence of the definit...
nf5ri 2053 Consequence of the definit...
nf5rd 2054 Consequence of the definit...
nfim1 2055 A closed form of ~ nfim . ...
nfan1 2056 A closed form of ~ nfan . ...
19.3 2057 A wff may be quantified wi...
19.9d 2058 A deduction version of one...
19.9t 2059 A closed version of ~ 19.9...
19.9 2060 A wff may be existentially...
19.21t 2061 Closed form of Theorem 19....
19.21 2062 Theorem 19.21 of [Margaris...
stdpc5 2063 An axiom scheme of standar...
stdpc5OLD 2064 Obsolete proof of ~ stdpc5...
19.21-2 2065 Version of ~ 19.21 with tw...
19.23t 2066 Closed form of Theorem 197...
19.23 2067 Theorem 19.23 of [Margaris...
alimd 2068 Deduction form of Theorem ...
alrimi 2069 Inference form of Theorem ...
alrimdd 2070 Deduction form of Theorem ...
alrimd 2071 Deduction form of Theorem ...
eximd 2072 Deduction form of Theorem ...
exlimi 2073 Inference associated with ...
exlimd 2074 Deduction form of Theorem ...
exlimdd 2075 Existential elimination ru...
nexd 2076 Deduction for generalizati...
albid 2077 Formula-building rule for ...
exbid 2078 Formula-building rule for ...
nfbidf 2079 An equality theorem for ef...
19.16 2080 Theorem 19.16 of [Margaris...
19.17 2081 Theorem 19.17 of [Margaris...
19.27 2082 Theorem 19.27 of [Margaris...
19.28 2083 Theorem 19.28 of [Margaris...
19.19 2084 Theorem 19.19 of [Margaris...
19.36 2085 Theorem 19.36 of [Margaris...
19.36i 2086 Inference associated with ...
19.37 2087 Theorem 19.37 of [Margaris...
19.32 2088 Theorem 19.32 of [Margaris...
19.31 2089 Theorem 19.31 of [Margaris...
19.41 2090 Theorem 19.41 of [Margaris...
19.42-1 2091 One direction of ~ 19.42 ....
19.42 2092 Theorem 19.42 of [Margaris...
19.44 2093 Theorem 19.44 of [Margaris...
19.45 2094 Theorem 19.45 of [Margaris...
equsexv 2095 Version of ~ equsex with a...
sbequ1 2096 An equality theorem for su...
sbequ12 2097 An equality theorem for su...
sbequ12r 2098 An equality theorem for su...
sbequ12a 2099 An equality theorem for su...
sbid 2100 An identity theorem for su...
spimv1 2101 Version of ~ spim with a d...
nf5 2102 Alternate definition of ~ ...
nf6 2103 An alternate definition of...
nf5d 2104 Deduce that ` x ` is not f...
nf5di 2105 Since the converse holds b...
19.9h 2106 A wff may be existentially...
19.21h 2107 Theorem 19.21 of [Margaris...
19.23h 2108 Theorem 19.23 of [Margaris...
equsalhw 2109 Weaker version of ~ equsal...
hbim1 2110 A closed form of ~ hbim . ...
hbimd 2111 Deduction form of bound-va...
hbim 2112 If ` x ` is not free in ` ...
hban 2113 If ` x ` is not free in ` ...
hb3an 2114 If ` x ` is not free in ` ...
axc4 2115 Show that the original axi...
axc4i 2116 Inference version of ~ axc...
axc7 2117 Show that the original axi...
axc7e 2118 Abbreviated version of ~ a...
axc16g 2119 Generalization of ~ axc16 ...
axc16 2120 Proof of older axiom ~ ax-...
axc16gb 2121 Biconditional strengthenin...
axc16nf 2122 If ~ dtru is false, then t...
axc11v 2123 Version of ~ axc11 with a ...
axc11rv 2124 Version of ~ axc11r with a...
axc11rvOLD 2125 Obsolete proof of ~ axc11r...
axc11vOLD 2126 Obsolete proof of ~ axc11v...
modal-b 2127 The analogue in our predic...
19.9ht 2128 A closed version of ~ 19.9...
hbnt 2129 Closed theorem version of ...
hbntOLD 2130 Obsolete proof of ~ hbnt a...
hbn 2131 If ` x ` is not free in ` ...
hbnd 2132 Deduction form of bound-va...
exlimih 2133 Inference associated with ...
exlimdh 2134 Deduction form of Theorem ...
equsexhv 2135 Version of ~ equsexh with ...
sb56 2136 Two equivalent ways of exp...
hba1 2137 The setvar ` x ` is not fr...
hbexOLD 2138 Obsolete proof of ~ hbex a...
nfal 2139 If ` x ` is not free in ` ...
nfex 2140 If ` x ` is not free in ` ...
nfexOLD 2141 Obsolete proof of ~ nfex a...
hbex 2142 If ` x ` is not free in ` ...
nfa1OLD 2143 Obsolete proof of ~ nfa1 a...
nfnf 2144 If ` x ` is not free in ` ...
nfnf1OLD 2145 Obsolete proof of ~ nfnf1 ...
axc11nlemOLD 2146 Obsolete proof of ~ axc11n...
axc16gOLD 2147 Obsolete proof of ~ axc16g...
aevOLD 2148 Obsolete proof of ~ aev as...
axc16nfOLD 2149 Obsolete proof of ~ axc16n...
19.12 2150 Theorem 19.12 of [Margaris...
nfald 2151 Deduction form of ~ nfal ....
nfaldOLD 2152 Obsolete proof of ~ nfald ...
nfexd 2153 If ` x ` is not free in ` ...
nfa2OLD 2154 Obsolete proof of ~ nfa2 a...
exanOLDOLD 2155 Obsolete proof of ~ exan a...
aaan 2156 Rearrange universal quanti...
eeor 2157 Rearrange existential quan...
cbv3v 2158 Version of ~ cbv3 with a d...
dvelimhw 2159 Proof of ~ dvelimh without...
cbv3hv 2160 Version of ~ cbv3h with a ...
cbv3hvOLD 2161 Obsolete proof of ~ cbv3hv...
cbv3hvOLDOLD 2162 Obsolete proof of ~ cbv3hv...
cbvalv1 2163 Version of ~ cbval with a ...
cbvexv1 2164 Version of ~ cbvex with a ...
equs5aALT 2165 Alternate proof of ~ equs5...
equs5eALT 2166 Alternate proof of ~ equs5...
pm11.53 2167 Theorem *11.53 in [Whitehe...
19.12vv 2168 Special case of ~ 19.12 wh...
eean 2169 Rearrange existential quan...
eeanv 2170 Rearrange existential quan...
eeeanv 2171 Rearrange existential quan...
ee4anv 2172 Rearrange existential quan...
cleljustALT 2173 Alternate proof of ~ clelj...
cleljustALT2 2174 Alternate proof of ~ clelj...
axc11r 2175 Same as ~ axc11 but with r...
nfrOLD 2176 Obsolete proof of ~ nf5r a...
nfriOLD 2177 Obsolete proof of ~ nf5ri ...
nfrdOLD 2178 Obsolete proof of ~ nf5rd ...
alimdOLD 2179 Obsolete proof of ~ alimd ...
alrimiOLD 2180 Obsolete proof of ~ alrimi...
nfdOLD 2181 Obsolete proof of ~ nf5d a...
nfdhOLD 2182 Obsolete proof of ~ nf5dh ...
alrimddOLD 2183 Obsolete proof of ~ alrimd...
alrimdOLD 2184 Obsolete proof of ~ alrimd...
eximdOLD 2185 Obsolete proof of ~ eximd ...
nexdOLD 2186 Obsolete proof of ~ nexd a...
albidOLD 2187 Obsolete proof of ~ albid ...
exbidOLD 2188 Obsolete proof of ~ exbid ...
nfbidfOLD 2189 Obsolete proof of ~ nfbidf...
19.3OLD 2190 Obsolete proof of ~ 19.3 a...
19.9dOLD 2191 Obsolete proof of ~ 19.9d ...
19.9tOLD 2192 Obsolete proof of ~ 19.9t ...
19.9OLD 2193 Obsolete proof of ~ 19.9 a...
19.9hOLD 2194 Obsolete proof of ~ 19.9h ...
nfa1OLDOLD 2195 Obsolete proof of ~ nfa1 a...
nfnf1OLDOLD 2196 Obsolete proof of ~ nfnf1 ...
nfntOLD 2197 Obsolete proof of ~ nfnt a...
nfnOLD 2198 Obsolete proof of ~ nfn as...
nfndOLD 2199 Obsolete proof of ~ nfnd a...
19.21t-1OLD 2200 One direction of the bi-co...
19.21tOLD 2201 Obsolete proof of ~ 19.21t...
19.21OLD 2202 Obsolete proof of ~ 19.21 ...
19.21-2OLD 2203 Obsolete proof of ~ 19.21-...
19.21hOLD 2204 Obsolete proof of ~ 19.21h...
stdpc5OLDOLD 2205 Obsolete proof of ~ stdpc5...
19.23tOLD 2206 Obsolete proof of ~ 19.23t...
19.23OLD 2207 Obsolete proof of ~ 19.23 ...
19.23hOLD 2208 Obsolete proof of ~ 19.23h...
exlimiOLD 2209 Obsolete proof of ~ exlimi...
exlimihOLD 2210 Obsolete proof of ~ exlimi...
exlimdOLD 2211 Obsolete proof of ~ exlimd...
exlimdhOLD 2212 Obsolete proof of ~ exlimd...
nfdiOLD 2213 Obsolete proof of ~ nf5di ...
nfimdOLD 2214 Obsolete proof of ~ nfimd ...
hbim1OLD 2215 Obsolete proof of ~ hbim a...
nfim1OLD 2216 Obsolete proof of ~ nfim1 ...
nfimOLD 2217 Obsolete proof of ~ nfim a...
hbimdOLD 2218 Obsolete proof of ~ hbimd ...
hbimOLD 2219 Obsolete proof of ~ hbim a...
nfandOLD 2220 Obsolete proof of ~ nfand ...
nf3andOLD 2221 Obsolete proof of ~ nf3and...
19.27OLD 2222 Obsolete proof of ~ 19.27 ...
19.28OLD 2223 Obsolete proof of ~ 19.28 ...
nfan1OLD 2224 Obsolete proof of ~ nfan1 ...
nfanOLDOLD 2225 Obsolete proof of ~ nfan a...
nfnanOLD 2226 Obsolete proof of ~ nfnan ...
nf3anOLD 2227 Obsolete proof of ~ nf3an ...
hbanOLD 2228 Obsolete proof of ~ hban a...
hb3anOLD 2229 Obsolete proof of ~ hb3an ...
nfbidOLD 2230 Obsolete proof of ~ nfbid ...
nfbiOLD 2231 Obsolete proof of ~ nfbi a...
nforOLD 2232 Obsolete proof of ~ nfor a...
nf3orOLD 2233 Obsolete proof of ~ nf3or ...
ax13v 2235 A weaker version of ~ ax-1...
ax13lem1 2236 A version of ~ ax13v with ...
ax13 2237 Derive ~ ax-13 from ~ ax13...
ax6e 2238 At least one individual ex...
ax6 2239 Theorem showing that ~ ax-...
axc10 2240 Show that the original axi...
spimt 2241 Closed theorem form of ~ s...
spim 2242 Specialization, using impl...
spimed 2243 Deduction version of ~ spi...
spime 2244 Existential introduction, ...
spimv 2245 A version of ~ spim with a...
spimvALT 2246 Alternate proof of ~ spimv...
spimev 2247 Distinct-variable version ...
spv 2248 Specialization, using impl...
spei 2249 Inference from existential...
chvar 2250 Implicit substitution of `...
chvarv 2251 Implicit substitution of `...
chvarvOLD 2252 Obsolete proof of ~ chvarv...
cbv3 2253 Rule used to change bound ...
cbv3h 2254 Rule used to change bound ...
cbv1 2255 Rule used to change bound ...
cbv1h 2256 Rule used to change bound ...
cbv2h 2257 Rule used to change bound ...
cbv2 2258 Rule used to change bound ...
cbval 2259 Rule used to change bound ...
cbvex 2260 Rule used to change bound ...
cbvalv 2261 Rule used to change bound ...
cbvalvOLD 2262 Obsolete proof of ~ cbvalv...
cbvexv 2263 Rule used to change bound ...
cbvexvOLD 2264 Obsolete proof of ~ cbvexv...
cbvald 2265 Deduction used to change b...
cbvexd 2266 Deduction used to change b...
cbval2 2267 Rule used to change bound ...
cbvex2 2268 Rule used to change bound ...
cbvaldva 2269 Rule used to change the bo...
cbvaldvaOLD 2270 Obsolete proof of ~ cbvald...
cbvexdva 2271 Rule used to change the bo...
cbvexdvaOLD 2272 Obsolete proof of ~ cbvexd...
cbval2v 2273 Rule used to change bound ...
cbval2vOLD 2274 Obsolete proof of ~ cbval2...
cbvex2v 2275 Rule used to change bound ...
cbvex2vOLD 2276 Obsolete proof of ~ cbvex2...
cbvex4v 2277 Rule used to change bound ...
equs4 2278 Lemma used in proofs of im...
equsal 2279 An equivalence related to ...
equsalh 2280 An equivalence related to ...
equsex 2281 An equivalence related to ...
equsexALT 2282 Alternate proof of ~ equse...
equsexh 2283 An equivalence related to ...
ax13lem2 2284 Lemma for ~ nfeqf2 . This...
nfeqf2 2285 An equation between setvar...
dveeq2 2286 Quantifier introduction wh...
nfeqf1 2287 An equation between setvar...
dveeq1 2288 Quantifier introduction wh...
nfeqf 2289 A variable is effectively ...
axc9 2290 Derive set.mm's original ~...
axc15 2291 Derivation of set.mm's ori...
ax12 2292 Rederivation of axiom ~ ax...
ax13OLD 2293 Obsolete proof of ~ ax13 a...
axc11nlemALT 2294 Alternate version of ~ axc...
axc11n 2295 Derive set.mm's original ~...
axc11nOLD 2296 Obsolete proof of ~ axc11n...
axc11nOLDOLD 2297 Old proof of ~ axc11n . O...
axc11nALT 2298 Alternate proof of ~ axc11...
aecom 2299 Commutation law for identi...
aecoms 2300 A commutation rule for ide...
naecoms 2301 A commutation rule for dis...
axc11 2302 Show that ~ ax-c11 can be ...
hbae 2303 All variables are effectiv...
nfae 2304 All variables are effectiv...
hbnae 2305 All variables are effectiv...
nfnae 2306 All variables are effectiv...
hbnaes 2307 Rule that applies ~ hbnae ...
aevlemALTOLD 2308 Older alternate version of...
aevALTOLD 2309 Older alternate proof of ~...
axc16i 2310 Inference with ~ axc16 as ...
axc16nfALT 2311 Alternate proof of ~ axc16...
dral2 2312 Formula-building lemma for...
dral1 2313 Formula-building lemma for...
dral1ALT 2314 Alternate proof of ~ dral1...
drex1 2315 Formula-building lemma for...
drex2 2316 Formula-building lemma for...
drnf1 2317 Formula-building lemma for...
drnf2 2318 Formula-building lemma for...
nfald2 2319 Variation on ~ nfald which...
nfexd2 2320 Variation on ~ nfexd which...
exdistrf 2321 Distribution of existentia...
dvelimf 2322 Version of ~ dvelimv witho...
dvelimdf 2323 Deduction form of ~ dvelim...
dvelimh 2324 Version of ~ dvelim withou...
dvelim 2325 This theorem can be used t...
dvelimv 2326 Similar to ~ dvelim with f...
dvelimnf 2327 Version of ~ dvelim using ...
dveeq2ALT 2328 Alternate proof of ~ dveeq...
ax12OLD 2329 Obsolete proof of ~ ax12 a...
ax12v2OLD 2330 Obsolete proof of ~ ax12v ...
ax12a2OLD 2331 Obsolete proof of ~ ax12v ...
axc15OLD 2332 Obsolete proof of ~ axc15 ...
ax12b 2333 A bidirectional version of...
equvini 2334 A variable introduction la...
equvel 2335 A variable elimination law...
equs5a 2336 A property related to subs...
equs5e 2337 A property related to subs...
equs45f 2338 Two ways of expressing sub...
equs5 2339 Lemma used in proofs of su...
sb2 2340 One direction of a simplif...
stdpc4 2341 The specialization axiom o...
2stdpc4 2342 A double specialization us...
sb3 2343 One direction of a simplif...
sb4 2344 One direction of a simplif...
sb4a 2345 A version of ~ sb4 that do...
sb4b 2346 Simplified definition of s...
hbsb2 2347 Bound-variable hypothesis ...
nfsb2 2348 Bound-variable hypothesis ...
hbsb2a 2349 Special case of a bound-va...
sb4e 2350 One direction of a simplif...
hbsb2e 2351 Special case of a bound-va...
hbsb3 2352 If ` y ` is not free in ` ...
nfs1 2353 If ` y ` is not free in ` ...
axc16ALT 2354 Alternate proof of ~ axc16...
axc16gALT 2355 Alternate proof of ~ axc16...
equsb1 2356 Substitution applied to an...
equsb2 2357 Substitution applied to an...
dveel1 2358 Quantifier introduction wh...
dveel2 2359 Quantifier introduction wh...
axc14 2360 Axiom ~ ax-c14 is redundan...
dfsb2 2361 An alternate definition of...
dfsb3 2362 An alternate definition of...
sbequi 2363 An equality theorem for su...
sbequ 2364 An equality theorem for su...
drsb1 2365 Formula-building lemma for...
drsb2 2366 Formula-building lemma for...
sbft 2367 Substitution has no effect...
sbf 2368 Substitution for a variabl...
sbh 2369 Substitution for a variabl...
sbf2 2370 Substitution has no effect...
nfs1f 2371 If ` x ` is not free in ` ...
sb6x 2372 Equivalence involving subs...
sb6f 2373 Equivalence for substituti...
sb5f 2374 Equivalence for substituti...
sbequ5 2375 Substitution does not chan...
sbequ6 2376 Substitution does not chan...
nfsb4t 2377 A variable not free remain...
nfsb4 2378 A variable not free remain...
sbn 2379 Negation inside and outsid...
sbi1 2380 Removal of implication fro...
sbi2 2381 Introduction of implicatio...
spsbim 2382 Specialization of implicat...
sbim 2383 Implication inside and out...
sbrim 2384 Substitution with a variab...
sblim 2385 Substitution with a variab...
sbor 2386 Logical OR inside and outs...
sban 2387 Conjunction inside and out...
sb3an 2388 Conjunction inside and out...
sbbi 2389 Equivalence inside and out...
spsbbi 2390 Specialization of bicondit...
sbbid 2391 Deduction substituting bot...
sblbis 2392 Introduce left bicondition...
sbrbis 2393 Introduce right biconditio...
sbrbif 2394 Introduce right biconditio...
sbequ8ALT 2395 Alternate proof of ~ sbequ...
sbie 2396 Conversion of implicit sub...
sbied 2397 Conversion of implicit sub...
sbiedv 2398 Conversion of implicit sub...
sbcom3 2399 Substituting ` y ` for ` x...
sbco 2400 A composition law for subs...
sbid2 2401 An identity law for substi...
sbidm 2402 An idempotent law for subs...
sbco2 2403 A composition law for subs...
sbco2d 2404 A composition law for subs...
sbco3 2405 A composition law for subs...
sbcom 2406 A commutativity law for su...
sbt 2407 A substitution into a theo...
sbtrt 2408 Partially closed form of ~...
sbtr 2409 A partial converse to ~ sb...
sb5rf 2410 Reversed substitution. (C...
sb6rf 2411 Reversed substitution. (C...
sb8 2412 Substitution of variable i...
sb8e 2413 Substitution of variable i...
sb9 2414 Commutation of quantificat...
sb9i 2415 Commutation of quantificat...
ax12vALT 2416 Alternate proof of ~ ax12v...
sb6 2417 Equivalence for substituti...
sb5 2418 Equivalence for substituti...
equsb3lem 2419 Lemma for ~ equsb3 . (Con...
equsb3 2420 Substitution applied to an...
equsb3ALT 2421 Alternate proof of ~ equsb...
elsb3 2422 Substitution applied to an...
elsb4 2423 Substitution applied to an...
hbs1 2424 The setvar ` x ` is not fr...
nfs1v 2425 The setvar ` x ` is not fr...
sbhb 2426 Two ways of expressing " `...
sbnf2 2427 Two ways of expressing " `...
nfsb 2428 If ` z ` is not free in ` ...
hbsb 2429 If ` z ` is not free in ` ...
nfsbd 2430 Deduction version of ~ nfs...
2sb5 2431 Equivalence for double sub...
2sb6 2432 Equivalence for double sub...
sbcom2 2433 Commutativity law for subs...
sbcom4 2434 Commutativity law for subs...
pm11.07 2435 Axiom *11.07 in [Whitehead...
sb6a 2436 Equivalence for substituti...
2ax6elem 2437 We can always find values ...
2ax6e 2438 We can always find values ...
2sb5rf 2439 Reversed double substituti...
2sb6rf 2440 Reversed double substituti...
sb7f 2441 This version of ~ dfsb7 do...
sb7h 2442 This version of ~ dfsb7 do...
dfsb7 2443 An alternate definition of...
sb10f 2444 Hao Wang's identity axiom ...
sbid2v 2445 An identity law for substi...
sbelx 2446 Elimination of substitutio...
sbel2x 2447 Elimination of double subs...
sbal1 2448 A theorem used in eliminat...
sbal2 2449 Move quantifier in and out...
sbal 2450 Move universal quantifier ...
sbex 2451 Move existential quantifie...
sbalv 2452 Quantify with new variable...
sbco4lem 2453 Lemma for ~ sbco4 . It re...
sbco4 2454 Two ways of exchanging two...
2sb8e 2455 An equivalent expression f...
exsb 2456 An equivalent expression f...
2exsb 2457 An equivalent expression f...
eujust 2460 A soundness justification ...
eujustALT 2461 Alternate proof of ~ eujus...
euequ1 2464 Equality has existential u...
mo2v 2465 Alternate definition of "a...
euf 2466 A version of the existenti...
mo2 2467 Alternate definition of "a...
nfeu1 2468 Bound-variable hypothesis ...
nfmo1 2469 Bound-variable hypothesis ...
nfeud2 2470 Bound-variable hypothesis ...
nfmod2 2471 Bound-variable hypothesis ...
nfeud 2472 Deduction version of ~ nfe...
nfmod 2473 Bound-variable hypothesis ...
nfeu 2474 Bound-variable hypothesis ...
nfmo 2475 Bound-variable hypothesis ...
eubid 2476 Formula-building rule for ...
mobid 2477 Formula-building rule for ...
eubidv 2478 Formula-building rule for ...
mobidv 2479 Formula-building rule for ...
eubii 2480 Introduce uniqueness quant...
mobii 2481 Formula-building rule for ...
euex 2482 Existential uniqueness imp...
exmo 2483 Something exists or at mos...
eu5 2484 Uniqueness in terms of "at...
exmoeu2 2485 Existence implies "at most...
eu3v 2486 An alternate way to expres...
eumo 2487 Existential uniqueness imp...
eumoi 2488 "At most one" inferred fro...
moabs 2489 Absorption of existence co...
exmoeu 2490 Existence in terms of "at ...
sb8eu 2491 Variable substitution in u...
sb8mo 2492 Variable substitution for ...
cbveu 2493 Rule used to change bound ...
cbvmo 2494 Rule used to change bound ...
mo3 2495 Alternate definition of "a...
mo 2496 Equivalent definitions of ...
eu2 2497 An alternate way of defini...
eu1 2498 An alternate way to expres...
euexALT 2499 Alternate proof of ~ euex ...
euor 2500 Introduce a disjunct into ...
euorv 2501 Introduce a disjunct into ...
euor2 2502 Introduce or eliminate a d...
sbmo 2503 Substitution into "at most...
mo4f 2504 "At most one" expressed us...
mo4 2505 "At most one" expressed us...
eu4 2506 Uniqueness using implicit ...
moim 2507 "At most one" reverses imp...
moimi 2508 "At most one" reverses imp...
moa1 2509 If an implication holds fo...
euimmo 2510 Uniqueness implies "at mos...
euim 2511 Add existential uniqueness...
moan 2512 "At most one" is still the...
moani 2513 "At most one" is still tru...
moor 2514 "At most one" is still the...
mooran1 2515 "At most one" imports disj...
mooran2 2516 "At most one" exports disj...
moanim 2517 Introduction of a conjunct...
euan 2518 Introduction of a conjunct...
moanimv 2519 Introduction of a conjunct...
moanmo 2520 Nested "at most one" quant...
moaneu 2521 Nested "at most one" and u...
euanv 2522 Introduction of a conjunct...
mopick 2523 "At most one" picks a vari...
eupick 2524 Existential uniqueness "pi...
eupicka 2525 Version of ~ eupick with c...
eupickb 2526 Existential uniqueness "pi...
eupickbi 2527 Theorem *14.26 in [Whitehe...
mopick2 2528 "At most one" can show the...
moexex 2529 "At most one" double quant...
moexexv 2530 "At most one" double quant...
2moex 2531 Double quantification with...
2euex 2532 Double quantification with...
2eumo 2533 Double quantification with...
2eu2ex 2534 Double existential uniquen...
2moswap 2535 A condition allowing swap ...
2euswap 2536 A condition allowing swap ...
2exeu 2537 Double existential uniquen...
2mo2 2538 This theorem extends the i...
2mo 2539 Two equivalent expressions...
2mos 2540 Double "exists at most one...
2eu1 2541 Double existential uniquen...
2eu2 2542 Double existential uniquen...
2eu3 2543 Double existential uniquen...
2eu4 2544 This theorem provides us w...
2eu5 2545 An alternate definition of...
2eu6 2546 Two equivalent expressions...
2eu7 2547 Two equivalent expressions...
2eu8 2548 Two equivalent expressions...
exists1 2549 Two ways to express "only ...
exists2 2550 A condition implying that ...
barbara 2551 "Barbara", one of the fund...
celarent 2552 "Celarent", one of the syl...
darii 2553 "Darii", one of the syllog...
ferio 2554 "Ferio" ("Ferioque"), one ...
barbari 2555 "Barbari", one of the syll...
celaront 2556 "Celaront", one of the syl...
cesare 2557 "Cesare", one of the syllo...
camestres 2558 "Camestres", one of the sy...
festino 2559 "Festino", one of the syll...
baroco 2560 "Baroco", one of the syllo...
cesaro 2561 "Cesaro", one of the syllo...
camestros 2562 "Camestros", one of the sy...
datisi 2563 "Datisi", one of the syllo...
disamis 2564 "Disamis", one of the syll...
ferison 2565 "Ferison", one of the syll...
bocardo 2566 "Bocardo", one of the syll...
felapton 2567 "Felapton", one of the syl...
darapti 2568 "Darapti", one of the syll...
calemes 2569 "Calemes", one of the syll...
dimatis 2570 "Dimatis", one of the syll...
fresison 2571 "Fresison", one of the syl...
calemos 2572 "Calemos", one of the syll...
fesapo 2573 "Fesapo", one of the syllo...
bamalip 2574 "Bamalip", one of the syll...
axia1 2575 Left 'and' elimination (in...
axia2 2576 Right 'and' elimination (i...
axia3 2577 'And' introduction (intuit...
axin1 2578 'Not' introduction (intuit...
axin2 2579 'Not' elimination (intuiti...
axio 2580 Definition of 'or' (intuit...
axi4 2581 Specialization (intuitioni...
axi5r 2582 Converse of ax-c4 (intuiti...
axial 2583 The setvar ` x ` is not fr...
axie1 2584 The setvar ` x ` is not fr...
axie2 2585 A key property of existent...
axi9 2586 Axiom of existence (intuit...
axi10 2587 Axiom of Quantifier Substi...
axi12 2588 Axiom of Quantifier Introd...
axbnd 2589 Axiom of Bundling (intuiti...
axext2 2591 The Axiom of Extensionalit...
axext3 2592 A generalization of the Ax...
axext3ALT 2593 Alternate proof of ~ axext...
axext4 2594 A bidirectional version of...
bm1.1 2595 Any set defined by a prope...
abid 2598 Simplification of class ab...
hbab1 2599 Bound-variable hypothesis ...
nfsab1 2600 Bound-variable hypothesis ...
hbab 2601 Bound-variable hypothesis ...
nfsab 2602 Bound-variable hypothesis ...
dfcleq 2604 The same as ~ df-cleq with...
cvjust 2605 Every set is a class. Pro...
eqriv 2607 Infer equality of classes ...
eqrdv 2608 Deduce equality of classes...
eqrdav 2609 Deduce equality of classes...
eqid 2610 Law of identity (reflexivi...
eqidd 2611 Class identity law with an...
eqeq1d 2612 Deduction from equality to...
eqeq1dALT 2613 Shorter proof of ~ eqeq1d ...
eqeq1 2614 Equality implies equivalen...
eqeq1i 2615 Inference from equality to...
eqcomd 2616 Deduction from commutative...
eqcom 2617 Commutative law for class ...
eqcoms 2618 Inference applying commuta...
eqcomi 2619 Inference from commutative...
eqeq2d 2620 Deduction from equality to...
eqeq2 2621 Equality implies equivalen...
eqeq2i 2622 Inference from equality to...
eqeq12 2623 Equality relationship amon...
eqeq12i 2624 A useful inference for sub...
eqeq12d 2625 A useful inference for sub...
eqeqan12d 2626 A useful inference for sub...
eqeqan12dALT 2627 Alternate proof of ~ eqeqa...
eqeqan12rd 2628 A useful inference for sub...
eqtr 2629 Transitive law for class e...
eqtr2 2630 A transitive law for class...
eqtr3 2631 A transitive law for class...
eqtri 2632 An equality transitivity i...
eqtr2i 2633 An equality transitivity i...
eqtr3i 2634 An equality transitivity i...
eqtr4i 2635 An equality transitivity i...
3eqtri 2636 An inference from three ch...
3eqtrri 2637 An inference from three ch...
3eqtr2i 2638 An inference from three ch...
3eqtr2ri 2639 An inference from three ch...
3eqtr3i 2640 An inference from three ch...
3eqtr3ri 2641 An inference from three ch...
3eqtr4i 2642 An inference from three ch...
3eqtr4ri 2643 An inference from three ch...
eqtrd 2644 An equality transitivity d...
eqtr2d 2645 An equality transitivity d...
eqtr3d 2646 An equality transitivity e...
eqtr4d 2647 An equality transitivity e...
3eqtrd 2648 A deduction from three cha...
3eqtrrd 2649 A deduction from three cha...
3eqtr2d 2650 A deduction from three cha...
3eqtr2rd 2651 A deduction from three cha...
3eqtr3d 2652 A deduction from three cha...
3eqtr3rd 2653 A deduction from three cha...
3eqtr4d 2654 A deduction from three cha...
3eqtr4rd 2655 A deduction from three cha...
syl5eq 2656 An equality transitivity d...
syl5req 2657 An equality transitivity d...
syl5eqr 2658 An equality transitivity d...
syl5reqr 2659 An equality transitivity d...
syl6eq 2660 An equality transitivity d...
syl6req 2661 An equality transitivity d...
syl6eqr 2662 An equality transitivity d...
syl6reqr 2663 An equality transitivity d...
sylan9eq 2664 An equality transitivity d...
sylan9req 2665 An equality transitivity d...
sylan9eqr 2666 An equality transitivity d...
3eqtr3g 2667 A chained equality inferen...
3eqtr3a 2668 A chained equality inferen...
3eqtr4g 2669 A chained equality inferen...
3eqtr4a 2670 A chained equality inferen...
eq2tri 2671 A compound transitive infe...
eleq1d 2672 Deduction from equality to...
eleq2d 2673 Deduction from equality to...
eleq2dOLD 2674 Obsolete proof of ~ eleq2d...
eleq2dALT 2675 Alternate proof of ~ eleq2...
eleq1 2676 Equality implies equivalen...
eleq2 2677 Equality implies equivalen...
eleq12 2678 Equality implies equivalen...
eleq1i 2679 Inference from equality to...
eleq2i 2680 Inference from equality to...
eleq12i 2681 Inference from equality to...
eleq12d 2682 Deduction from equality to...
eleq1a 2683 A transitive-type law rela...
eqeltri 2684 Substitution of equal clas...
eqeltrri 2685 Substitution of equal clas...
eleqtri 2686 Substitution of equal clas...
eleqtrri 2687 Substitution of equal clas...
eqeltrd 2688 Substitution of equal clas...
eqeltrrd 2689 Deduction that substitutes...
eleqtrd 2690 Deduction that substitutes...
eleqtrrd 2691 Deduction that substitutes...
syl5eqel 2692 A membership and equality ...
syl5eqelr 2693 A membership and equality ...
syl5eleq 2694 A membership and equality ...
syl5eleqr 2695 A membership and equality ...
syl6eqel 2696 A membership and equality ...
syl6eqelr 2697 A membership and equality ...
syl6eleq 2698 A membership and equality ...
syl6eleqr 2699 A membership and equality ...
3eltr3i 2700 Substitution of equal clas...
3eltr4i 2701 Substitution of equal clas...
3eltr3d 2702 Substitution of equal clas...
3eltr4d 2703 Substitution of equal clas...
3eltr3g 2704 Substitution of equal clas...
3eltr4g 2705 Substitution of equal clas...
eleq2s 2706 Substitution of equal clas...
eqneltrd 2707 If a class is not an eleme...
eqneltrrd 2708 If a class is not an eleme...
neleqtrd 2709 If a class is not an eleme...
neleqtrrd 2710 If a class is not an eleme...
cleqh 2711 Establish equality between...
nelneq 2712 A way of showing two class...
nelneq2 2713 A way of showing two class...
eqsb3lem 2714 Lemma for ~ eqsb3 . (Cont...
eqsb3 2715 Substitution applied to an...
clelsb3 2716 Substitution applied to an...
hbxfreq 2717 A utility lemma to transfe...
hblem 2718 Change the free variable o...
abeq2 2719 Equality of a class variab...
abeq1 2720 Equality of a class variab...
abeq2d 2721 Equality of a class variab...
abeq2i 2722 Equality of a class variab...
abeq1i 2723 Equality of a class variab...
abbi 2724 Equivalent wff's correspon...
abbi2i 2725 Equality of a class variab...
abbii 2726 Equivalent wff's yield equ...
abbid 2727 Equivalent wff's yield equ...
abbidv 2728 Equivalent wff's yield equ...
abbi2dv 2729 Deduction from a wff to a ...
abbi1dv 2730 Deduction from a wff to a ...
abid1 2731 Every class is equal to a ...
abid2 2732 A simplification of class ...
cbvab 2733 Rule used to change bound ...
cbvabv 2734 Rule used to change bound ...
clelab 2735 Membership of a class vari...
clabel 2736 Membership of a class abst...
sbab 2737 The right-hand side of the...
nfcjust 2739 Justification theorem for ...
nfci 2741 Deduce that a class ` A ` ...
nfcii 2742 Deduce that a class ` A ` ...
nfcr 2743 Consequence of the not-fre...
nfcrii 2744 Consequence of the not-fre...
nfcri 2745 Consequence of the not-fre...
nfcd 2746 Deduce that a class ` A ` ...
nfceqdf 2747 An equality theorem for ef...
nfceqi 2748 Equality theorem for class...
nfcxfr 2749 A utility lemma to transfe...
nfcxfrd 2750 A utility lemma to transfe...
nfcv 2751 If ` x ` is disjoint from ...
nfcvd 2752 If ` x ` is disjoint from ...
nfab1 2753 Bound-variable hypothesis ...
nfnfc1 2754 The setvar ` x ` is bound ...
nfab 2755 Bound-variable hypothesis ...
nfaba1 2756 Bound-variable hypothesis ...
nfcrd 2757 Consequence of the not-fre...
nfeqd 2758 Hypothesis builder for equ...
nfeld 2759 Hypothesis builder for ele...
nfnfc 2760 Hypothesis builder for ` F...
nfnfcALT 2761 Alternate proof of ~ nfnfc...
nfeq 2762 Hypothesis builder for equ...
nfel 2763 Hypothesis builder for ele...
nfeq1 2764 Hypothesis builder for equ...
nfel1 2765 Hypothesis builder for ele...
nfeq2 2766 Hypothesis builder for equ...
nfel2 2767 Hypothesis builder for ele...
drnfc1 2768 Formula-building lemma for...
drnfc2 2769 Formula-building lemma for...
nfabd2 2770 Bound-variable hypothesis ...
nfabd 2771 Bound-variable hypothesis ...
dvelimdc 2772 Deduction form of ~ dvelim...
dvelimc 2773 Version of ~ dvelim for cl...
nfcvf 2774 If ` x ` and ` y ` are dis...
nfcvf2 2775 If ` x ` and ` y ` are dis...
cleqf 2776 Establish equality between...
abid2f 2777 A simplification of class ...
abeq2f 2778 Equality of a class variab...
sbabel 2779 Theorem to move a substitu...
neii 2784 Inference associated with ...
neir 2785 Inference associated with ...
nne 2786 Negation of inequality. (...
neneqd 2787 Deduction eliminating ineq...
neneq 2788 From inequality to non equ...
neqned 2789 If it is not the case that...
neqne 2790 From non equality to inequ...
neirr 2791 No class is unequal to its...
exmidne 2792 Excluded middle with equal...
eqneqall 2793 A contradiction concerning...
nonconne 2794 Law of noncontradiction wi...
necon3ad 2795 Contrapositive law deducti...
necon3bd 2796 Contrapositive law deducti...
necon2ad 2797 Contrapositive inference f...
necon2bd 2798 Contrapositive inference f...
necon1ad 2799 Contrapositive deduction f...
necon1bd 2800 Contrapositive deduction f...
necon4ad 2801 Contrapositive inference f...
necon4bd 2802 Contrapositive inference f...
necon3d 2803 Contrapositive law deducti...
necon1d 2804 Contrapositive law deducti...
necon2d 2805 Contrapositive inference f...
necon4d 2806 Contrapositive inference f...
necon3ai 2807 Contrapositive inference f...
necon3bi 2808 Contrapositive inference f...
necon1ai 2809 Contrapositive inference f...
necon1bi 2810 Contrapositive inference f...
necon2ai 2811 Contrapositive inference f...
necon2bi 2812 Contrapositive inference f...
necon4ai 2813 Contrapositive inference f...
necon3i 2814 Contrapositive inference f...
necon1i 2815 Contrapositive inference f...
necon2i 2816 Contrapositive inference f...
necon4i 2817 Contrapositive inference f...
necon3abid 2818 Deduction from equality to...
necon3bbid 2819 Deduction from equality to...
necon1abid 2820 Contrapositive deduction f...
necon1bbid 2821 Contrapositive inference f...
necon4abid 2822 Contrapositive law deducti...
necon4bbid 2823 Contrapositive law deducti...
necon2abid 2824 Contrapositive deduction f...
necon2bbid 2825 Contrapositive deduction f...
necon3bid 2826 Deduction from equality to...
necon4bid 2827 Contrapositive law deducti...
necon3abii 2828 Deduction from equality to...
necon3bbii 2829 Deduction from equality to...
necon1abii 2830 Contrapositive inference f...
necon1bbii 2831 Contrapositive inference f...
necon2abii 2832 Contrapositive inference f...
necon2bbii 2833 Contrapositive inference f...
necon3bii 2834 Inference from equality to...
necom 2835 Commutation of inequality....
necomi 2836 Inference from commutative...
necomd 2837 Deduction from commutative...
nesym 2838 Characterization of inequa...
nesymi 2839 Inference associated with ...
nesymir 2840 Inference associated with ...
neeq1d 2841 Deduction for inequality. ...
neeq2d 2842 Deduction for inequality. ...
neeq12d 2843 Deduction for inequality. ...
neeq1 2844 Equality theorem for inequ...
neeq2 2845 Equality theorem for inequ...
neeq1i 2846 Inference for inequality. ...
neeq2i 2847 Inference for inequality. ...
neeq12i 2848 Inference for inequality. ...
eqnetrd 2849 Substitution of equal clas...
eqnetrrd 2850 Substitution of equal clas...
neeqtrd 2851 Substitution of equal clas...
eqnetri 2852 Substitution of equal clas...
eqnetrri 2853 Substitution of equal clas...
neeqtri 2854 Substitution of equal clas...
neeqtrri 2855 Substitution of equal clas...
neeqtrrd 2856 Substitution of equal clas...
syl5eqner 2857 A chained equality inferen...
3netr3d 2858 Substitution of equality i...
3netr4d 2859 Substitution of equality i...
3netr3g 2860 Substitution of equality i...
3netr4g 2861 Substitution of equality i...
nebi 2862 Contraposition law for ine...
pm13.18 2863 Theorem *13.18 in [Whitehe...
pm13.181 2864 Theorem *13.181 in [Whiteh...
pm2.61ine 2865 Inference eliminating an i...
pm2.21ddne 2866 A contradiction implies an...
pm2.61ne 2867 Deduction eliminating an i...
pm2.61dne 2868 Deduction eliminating an i...
pm2.61dane 2869 Deduction eliminating an i...
pm2.61da2ne 2870 Deduction eliminating two ...
pm2.61da3ne 2871 Deduction eliminating thre...
pm2.61iine 2872 Equality version of ~ pm2....
neor 2873 Logical OR with an equalit...
neanior 2874 A De Morgan's law for ineq...
ne3anior 2875 A De Morgan's law for ineq...
neorian 2876 A De Morgan's law for ineq...
nemtbir 2877 An inference from an inequ...
nelne1 2878 Two classes are different ...
nelne2 2879 Two classes are different ...
nelelne 2880 Two classes are different ...
neneor 2881 If two classes are differe...
nfne 2882 Bound-variable hypothesis ...
nfned 2883 Bound-variable hypothesis ...
nabbi 2884 Not equivalent wff's corre...
neli 2885 Inference associated with ...
nelir 2886 Inference associated with ...
neleq12d 2887 Equality theorem for negat...
neleq1 2888 Equality theorem for negat...
neleq2 2889 Equality theorem for negat...
nfnel 2890 Bound-variable hypothesis ...
nfneld 2891 Bound-variable hypothesis ...
nnel 2892 Negation of negated member...
elnelne1 2893 Two classes are different ...
elnelne2 2894 Two classes are different ...
nelcon3d 2895 Contrapositive law deducti...
rgen 2906 Generalization rule for re...
ralel 2907 All elements of a class ar...
rgenw 2908 Generalization rule for re...
rgen2w 2909 Generalization rule for re...
mprg 2910 Modus ponens combined with...
mprgbir 2911 Modus ponens on biconditio...
alral 2912 Universal quantification i...
rsp 2913 Restricted specialization....
rspa 2914 Restricted specialization....
rspec 2915 Specialization rule for re...
r19.21bi 2916 Inference from Theorem 19....
r19.21be 2917 Inference from Theorem 19....
rspec2 2918 Specialization rule for re...
rspec3 2919 Specialization rule for re...
rsp2 2920 Restricted specialization,...
r2allem 2921 Lemma factoring out common...
r2alf 2922 Double restricted universa...
r2al 2923 Double restricted universa...
r3al 2924 Triple restricted universa...
nfra1 2925 The setvar ` x ` is not fr...
hbra1 2926 The setvar ` x ` is not fr...
hbral 2927 Bound-variable hypothesis ...
nfrald 2928 Deduction version of ~ nfr...
nfral 2929 Bound-variable hypothesis ...
nfra2 2930 Similar to Lemma 24 of [Mo...
ral2imi 2931 Inference quantifying ante...
ralim 2932 Distribution of restricted...
ralimi2 2933 Inference quantifying both...
ralimia 2934 Inference quantifying both...
ralimiaa 2935 Inference quantifying both...
ralimi 2936 Inference quantifying both...
hbralrimi 2937 Inference from Theorem 19....
r19.21t 2938 Restricted quantifier vers...
r19.21 2939 Restricted quantifier vers...
ralrimi 2940 Inference from Theorem 19....
ralimdaa 2941 Deduction quantifying both...
ralrimd 2942 Inference from Theorem 19....
r19.21v 2943 Restricted quantifier vers...
ralimdv2 2944 Inference quantifying both...
ralimdva 2945 Deduction quantifying both...
ralimdv 2946 Deduction quantifying both...
ralimdvva 2947 Deduction doubly quantifyi...
ralrimiv 2948 Inference from Theorem 19....
ralrimiva 2949 Inference from Theorem 19....
ralrimivw 2950 Inference from Theorem 19....
ralrimdv 2951 Inference from Theorem 19....
ralrimdva 2952 Inference from Theorem 19....
ralrimivv 2953 Inference from Theorem 19....
ralrimivva 2954 Inference from Theorem 19....
ralrimivvva 2955 Inference from Theorem 19....
ralrimdvv 2956 Inference from Theorem 19....
ralrimdvva 2957 Inference from Theorem 19....
rgen2 2958 Generalization rule for re...
rgen3 2959 Generalization rule for re...
rgen2a 2960 Generalization rule for re...
ralbii2 2961 Inference adding different...
ralbiia 2962 Inference adding restricte...
ralbii 2963 Inference adding restricte...
2ralbii 2964 Inference adding two restr...
ralbida 2965 Formula-building rule for ...
ralbid 2966 Formula-building rule for ...
ralbidv2 2967 Formula-building rule for ...
ralbidva 2968 Formula-building rule for ...
ralbidv 2969 Formula-building rule for ...
2ralbida 2970 Formula-building rule for ...
2ralbidva 2971 Formula-building rule for ...
2ralbidv 2972 Formula-building rule for ...
raleqbii 2973 Equality deduction for res...
raln 2974 Restricted universally qua...
ralnex 2975 Relationship between restr...
ralnexOLD 2976 Obsolete proof of ~ ralnex...
dfral2 2977 Relationship between restr...
rexnal 2978 Relationship between restr...
dfrex2 2979 Relationship between restr...
ralinexa 2980 A transformation of restri...
rexanali 2981 A transformation of restri...
nrexralim 2982 Negation of a complex pred...
nrex 2983 Inference adding restricte...
nrexdv 2984 Deduction adding restricte...
rexex 2985 Restricted existence impli...
rspe 2986 Restricted specialization....
rsp2e 2987 Restricted specialization....
nfre1 2988 The setvar ` x ` is not fr...
nfrexd 2989 Deduction version of ~ nfr...
nfrex 2990 Bound-variable hypothesis ...
rexim 2991 Theorem 19.22 of [Margaris...
reximia 2992 Inference quantifying both...
reximi2 2993 Inference quantifying both...
reximi 2994 Inference quantifying both...
reximdai 2995 Deduction from Theorem 19....
reximd2a 2996 Deduction quantifying both...
reximdv2 2997 Deduction quantifying both...
reximdvai 2998 Deduction quantifying both...
reximdv 2999 Deduction from Theorem 19....
reximdva 3000 Deduction quantifying both...
reximddv 3001 Deduction from Theorem 19....
reximddv2 3002 Double deduction from Theo...
r19.23t 3003 Closed theorem form of ~ r...
r19.23 3004 Restricted quantifier vers...
r19.23v 3005 Restricted quantifier vers...
rexlimi 3006 Restricted quantifier vers...
rexlimd2 3007 Version of ~ rexlimd with ...
rexlimd 3008 Deduction form of ~ rexlim...
rexlimiv 3009 Inference from Theorem 19....
rexlimiva 3010 Inference from Theorem 19....
rexlimivw 3011 Weaker version of ~ rexlim...
rexlimdv 3012 Inference from Theorem 19....
rexlimdva 3013 Inference from Theorem 19....
rexlimdvaa 3014 Inference from Theorem 19....
rexlimdv3a 3015 Inference from Theorem 19....
rexlimdvw 3016 Inference from Theorem 19....
rexlimddv 3017 Restricted existential eli...
rexlimivv 3018 Inference from Theorem 19....
rexlimdvv 3019 Inference from Theorem 19....
rexlimdvva 3020 Inference from Theorem 19....
rexbii2 3021 Inference adding different...
rexbiia 3022 Inference adding restricte...
rexbii 3023 Inference adding restricte...
2rexbii 3024 Inference adding two restr...
rexnal2 3025 Relationship between two r...
rexnal3 3026 Relationship between three...
ralnex2 3027 Relationship between two r...
ralnex3 3028 Relationship between three...
rexbida 3029 Formula-building rule for ...
rexbidv2 3030 Formula-building rule for ...
rexbidva 3031 Formula-building rule for ...
rexbidvaALT 3032 Alternate proof of ~ rexbi...
rexbid 3033 Formula-building rule for ...
rexbidv 3034 Formula-building rule for ...
rexbidvALT 3035 Alternate proof of ~ rexbi...
rexeqbii 3036 Equality deduction for res...
2rexbiia 3037 Inference adding two restr...
2rexbidva 3038 Formula-building rule for ...
2rexbidv 3039 Formula-building rule for ...
rexralbidv 3040 Formula-building rule for ...
r2exlem 3041 Lemma factoring out common...
r2exf 3042 Double restricted existent...
r2ex 3043 Double restricted existent...
risset 3044 Two ways to say " ` A ` be...
r19.12 3045 Restricted quantifier vers...
r19.26 3046 Restricted quantifier vers...
r19.26-2 3047 Restricted quantifier vers...
r19.26-3 3048 Version of ~ r19.26 with t...
r19.26m 3049 Version of ~ 19.26 and ~ r...
ralbi 3050 Distribute a restricted un...
ralbiim 3051 Split a biconditional and ...
r19.27v 3052 Restricted quantitifer ver...
r19.28v 3053 Restricted quantifier vers...
r19.29 3054 Restricted quantifier vers...
r19.29r 3055 Restricted quantifier vers...
r19.29imd 3056 Theorem 19.29 of [Margaris...
r19.29af2 3057 A commonly used pattern ba...
r19.29af 3058 A commonly used pattern ba...
r19.29an 3059 A commonly used pattern ba...
r19.29a 3060 A commonly used pattern ba...
r19.29d2r 3061 Theorem 19.29 of [Margaris...
r19.29vva 3062 A commonly used pattern ba...
r19.30 3063 Restricted quantifier vers...
r19.32v 3064 Restricted quantifier vers...
r19.35 3065 Restricted quantifier vers...
r19.36v 3066 Restricted quantifier vers...
r19.37 3067 Restricted quantifier vers...
r19.37v 3068 Restricted quantifier vers...
r19.40 3069 Restricted quantifier vers...
r19.41v 3070 Restricted quantifier vers...
r19.41 3071 Restricted quantifier vers...
r19.41vv 3072 Version of ~ r19.41v with ...
r19.42v 3073 Restricted quantifier vers...
r19.43 3074 Restricted quantifier vers...
r19.44v 3075 One direction of a restric...
r19.45v 3076 Restricted quantifier vers...
ralcomf 3077 Commutation of restricted ...
rexcomf 3078 Commutation of restricted ...
ralcom 3079 Commutation of restricted ...
rexcom 3080 Commutation of restricted ...
rexcom13 3081 Swap first and third restr...
rexrot4 3082 Rotate four restricted exi...
ralcom2 3083 Commutation of restricted ...
ralcom3 3084 A commutation law for rest...
reean 3085 Rearrange restricted exist...
reeanv 3086 Rearrange restricted exist...
3reeanv 3087 Rearrange three restricted...
2ralor 3088 Distribute restricted univ...
nfreu1 3089 The setvar ` x ` is not fr...
nfrmo1 3090 The setvar ` x ` is not fr...
nfreud 3091 Deduction version of ~ nfr...
nfrmod 3092 Deduction version of ~ nfr...
nfreu 3093 Bound-variable hypothesis ...
nfrmo 3094 Bound-variable hypothesis ...
rabid 3095 An "identity" law of concr...
rabid2 3096 An "identity" law for rest...
rabbi 3097 Equivalent wff's correspon...
rabswap 3098 Swap with a membership rel...
nfrab1 3099 The abstraction variable i...
nfrab 3100 A variable not free in a w...
reubida 3101 Formula-building rule for ...
reubidva 3102 Formula-building rule for ...
reubidv 3103 Formula-building rule for ...
reubiia 3104 Formula-building rule for ...
reubii 3105 Formula-building rule for ...
rmobida 3106 Formula-building rule for ...
rmobidva 3107 Formula-building rule for ...
rmobidv 3108 Formula-building rule for ...
rmobiia 3109 Formula-building rule for ...
rmobii 3110 Formula-building rule for ...
raleqf 3111 Equality theorem for restr...
rexeqf 3112 Equality theorem for restr...
reueq1f 3113 Equality theorem for restr...
rmoeq1f 3114 Equality theorem for restr...
raleq 3115 Equality theorem for restr...
rexeq 3116 Equality theorem for restr...
reueq1 3117 Equality theorem for restr...
rmoeq1 3118 Equality theorem for restr...
raleqi 3119 Equality inference for res...
rexeqi 3120 Equality inference for res...
raleqdv 3121 Equality deduction for res...
rexeqdv 3122 Equality deduction for res...
raleqbi1dv 3123 Equality deduction for res...
rexeqbi1dv 3124 Equality deduction for res...
reueqd 3125 Equality deduction for res...
rmoeqd 3126 Equality deduction for res...
raleqbid 3127 Equality deduction for res...
rexeqbid 3128 Equality deduction for res...
raleqbidv 3129 Equality deduction for res...
rexeqbidv 3130 Equality deduction for res...
raleqbidva 3131 Equality deduction for res...
rexeqbidva 3132 Equality deduction for res...
raleleq 3133 All elements of a class ar...
raleleqALT 3134 Alternate proof of ~ ralel...
mormo 3135 Unrestricted "at most one"...
reu5 3136 Restricted uniqueness in t...
reurex 3137 Restricted unique existenc...
reurmo 3138 Restricted existential uni...
rmo5 3139 Restricted "at most one" i...
nrexrmo 3140 Nonexistence implies restr...
cbvralf 3141 Rule used to change bound ...
cbvrexf 3142 Rule used to change bound ...
cbvral 3143 Rule used to change bound ...
cbvrex 3144 Rule used to change bound ...
cbvreu 3145 Change the bound variable ...
cbvrmo 3146 Change the bound variable ...
cbvralv 3147 Change the bound variable ...
cbvrexv 3148 Change the bound variable ...
cbvreuv 3149 Change the bound variable ...
cbvrmov 3150 Change the bound variable ...
cbvraldva2 3151 Rule used to change the bo...
cbvrexdva2 3152 Rule used to change the bo...
cbvraldva 3153 Rule used to change the bo...
cbvrexdva 3154 Rule used to change the bo...
cbvral2v 3155 Change bound variables of ...
cbvrex2v 3156 Change bound variables of ...
cbvral3v 3157 Change bound variables of ...
cbvralsv 3158 Change bound variable by u...
cbvrexsv 3159 Change bound variable by u...
sbralie 3160 Implicit to explicit subst...
rabbiia 3161 Equivalent wff's yield equ...
rabbidva2 3162 Equivalent wff's yield equ...
rabbidva 3163 Equivalent wff's yield equ...
rabbidv 3164 Equivalent wff's yield equ...
rabeqf 3165 Equality theorem for restr...
rabeq 3166 Equality theorem for restr...
rabeqdv 3167 Equality of restricted cla...
rabeqbidv 3168 Equality of restricted cla...
rabeqbidva 3169 Equality of restricted cla...
rabeq2i 3170 Inference rule from equali...
cbvrab 3171 Rule to change the bound v...
cbvrabv 3172 Rule to change the bound v...
vjust 3174 Soundness justification th...
vex 3176 All setvar variables are s...
eqvf 3177 The universe contains ever...
eqv 3178 The universe contains ever...
abv 3179 The class of sets verifyin...
isset 3180 Two ways to say " ` A ` is...
issetf 3181 A version of ~ isset that ...
isseti 3182 A way to say " ` A ` is a ...
issetri 3183 A way to say " ` A ` is a ...
eqvisset 3184 A class equal to a variabl...
elex 3185 If a class is a member of ...
elexi 3186 If a class is a member of ...
elexd 3187 If a class is a member of ...
elisset 3188 An element of a class exis...
elex2 3189 If a class contains anothe...
elex22 3190 If two classes each contai...
prcnel 3191 A proper class doesn't bel...
ralv 3192 A universal quantifier res...
rexv 3193 An existential quantifier ...
reuv 3194 A uniqueness quantifier re...
rmov 3195 A uniqueness quantifier re...
rabab 3196 A class abstraction restri...
ralcom4 3197 Commutation of restricted ...
rexcom4 3198 Commutation of restricted ...
rexcom4a 3199 Specialized existential co...
rexcom4b 3200 Specialized existential co...
ceqsalt 3201 Closed theorem version of ...
ceqsralt 3202 Restricted quantifier vers...
ceqsalg 3203 A representation of explic...
ceqsalgALT 3204 Alternate proof of ~ ceqsa...
ceqsal 3205 A representation of explic...
ceqsalv 3206 A representation of explic...
ceqsralv 3207 Restricted quantifier vers...
gencl 3208 Implicit substitution for ...
2gencl 3209 Implicit substitution for ...
3gencl 3210 Implicit substitution for ...
cgsexg 3211 Implicit substitution infe...
cgsex2g 3212 Implicit substitution infe...
cgsex4g 3213 An implicit substitution i...
ceqsex 3214 Elimination of an existent...
ceqsexv 3215 Elimination of an existent...
ceqsexv2d 3216 Elimination of an existent...
ceqsex2 3217 Elimination of two existen...
ceqsex2v 3218 Elimination of two existen...
ceqsex3v 3219 Elimination of three exist...
ceqsex4v 3220 Elimination of four existe...
ceqsex6v 3221 Elimination of six existen...
ceqsex8v 3222 Elimination of eight exist...
gencbvex 3223 Change of bound variable u...
gencbvex2 3224 Restatement of ~ gencbvex ...
gencbval 3225 Change of bound variable u...
sbhypf 3226 Introduce an explicit subs...
vtoclgft 3227 Closed theorem form of ~ v...
vtoclgftOLD 3228 Obsolete proof of ~ vtoclg...
vtocldf 3229 Implicit substitution of a...
vtocld 3230 Implicit substitution of a...
vtoclf 3231 Implicit substitution of a...
vtocl 3232 Implicit substitution of a...
vtoclALT 3233 Alternate proof of ~ vtocl...
vtocl2 3234 Implicit substitution of c...
vtocl3 3235 Implicit substitution of c...
vtoclb 3236 Implicit substitution of a...
vtoclgf 3237 Implicit substitution of a...
vtoclg1f 3238 Version of ~ vtoclgf with ...
vtoclg 3239 Implicit substitution of a...
vtoclbg 3240 Implicit substitution of a...
vtocl2gf 3241 Implicit substitution of a...
vtocl3gf 3242 Implicit substitution of a...
vtocl2g 3243 Implicit substitution of 2...
vtoclgaf 3244 Implicit substitution of a...
vtoclga 3245 Implicit substitution of a...
vtocl2gaf 3246 Implicit substitution of 2...
vtocl2ga 3247 Implicit substitution of 2...
vtocl3gaf 3248 Implicit substitution of 3...
vtocl3ga 3249 Implicit substitution of 3...
vtocl4g 3250 Implicit substitution of 4...
vtocl4ga 3251 Implicit substitution of 4...
vtocleg 3252 Implicit substitution of a...
vtoclegft 3253 Implicit substitution of a...
vtoclef 3254 Implicit substitution of a...
vtocle 3255 Implicit substitution of a...
vtoclri 3256 Implicit substitution of a...
spcimgft 3257 A closed version of ~ spci...
spcgft 3258 A closed version of ~ spcg...
spcimgf 3259 Rule of specialization, us...
spcimegf 3260 Existential specialization...
spcgf 3261 Rule of specialization, us...
spcegf 3262 Existential specialization...
spcimdv 3263 Restricted specialization,...
spcdv 3264 Rule of specialization, us...
spcimedv 3265 Restricted existential spe...
spcgv 3266 Rule of specialization, us...
spcegv 3267 Existential specialization...
spc2egv 3268 Existential specialization...
spc2gv 3269 Specialization with two qu...
spc3egv 3270 Existential specialization...
spc3gv 3271 Specialization with three ...
spcv 3272 Rule of specialization, us...
spcev 3273 Existential specialization...
spc2ev 3274 Existential specialization...
rspct 3275 A closed version of ~ rspc...
rspc 3276 Restricted specialization,...
rspce 3277 Restricted existential spe...
rspcv 3278 Restricted specialization,...
rspccv 3279 Restricted specialization,...
rspcva 3280 Restricted specialization,...
rspccva 3281 Restricted specialization,...
rspcev 3282 Restricted existential spe...
rspcimdv 3283 Restricted specialization,...
rspcimedv 3284 Restricted existential spe...
rspcdv 3285 Restricted specialization,...
rspcedv 3286 Restricted existential spe...
rspcda 3287 Restricted specialization,...
rspcdva 3288 Restricted specialization,...
rspcedvd 3289 Restricted existential spe...
rspcedeq1vd 3290 Restricted existential spe...
rspcedeq2vd 3291 Restricted existential spe...
rspc2 3292 2-variable restricted spec...
rspc2v 3293 2-variable restricted spec...
rspc2va 3294 2-variable restricted spec...
rspc2ev 3295 2-variable restricted exis...
rspc3v 3296 3-variable restricted spec...
rspc3ev 3297 3-variable restricted exis...
ralxpxfr2d 3298 Transfer a universal quant...
rexraleqim 3299 Statement following from e...
eqvinc 3300 A variable introduction la...
eqvincf 3301 A variable introduction la...
alexeqg 3302 Two ways to express substi...
ceqex 3303 Equality implies equivalen...
ceqsexg 3304 A representation of explic...
ceqsexgv 3305 Elimination of an existent...
ceqsrexv 3306 Elimination of a restricte...
ceqsrexbv 3307 Elimination of a restricte...
ceqsrex2v 3308 Elimination of a restricte...
clel2 3309 An alternate definition of...
clel3g 3310 An alternate definition of...
clel3 3311 An alternate definition of...
clel4 3312 An alternate definition of...
pm13.183 3313 Compare theorem *13.183 in...
rr19.3v 3314 Restricted quantifier vers...
rr19.28v 3315 Restricted quantifier vers...
elabgt 3316 Membership in a class abst...
elabgf 3317 Membership in a class abst...
elabf 3318 Membership in a class abst...
elab 3319 Membership in a class abst...
elabg 3320 Membership in a class abst...
elabd 3321 Explicit demonstration the...
elab2g 3322 Membership in a class abst...
elab2 3323 Membership in a class abst...
elab4g 3324 Membership in a class abst...
elab3gf 3325 Membership in a class abst...
elab3g 3326 Membership in a class abst...
elab3 3327 Membership in a class abst...
elrabi 3328 Implication for the member...
elrabf 3329 Membership in a restricted...
elrab3t 3330 Membership in a restricted...
elrab 3331 Membership in a restricted...
elrab3 3332 Membership in a restricted...
elrab2 3333 Membership in a class abst...
ralab 3334 Universal quantification o...
ralrab 3335 Universal quantification o...
rexab 3336 Existential quantification...
rexrab 3337 Existential quantification...
ralab2 3338 Universal quantification o...
ralrab2 3339 Universal quantification o...
rexab2 3340 Existential quantification...
rexrab2 3341 Existential quantification...
abidnf 3342 Identity used to create cl...
dedhb 3343 A deduction theorem for co...
eqeu 3344 A condition which implies ...
eueq 3345 Equality has existential u...
eueq1 3346 Equality has existential u...
eueq2 3347 Equality has existential u...
eueq3 3348 Equality has existential u...
moeq 3349 There is at most one set e...
moeq3 3350 "At most one" property of ...
mosub 3351 "At most one" remains true...
mo2icl 3352 Theorem for inferring "at ...
mob2 3353 Consequence of "at most on...
moi2 3354 Consequence of "at most on...
mob 3355 Equality implied by "at mo...
moi 3356 Equality implied by "at mo...
morex 3357 Derive membership from uni...
euxfr2 3358 Transfer existential uniqu...
euxfr 3359 Transfer existential uniqu...
euind 3360 Existential uniqueness via...
reu2 3361 A way to express restricte...
reu6 3362 A way to express restricte...
reu3 3363 A way to express restricte...
reu6i 3364 A condition which implies ...
eqreu 3365 A condition which implies ...
rmo4 3366 Restricted "at most one" u...
reu4 3367 Restricted uniqueness usin...
reu7 3368 Restricted uniqueness usin...
reu8 3369 Restricted uniqueness usin...
reu2eqd 3370 Deduce equality from restr...
reueq 3371 Equality has existential u...
rmoeq 3372 Equality's restricted exis...
rmoan 3373 Restricted "at most one" s...
rmoim 3374 Restricted "at most one" i...
rmoimia 3375 Restricted "at most one" i...
rmoimi2 3376 Restricted "at most one" i...
2reuswap 3377 A condition allowing swap ...
reuind 3378 Existential uniqueness via...
2rmorex 3379 Double restricted quantifi...
2reu5lem1 3380 Lemma for ~ 2reu5 . Note ...
2reu5lem2 3381 Lemma for ~ 2reu5 . (Cont...
2reu5lem3 3382 Lemma for ~ 2reu5 . This ...
2reu5 3383 Double restricted existent...
nelrdva 3384 Deduce negative membership...
cdeqi 3387 Deduce conditional equalit...
cdeqri 3388 Property of conditional eq...
cdeqth 3389 Deduce conditional equalit...
cdeqnot 3390 Distribute conditional equ...
cdeqal 3391 Distribute conditional equ...
cdeqab 3392 Distribute conditional equ...
cdeqal1 3393 Distribute conditional equ...
cdeqab1 3394 Distribute conditional equ...
cdeqim 3395 Distribute conditional equ...
cdeqcv 3396 Conditional equality for s...
cdeqeq 3397 Distribute conditional equ...
cdeqel 3398 Distribute conditional equ...
nfcdeq 3399 If we have a conditional e...
nfccdeq 3400 Variation of ~ nfcdeq for ...
ru 3401 Russell's Paradox. Propos...
dfsbcq 3404 Proper substitution of a c...
dfsbcq2 3405 This theorem, which is sim...
sbsbc 3406 Show that ~ df-sb and ~ df...
sbceq1d 3407 Equality theorem for class...
sbceq1dd 3408 Equality theorem for class...
sbceqbid 3409 Equality theorem for class...
sbc8g 3410 This is the closest we can...
sbc2or 3411 The disjunction of two equ...
sbcex 3412 By our definition of prope...
sbceq1a 3413 Equality theorem for class...
sbceq2a 3414 Equality theorem for class...
spsbc 3415 Specialization: if a formu...
spsbcd 3416 Specialization: if a formu...
sbcth 3417 A substitution into a theo...
sbcthdv 3418 Deduction version of ~ sbc...
sbcid 3419 An identity theorem for su...
nfsbc1d 3420 Deduction version of ~ nfs...
nfsbc1 3421 Bound-variable hypothesis ...
nfsbc1v 3422 Bound-variable hypothesis ...
nfsbcd 3423 Deduction version of ~ nfs...
nfsbc 3424 Bound-variable hypothesis ...
sbcco 3425 A composition law for clas...
sbcco2 3426 A composition law for clas...
sbc5 3427 An equivalence for class s...
sbc6g 3428 An equivalence for class s...
sbc6 3429 An equivalence for class s...
sbc7 3430 An equivalence for class s...
cbvsbc 3431 Change bound variables in ...
cbvsbcv 3432 Change the bound variable ...
sbciegft 3433 Conversion of implicit sub...
sbciegf 3434 Conversion of implicit sub...
sbcieg 3435 Conversion of implicit sub...
sbcie2g 3436 Conversion of implicit sub...
sbcie 3437 Conversion of implicit sub...
sbciedf 3438 Conversion of implicit sub...
sbcied 3439 Conversion of implicit sub...
sbcied2 3440 Conversion of implicit sub...
elrabsf 3441 Membership in a restricted...
eqsbc3 3442 Substitution applied to an...
sbcng 3443 Move negation in and out o...
sbcimg 3444 Distribution of class subs...
sbcan 3445 Distribution of class subs...
sbcor 3446 Distribution of class subs...
sbcbig 3447 Distribution of class subs...
sbcn1 3448 Move negation in and out o...
sbcim1 3449 Distribution of class subs...
sbcbi1 3450 Distribution of class subs...
sbcbi2 3451 Substituting into equivale...
sbcal 3452 Move universal quantifier ...
sbcex2 3453 Move existential quantifie...
sbceqal 3454 Set theory version of ~ sb...
sbeqalb 3455 Theorem *14.121 in [Whiteh...
sbcbid 3456 Formula-building deduction...
sbcbidv 3457 Formula-building deduction...
sbcbii 3458 Formula-building inference...
eqsbc3r 3459 ~ eqsbc3 with setvar varia...
eqsbc3rOLD 3460 Obsolete proof of ~ eqsbc3...
sbc3an 3461 Distribution of class subs...
sbcel1v 3462 Class substitution into a ...
sbcel2gv 3463 Class substitution into a ...
sbcel21v 3464 Class substitution into a ...
sbcimdv 3465 Substitution analogue of T...
sbcimdvOLD 3466 Obsolete proof of ~ sbcimd...
sbctt 3467 Substitution for a variabl...
sbcgf 3468 Substitution for a variabl...
sbc19.21g 3469 Substitution for a variabl...
sbcg 3470 Substitution for a variabl...
sbc2iegf 3471 Conversion of implicit sub...
sbc2ie 3472 Conversion of implicit sub...
sbc2iedv 3473 Conversion of implicit sub...
sbc3ie 3474 Conversion of implicit sub...
sbccomlem 3475 Lemma for ~ sbccom . (Con...
sbccom 3476 Commutative law for double...
sbcralt 3477 Interchange class substitu...
sbcrext 3478 Interchange class substitu...
sbcrextOLD 3479 Obsolete proof of ~ sbcrex...
sbcralg 3480 Interchange class substitu...
sbcrex 3481 Interchange class substitu...
sbcreu 3482 Interchange class substitu...
sbcabel 3483 Interchange class substitu...
rspsbc 3484 Restricted quantifier vers...
rspsbca 3485 Restricted quantifier vers...
rspesbca 3486 Existence form of ~ rspsbc...
spesbc 3487 Existence form of ~ spsbc ...
spesbcd 3488 form of ~ spsbc . (Contri...
sbcth2 3489 A substitution into a theo...
ra4v 3490 Version of ~ ra4 with a dv...
ra4 3491 Restricted quantifier vers...
rmo2 3492 Alternate definition of re...
rmo2i 3493 Condition implying restric...
rmo3 3494 Restricted "at most one" u...
rmob 3495 Consequence of "at most on...
rmoi 3496 Consequence of "at most on...
rmob2 3497 Consequence of "restricted...
rmoi2 3498 Consequence of "restricted...
csb2 3501 Alternate expression for t...
csbeq1 3502 Analogue of ~ dfsbcq for p...
csbeq2 3503 Substituting into equivale...
cbvcsb 3504 Change bound variables in ...
cbvcsbv 3505 Change the bound variable ...
csbeq1d 3506 Equality deduction for pro...
csbid 3507 Analogue of ~ sbid for pro...
csbeq1a 3508 Equality theorem for prope...
csbco 3509 Composition law for chaine...
csbtt 3510 Substitution doesn't affec...
csbconstgf 3511 Substitution doesn't affec...
csbconstg 3512 Substitution doesn't affec...
nfcsb1d 3513 Bound-variable hypothesis ...
nfcsb1 3514 Bound-variable hypothesis ...
nfcsb1v 3515 Bound-variable hypothesis ...
nfcsbd 3516 Deduction version of ~ nfc...
nfcsb 3517 Bound-variable hypothesis ...
csbhypf 3518 Introduce an explicit subs...
csbiebt 3519 Conversion of implicit sub...
csbiedf 3520 Conversion of implicit sub...
csbieb 3521 Bidirectional conversion b...
csbiebg 3522 Bidirectional conversion b...
csbiegf 3523 Conversion of implicit sub...
csbief 3524 Conversion of implicit sub...
csbie 3525 Conversion of implicit sub...
csbied 3526 Conversion of implicit sub...
csbied2 3527 Conversion of implicit sub...
csbie2t 3528 Conversion of implicit sub...
csbie2 3529 Conversion of implicit sub...
csbie2g 3530 Conversion of implicit sub...
cbvralcsf 3531 A more general version of ...
cbvrexcsf 3532 A more general version of ...
cbvreucsf 3533 A more general version of ...
cbvrabcsf 3534 A more general version of ...
cbvralv2 3535 Rule used to change the bo...
cbvrexv2 3536 Rule used to change the bo...
difjust 3542 Soundness justification th...
unjust 3544 Soundness justification th...
injust 3546 Soundness justification th...
dfin5 3548 Alternate definition for t...
dfdif2 3549 Alternate definition of cl...
eldif 3550 Expansion of membership in...
eldifd 3551 If a class is in one class...
eldifad 3552 If a class is in the diffe...
eldifbd 3553 If a class is in the diffe...
dfss 3555 Variant of subclass defini...
dfss2 3557 Alternate definition of th...
dfss3 3558 Alternate definition of su...
dfss2f 3559 Equivalence for subclass r...
dfss3f 3560 Equivalence for subclass r...
nfss 3561 If ` x ` is not free in ` ...
ssel 3562 Membership relationships f...
ssel2 3563 Membership relationships f...
sseli 3564 Membership inference from ...
sselii 3565 Membership inference from ...
sseldi 3566 Membership inference from ...
sseld 3567 Membership deduction from ...
sselda 3568 Membership deduction from ...
sseldd 3569 Membership inference from ...
ssneld 3570 If a class is not in anoth...
ssneldd 3571 If an element is not in a ...
ssriv 3572 Inference rule based on su...
ssrd 3573 Deduction rule based on su...
ssrdv 3574 Deduction rule based on su...
sstr2 3575 Transitivity of subclasses...
sstr 3576 Transitivity of subclasses...
sstri 3577 Subclass transitivity infe...
sstrd 3578 Subclass transitivity dedu...
syl5ss 3579 Subclass transitivity dedu...
syl6ss 3580 Subclass transitivity dedu...
sylan9ss 3581 A subclass transitivity de...
sylan9ssr 3582 A subclass transitivity de...
eqss 3583 The subclass relationship ...
eqssi 3584 Infer equality from two su...
eqssd 3585 Equality deduction from tw...
eqrd 3586 Deduce equality of classes...
ssid 3587 Any class is a subclass of...
ssv 3588 Any class is a subclass of...
sseq1 3589 Equality theorem for subcl...
sseq2 3590 Equality theorem for the s...
sseq12 3591 Equality theorem for the s...
sseq1i 3592 An equality inference for ...
sseq2i 3593 An equality inference for ...
sseq12i 3594 An equality inference for ...
sseq1d 3595 An equality deduction for ...
sseq2d 3596 An equality deduction for ...
sseq12d 3597 An equality deduction for ...
eqsstri 3598 Substitution of equality i...
eqsstr3i 3599 Substitution of equality i...
sseqtri 3600 Substitution of equality i...
sseqtr4i 3601 Substitution of equality i...
eqsstrd 3602 Substitution of equality i...
eqsstr3d 3603 Substitution of equality i...
sseqtrd 3604 Substitution of equality i...
sseqtr4d 3605 Substitution of equality i...
3sstr3i 3606 Substitution of equality i...
3sstr4i 3607 Substitution of equality i...
3sstr3g 3608 Substitution of equality i...
3sstr4g 3609 Substitution of equality i...
3sstr3d 3610 Substitution of equality i...
3sstr4d 3611 Substitution of equality i...
syl5eqss 3612 A chained subclass and equ...
syl5eqssr 3613 A chained subclass and equ...
syl6sseq 3614 A chained subclass and equ...
syl6sseqr 3615 A chained subclass and equ...
syl5sseq 3616 Subclass transitivity dedu...
syl5sseqr 3617 Subclass transitivity dedu...
syl6eqss 3618 A chained subclass and equ...
syl6eqssr 3619 A chained subclass and equ...
eqimss 3620 Equality implies the subcl...
eqimss2 3621 Equality implies the subcl...
eqimssi 3622 Infer subclass relationshi...
eqimss2i 3623 Infer subclass relationshi...
nssne1 3624 Two classes are different ...
nssne2 3625 Two classes are different ...
nss 3626 Negation of subclass relat...
nelss 3627 Demonstrate by witnesses t...
ssrexf 3628 restricted existential qua...
ssralv 3629 Quantification restricted ...
ssrexv 3630 Existential quantification...
ralss 3631 Restricted universal quant...
rexss 3632 Restricted existential qua...
ss2ab 3633 Class abstractions in a su...
abss 3634 Class abstraction in a sub...
ssab 3635 Subclass of a class abstra...
ssabral 3636 The relation for a subclas...
ss2abi 3637 Inference of abstraction s...
ss2abdv 3638 Deduction of abstraction s...
abssdv 3639 Deduction of abstraction s...
abssi 3640 Inference of abstraction s...
ss2rab 3641 Restricted abstraction cla...
rabss 3642 Restricted class abstracti...
ssrab 3643 Subclass of a restricted c...
ssrabdv 3644 Subclass of a restricted c...
rabssdv 3645 Subclass of a restricted c...
ss2rabdv 3646 Deduction of restricted ab...
ss2rabi 3647 Inference of restricted ab...
rabss2 3648 Subclass law for restricte...
ssab2 3649 Subclass relation for the ...
ssrab2 3650 Subclass relation for a re...
ssrabeq 3651 If the restricting class o...
rabssab 3652 A restricted class is a su...
uniiunlem 3653 A subset relationship usef...
dfpss2 3654 Alternate definition of pr...
dfpss3 3655 Alternate definition of pr...
psseq1 3656 Equality theorem for prope...
psseq2 3657 Equality theorem for prope...
psseq1i 3658 An equality inference for ...
psseq2i 3659 An equality inference for ...
psseq12i 3660 An equality inference for ...
psseq1d 3661 An equality deduction for ...
psseq2d 3662 An equality deduction for ...
psseq12d 3663 An equality deduction for ...
pssss 3664 A proper subclass is a sub...
pssne 3665 Two classes in a proper su...
pssssd 3666 Deduce subclass from prope...
pssned 3667 Proper subclasses are uneq...
sspss 3668 Subclass in terms of prope...
pssirr 3669 Proper subclass is irrefle...
pssn2lp 3670 Proper subclass has no 2-c...
sspsstri 3671 Two ways of stating tricho...
ssnpss 3672 Partial trichotomy law for...
psstr 3673 Transitive law for proper ...
sspsstr 3674 Transitive law for subclas...
psssstr 3675 Transitive law for subclas...
psstrd 3676 Proper subclass inclusion ...
sspsstrd 3677 Transitivity involving sub...
psssstrd 3678 Transitivity involving sub...
npss 3679 A class is not a proper su...
ssnelpss 3680 A subclass missing a membe...
ssnelpssd 3681 Subclass inclusion with on...
ssexnelpss 3682 If there is an element of ...
difeq1 3683 Equality theorem for class...
difeq2 3684 Equality theorem for class...
difeq12 3685 Equality theorem for class...
difeq1i 3686 Inference adding differenc...
difeq2i 3687 Inference adding differenc...
difeq12i 3688 Equality inference for cla...
difeq1d 3689 Deduction adding differenc...
difeq2d 3690 Deduction adding differenc...
difeq12d 3691 Equality deduction for cla...
difeqri 3692 Inference from membership ...
nfdif 3693 Bound-variable hypothesis ...
eldifi 3694 Implication of membership ...
eldifn 3695 Implication of membership ...
elndif 3696 A set does not belong to a...
neldif 3697 Implication of membership ...
difdif 3698 Double class difference. ...
difss 3699 Subclass relationship for ...
difssd 3700 A difference of two classe...
difss2 3701 If a class is contained in...
difss2d 3702 If a class is contained in...
ssdifss 3703 Preservation of a subclass...
ddif 3704 Double complement under un...
ssconb 3705 Contraposition law for sub...
sscon 3706 Contraposition law for sub...
ssdif 3707 Difference law for subsets...
ssdifd 3708 If ` A ` is contained in `...
sscond 3709 If ` A ` is contained in `...
ssdifssd 3710 If ` A ` is contained in `...
ssdif2d 3711 If ` A ` is contained in `...
raldifb 3712 Restricted universal quant...
complss 3713 Complementation reverses i...
compleq 3714 Two classes are equal if a...
elun 3715 Expansion of membership in...
elunnel1 3716 A member of a union that i...
uneqri 3717 Inference from membership ...
unidm 3718 Idempotent law for union o...
uncom 3719 Commutative law for union ...
equncom 3720 If a class equals the unio...
equncomi 3721 Inference form of ~ equnco...
uneq1 3722 Equality theorem for the u...
uneq2 3723 Equality theorem for the u...
uneq12 3724 Equality theorem for the u...
uneq1i 3725 Inference adding union to ...
uneq2i 3726 Inference adding union to ...
uneq12i 3727 Equality inference for the...
uneq1d 3728 Deduction adding union to ...
uneq2d 3729 Deduction adding union to ...
uneq12d 3730 Equality deduction for the...
nfun 3731 Bound-variable hypothesis ...
unass 3732 Associative law for union ...
un12 3733 A rearrangement of union. ...
un23 3734 A rearrangement of union. ...
un4 3735 A rearrangement of the uni...
unundi 3736 Union distributes over its...
unundir 3737 Union distributes over its...
ssun1 3738 Subclass relationship for ...
ssun2 3739 Subclass relationship for ...
ssun3 3740 Subclass law for union of ...
ssun4 3741 Subclass law for union of ...
elun1 3742 Membership law for union o...
elun2 3743 Membership law for union o...
unss1 3744 Subclass law for union of ...
ssequn1 3745 A relationship between sub...
unss2 3746 Subclass law for union of ...
unss12 3747 Subclass law for union of ...
ssequn2 3748 A relationship between sub...
unss 3749 The union of two subclasse...
unssi 3750 An inference showing the u...
unssd 3751 A deduction showing the un...
unssad 3752 If ` ( A u. B ) ` is conta...
unssbd 3753 If ` ( A u. B ) ` is conta...
ssun 3754 A condition that implies i...
rexun 3755 Restricted existential qua...
ralunb 3756 Restricted quantification ...
ralun 3757 Restricted quantification ...
elin 3758 Expansion of membership in...
elini 3759 Membership in an intersect...
elind 3760 Deduce membership in an in...
elinel1 3761 Membership in an intersect...
elinel2 3762 Membership in an intersect...
elin2 3763 Membership in a class defi...
elin1d 3764 Elementhood in the first s...
elin2d 3765 Elementhood in the first s...
elin3 3766 Membership in a class defi...
incom 3767 Commutative law for inters...
ineqri 3768 Inference from membership ...
ineq1 3769 Equality theorem for inter...
ineq2 3770 Equality theorem for inter...
ineq12 3771 Equality theorem for inter...
ineq1i 3772 Equality inference for int...
ineq2i 3773 Equality inference for int...
ineq12i 3774 Equality inference for int...
ineq1d 3775 Equality deduction for int...
ineq2d 3776 Equality deduction for int...
ineq12d 3777 Equality deduction for int...
ineqan12d 3778 Equality deduction for int...
sseqin2 3779 A relationship between sub...
dfss1OLD 3780 Obsolete as of 22-Jul-2021...
dfss5OLD 3781 Obsolete as of 22-Jul-2021...
nfin 3782 Bound-variable hypothesis ...
rabbi2dva 3783 Deduction from a wff to a ...
inidm 3784 Idempotent law for interse...
inass 3785 Associative law for inters...
in12 3786 A rearrangement of interse...
in32 3787 A rearrangement of interse...
in13 3788 A rearrangement of interse...
in31 3789 A rearrangement of interse...
inrot 3790 Rotate the intersection of...
in4 3791 Rearrangement of intersect...
inindi 3792 Intersection distributes o...
inindir 3793 Intersection distributes o...
sseqin2OLD 3794 Obsolete proof of ~ sseqin...
inss1 3795 The intersection of two cl...
inss2 3796 The intersection of two cl...
ssin 3797 Subclass of intersection. ...
ssini 3798 An inference showing that ...
ssind 3799 A deduction showing that a...
ssrin 3800 Add right intersection to ...
sslin 3801 Add left intersection to s...
ss2in 3802 Intersection of subclasses...
ssinss1 3803 Intersection preserves sub...
inss 3804 Inclusion of an intersecti...
symdifcom 3807 Symmetric difference commu...
symdifeq1 3808 Equality theorem for symme...
symdifeq2 3809 Equality theorem for symme...
nfsymdif 3810 Hypothesis builder for sym...
elsymdif 3811 Membership in a symmetric ...
elsymdifxor 3812 Membership in a symmetric ...
dfsymdif2 3813 Alternate definition of th...
symdif2 3814 Two ways to express symmet...
symdifass 3815 Symmetric difference assoc...
unabs 3816 Absorption law for union. ...
inabs 3817 Absorption law for interse...
nssinpss 3818 Negation of subclass expre...
nsspssun 3819 Negation of subclass expre...
dfss4 3820 Subclass defined in terms ...
dfun2 3821 An alternate definition of...
dfin2 3822 An alternate definition of...
difin 3823 Difference with intersecti...
dfun3 3824 Union defined in terms of ...
dfin3 3825 Intersection defined in te...
dfin4 3826 Alternate definition of th...
invdif 3827 Intersection with universa...
indif 3828 Intersection with class di...
indif2 3829 Bring an intersection in a...
indif1 3830 Bring an intersection in a...
indifcom 3831 Commutation law for inters...
indi 3832 Distributive law for inter...
undi 3833 Distributive law for union...
indir 3834 Distributive law for inter...
undir 3835 Distributive law for union...
unineq 3836 Infer equality from equali...
uneqin 3837 Equality of union and inte...
difundi 3838 Distributive law for class...
difundir 3839 Distributive law for class...
difindi 3840 Distributive law for class...
difindir 3841 Distributive law for class...
indifdir 3842 Distribute intersection ov...
difdif2 3843 Class difference by a clas...
undm 3844 De Morgan's law for union....
indm 3845 De Morgan's law for inters...
difun1 3846 A relationship involving d...
undif3 3847 An equality involving clas...
undif3OLD 3848 Obsolete proof of ~ undif3...
difin2 3849 Represent a class differen...
dif32 3850 Swap second and third argu...
difabs 3851 Absorption-like law for cl...
dfsymdif3 3852 Alternate definition of th...
unab 3853 Union of two class abstrac...
inab 3854 Intersection of two class ...
difab 3855 Difference of two class ab...
notab 3856 A class builder defined by...
unrab 3857 Union of two restricted cl...
inrab 3858 Intersection of two restri...
inrab2 3859 Intersection with a restri...
difrab 3860 Difference of two restrict...
dfrab3 3861 Alternate definition of re...
dfrab2 3862 Alternate definition of re...
notrab 3863 Complementation of restric...
dfrab3ss 3864 Restricted class abstracti...
rabun2 3865 Abstraction restricted to ...
reuss2 3866 Transfer uniqueness to a s...
reuss 3867 Transfer uniqueness to a s...
reuun1 3868 Transfer uniqueness to a s...
reuun2 3869 Transfer uniqueness to a s...
reupick 3870 Restricted uniqueness "pic...
reupick3 3871 Restricted uniqueness "pic...
reupick2 3872 Restricted uniqueness "pic...
euelss 3873 Transfer uniqueness of an ...
dfnul2 3876 Alternate definition of th...
dfnul3 3877 Alternate definition of th...
noel 3878 The empty set has no eleme...
n0i 3879 If a set has elements, the...
ne0i 3880 If a set has elements, the...
n0ii 3881 If a set has elements, the...
ne0ii 3882 If a set has elements, the...
vn0 3883 The universal class is not...
eq0f 3884 The empty set has no eleme...
neq0f 3885 A nonempty class has at le...
n0f 3886 A nonempty class has at le...
n0fOLD 3887 Obsolete proof of ~ n0f as...
eq0 3888 The empty set has no eleme...
neq0 3889 A nonempty class has at le...
n0 3890 A nonempty class has at le...
reximdva0 3891 Restricted existence deduc...
rspn0 3892 Specialization for restric...
n0moeu 3893 A case of equivalence of "...
rex0 3894 Vacuous existential quanti...
0el 3895 Membership of the empty se...
ssdif0 3896 Subclass expressed in term...
difn0 3897 If the difference of two s...
pssdifn0 3898 A proper subclass has a no...
pssdif 3899 A proper subclass has a no...
difin0ss 3900 Difference, intersection, ...
inssdif0 3901 Intersection, subclass, an...
difid 3902 The difference between a c...
difidALT 3903 Alternate proof of ~ difid...
dif0 3904 The difference between a c...
ab0 3905 The class of sets verifyin...
dfnf5 3906 Characterization of non-fr...
ab0orv 3907 The class builder of a wff...
abn0 3908 Nonempty class abstraction...
rab0 3909 Any restricted class abstr...
rab0OLD 3910 Obsolete proof of ~ rab0 a...
rabeq0 3911 Condition for a restricted...
rabn0 3912 Nonempty restricted class ...
rabn0OLD 3913 Obsolete proof of ~ rabn0 ...
rabeq0OLD 3914 Obsolete proof of ~ rabeq0...
rabxm 3915 Law of excluded middle, in...
rabnc 3916 Law of noncontradiction, i...
elneldisj 3917 The set of elements contai...
elnelun 3918 The union of the set of el...
un0 3919 The union of a class with ...
in0 3920 The intersection of a clas...
0in 3921 The intersection of the em...
inv1 3922 The intersection of a clas...
unv 3923 The union of a class with ...
0ss 3924 The null set is a subset o...
ss0b 3925 Any subset of the empty se...
ss0 3926 Any subset of the empty se...
sseq0 3927 A subclass of an empty cla...
ssn0 3928 A class with a nonempty su...
0dif 3929 The difference between the...
abf 3930 A class builder with a fal...
eq0rdv 3931 Deduction rule for equalit...
csbprc 3932 The proper substitution of...
csbprcOLD 3933 Obsolete proof of ~ csbprc...
csb0 3934 The proper substitution of...
sbcel12 3935 Distribute proper substitu...
sbceqg 3936 Distribute proper substitu...
sbcnel12g 3937 Distribute proper substitu...
sbcne12 3938 Distribute proper substitu...
sbcel1g 3939 Move proper substitution i...
sbceq1g 3940 Move proper substitution t...
sbcel2 3941 Move proper substitution i...
sbceq2g 3942 Move proper substitution t...
csbeq2d 3943 Formula-building deduction...
csbeq2dv 3944 Formula-building deduction...
csbeq2i 3945 Formula-building inference...
csbcom 3946 Commutative law for double...
sbcnestgf 3947 Nest the composition of tw...
csbnestgf 3948 Nest the composition of tw...
sbcnestg 3949 Nest the composition of tw...
csbnestg 3950 Nest the composition of tw...
sbcco3g 3951 Composition of two substit...
csbco3g 3952 Composition of two class s...
csbnest1g 3953 Nest the composition of tw...
csbidm 3954 Idempotent law for class s...
csbvarg 3955 The proper substitution of...
sbccsb 3956 Substitution into a wff ex...
sbccsb2 3957 Substitution into a wff ex...
rspcsbela 3958 Special case related to ~ ...
sbnfc2 3959 Two ways of expressing " `...
csbab 3960 Move substitution into a c...
csbun 3961 Distribution of class subs...
csbin 3962 Distribute proper substitu...
un00 3963 Two classes are empty iff ...
vss 3964 Only the universal class h...
0pss 3965 The null set is a proper s...
npss0 3966 No set is a proper subset ...
npss0OLD 3967 Obsolete proof of ~ npss0 ...
pssv 3968 Any non-universal class is...
disj 3969 Two ways of saying that tw...
disjr 3970 Two ways of saying that tw...
disj1 3971 Two ways of saying that tw...
reldisj 3972 Two ways of saying that tw...
disj3 3973 Two ways of saying that tw...
disjne 3974 Members of disjoint sets a...
disjel 3975 A set can't belong to both...
disj2 3976 Two ways of saying that tw...
disj4 3977 Two ways of saying that tw...
ssdisj 3978 Intersection with a subcla...
ssdisjOLD 3979 Obsolete proof of ~ ssdisj...
disjpss 3980 A class is a proper subset...
undisj1 3981 The union of disjoint clas...
undisj2 3982 The union of disjoint clas...
ssindif0 3983 Subclass expressed in term...
inelcm 3984 The intersection of classe...
minel 3985 A minimum element of a cla...
minelOLD 3986 Obsolete proof of ~ minel ...
undif4 3987 Distribute union over diff...
disjssun 3988 Subset relation for disjoi...
vdif0 3989 Universal class equality i...
difrab0eq 3990 If the difference between ...
pssnel 3991 A proper subclass has a me...
disjdif 3992 A class and its relative c...
difin0 3993 The difference of a class ...
unvdif 3994 The union of a class and i...
undif1 3995 Absorption of difference b...
undif2 3996 Absorption of difference b...
undifabs 3997 Absorption of difference b...
inundif 3998 The intersection and class...
disjdif2 3999 The difference of a class ...
difun2 4000 Absorption of union by dif...
undif 4001 Union of complementary par...
ssdifin0 4002 A subset of a difference d...
ssdifeq0 4003 A class is a subclass of i...
ssundif 4004 A condition equivalent to ...
difcom 4005 Swap the arguments of a cl...
pssdifcom1 4006 Two ways to express overla...
pssdifcom2 4007 Two ways to express non-co...
difdifdir 4008 Distributive law for class...
uneqdifeq 4009 Two ways to say that ` A `...
uneqdifeqOLD 4010 Obsolete proof of ~ uneqdi...
raldifeq 4011 Equality theorem for restr...
r19.2z 4012 Theorem 19.2 of [Margaris]...
r19.2zb 4013 A response to the notion t...
r19.3rz 4014 Restricted quantification ...
r19.28z 4015 Restricted quantifier vers...
r19.3rzv 4016 Restricted quantification ...
r19.9rzv 4017 Restricted quantification ...
r19.28zv 4018 Restricted quantifier vers...
r19.37zv 4019 Restricted quantifier vers...
r19.45zv 4020 Restricted version of Theo...
r19.44zv 4021 Restricted version of Theo...
r19.27z 4022 Restricted quantifier vers...
r19.27zv 4023 Restricted quantifier vers...
r19.36zv 4024 Restricted quantifier vers...
rzal 4025 Vacuous quantification is ...
rexn0 4026 Restricted existential qua...
ralidm 4027 Idempotent law for restric...
ral0 4028 Vacuous universal quantifi...
rgenzOLD 4029 Obsolete as of 22-Jul-2021...
ralf0 4030 The quantification of a fa...
ralf0OLD 4031 Obsolete proof of ~ ralf0 ...
raaan 4032 Rearrange restricted quant...
raaanv 4033 Rearrange restricted quant...
sbss 4034 Set substitution into the ...
sbcssg 4035 Distribute proper substitu...
dfif2 4038 An alternate definition of...
dfif6 4039 An alternate definition of...
ifeq1 4040 Equality theorem for condi...
ifeq2 4041 Equality theorem for condi...
iftrue 4042 Value of the conditional o...
iftruei 4043 Inference associated with ...
iftrued 4044 Value of the conditional o...
iffalse 4045 Value of the conditional o...
iffalsei 4046 Inference associated with ...
iffalsed 4047 Value of the conditional o...
ifnefalse 4048 When values are unequal, b...
ifsb 4049 Distribute a function over...
dfif3 4050 Alternate definition of th...
dfif4 4051 Alternate definition of th...
dfif5 4052 Alternate definition of th...
ifeq12 4053 Equality theorem for condi...
ifeq1d 4054 Equality deduction for con...
ifeq2d 4055 Equality deduction for con...
ifeq12d 4056 Equality deduction for con...
ifbi 4057 Equivalence theorem for co...
ifbid 4058 Equivalence deduction for ...
ifbieq1d 4059 Equivalence/equality deduc...
ifbieq2i 4060 Equivalence/equality infer...
ifbieq2d 4061 Equivalence/equality deduc...
ifbieq12i 4062 Equivalence deduction for ...
ifbieq12d 4063 Equivalence deduction for ...
nfifd 4064 Deduction version of ~ nfi...
nfif 4065 Bound-variable hypothesis ...
ifeq1da 4066 Conditional equality. (Co...
ifeq2da 4067 Conditional equality. (Co...
ifeq12da 4068 Equivalence deduction for ...
ifbieq12d2 4069 Equivalence deduction for ...
ifclda 4070 Conditional closure. (Con...
ifeqda 4071 Separation of the values o...
elimif 4072 Elimination of a condition...
ifbothda 4073 A wff ` th ` containing a ...
ifboth 4074 A wff ` th ` containing a ...
ifid 4075 Identical true and false a...
eqif 4076 Expansion of an equality w...
ifval 4077 Another expression of the ...
elif 4078 Membership in a conditiona...
ifel 4079 Membership of a conditiona...
ifcl 4080 Membership (closure) of a ...
ifcld 4081 Membership (closure) of a ...
ifeqor 4082 The possible values of a c...
ifnot 4083 Negating the first argumen...
ifan 4084 Rewrite a conjunction in a...
ifor 4085 Rewrite a disjunction in a...
2if2 4086 Resolve two nested conditi...
ifcomnan 4087 Commute the conditions in ...
csbif 4088 Distribute proper substitu...
dedth 4089 Weak deduction theorem tha...
dedth2h 4090 Weak deduction theorem eli...
dedth3h 4091 Weak deduction theorem eli...
dedth4h 4092 Weak deduction theorem eli...
dedth2v 4093 Weak deduction theorem for...
dedth3v 4094 Weak deduction theorem for...
dedth4v 4095 Weak deduction theorem for...
elimhyp 4096 Eliminate a hypothesis con...
elimhyp2v 4097 Eliminate a hypothesis con...
elimhyp3v 4098 Eliminate a hypothesis con...
elimhyp4v 4099 Eliminate a hypothesis con...
elimel 4100 Eliminate a membership hyp...
elimdhyp 4101 Version of ~ elimhyp where...
keephyp 4102 Transform a hypothesis ` p...
keephyp2v 4103 Keep a hypothesis containi...
keephyp3v 4104 Keep a hypothesis containi...
keepel 4105 Keep a membership hypothes...
ifex 4106 Conditional operator exist...
ifexg 4107 Conditional operator exist...
pwjust 4109 Soundness justification th...
pweq 4111 Equality theorem for power...
pweqi 4112 Equality inference for pow...
pweqd 4113 Equality deduction for pow...
elpw 4114 Membership in a power clas...
selpw 4115 Setvar variable membership...
elpwg 4116 Membership in a power clas...
elpwi 4117 Subset relation implied by...
elpwid 4118 An element of a power clas...
elelpwi 4119 If ` A ` belongs to a part...
nfpw 4120 Bound-variable hypothesis ...
pwidg 4121 Membership of the original...
pwid 4122 A set is a member of its p...
pwss 4123 Subclass relationship for ...
snjust 4124 Soundness justification th...
sneq 4135 Equality theorem for singl...
sneqi 4136 Equality inference for sin...
sneqd 4137 Equality deduction for sin...
dfsn2 4138 Alternate definition of si...
elsng 4139 There is exactly one eleme...
elsn 4140 There is exactly one eleme...
velsn 4141 There is only one element ...
elsni 4142 There is only one element ...
dfpr2 4143 Alternate definition of un...
elprg 4144 A member of an unordered p...
elpri 4145 If a class is an element o...
elpr 4146 A member of an unordered p...
elpr2 4147 A member of an unordered p...
elpr2OLD 4148 Obsolete proof of ~ elpr2 ...
nelpri 4149 If an element doesn't matc...
prneli 4150 If an element doesn't matc...
nelprd 4151 If an element doesn't matc...
eldifpr 4152 Membership in a set with t...
snidg 4153 A set is a member of its s...
snidb 4154 A class is a set iff it is...
snid 4155 A set is a member of its s...
vsnid 4156 A setvar variable is a mem...
elsn2g 4157 There is exactly one eleme...
elsn2 4158 There is exactly one eleme...
nelsn 4159 If a class is not equal to...
nelsnOLD 4160 Obsolete proof of ~ nelsn ...
rabeqsn 4161 Conditions for a restricte...
rabsssn 4162 Conditions for a restricte...
ralsnsg 4163 Substitution expressed in ...
rexsns 4164 Restricted existential qua...
ralsng 4165 Substitution expressed in ...
rexsng 4166 Restricted existential qua...
2ralsng 4167 Substitution expressed in ...
exsnrex 4168 There is a set being the e...
ralsn 4169 Convert a quantification o...
rexsn 4170 Restricted existential qua...
elpwunsn 4171 Membership in an extension...
eqoreldif 4172 An element of a set is eit...
eqoreldifOLD 4173 Obsolete proof of ~ eqorel...
eltpg 4174 Members of an unordered tr...
eldiftp 4175 Membership in a set with t...
eltpi 4176 A member of an unordered t...
eltp 4177 A member of an unordered t...
dftp2 4178 Alternate definition of un...
nfpr 4179 Bound-variable hypothesis ...
ifpr 4180 Membership of a conditiona...
ralprg 4181 Convert a quantification o...
rexprg 4182 Convert a quantification o...
raltpg 4183 Convert a quantification o...
rextpg 4184 Convert a quantification o...
ralpr 4185 Convert a quantification o...
rexpr 4186 Convert an existential qua...
raltp 4187 Convert a quantification o...
rextp 4188 Convert a quantification o...
nfsn 4189 Bound-variable hypothesis ...
csbsng 4190 Distribute proper substitu...
csbprg 4191 Distribute proper substitu...
disjsn 4192 Intersection with the sing...
disjsn2 4193 Intersection of distinct s...
disjpr2 4194 The intersection of distin...
disjpr2OLD 4195 Obsolete proof of ~ disjpr...
disjprsn 4196 The disjoint intersection ...
snprc 4197 The singleton of a proper ...
snnzb 4198 A singleton is nonempty if...
r19.12sn 4199 Special case of ~ r19.12 w...
rabsn 4200 Condition where a restrict...
rabsnifsb 4201 A restricted class abstrac...
rabsnif 4202 A restricted class abstrac...
rabrsn 4203 A restricted class abstrac...
euabsn2 4204 Another way to express exi...
euabsn 4205 Another way to express exi...
reusn 4206 A way to express restricte...
absneu 4207 Restricted existential uni...
rabsneu 4208 Restricted existential uni...
eusn 4209 Two ways to express " ` A ...
rabsnt 4210 Truth implied by equality ...
prcom 4211 Commutative law for unorde...
preq1 4212 Equality theorem for unord...
preq2 4213 Equality theorem for unord...
preq12 4214 Equality theorem for unord...
preq1i 4215 Equality inference for uno...
preq2i 4216 Equality inference for uno...
preq12i 4217 Equality inference for uno...
preq1d 4218 Equality deduction for uno...
preq2d 4219 Equality deduction for uno...
preq12d 4220 Equality deduction for uno...
tpeq1 4221 Equality theorem for unord...
tpeq2 4222 Equality theorem for unord...
tpeq3 4223 Equality theorem for unord...
tpeq1d 4224 Equality theorem for unord...
tpeq2d 4225 Equality theorem for unord...
tpeq3d 4226 Equality theorem for unord...
tpeq123d 4227 Equality theorem for unord...
tprot 4228 Rotation of the elements o...
tpcoma 4229 Swap 1st and 2nd members o...
tpcomb 4230 Swap 2nd and 3rd members o...
tpass 4231 Split off the first elemen...
qdass 4232 Two ways to write an unord...
qdassr 4233 Two ways to write an unord...
tpidm12 4234 Unordered triple ` { A , A...
tpidm13 4235 Unordered triple ` { A , B...
tpidm23 4236 Unordered triple ` { A , B...
tpidm 4237 Unordered triple ` { A , A...
tppreq3 4238 An unordered triple is an ...
prid1g 4239 An unordered pair contains...
prid2g 4240 An unordered pair contains...
prid1 4241 An unordered pair contains...
prid2 4242 An unordered pair contains...
prprc1 4243 A proper class vanishes in...
prprc2 4244 A proper class vanishes in...
prprc 4245 An unordered pair containi...
tpid1 4246 One of the three elements ...
tpid2 4247 One of the three elements ...
tpid3g 4248 Closed theorem form of ~ t...
tpid3gOLD 4249 Obsolete proof of ~ tpid3g...
tpid3 4250 One of the three elements ...
snnzg 4251 The singleton of a set is ...
snnz 4252 The singleton of a set is ...
prnz 4253 A pair containing a set is...
prnzg 4254 A pair containing a set is...
prnzgOLD 4255 Obsolete proof of ~ prnzg ...
tpnz 4256 A triplet containing a set...
tpnzd 4257 A triplet containing a set...
raltpd 4258 Convert a quantification o...
snss 4259 The singleton of an elemen...
eldifsn 4260 Membership in a set with a...
eldifsni 4261 Membership in a set with a...
neldifsn 4262 The class ` A ` is not in ...
neldifsnd 4263 The class ` A ` is not in ...
rexdifsn 4264 Restricted existential qua...
raldifsni 4265 Rearrangement of a propert...
raldifsnb 4266 Restricted universal quant...
eldifvsn 4267 A set is an element of the...
snssg 4268 The singleton of an elemen...
difsn 4269 An element not in a set ca...
difprsnss 4270 Removal of a singleton fro...
difprsn1 4271 Removal of a singleton fro...
difprsn2 4272 Removal of a singleton fro...
diftpsn3 4273 Removal of a singleton fro...
diftpsn3OLD 4274 Obsolete proof of ~ diftps...
difpr 4275 Removing two elements as p...
tpprceq3 4276 An unordered triple is an ...
tppreqb 4277 An unordered triple is an ...
difsnb 4278 ` ( B \ { A } ) ` equals `...
difsnpss 4279 ` ( B \ { A } ) ` is a pro...
snssi 4280 The singleton of an elemen...
snssd 4281 The singleton of an elemen...
difsnid 4282 If we remove a single elem...
pw0 4283 Compute the power set of t...
pwpw0 4284 Compute the power set of t...
snsspr1 4285 A singleton is a subset of...
snsspr2 4286 A singleton is a subset of...
snsstp1 4287 A singleton is a subset of...
snsstp2 4288 A singleton is a subset of...
snsstp3 4289 A singleton is a subset of...
prssg 4290 A pair of elements of a cl...
prss 4291 A pair of elements of a cl...
prssOLD 4292 Obsolete proof of ~ prss a...
prssi 4293 A pair of elements of a cl...
prssd 4294 Deduction version of ~ prs...
prsspwg 4295 An unordered pair belongs ...
ssprss 4296 A pair as subset of a pair...
ssprsseq 4297 A proper pair is a subset ...
sssn 4298 The subsets of a singleton...
ssunsn2 4299 The property of being sand...
ssunsn 4300 Possible values for a set ...
eqsn 4301 Two ways to express that a...
eqsnOLD 4302 Obsolete proof of ~ eqsn a...
issn 4303 A sufficient condition for...
n0snor2el 4304 A nonempty set is either a...
ssunpr 4305 Possible values for a set ...
sspr 4306 The subsets of a pair. (C...
sstp 4307 The subsets of a triple. ...
tpss 4308 A triplet of elements of a...
tpssi 4309 A triple of elements of a ...
sneqrg 4310 Closed form of ~ sneqr . ...
sneqr 4311 If the singletons of two s...
snsssn 4312 If a singleton is a subset...
sneqrgOLD 4313 Obsolete proof of ~ sneqrg...
sneqbg 4314 Two singletons of sets are...
snsspw 4315 The singleton of a class i...
prsspw 4316 An unordered pair belongs ...
preq1b 4317 Biconditional equality lem...
preq2b 4318 Biconditional equality lem...
preqr1 4319 Reverse equality lemma for...
preqr1OLD 4320 Reverse equality lemma for...
preqr2 4321 Reverse equality lemma for...
preq12b 4322 Equality relationship for ...
prel12 4323 Equality of two unordered ...
opthpr 4324 An unordered pair has the ...
preqr1g 4325 Reverse equality lemma for...
preq12bg 4326 Closed form of ~ preq12b ....
prel12g 4327 Closed form of ~ prel12 . ...
prneimg 4328 Two pairs are not equal if...
prnebg 4329 A (proper) pair is not equ...
preqsnd 4330 Equivalence for a pair equ...
preqsn 4331 Equivalence for a pair equ...
preqsnOLD 4332 Obsolete proof of ~ preqsn...
elpreqprlem 4333 Lemma for ~ elpreqpr . (C...
elpreqpr 4334 Equality and membership ru...
elpreqprb 4335 A set is an element of an ...
elpr2elpr 4336 For an element of an unord...
dfopif 4337 Rewrite ~ df-op using ` if...
dfopg 4338 Value of the ordered pair ...
dfop 4339 Value of an ordered pair w...
opeq1 4340 Equality theorem for order...
opeq2 4341 Equality theorem for order...
opeq12 4342 Equality theorem for order...
opeq1i 4343 Equality inference for ord...
opeq2i 4344 Equality inference for ord...
opeq12i 4345 Equality inference for ord...
opeq1d 4346 Equality deduction for ord...
opeq2d 4347 Equality deduction for ord...
opeq12d 4348 Equality deduction for ord...
oteq1 4349 Equality theorem for order...
oteq2 4350 Equality theorem for order...
oteq3 4351 Equality theorem for order...
oteq1d 4352 Equality deduction for ord...
oteq2d 4353 Equality deduction for ord...
oteq3d 4354 Equality deduction for ord...
oteq123d 4355 Equality deduction for ord...
nfop 4356 Bound-variable hypothesis ...
nfopd 4357 Deduction version of bound...
csbopg 4358 Distribution of class subs...
opid 4359 The ordered pair ` <. A , ...
ralunsn 4360 Restricted quantification ...
2ralunsn 4361 Double restricted quantifi...
opprc 4362 Expansion of an ordered pa...
opprc1 4363 Expansion of an ordered pa...
opprc2 4364 Expansion of an ordered pa...
oprcl 4365 If an ordered pair has an ...
pwsn 4366 The power set of a singlet...
pwsnALT 4367 Alternate proof of ~ pwsn ...
pwpr 4368 The power set of an unorde...
pwtp 4369 The power set of an unorde...
pwpwpw0 4370 Compute the power set of t...
pwv 4371 The power class of the uni...
dfuni2 4374 Alternate definition of cl...
eluni 4375 Membership in class union....
eluni2 4376 Membership in class union....
elunii 4377 Membership in class union....
nfuni 4378 Bound-variable hypothesis ...
nfunid 4379 Deduction version of ~ nfu...
unieq 4380 Equality theorem for class...
unieqi 4381 Inference of equality of t...
unieqd 4382 Deduction of equality of t...
eluniab 4383 Membership in union of a c...
elunirab 4384 Membership in union of a c...
unipr 4385 The union of a pair is the...
uniprg 4386 The union of a pair is the...
unisn 4387 A set equals the union of ...
unisng 4388 A set equals the union of ...
unisn3 4389 Union of a singleton in th...
dfnfc2 4390 An alternative statement o...
dfnfc2OLD 4391 Obsolete proof of ~ dfnfc2...
uniun 4392 The class union of the uni...
uniin 4393 The class union of the int...
uniss 4394 Subclass relationship for ...
ssuni 4395 Subclass relationship for ...
ssuniOLD 4396 Obsolete proof of ~ ssuni ...
unissi 4397 Subclass relationship for ...
unissd 4398 Subclass relationship for ...
uni0b 4399 The union of a set is empt...
uni0c 4400 The union of a set is empt...
uni0 4401 The union of the empty set...
csbuni 4402 Distribute proper substitu...
elssuni 4403 An element of a class is a...
unissel 4404 Condition turning a subcla...
unissb 4405 Relationship involving mem...
uniss2 4406 A subclass condition on th...
unidif 4407 If the difference ` A \ B ...
ssunieq 4408 Relationship implying unio...
unimax 4409 Any member of a class is t...
dfint2 4412 Alternate definition of cl...
inteq 4413 Equality law for intersect...
inteqi 4414 Equality inference for cla...
inteqd 4415 Equality deduction for cla...
elint 4416 Membership in class inters...
elint2 4417 Membership in class inters...
elintg 4418 Membership in class inters...
elintgOLD 4419 Obsolete proof of ~ elintg...
elinti 4420 Membership in class inters...
nfint 4421 Bound-variable hypothesis ...
elintab 4422 Membership in the intersec...
elintrab 4423 Membership in the intersec...
elintrabg 4424 Membership in the intersec...
int0 4425 The intersection of the em...
int0OLD 4426 Obsolete proof of ~ int0 a...
intss1 4427 An element of a class incl...
ssint 4428 Subclass of a class inters...
ssintab 4429 Subclass of the intersecti...
ssintub 4430 Subclass of the least uppe...
ssmin 4431 Subclass of the minimum va...
intmin 4432 Any member of a class is t...
intss 4433 Intersection of subclasses...
intssuni 4434 The intersection of a none...
ssintrab 4435 Subclass of the intersecti...
unissint 4436 If the union of a class is...
intssuni2 4437 Subclass relationship for ...
intminss 4438 Under subset ordering, the...
intmin2 4439 Any set is the smallest of...
intmin3 4440 Under subset ordering, the...
intmin4 4441 Elimination of a conjunct ...
intab 4442 The intersection of a spec...
int0el 4443 The intersection of a clas...
intun 4444 The class intersection of ...
intpr 4445 The intersection of a pair...
intprg 4446 The intersection of a pair...
intsng 4447 Intersection of a singleto...
intsn 4448 The intersection of a sing...
uniintsn 4449 Two ways to express " ` A ...
uniintab 4450 The union and the intersec...
intunsn 4451 Theorem joining a singleto...
rint0 4452 Relative intersection of a...
elrint 4453 Membership in a restricted...
elrint2 4454 Membership in a restricted...
rabasiun 4459 A class abstraction with a...
eliun 4460 Membership in indexed unio...
eliin 4461 Membership in indexed inte...
eliuni 4462 Membership in an indexed u...
iuncom 4463 Commutation of indexed uni...
iuncom4 4464 Commutation of union with ...
iunconst 4465 Indexed union of a constan...
iinconst 4466 Indexed intersection of a ...
iuniin 4467 Law combining indexed unio...
iunss1 4468 Subclass theorem for index...
iinss1 4469 Subclass theorem for index...
iuneq1 4470 Equality theorem for index...
iineq1 4471 Equality theorem for index...
ss2iun 4472 Subclass theorem for index...
iuneq2 4473 Equality theorem for index...
iineq2 4474 Equality theorem for index...
iuneq2i 4475 Equality inference for ind...
iineq2i 4476 Equality inference for ind...
iineq2d 4477 Equality deduction for ind...
iuneq2dv 4478 Equality deduction for ind...
iineq2dv 4479 Equality deduction for ind...
iuneq12df 4480 Equality deduction for ind...
iuneq1d 4481 Equality theorem for index...
iuneq12d 4482 Equality deduction for ind...
iuneq2d 4483 Equality deduction for ind...
nfiun 4484 Bound-variable hypothesis ...
nfiin 4485 Bound-variable hypothesis ...
nfiu1 4486 Bound-variable hypothesis ...
nfii1 4487 Bound-variable hypothesis ...
dfiun2g 4488 Alternate definition of in...
dfiin2g 4489 Alternate definition of in...
dfiun2 4490 Alternate definition of in...
dfiin2 4491 Alternate definition of in...
dfiunv2 4492 Define double indexed unio...
cbviun 4493 Rule used to change the bo...
cbviin 4494 Change bound variables in ...
cbviunv 4495 Rule used to change the bo...
cbviinv 4496 Change bound variables in ...
iunss 4497 Subset theorem for an inde...
ssiun 4498 Subset implication for an ...
ssiun2 4499 Identity law for subset of...
ssiun2s 4500 Subset relationship for an...
iunss2 4501 A subclass condition on th...
iunab 4502 The indexed union of a cla...
iunrab 4503 The indexed union of a res...
iunxdif2 4504 Indexed union with a class...
ssiinf 4505 Subset theorem for an inde...
ssiin 4506 Subset theorem for an inde...
iinss 4507 Subset implication for an ...
iinss2 4508 An indexed intersection is...
uniiun 4509 Class union in terms of in...
intiin 4510 Class intersection in term...
iunid 4511 An indexed union of single...
iun0 4512 An indexed union of the em...
0iun 4513 An empty indexed union is ...
0iin 4514 An empty indexed intersect...
viin 4515 Indexed intersection with ...
iunn0 4516 There is a nonempty class ...
iinab 4517 Indexed intersection of a ...
iinrab 4518 Indexed intersection of a ...
iinrab2 4519 Indexed intersection of a ...
iunin2 4520 Indexed union of intersect...
iunin1 4521 Indexed union of intersect...
iinun2 4522 Indexed intersection of un...
iundif2 4523 Indexed union of class dif...
2iunin 4524 Rearrange indexed unions o...
iindif2 4525 Indexed intersection of cl...
iinin2 4526 Indexed intersection of in...
iinin1 4527 Indexed intersection of in...
iinvdif 4528 The indexed intersection o...
elriin 4529 Elementhood in a relative ...
riin0 4530 Relative intersection of a...
riinn0 4531 Relative intersection of a...
riinrab 4532 Relative intersection of a...
symdif0 4533 Symmetric difference with ...
symdifv 4534 Symmetric difference with ...
symdifid 4535 Symmetric difference with ...
iinxsng 4536 A singleton index picks ou...
iinxprg 4537 Indexed intersection with ...
iunxsng 4538 A singleton index picks ou...
iunxsn 4539 A singleton index picks ou...
iunun 4540 Separate a union in an ind...
iunxun 4541 Separate a union in the in...
iunxdif3 4542 An indexed union where som...
iunxprg 4543 A pair index picks out two...
iunxiun 4544 Separate an indexed union ...
iinuni 4545 A relationship involving u...
iununi 4546 A relationship involving u...
sspwuni 4547 Subclass relationship for ...
pwssb 4548 Two ways to express a coll...
elpwuni 4549 Relationship for power cla...
iinpw 4550 The power class of an inte...
iunpwss 4551 Inclusion of an indexed un...
rintn0 4552 Relative intersection of a...
dfdisj2 4555 Alternate definition for d...
disjss2 4556 If each element of a colle...
disjeq2 4557 Equality theorem for disjo...
disjeq2dv 4558 Equality deduction for dis...
disjss1 4559 A subset of a disjoint col...
disjeq1 4560 Equality theorem for disjo...
disjeq1d 4561 Equality theorem for disjo...
disjeq12d 4562 Equality theorem for disjo...
cbvdisj 4563 Change bound variables in ...
cbvdisjv 4564 Change bound variables in ...
nfdisj 4565 Bound-variable hypothesis ...
nfdisj1 4566 Bound-variable hypothesis ...
disjor 4567 Two ways to say that a col...
disjors 4568 Two ways to say that a col...
disji2 4569 Property of a disjoint col...
disji 4570 Property of a disjoint col...
invdisj 4571 If there is a function ` C...
invdisjrab 4572 The restricted class abstr...
disjiun 4573 A disjoint collection yiel...
sndisj 4574 Any collection of singleto...
0disj 4575 Any collection of empty se...
disjxsn 4576 A singleton collection is ...
disjx0 4577 An empty collection is dis...
disjprg 4578 A pair collection is disjo...
disjxiun 4579 An indexed union of a disj...
disjxiunOLD 4580 Obsolete proof of ~ disjxi...
disjxun 4581 The union of two disjoint ...
disjss3 4582 Expand a disjoint collecti...
breq 4585 Equality theorem for binar...
breq1 4586 Equality theorem for a bin...
breq2 4587 Equality theorem for a bin...
breq12 4588 Equality theorem for a bin...
breqi 4589 Equality inference for bin...
breq1i 4590 Equality inference for a b...
breq2i 4591 Equality inference for a b...
breq12i 4592 Equality inference for a b...
breq1d 4593 Equality deduction for a b...
breqd 4594 Equality deduction for a b...
breq2d 4595 Equality deduction for a b...
breq12d 4596 Equality deduction for a b...
breq123d 4597 Equality deduction for a b...
breqdi 4598 Equality deduction for a b...
breqan12d 4599 Equality deduction for a b...
breqan12rd 4600 Equality deduction for a b...
eqnbrtrd 4601 Substitution of equal clas...
nbrne1 4602 Two classes are different ...
nbrne2 4603 Two classes are different ...
eqbrtri 4604 Substitution of equal clas...
eqbrtrd 4605 Substitution of equal clas...
eqbrtrri 4606 Substitution of equal clas...
eqbrtrrd 4607 Substitution of equal clas...
breqtri 4608 Substitution of equal clas...
breqtrd 4609 Substitution of equal clas...
breqtrri 4610 Substitution of equal clas...
breqtrrd 4611 Substitution of equal clas...
3brtr3i 4612 Substitution of equality i...
3brtr4i 4613 Substitution of equality i...
3brtr3d 4614 Substitution of equality i...
3brtr4d 4615 Substitution of equality i...
3brtr3g 4616 Substitution of equality i...
3brtr4g 4617 Substitution of equality i...
syl5eqbr 4618 A chained equality inferen...
syl5eqbrr 4619 A chained equality inferen...
syl5breq 4620 A chained equality inferen...
syl5breqr 4621 A chained equality inferen...
syl6eqbr 4622 A chained equality inferen...
syl6eqbrr 4623 A chained equality inferen...
syl6breq 4624 A chained equality inferen...
syl6breqr 4625 A chained equality inferen...
ssbrd 4626 Deduction from a subclass ...
ssbri 4627 Inference from a subclass ...
nfbrd 4628 Deduction version of bound...
nfbr 4629 Bound-variable hypothesis ...
brab1 4630 Relationship between a bin...
br0 4631 The empty binary relation ...
brne0 4632 If two sets are in a binar...
brun 4633 The union of two binary re...
brin 4634 The intersection of two re...
brdif 4635 The difference of two bina...
sbcbr123 4636 Move substitution in and o...
sbcbr 4637 Move substitution in and o...
sbcbr12g 4638 Move substitution in and o...
sbcbr1g 4639 Move substitution in and o...
sbcbr2g 4640 Move substitution in and o...
brsymdif 4641 The binary relationship of...
opabss 4646 The collection of ordered ...
opabbid 4647 Equivalent wff's yield equ...
opabbidv 4648 Equivalent wff's yield equ...
opabbii 4649 Equivalent wff's yield equ...
nfopab 4650 Bound-variable hypothesis ...
nfopab1 4651 The first abstraction vari...
nfopab2 4652 The second abstraction var...
cbvopab 4653 Rule used to change bound ...
cbvopabv 4654 Rule used to change bound ...
cbvopab1 4655 Change first bound variabl...
cbvopab2 4656 Change second bound variab...
cbvopab1s 4657 Change first bound variabl...
cbvopab1v 4658 Rule used to change the fi...
cbvopab2v 4659 Rule used to change the se...
unopab 4660 Union of two ordered pair ...
mpteq12f 4661 An equality theorem for th...
mpteq12dva 4662 An equality inference for ...
mpteq12dv 4663 An equality inference for ...
mpteq12 4664 An equality theorem for th...
mpteq1 4665 An equality theorem for th...
mpteq1d 4666 An equality theorem for th...
mpteq1i 4667 An equality theorem for th...
mpteq2ia 4668 An equality inference for ...
mpteq2i 4669 An equality inference for ...
mpteq12i 4670 An equality inference for ...
mpteq2da 4671 Slightly more general equa...
mpteq2dva 4672 Slightly more general equa...
mpteq2dv 4673 An equality inference for ...
nfmpt 4674 Bound-variable hypothesis ...
nfmpt1 4675 Bound-variable hypothesis ...
cbvmptf 4676 Rule to change the bound v...
cbvmpt 4677 Rule to change the bound v...
cbvmptv 4678 Rule to change the bound v...
mptv 4679 Function with universal do...
dftr2 4682 An alternate way of defini...
dftr5 4683 An alternate way of defini...
dftr3 4684 An alternate way of defini...
dftr4 4685 An alternate way of defini...
treq 4686 Equality theorem for the t...
trel 4687 In a transitive class, the...
trel3 4688 In a transitive class, the...
trss 4689 An element of a transitive...
trssOLD 4690 Obsolete proof of ~ trss a...
trin 4691 The intersection of transi...
tr0 4692 The empty set is transitiv...
trv 4693 The universe is transitive...
triun 4694 The indexed union of a cla...
truni 4695 The union of a class of tr...
trint 4696 The intersection of a clas...
trintss 4697 If ` A ` is transitive and...
trint0 4698 Any nonempty transitive cl...
axrep1 4700 The version of the Axiom o...
axrep2 4701 Axiom of Replacement expre...
axrep3 4702 Axiom of Replacement sligh...
axrep4 4703 A more traditional version...
axrep5 4704 Axiom of Replacement (simi...
zfrepclf 4705 An inference rule based on...
zfrep3cl 4706 An inference rule based on...
zfrep4 4707 A version of Replacement u...
axsep 4708 Separation Scheme, which i...
axsep2 4710 A less restrictive version...
zfauscl 4711 Separation Scheme (Aussond...
bm1.3ii 4712 Convert implication to equ...
ax6vsep 4713 Derive a weakened version ...
zfnuleu 4714 Show the uniqueness of the...
axnulALT 4715 Alternate proof of ~ axnul...
axnul 4716 The Null Set Axiom of ZF s...
0ex 4718 The Null Set Axiom of ZF s...
sseliALT 4719 Alternate proof of ~ sseli...
csbexg 4720 The existence of proper su...
csbex 4721 The existence of proper su...
unisn2 4722 A version of ~ unisn witho...
nalset 4723 No set contains all sets. ...
vprc 4724 The universal class is not...
nvel 4725 The universal class doesn'...
vnex 4726 The universal class does n...
inex1 4727 Separation Scheme (Aussond...
inex2 4728 Separation Scheme (Aussond...
inex1g 4729 Closed-form, generalized S...
ssex 4730 The subset of a set is als...
ssexi 4731 The subset of a set is als...
ssexg 4732 The subset of a set is als...
ssexd 4733 A subclass of a set is a s...
sselpwd 4734 Elementhood to a power set...
difexg 4735 Existence of a difference....
difexi 4736 Existence of a difference,...
difexOLD 4737 Obsolete version of ~ dife...
zfausab 4738 Separation Scheme (Aussond...
rabexg 4739 Separation Scheme in terms...
rabex 4740 Separation Scheme in terms...
rabexd 4741 Separation Scheme in terms...
rabex2 4742 Separation Scheme in terms...
rab2ex 4743 A class abstraction based ...
rabex2OLD 4744 Obsolete version of ~ rabe...
rab2exOLD 4745 Obsolete version of ~ rabe...
elssabg 4746 Membership in a class abst...
intex 4747 The intersection of a none...
intnex 4748 If a class intersection is...
intexab 4749 The intersection of a none...
intexrab 4750 The intersection of a none...
iinexg 4751 The existence of a class i...
intabs 4752 Absorption of a redundant ...
inuni 4753 The intersection of a unio...
elpw2g 4754 Membership in a power clas...
elpw2 4755 Membership in a power clas...
pwnss 4756 The power set of a set is ...
pwne 4757 No set equals its power se...
class2set 4758 Construct, from any class ...
class2seteq 4759 Equality theorem based on ...
0elpw 4760 Every power class contains...
pwne0 4761 A power class is never emp...
0nep0 4762 The empty set and its powe...
0inp0 4763 Something cannot be equal ...
unidif0 4764 The removal of the empty s...
iin0 4765 An indexed intersection of...
notzfaus 4766 In the Separation Scheme ~...
intv 4767 The intersection of the un...
axpweq 4768 Two equivalent ways to exp...
zfpow 4770 Axiom of Power Sets expres...
axpow2 4771 A variant of the Axiom of ...
axpow3 4772 A variant of the Axiom of ...
el 4773 Every set is an element of...
pwex 4774 Power set axiom expressed ...
vpwex 4775 The powerset of a setvar i...
pwexg 4776 Power set axiom expressed ...
abssexg 4777 Existence of a class of su...
snexALT 4778 Alternate proof of ~ snex ...
p0ex 4779 The power set of the empty...
p0exALT 4780 Alternate proof of ~ p0ex ...
pp0ex 4781 The power set of the power...
ord3ex 4782 The ordinal number 3 is a ...
dtru 4783 At least two sets exist (o...
axc16b 4784 This theorem shows that ax...
eunex 4785 Existential uniqueness imp...
eusv1 4786 Two ways to express single...
eusvnf 4787 Even if ` x ` is free in `...
eusvnfb 4788 Two ways to say that ` A (...
eusv2i 4789 Two ways to express single...
eusv2nf 4790 Two ways to express single...
eusv2 4791 Two ways to express single...
reusv1 4792 Two ways to express single...
reusv1OLD 4793 Obsolete proof of ~ reusv1...
reusv2lem1 4794 Lemma for ~ reusv2 . (Con...
reusv2lem2 4795 Lemma for ~ reusv2 . (Con...
reusv2lem2OLD 4796 Obsolete proof of ~ reusv2...
reusv2lem3 4797 Lemma for ~ reusv2 . (Con...
reusv2lem4 4798 Lemma for ~ reusv2 . (Con...
reusv2lem5 4799 Lemma for ~ reusv2 . (Con...
reusv2 4800 Two ways to express single...
reusv3i 4801 Two ways of expressing exi...
reusv3 4802 Two ways to express single...
eusv4 4803 Two ways to express single...
alxfr 4804 Transfer universal quantif...
ralxfrd 4805 Transfer universal quantif...
ralxfrdOLD 4806 Obsolete proof of ~ ralxfr...
rexxfrd 4807 Transfer universal quantif...
ralxfr2d 4808 Transfer universal quantif...
rexxfr2d 4809 Transfer universal quantif...
ralxfrd2 4810 Transfer universal quantif...
rexxfrd2 4811 Transfer existence from a ...
ralxfr 4812 Transfer universal quantif...
ralxfrALT 4813 Alternate proof of ~ ralxf...
rexxfr 4814 Transfer existence from a ...
rabxfrd 4815 Class builder membership a...
rabxfr 4816 Class builder membership a...
reuxfr2d 4817 Transfer existential uniqu...
reuxfr2 4818 Transfer existential uniqu...
reuxfrd 4819 Transfer existential uniqu...
reuxfr 4820 Transfer existential uniqu...
reuhypd 4821 A theorem useful for elimi...
reuhyp 4822 A theorem useful for elimi...
nfnid 4823 A setvar variable is not f...
nfcvb 4824 The "distinctor" expressio...
pwuni 4825 A class is a subclass of t...
dtruALT 4826 Alternate proof of ~ dtru ...
dtrucor 4827 Corollary of ~ dtru . Thi...
dtrucor2 4828 The theorem form of the de...
dvdemo1 4829 Demonstration of a theorem...
dvdemo2 4830 Demonstration of a theorem...
zfpair 4831 The Axiom of Pairing of Ze...
axpr 4832 Unabbreviated version of t...
zfpair2 4834 Derive the abbreviated ver...
snex 4835 A singleton is a set. The...
prex 4836 The Axiom of Pairing using...
elALT 4837 Alternate proof of ~ el , ...
dtruALT2 4838 Alternate proof of ~ dtru ...
snelpwi 4839 A singleton of a set belon...
snelpw 4840 A singleton of a set belon...
prelpw 4841 A pair of two sets belongs...
prelpwi 4842 A pair of two sets belongs...
rext 4843 A theorem similar to exten...
sspwb 4844 Classes are subclasses if ...
unipw 4845 A class equals the union o...
univ 4846 The union of the universe ...
pwel 4847 Membership of a power clas...
pwtr 4848 A class is transitive iff ...
ssextss 4849 An extensionality-like pri...
ssext 4850 An extensionality-like pri...
nssss 4851 Negation of subclass relat...
pweqb 4852 Classes are equal if and o...
intid 4853 The intersection of all se...
moabex 4854 "At most one" existence im...
rmorabex 4855 Restricted "at most one" e...
euabex 4856 The abstraction of a wff w...
nnullss 4857 A nonempty class (even if ...
exss 4858 Restricted existence in a ...
opex 4859 An ordered pair of classes...
otex 4860 An ordered triple of class...
elopg 4861 Characterization of the el...
elop 4862 Characterization of the el...
elopOLD 4863 Obsolete version of ~ elop...
opi1 4864 One of the two elements in...
opi2 4865 One of the two elements of...
opeluu 4866 Each member of an ordered ...
op1stb 4867 Extract the first member o...
opnz 4868 An ordered pair is nonempt...
opnzi 4869 An ordered pair is nonempt...
opth1 4870 Equality of the first memb...
opth 4871 The ordered pair theorem. ...
opthg 4872 Ordered pair theorem. ` C ...
opth1g 4873 Equality of the first memb...
opthg2 4874 Ordered pair theorem. (Co...
opth2 4875 Ordered pair theorem. (Co...
opthneg 4876 Two ordered pairs are not ...
opthne 4877 Two ordered pairs are not ...
otth2 4878 Ordered triple theorem, wi...
otth 4879 Ordered triple theorem. (...
otthg 4880 Ordered triple theorem, cl...
eqvinop 4881 A variable introduction la...
copsexg 4882 Substitution of class ` A ...
copsex2t 4883 Closed theorem form of ~ c...
copsex2g 4884 Implicit substitution infe...
copsex4g 4885 An implicit substitution i...
0nelop 4886 A property of ordered pair...
opeqex 4887 Equivalence of existence i...
oteqex2 4888 Equivalence of existence i...
oteqex 4889 Equivalence of existence i...
opcom 4890 An ordered pair commutes i...
moop2 4891 "At most one" property of ...
opeqsn 4892 Equivalence for an ordered...
opeqpr 4893 Equivalence for an ordered...
snopeqop 4894 Equivalence for an ordered...
propeqop 4895 Equivalence for an ordered...
propssopi 4896 If a pair of ordered pairs...
mosubopt 4897 "At most one" remains true...
mosubop 4898 "At most one" remains true...
euop2 4899 Transfer existential uniqu...
euotd 4900 Prove existential uniquene...
opthwiener 4901 Justification theorem for ...
uniop 4902 The union of an ordered pa...
uniopel 4903 Ordered pair membership is...
otsndisj 4904 The singletons consisting ...
otiunsndisj 4905 The union of singletons co...
iunopeqop 4906 Implication of an ordered ...
opabid 4907 The law of concretion. Sp...
elopab 4908 Membership in a class abst...
opelopabsbALT 4909 The law of concretion in t...
opelopabsb 4910 The law of concretion in t...
brabsb 4911 The law of concretion in t...
opelopabt 4912 Closed theorem form of ~ o...
opelopabga 4913 The law of concretion. Th...
brabga 4914 The law of concretion for ...
opelopab2a 4915 Ordered pair membership in...
opelopaba 4916 The law of concretion. Th...
braba 4917 The law of concretion for ...
opelopabg 4918 The law of concretion. Th...
brabg 4919 The law of concretion for ...
opelopabgf 4920 The law of concretion. Th...
opelopab2 4921 Ordered pair membership in...
opelopab 4922 The law of concretion. Th...
brab 4923 The law of concretion for ...
opelopabaf 4924 The law of concretion. Th...
opelopabf 4925 The law of concretion. Th...
ssopab2 4926 Equivalence of ordered pai...
ssopab2b 4927 Equivalence of ordered pai...
ssopab2i 4928 Inference of ordered pair ...
ssopab2dv 4929 Inference of ordered pair ...
eqopab2b 4930 Equivalence of ordered pai...
opabn0 4931 Nonempty ordered pair clas...
csbopab 4932 Move substitution into a c...
csbopabgALT 4933 Move substitution into a c...
csbmpt12 4934 Move substitution into a m...
csbmpt2 4935 Move substitution into the...
iunopab 4936 Move indexed union inside ...
elopabr 4937 Membership in a class abst...
elopabran 4938 Membership in a class abst...
rbropapd 4939 Properties of a pair in an...
rbropap 4940 Properties of a pair in a ...
2rbropap 4941 Properties of a pair in a ...
pwin 4942 The power class of the int...
pwunss 4943 The power class of the uni...
pwssun 4944 The power class of the uni...
pwundif 4945 Break up the power class o...
pwun 4946 The power class of the uni...
epelg 4950 The epsilon relation and m...
epelc 4951 The epsilon relationship a...
epel 4952 The epsilon relation and t...
dfid3 4954 A stronger version of ~ df...
dfid4 4955 The identity function usin...
dfid2 4956 Alternate definition of th...
poss 4961 Subset theorem for the par...
poeq1 4962 Equality theorem for parti...
poeq2 4963 Equality theorem for parti...
nfpo 4964 Bound-variable hypothesis ...
nfso 4965 Bound-variable hypothesis ...
pocl 4966 Properties of partial orde...
ispod 4967 Sufficient conditions for ...
swopolem 4968 Perform the substitutions ...
swopo 4969 A strict weak order is a p...
poirr 4970 A partial order relation i...
potr 4971 A partial order relation i...
po2nr 4972 A partial order relation h...
po3nr 4973 A partial order relation h...
po0 4974 Any relation is a partial ...
pofun 4975 A function preserves a par...
sopo 4976 A strict linear order is a...
soss 4977 Subset theorem for the str...
soeq1 4978 Equality theorem for the s...
soeq2 4979 Equality theorem for the s...
sonr 4980 A strict order relation is...
sotr 4981 A strict order relation is...
solin 4982 A strict order relation is...
so2nr 4983 A strict order relation ha...
so3nr 4984 A strict order relation ha...
sotric 4985 A strict order relation sa...
sotrieq 4986 Trichotomy law for strict ...
sotrieq2 4987 Trichotomy law for strict ...
sotr2 4988 A transitivity relation. ...
issod 4989 An irreflexive, transitive...
issoi 4990 An irreflexive, transitive...
isso2i 4991 Deduce strict ordering fro...
so0 4992 Any relation is a strict o...
somo 4993 A totally ordered set has ...
fri 5000 Property of well-founded r...
seex 5001 The ` R ` -preimage of an ...
exse 5002 Any relation on a set is s...
dffr2 5003 Alternate definition of we...
frc 5004 Property of well-founded r...
frss 5005 Subset theorem for the wel...
sess1 5006 Subset theorem for the set...
sess2 5007 Subset theorem for the set...
freq1 5008 Equality theorem for the w...
freq2 5009 Equality theorem for the w...
seeq1 5010 Equality theorem for the s...
seeq2 5011 Equality theorem for the s...
nffr 5012 Bound-variable hypothesis ...
nfse 5013 Bound-variable hypothesis ...
nfwe 5014 Bound-variable hypothesis ...
frirr 5015 A well-founded relation is...
fr2nr 5016 A well-founded relation ha...
fr0 5017 Any relation is well-found...
frminex 5018 If an element of a well-fo...
efrirr 5019 Irreflexivity of the epsil...
efrn2lp 5020 A set founded by epsilon c...
epse 5021 The epsilon relation is se...
tz7.2 5022 Similar to Theorem 7.2 of ...
dfepfr 5023 An alternate way of saying...
epfrc 5024 A subset of an epsilon-fou...
wess 5025 Subset theorem for the wel...
weeq1 5026 Equality theorem for the w...
weeq2 5027 Equality theorem for the w...
wefr 5028 A well-ordering is well-fo...
weso 5029 A well-ordering is a stric...
wecmpep 5030 The elements of an epsilon...
wetrep 5031 An epsilon well-ordering i...
wefrc 5032 A nonempty (possibly prope...
we0 5033 Any relation is a well-ord...
wereu 5034 A subset of a well-ordered...
wereu2 5035 All nonempty (possibly pro...
xpeq1 5052 Equality theorem for Carte...
xpeq2 5053 Equality theorem for Carte...
elxpi 5054 Membership in a Cartesian ...
elxp 5055 Membership in a Cartesian ...
elxp2 5056 Membership in a Cartesian ...
elxp2OLD 5057 Obsolete proof of ~ elxp2 ...
xpeq12 5058 Equality theorem for Carte...
xpeq1i 5059 Equality inference for Car...
xpeq2i 5060 Equality inference for Car...
xpeq12i 5061 Equality inference for Car...
xpeq1d 5062 Equality deduction for Car...
xpeq2d 5063 Equality deduction for Car...
xpeq12d 5064 Equality deduction for Car...
sqxpeqd 5065 Equality deduction for a C...
nfxp 5066 Bound-variable hypothesis ...
0nelxp 5067 The empty set is not a mem...
0nelxpOLD 5068 Obsolete proof of ~ 0nelxp...
0nelelxp 5069 A member of a Cartesian pr...
opelxp 5070 Ordered pair membership in...
brxp 5071 Binary relation on a Carte...
opelxpi 5072 Ordered pair membership in...
opelxpd 5073 Ordered pair membership in...
opelxp1 5074 The first member of an ord...
opelxp2 5075 The second member of an or...
otelxp1 5076 The first member of an ord...
otel3xp 5077 An ordered triple is an el...
rabxp 5078 Membership in a class buil...
brrelex12 5079 A true binary relation on ...
brrelex 5080 A true binary relation on ...
brrelex2 5081 A true binary relation on ...
brrelexi 5082 The first argument of a bi...
brrelex2i 5083 The second argument of a b...
nprrel 5084 No proper class is related...
fconstmpt 5085 Representation of a consta...
vtoclr 5086 Variable to class conversi...
opelvvg 5087 Ordered pair membership in...
opelvv 5088 Ordered pair membership in...
opthprc 5089 Justification theorem for ...
brel 5090 Two things in a binary rel...
brab2a 5091 Ordered pair membership in...
elxp3 5092 Membership in a Cartesian ...
opeliunxp 5093 Membership in a union of C...
xpundi 5094 Distributive law for Carte...
xpundir 5095 Distributive law for Carte...
xpiundi 5096 Distributive law for Carte...
xpiundir 5097 Distributive law for Carte...
iunxpconst 5098 Membership in a union of C...
xpun 5099 The Cartesian product of t...
elvv 5100 Membership in universal cl...
elvvv 5101 Membership in universal cl...
elvvuni 5102 An ordered pair contains i...
brinxp2 5103 Intersection of binary rel...
brinxp 5104 Intersection of binary rel...
poinxp 5105 Intersection of partial or...
soinxp 5106 Intersection of total orde...
frinxp 5107 Intersection of well-found...
seinxp 5108 Intersection of set-like r...
weinxp 5109 Intersection of well-order...
posn 5110 Partial ordering of a sing...
sosn 5111 Strict ordering on a singl...
frsn 5112 Founded relation on a sing...
wesn 5113 Well-ordering of a singlet...
elopaelxp 5114 Membership in an ordered p...
bropaex12 5115 Two classes related by an ...
opabssxp 5116 An abstraction relation is...
brab2ga 5117 The law of concretion for ...
optocl 5118 Implicit substitution of c...
2optocl 5119 Implicit substitution of c...
3optocl 5120 Implicit substitution of c...
opbrop 5121 Ordered pair membership in...
0xp 5122 The Cartesian product with...
csbxp 5123 Distribute proper substitu...
releq 5124 Equality theorem for the r...
releqi 5125 Equality inference for the...
releqd 5126 Equality deduction for the...
nfrel 5127 Bound-variable hypothesis ...
sbcrel 5128 Distribute proper substitu...
relss 5129 Subclass theorem for relat...
ssrel 5130 A subclass relationship de...
ssrelOLD 5131 Obsolete proof of ~ ssrel ...
eqrel 5132 Extensionality principle f...
ssrel2 5133 A subclass relationship de...
relssi 5134 Inference from subclass pr...
relssdv 5135 Deduction from subclass pr...
eqrelriv 5136 Inference from extensional...
eqrelriiv 5137 Inference from extensional...
eqbrriv 5138 Inference from extensional...
eqrelrdv 5139 Deduce equality of relatio...
eqbrrdv 5140 Deduction from extensional...
eqbrrdiv 5141 Deduction from extensional...
eqrelrdv2 5142 A version of ~ eqrelrdv . ...
ssrelrel 5143 A subclass relationship de...
eqrelrel 5144 Extensionality principle f...
elrel 5145 A member of a relation is ...
relsn 5146 A singleton is a relation ...
relsnop 5147 A singleton of an ordered ...
xpss12 5148 Subset theorem for Cartesi...
xpss 5149 A Cartesian product is inc...
relxp 5150 A Cartesian product is a r...
xpss1 5151 Subset relation for Cartes...
xpss2 5152 Subset relation for Cartes...
copsex2gb 5153 Implicit substitution infe...
copsex2ga 5154 Implicit substitution infe...
elopaba 5155 Membership in an ordered p...
xpsspw 5156 A Cartesian product is inc...
unixpss 5157 The double class union of ...
relun 5158 The union of two relations...
relin1 5159 The intersection with a re...
relin2 5160 The intersection with a re...
reldif 5161 A difference cutting down ...
reliun 5162 An indexed union is a rela...
reliin 5163 An indexed intersection is...
reluni 5164 The union of a class is a ...
relint 5165 The intersection of a clas...
rel0 5166 The empty set is a relatio...
relopabi 5167 A class of ordered pairs i...
relopabiALT 5168 Alternate proof of ~ relop...
relopab 5169 A class of ordered pairs i...
mptrel 5170 The maps-to notation alway...
reli 5171 The identity relation is a...
rele 5172 The membership relation is...
opabid2 5173 A relation expressed as an...
inopab 5174 Intersection of two ordere...
difopab 5175 The difference of two orde...
inxp 5176 The intersection of two Ca...
xpindi 5177 Distributive law for Carte...
xpindir 5178 Distributive law for Carte...
xpiindi 5179 Distributive law for Carte...
xpriindi 5180 Distributive law for Carte...
eliunxp 5181 Membership in a union of C...
opeliunxp2 5182 Membership in a union of C...
raliunxp 5183 Write a double restricted ...
rexiunxp 5184 Write a double restricted ...
ralxp 5185 Universal quantification r...
rexxp 5186 Existential quantification...
exopxfr 5187 Transfer ordered-pair exis...
exopxfr2 5188 Transfer ordered-pair exis...
djussxp 5189 Disjoint union is a subset...
ralxpf 5190 Version of ~ ralxp with bo...
rexxpf 5191 Version of ~ rexxp with bo...
iunxpf 5192 Indexed union on a Cartesi...
opabbi2dv 5193 Deduce equality of a relat...
relop 5194 A necessary and sufficient...
ideqg 5195 For sets, the identity rel...
ideq 5196 For sets, the identity rel...
ididg 5197 A set is identical to itse...
issetid 5198 Two ways of expressing set...
coss1 5199 Subclass theorem for compo...
coss2 5200 Subclass theorem for compo...
coeq1 5201 Equality theorem for compo...
coeq2 5202 Equality theorem for compo...
coeq1i 5203 Equality inference for com...
coeq2i 5204 Equality inference for com...
coeq1d 5205 Equality deduction for com...
coeq2d 5206 Equality deduction for com...
coeq12i 5207 Equality inference for com...
coeq12d 5208 Equality deduction for com...
nfco 5209 Bound-variable hypothesis ...
brcog 5210 Ordered pair membership in...
opelco2g 5211 Ordered pair membership in...
brcogw 5212 Ordered pair membership in...
eqbrrdva 5213 Deduction from extensional...
brco 5214 Binary relation on a compo...
opelco 5215 Ordered pair membership in...
cnvss 5216 Subset theorem for convers...
cnvssOLD 5217 Obsolete proof of ~ cnvss ...
cnveq 5218 Equality theorem for conve...
cnveqi 5219 Equality inference for con...
cnveqd 5220 Equality deduction for con...
elcnv 5221 Membership in a converse. ...
elcnv2 5222 Membership in a converse. ...
nfcnv 5223 Bound-variable hypothesis ...
opelcnvg 5224 Ordered-pair membership in...
brcnvg 5225 The converse of a binary r...
opelcnv 5226 Ordered-pair membership in...
brcnv 5227 The converse of a binary r...
csbcnv 5228 Move class substitution in...
csbcnvgALT 5229 Move class substitution in...
cnvco 5230 Distributive law of conver...
cnvuni 5231 The converse of a class un...
dfdm3 5232 Alternate definition of do...
dfrn2 5233 Alternate definition of ra...
dfrn3 5234 Alternate definition of ra...
elrn2g 5235 Membership in a range. (C...
elrng 5236 Membership in a range. (C...
ssrelrn 5237 If a relation is a subset ...
dfdm4 5238 Alternate definition of do...
dfdmf 5239 Definition of domain, usin...
csbdm 5240 Distribute proper substitu...
eldmg 5241 Domain membership. Theore...
eldm2g 5242 Domain membership. Theore...
eldm 5243 Membership in a domain. T...
eldm2 5244 Membership in a domain. T...
dmss 5245 Subset theorem for domain....
dmeq 5246 Equality theorem for domai...
dmeqi 5247 Equality inference for dom...
dmeqd 5248 Equality deduction for dom...
opeldmd 5249 Membership of first of an ...
opeldm 5250 Membership of first of an ...
breldm 5251 Membership of first of a b...
breldmg 5252 Membership of first of a b...
dmun 5253 The domain of a union is t...
dmin 5254 The domain of an intersect...
dmiun 5255 The domain of an indexed u...
dmuni 5256 The domain of a union. Pa...
dmopab 5257 The domain of a class of o...
dmopabss 5258 Upper bound for the domain...
dmopab3 5259 The domain of a restricted...
dm0 5260 The domain of the empty se...
dmi 5261 The domain of the identity...
dmv 5262 The domain of the universe...
dm0rn0 5263 An empty domain implies an...
reldm0 5264 A relation is empty iff it...
dmxp 5265 The domain of a Cartesian ...
dmxpid 5266 The domain of a square Car...
dmxpin 5267 The domain of the intersec...
xpid11 5268 The Cartesian product of a...
dmcnvcnv 5269 The domain of the double c...
rncnvcnv 5270 The range of the double co...
elreldm 5271 The first member of an ord...
rneq 5272 Equality theorem for range...
rneqi 5273 Equality inference for ran...
rneqd 5274 Equality deduction for ran...
rnss 5275 Subset theorem for range. ...
brelrng 5276 The second argument of a b...
brelrn 5277 The second argument of a b...
opelrn 5278 Membership of second membe...
releldm 5279 The first argument of a bi...
relelrn 5280 The second argument of a b...
releldmb 5281 Membership in a domain. (...
relelrnb 5282 Membership in a range. (C...
releldmi 5283 The first argument of a bi...
relelrni 5284 The second argument of a b...
dfrnf 5285 Definition of range, using...
elrn2 5286 Membership in a range. (C...
elrn 5287 Membership in a range. (C...
nfdm 5288 Bound-variable hypothesis ...
nfrn 5289 Bound-variable hypothesis ...
dmiin 5290 Domain of an intersection....
rnopab 5291 The range of a class of or...
rnmpt 5292 The range of a function in...
elrnmpt 5293 The range of a function in...
elrnmpt1s 5294 Elementhood in an image se...
elrnmpt1 5295 Elementhood in an image se...
elrnmptg 5296 Membership in the range of...
elrnmpti 5297 Membership in the range of...
rn0 5298 The range of the empty set...
dfiun3g 5299 Alternate definition of in...
dfiin3g 5300 Alternate definition of in...
dfiun3 5301 Alternate definition of in...
dfiin3 5302 Alternate definition of in...
riinint 5303 Express a relative indexed...
relrn0 5304 A relation is empty iff it...
dmrnssfld 5305 The domain and range of a ...
dmcoss 5306 Domain of a composition. ...
rncoss 5307 Range of a composition. (...
dmcosseq 5308 Domain of a composition. ...
dmcoeq 5309 Domain of a composition. ...
rncoeq 5310 Range of a composition. (...
reseq1 5311 Equality theorem for restr...
reseq2 5312 Equality theorem for restr...
reseq1i 5313 Equality inference for res...
reseq2i 5314 Equality inference for res...
reseq12i 5315 Equality inference for res...
reseq1d 5316 Equality deduction for res...
reseq2d 5317 Equality deduction for res...
reseq12d 5318 Equality deduction for res...
nfres 5319 Bound-variable hypothesis ...
csbres 5320 Distribute proper substitu...
res0 5321 A restriction to the empty...
opelres 5322 Ordered pair membership in...
brres 5323 Binary relation on a restr...
opelresg 5324 Ordered pair membership in...
brresg 5325 Binary relation on a restr...
opres 5326 Ordered pair membership in...
resieq 5327 A restricted identity rela...
opelresi 5328 ` <. A , A >. ` belongs to...
resres 5329 The restriction of a restr...
resundi 5330 Distributive law for restr...
resundir 5331 Distributive law for restr...
resindi 5332 Class restriction distribu...
resindir 5333 Class restriction distribu...
inres 5334 Move intersection into cla...
resdifcom 5335 Commutative law for restri...
resiun1 5336 Distribution of restrictio...
resiun1OLD 5337 Obsolete proof of ~ resiun...
resiun2 5338 Distribution of restrictio...
dmres 5339 The domain of a restrictio...
ssdmres 5340 A domain restricted to a s...
dmresexg 5341 The domain of a restrictio...
resss 5342 A class includes its restr...
rescom 5343 Commutative law for restri...
ssres 5344 Subclass theorem for restr...
ssres2 5345 Subclass theorem for restr...
relres 5346 A restriction is a relatio...
resabs1 5347 Absorption law for restric...
resabs1d 5348 Absorption law for restric...
resabs2 5349 Absorption law for restric...
residm 5350 Idempotent law for restric...
resima 5351 A restriction to an image....
resima2 5352 Image under a restricted c...
resima2OLD 5353 Obsolete proof of ~ resima...
xpssres 5354 Restriction of a constant ...
elres 5355 Membership in a restrictio...
elsnres 5356 Membership in restriction ...
relssres 5357 Simplification law for res...
dmressnsn 5358 The domain of a restrictio...
eldmressnsn 5359 The element of the domain ...
eldmeldmressn 5360 An element of the domain (...
resdm 5361 A relation restricted to i...
resexg 5362 The restriction of a set i...
resex 5363 The restriction of a set i...
resindm 5364 Class restriction distribu...
resdmdfsn 5365 Restricting a function to ...
resopab 5366 Restriction of a class abs...
iss 5367 A subclass of the identity...
resopab2 5368 Restriction of a class abs...
resmpt 5369 Restriction of the mapping...
resmpt3 5370 Unconditional restriction ...
resmptd 5371 Restriction of the mapping...
dfres2 5372 Alternate definition of th...
mptss 5373 Sufficient condition for i...
opabresid 5374 The restricted identity ex...
mptresid 5375 The restricted identity ex...
dmresi 5376 The domain of a restricted...
restidsing 5377 Restriction of the identit...
restidsingOLD 5378 Obsolete proof of ~ restid...
resid 5379 Any relation restricted to...
imaeq1 5380 Equality theorem for image...
imaeq2 5381 Equality theorem for image...
imaeq1i 5382 Equality theorem for image...
imaeq2i 5383 Equality theorem for image...
imaeq1d 5384 Equality theorem for image...
imaeq2d 5385 Equality theorem for image...
imaeq12d 5386 Equality theorem for image...
dfima2 5387 Alternate definition of im...
dfima3 5388 Alternate definition of im...
elimag 5389 Membership in an image. T...
elima 5390 Membership in an image. T...
elima2 5391 Membership in an image. T...
elima3 5392 Membership in an image. T...
nfima 5393 Bound-variable hypothesis ...
nfimad 5394 Deduction version of bound...
imadmrn 5395 The image of the domain of...
imassrn 5396 The image of a class is a ...
imai 5397 Image under the identity r...
rnresi 5398 The range of the restricte...
resiima 5399 The image of a restriction...
ima0 5400 Image of the empty set. T...
0ima 5401 Image under the empty rela...
csbima12 5402 Move class substitution in...
imadisj 5403 A class whose image under ...
cnvimass 5404 A preimage under any class...
cnvimarndm 5405 The preimage of the range ...
imasng 5406 The image of a singleton. ...
relimasn 5407 The image of a singleton. ...
elrelimasn 5408 Elementhood in the image o...
elimasn 5409 Membership in an image of ...
elimasng 5410 Membership in an image of ...
elimasni 5411 Membership in an image of ...
args 5412 Two ways to express the cl...
eliniseg 5413 Membership in an initial s...
epini 5414 Any set is equal to its pr...
iniseg 5415 An idiom that signifies an...
inisegn0 5416 Nonemptyness of an initial...
dffr3 5417 Alternate definition of we...
dfse2 5418 Alternate definition of se...
imass1 5419 Subset theorem for image. ...
imass2 5420 Subset theorem for image. ...
ndmima 5421 The image of a singleton o...
relcnv 5422 A converse is a relation. ...
relbrcnvg 5423 When ` R ` is a relation, ...
eliniseg2 5424 Eliminate the class existe...
relbrcnv 5425 When ` R ` is a relation, ...
cotrg 5426 Two ways of saying that th...
cotr 5427 Two ways of saying a relat...
issref 5428 Two ways to state a relati...
cnvsym 5429 Two ways of saying a relat...
intasym 5430 Two ways of saying a relat...
asymref 5431 Two ways of saying a relat...
asymref2 5432 Two ways of saying a relat...
intirr 5433 Two ways of saying a relat...
brcodir 5434 Two ways of saying that tw...
codir 5435 Two ways of saying a relat...
qfto 5436 A quantifier-free way of e...
xpidtr 5437 A square Cartesian product...
trin2 5438 The intersection of two tr...
poirr2 5439 A partial order relation i...
trinxp 5440 The relation induced by a ...
soirri 5441 A strict order relation is...
sotri 5442 A strict order relation is...
son2lpi 5443 A strict order relation ha...
sotri2 5444 A transitivity relation. ...
sotri3 5445 A transitivity relation. ...
poleloe 5446 Express "less than or equa...
poltletr 5447 Transitive law for general...
somin1 5448 Property of a minimum in a...
somincom 5449 Commutativity of minimum i...
somin2 5450 Property of a minimum in a...
soltmin 5451 Being less than a minimum,...
cnvopab 5452 The converse of a class ab...
mptcnv 5453 The converse of a mapping ...
cnv0 5454 The converse of the empty ...
cnv0OLD 5455 Obsolete version of ~ cnv0...
cnvi 5456 The converse of the identi...
cnvun 5457 The converse of a union is...
cnvdif 5458 Distributive law for conve...
cnvin 5459 Distributive law for conve...
rnun 5460 Distributive law for range...
rnin 5461 The range of an intersecti...
rniun 5462 The range of an indexed un...
rnuni 5463 The range of a union. Par...
imaundi 5464 Distributive law for image...
imaundir 5465 The image of a union. (Co...
dminss 5466 An upper bound for interse...
imainss 5467 An upper bound for interse...
inimass 5468 The image of an intersecti...
inimasn 5469 The intersection of the im...
cnvxp 5470 The converse of a Cartesia...
xp0 5471 The Cartesian product with...
xpnz 5472 The Cartesian product of n...
xpeq0 5473 At least one member of an ...
xpdisj1 5474 Cartesian products with di...
xpdisj2 5475 Cartesian products with di...
xpsndisj 5476 Cartesian products with tw...
difxp 5477 Difference of Cartesian pr...
difxp1 5478 Difference law for Cartesi...
difxp2 5479 Difference law for Cartesi...
djudisj 5480 Disjoint unions with disjo...
xpdifid 5481 The set of distinct couple...
resdisj 5482 A double restriction to di...
rnxp 5483 The range of a Cartesian p...
dmxpss 5484 The domain of a Cartesian ...
rnxpss 5485 The range of a Cartesian p...
rnxpid 5486 The range of a square Cart...
ssxpb 5487 A Cartesian product subcla...
xp11 5488 The Cartesian product of n...
xpcan 5489 Cancellation law for Carte...
xpcan2 5490 Cancellation law for Carte...
ssrnres 5491 Subset of the range of a r...
rninxp 5492 Range of the intersection ...
dminxp 5493 Domain of the intersection...
imainrect 5494 Image of a relation restri...
xpima 5495 The image by a constant fu...
xpima1 5496 The image by a Cartesian p...
xpima2 5497 The image by a Cartesian p...
xpimasn 5498 The image of a singleton b...
sossfld 5499 The base set of a strict o...
sofld 5500 The base set of a nonempty...
cnvcnv3 5501 The set of all ordered pai...
dfrel2 5502 Alternate definition of re...
dfrel4v 5503 A relation can be expresse...
dfrel4 5504 A relation can be expresse...
cnvcnv 5505 The double converse of a c...
cnvcnv2 5506 The double converse of a c...
cnvcnvss 5507 The double converse of a c...
cnveqb 5508 Equality theorem for conve...
cnveq0 5509 A relation empty iff its c...
dfrel3 5510 Alternate definition of re...
dmresv 5511 The domain of a universal ...
rnresv 5512 The range of a universal r...
dfrn4 5513 Range defined in terms of ...
csbrn 5514 Distribute proper substitu...
rescnvcnv 5515 The restriction of the dou...
cnvcnvres 5516 The double converse of the...
imacnvcnv 5517 The image of the double co...
dmsnn0 5518 The domain of a singleton ...
rnsnn0 5519 The range of a singleton i...
dmsn0 5520 The domain of the singleto...
cnvsn0 5521 The converse of the single...
dmsn0el 5522 The domain of a singleton ...
relsn2 5523 A singleton is a relation ...
dmsnopg 5524 The domain of a singleton ...
dmsnopss 5525 The domain of a singleton ...
dmpropg 5526 The domain of an unordered...
dmsnop 5527 The domain of a singleton ...
dmprop 5528 The domain of an unordered...
dmtpop 5529 The domain of an unordered...
cnvcnvsn 5530 Double converse of a singl...
dmsnsnsn 5531 The domain of the singleto...
rnsnopg 5532 The range of a singleton o...
rnpropg 5533 The range of a pair of ord...
rnsnop 5534 The range of a singleton o...
op1sta 5535 Extract the first member o...
cnvsn 5536 Converse of a singleton of...
op2ndb 5537 Extract the second member ...
op2nda 5538 Extract the second member ...
cnvsng 5539 Converse of a singleton of...
opswap 5540 Swap the members of an ord...
cnvresima 5541 An image under the convers...
resdm2 5542 A class restricted to its ...
resdmres 5543 Restriction to the domain ...
imadmres 5544 The image of the domain of...
mptpreima 5545 The preimage of a function...
mptiniseg 5546 Converse singleton image o...
dmmpt 5547 The domain of the mapping ...
dmmptss 5548 The domain of a mapping is...
dmmptg 5549 The domain of the mapping ...
relco 5550 A composition is a relatio...
dfco2 5551 Alternate definition of a ...
dfco2a 5552 Generalization of ~ dfco2 ...
coundi 5553 Class composition distribu...
coundir 5554 Class composition distribu...
cores 5555 Restricted first member of...
resco 5556 Associative law for the re...
imaco 5557 Image of the composition o...
rnco 5558 The range of the compositi...
rnco2 5559 The range of the compositi...
dmco 5560 The domain of a compositio...
coeq0 5561 A composition of two relat...
coiun 5562 Composition with an indexe...
cocnvcnv1 5563 A composition is not affec...
cocnvcnv2 5564 A composition is not affec...
cores2 5565 Absorption of a reverse (p...
co02 5566 Composition with the empty...
co01 5567 Composition with the empty...
coi1 5568 Composition with the ident...
coi2 5569 Composition with the ident...
coires1 5570 Composition with a restric...
coass 5571 Associative law for class ...
relcnvtr 5572 A relation is transitive i...
relssdmrn 5573 A relation is included in ...
cnvssrndm 5574 The converse is a subset o...
cossxp 5575 Composition as a subset of...
relrelss 5576 Two ways to describe the s...
unielrel 5577 The membership relation fo...
relfld 5578 The double union of a rela...
relresfld 5579 Restriction of a relation ...
relcoi2 5580 Composition with the ident...
relcoi1 5581 Composition with the ident...
unidmrn 5582 The double union of the co...
relcnvfld 5583 if ` R ` is a relation, it...
dfdm2 5584 Alternate definition of do...
unixp 5585 The double class union of ...
unixp0 5586 A Cartesian product is emp...
unixpid 5587 Field of a square Cartesia...
ressn 5588 Restriction of a class to ...
cnviin 5589 The converse of an interse...
cnvpo 5590 The converse of a partial ...
cnvso 5591 The converse of a strict o...
xpco 5592 Composition of two Cartesi...
xpcoid 5593 Composition of two square ...
elsnxp 5594 Elementhood to a cartesian...
elsnxpOLD 5595 Obsolete proof of ~ elsnxp...
predeq123 5598 Equality theorem for the p...
predeq1 5599 Equality theorem for the p...
predeq2 5600 Equality theorem for the p...
predeq3 5601 Equality theorem for the p...
nfpred 5602 Bound-variable hypothesis ...
predpredss 5603 If ` A ` is a subset of ` ...
predss 5604 The predecessor class of `...
sspred 5605 Another subset/predecessor...
dfpred2 5606 An alternate definition of...
dfpred3 5607 An alternate definition of...
dfpred3g 5608 An alternate definition of...
elpredim 5609 Membership in a predecesso...
elpred 5610 Membership in a predecesso...
elpredg 5611 Membership in a predecesso...
predasetex 5612 The predecessor class exis...
dffr4 5613 Alternate definition of we...
predel 5614 Membership in the predeces...
predpo 5615 Property of the precessor ...
predso 5616 Property of the predecesso...
predbrg 5617 Closed form of ~ elpredim ...
setlikespec 5618 If ` R ` is set-like in ` ...
predidm 5619 Idempotent law for the pre...
predin 5620 Intersection law for prede...
predun 5621 Union law for predecessor ...
preddif 5622 Difference law for predece...
predep 5623 The predecessor under the ...
preddowncl 5624 A property of classes that...
predpoirr 5625 Given a partial ordering, ...
predfrirr 5626 Given a well-founded relat...
pred0 5627 The predecessor class over...
tz6.26 5628 All nonempty (possibly pro...
tz6.26i 5629 All nonempty (possibly pro...
wfi 5630 The Principle of Well-Foun...
wfii 5631 The Principle of Well-Foun...
wfisg 5632 Well-Founded Induction Sch...
wfis 5633 Well-Founded Induction Sch...
wfis2fg 5634 Well-Founded Induction Sch...
wfis2f 5635 Well Founded Induction sch...
wfis2g 5636 Well-Founded Induction Sch...
wfis2 5637 Well Founded Induction sch...
wfis3 5638 Well Founded Induction sch...
ordeq 5647 Equality theorem for the o...
elong 5648 An ordinal number is an or...
elon 5649 An ordinal number is an or...
eloni 5650 An ordinal number has the ...
elon2 5651 An ordinal number is an or...
limeq 5652 Equality theorem for the l...
ordwe 5653 Epsilon well-orders every ...
ordtr 5654 An ordinal class is transi...
ordfr 5655 Epsilon is well-founded on...
ordelss 5656 An element of an ordinal c...
trssord 5657 A transitive subclass of a...
ordirr 5658 Epsilon irreflexivity of o...
nordeq 5659 A member of an ordinal cla...
ordn2lp 5660 An ordinal class cannot be...
tz7.5 5661 A nonempty subclass of an ...
ordelord 5662 An element of an ordinal c...
tron 5663 The class of all ordinal n...
ordelon 5664 An element of an ordinal c...
onelon 5665 An element of an ordinal n...
tz7.7 5666 A transitive class belongs...
ordelssne 5667 For ordinal classes, membe...
ordelpss 5668 For ordinal classes, membe...
ordsseleq 5669 For ordinal classes, inclu...
ordin 5670 The intersection of two or...
onin 5671 The intersection of two or...
ordtri3or 5672 A trichotomy law for ordin...
ordtri1 5673 A trichotomy law for ordin...
ontri1 5674 A trichotomy law for ordin...
ordtri2 5675 A trichotomy law for ordin...
ordtri3 5676 A trichotomy law for ordin...
ordtri3OLD 5677 Obsolete proof of ~ ordtri...
ordtri4 5678 A trichotomy law for ordin...
orddisj 5679 An ordinal class and its s...
onfr 5680 The ordinal class is well-...
onelpss 5681 Relationship between membe...
onsseleq 5682 Relationship between subse...
onelss 5683 An element of an ordinal n...
ordtr1 5684 Transitive law for ordinal...
ordtr2 5685 Transitive law for ordinal...
ordtr3 5686 Transitive law for ordinal...
ordtr3OLD 5687 Obsolete proof of ~ ordtr3...
ontr1 5688 Transitive law for ordinal...
ontr2 5689 Transitive law for ordinal...
ordunidif 5690 The union of an ordinal st...
ordintdif 5691 If ` B ` is smaller than `...
onintss 5692 If a property is true for ...
oneqmini 5693 A way to show that an ordi...
ord0 5694 The empty set is an ordina...
0elon 5695 The empty set is an ordina...
ord0eln0 5696 A nonempty ordinal contain...
on0eln0 5697 An ordinal number contains...
dflim2 5698 An alternate definition of...
inton 5699 The intersection of the cl...
nlim0 5700 The empty set is not a lim...
limord 5701 A limit ordinal is ordinal...
limuni 5702 A limit ordinal is its own...
limuni2 5703 The union of a limit ordin...
0ellim 5704 A limit ordinal contains t...
limelon 5705 A limit ordinal class that...
onn0 5706 The class of all ordinal n...
suceq 5707 Equality of successors. (...
elsuci 5708 Membership in a successor....
elsucg 5709 Membership in a successor....
elsuc2g 5710 Variant of membership in a...
elsuc 5711 Membership in a successor....
elsuc2 5712 Membership in a successor....
nfsuc 5713 Bound-variable hypothesis ...
elelsuc 5714 Membership in a successor....
sucel 5715 Membership of a successor ...
suc0 5716 The successor of the empty...
sucprc 5717 A proper class is its own ...
unisuc 5718 A transitive class is equa...
sssucid 5719 A class is included in its...
sucidg 5720 Part of Proposition 7.23 o...
sucid 5721 A set belongs to its succe...
nsuceq0 5722 No successor is empty. (C...
eqelsuc 5723 A set belongs to the succe...
iunsuc 5724 Inductive definition for t...
suctr 5725 The successor of a transit...
suctrOLD 5726 Obsolete proof of ~ suctr ...
trsuc 5727 A set whose successor belo...
trsucss 5728 A member of the successor ...
ordsssuc 5729 A subset of an ordinal bel...
onsssuc 5730 A subset of an ordinal num...
ordsssuc2 5731 An ordinal subset of an or...
onmindif 5732 When its successor is subt...
ordnbtwn 5733 There is no set between an...
ordnbtwnOLD 5734 Obsolete proof of ~ ordnbt...
onnbtwn 5735 There is no set between an...
sucssel 5736 A set whose successor is a...
orddif 5737 Ordinal derived from its s...
orduniss 5738 An ordinal class includes ...
ordtri2or 5739 A trichotomy law for ordin...
ordtri2or2 5740 A trichotomy law for ordin...
ordtri2or3 5741 A consequence of total ord...
ordelinel 5742 The intersection of two or...
ordelinelOLD 5743 Obsolete proof of ~ ordeli...
ordssun 5744 Property of a subclass of ...
ordequn 5745 The maximum (i.e. union) o...
ordun 5746 The maximum (i.e. union) o...
ordunisssuc 5747 A subclass relationship fo...
suc11 5748 The successor operation be...
onordi 5749 An ordinal number is an or...
ontrci 5750 An ordinal number is a tra...
onirri 5751 An ordinal number is not a...
oneli 5752 A member of an ordinal num...
onelssi 5753 A member of an ordinal num...
onssneli 5754 An ordering law for ordina...
onssnel2i 5755 An ordering law for ordina...
onelini 5756 An element of an ordinal n...
oneluni 5757 An ordinal number equals i...
onunisuci 5758 An ordinal number is equal...
onsseli 5759 Subset is equivalent to me...
onun2i 5760 The union of two ordinal n...
unizlim 5761 An ordinal equal to its ow...
on0eqel 5762 An ordinal number either e...
snsn0non 5763 The singleton of the singl...
onxpdisj 5764 Ordinal numbers and ordere...
onnev 5765 The class of ordinal numbe...
iotajust 5767 Soundness justification th...
dfiota2 5769 Alternate definition for d...
nfiota1 5770 Bound-variable hypothesis ...
nfiotad 5771 Deduction version of ~ nfi...
nfiota 5772 Bound-variable hypothesis ...
cbviota 5773 Change bound variables in ...
cbviotav 5774 Change bound variables in ...
sb8iota 5775 Variable substitution in d...
iotaeq 5776 Equality theorem for descr...
iotabi 5777 Equivalence theorem for de...
uniabio 5778 Part of Theorem 8.17 in [Q...
iotaval 5779 Theorem 8.19 in [Quine] p....
iotauni 5780 Equivalence between two di...
iotaint 5781 Equivalence between two di...
iota1 5782 Property of iota. (Contri...
iotanul 5783 Theorem 8.22 in [Quine] p....
iotassuni 5784 The ` iota ` class is a su...
iotaex 5785 Theorem 8.23 in [Quine] p....
iota4 5786 Theorem *14.22 in [Whitehe...
iota4an 5787 Theorem *14.23 in [Whitehe...
iota5 5788 A method for computing iot...
iotabidv 5789 Formula-building deduction...
iotabii 5790 Formula-building deduction...
iotacl 5791 Membership law for descrip...
iota2df 5792 A condition that allows us...
iota2d 5793 A condition that allows us...
iota2 5794 The unique element such th...
sniota 5795 A class abstraction with a...
dfiota4 5796 The ` iota ` operation usi...
csbiota 5797 Class substitution within ...
dffun2 5814 Alternate definition of a ...
dffun3 5815 Alternate definition of fu...
dffun4 5816 Alternate definition of a ...
dffun5 5817 Alternate definition of fu...
dffun6f 5818 Definition of function, us...
dffun6 5819 Alternate definition of a ...
funmo 5820 A function has at most one...
funrel 5821 A function is a relation. ...
funss 5822 Subclass theorem for funct...
funeq 5823 Equality theorem for funct...
funeqi 5824 Equality inference for the...
funeqd 5825 Equality deduction for the...
nffun 5826 Bound-variable hypothesis ...
sbcfung 5827 Distribute proper substitu...
funeu 5828 There is exactly one value...
funeu2 5829 There is exactly one value...
dffun7 5830 Alternate definition of a ...
dffun8 5831 Alternate definition of a ...
dffun9 5832 Alternate definition of a ...
funfn 5833 An equivalence for the fun...
funi 5834 The identity relation is a...
nfunv 5835 The universe is not a func...
funopg 5836 A Kuratowski ordered pair ...
funopab 5837 A class of ordered pairs i...
funopabeq 5838 A class of ordered pairs o...
funopab4 5839 A class of ordered pairs o...
funmpt 5840 A function in maps-to nota...
funmpt2 5841 Functionality of a class g...
funco 5842 The composition of two fun...
funres 5843 A restriction of a functio...
funssres 5844 The restriction of a funct...
fun2ssres 5845 Equality of restrictions o...
funun 5846 The union of functions wit...
fununmo 5847 If the union of classes is...
fununfun 5848 If the union of classes is...
fundif 5849 A function with removed el...
funcnvsn 5850 The converse singleton of ...
funsng 5851 A singleton of an ordered ...
fnsng 5852 Functionality and domain o...
funsn 5853 A singleton of an ordered ...
funprg 5854 A set of two pairs is a fu...
funprgOLD 5855 Obsolete proof of ~ funprg...
funtpg 5856 A set of three pairs is a ...
funtpgOLD 5857 Obsolete proof of ~ funtpg...
funpr 5858 A function with a domain o...
funtp 5859 A function with a domain o...
fnsn 5860 Functionality and domain o...
fnprg 5861 Function with a domain of ...
fntpg 5862 Function with a domain of ...
fntp 5863 A function with a domain o...
funcnvpr 5864 The converse pair of order...
funcnvtp 5865 The converse triple of ord...
funcnvqp 5866 The converse quadruple of ...
funcnvqpOLD 5867 Obsolete proof of ~ funcnv...
fun0 5868 The empty set is a functio...
funcnv0 5869 The converse of the empty ...
funcnvcnv 5870 The double converse of a f...
funcnv2 5871 A simpler equivalence for ...
funcnv 5872 The converse of a class is...
funcnv3 5873 A condition showing a clas...
fun2cnv 5874 The double converse of a c...
svrelfun 5875 A single-valued relation i...
fncnv 5876 Single-rootedness (see ~ f...
fun11 5877 Two ways of stating that `...
fununi 5878 The union of a chain (with...
funin 5879 The intersection with a fu...
funres11 5880 The restriction of a one-t...
funcnvres 5881 The converse of a restrict...
cnvresid 5882 Converse of a restricted i...
funcnvres2 5883 The converse of a restrict...
funimacnv 5884 The image of the preimage ...
funimass1 5885 A kind of contraposition l...
funimass2 5886 A kind of contraposition l...
imadif 5887 The image of a difference ...
imain 5888 The image of an intersecti...
funimaexg 5889 Axiom of Replacement using...
funimaex 5890 The image of a set under a...
isarep1 5891 Part of a study of the Axi...
isarep2 5892 Part of a study of the Axi...
fneq1 5893 Equality theorem for funct...
fneq2 5894 Equality theorem for funct...
fneq1d 5895 Equality deduction for fun...
fneq2d 5896 Equality deduction for fun...
fneq12d 5897 Equality deduction for fun...
fneq12 5898 Equality theorem for funct...
fneq1i 5899 Equality inference for fun...
fneq2i 5900 Equality inference for fun...
nffn 5901 Bound-variable hypothesis ...
fnfun 5902 A function with domain is ...
fnrel 5903 A function with domain is ...
fndm 5904 The domain of a function. ...
funfni 5905 Inference to convert a fun...
fndmu 5906 A function has a unique do...
fnbr 5907 The first argument of bina...
fnop 5908 The first argument of an o...
fneu 5909 There is exactly one value...
fneu2 5910 There is exactly one value...
fnun 5911 The union of two functions...
fnunsn 5912 Extension of a function wi...
fnco 5913 Composition of two functio...
fnresdm 5914 A function does not change...
fnresdisj 5915 A function restricted to a...
2elresin 5916 Membership in two function...
fnssresb 5917 Restriction of a function ...
fnssres 5918 Restriction of a function ...
fnresin1 5919 Restriction of a function'...
fnresin2 5920 Restriction of a function'...
fnres 5921 An equivalence for functio...
fnresi 5922 Functionality and domain o...
fnima 5923 The image of a function's ...
fn0 5924 A function with empty doma...
fnimadisj 5925 A class that is disjoint w...
fnimaeq0 5926 Images under a function ne...
dfmpt3 5927 Alternate definition for t...
mptfnf 5928 The maps-to notation defin...
fnmptf 5929 The maps-to notation defin...
fnopabg 5930 Functionality and domain o...
fnopab 5931 Functionality and domain o...
mptfng 5932 The maps-to notation defin...
fnmpt 5933 The maps-to notation defin...
mpt0 5934 A mapping operation with e...
fnmpti 5935 Functionality and domain o...
dmmpti 5936 Domain of the mapping oper...
dmmptd 5937 The domain of the mapping ...
mptun 5938 Union of mappings which ar...
feq1 5939 Equality theorem for funct...
feq2 5940 Equality theorem for funct...
feq3 5941 Equality theorem for funct...
feq23 5942 Equality theorem for funct...
feq1d 5943 Equality deduction for fun...
feq2d 5944 Equality deduction for fun...
feq3d 5945 Equality deduction for fun...
feq12d 5946 Equality deduction for fun...
feq123d 5947 Equality deduction for fun...
feq123 5948 Equality theorem for funct...
feq1i 5949 Equality inference for fun...
feq2i 5950 Equality inference for fun...
feq12i 5951 Equality inference for fun...
feq23i 5952 Equality inference for fun...
feq23d 5953 Equality deduction for fun...
nff 5954 Bound-variable hypothesis ...
sbcfng 5955 Distribute proper substitu...
sbcfg 5956 Distribute proper substitu...
elimf 5957 Eliminate a mapping hypoth...
ffn 5958 A mapping is a function wi...
ffnd 5959 A mapping is a function wi...
dffn2 5960 Any function is a mapping ...
ffun 5961 A mapping is a function. ...
ffund 5962 A mapping is a function, d...
frel 5963 A mapping is a relation. ...
fdm 5964 The domain of a mapping. ...
fdmi 5965 The domain of a mapping. ...
frn 5966 The range of a mapping. (...
dffn3 5967 A function maps to its ran...
ffrn 5968 A function maps to its ran...
fss 5969 Expanding the codomain of ...
fssd 5970 Expanding the codomain of ...
fco 5971 Composition of two mapping...
fco2 5972 Functionality of a composi...
fssxp 5973 A mapping is a class of or...
funssxp 5974 Two ways of specifying a p...
ffdm 5975 A mapping is a partial fun...
ffdmd 5976 The domain of a function. ...
fdmrn 5977 A different way to write `...
opelf 5978 The members of an ordered ...
fun 5979 The union of two functions...
fun2 5980 The union of two functions...
fun2d 5981 The union of functions wit...
fnfco 5982 Composition of two functio...
fssres 5983 Restriction of a function ...
fssresd 5984 Restriction of a function ...
fssres2 5985 Restriction of a restricte...
fresin 5986 An identity for the mappin...
resasplit 5987 If two functions agree on ...
fresaun 5988 The union of two functions...
fresaunres2 5989 From the union of two func...
fresaunres1 5990 From the union of two func...
fcoi1 5991 Composition of a mapping a...
fcoi2 5992 Composition of restricted ...
feu 5993 There is exactly one value...
fimass 5994 The image of a class is a ...
fcnvres 5995 The converse of a restrict...
fimacnvdisj 5996 The preimage of a class di...
fint 5997 Function into an intersect...
fin 5998 Mapping into an intersecti...
f0 5999 The empty function. (Cont...
f00 6000 A class is a function with...
f0bi 6001 A function with empty doma...
f0dom0 6002 A function is empty iff it...
f0rn0 6003 If there is no element in ...
fconst 6004 A Cartesian product with a...
fconstg 6005 A Cartesian product with a...
fnconstg 6006 A Cartesian product with a...
fconst6g 6007 Constant function with loo...
fconst6 6008 A constant function as a m...
f1eq1 6009 Equality theorem for one-t...
f1eq2 6010 Equality theorem for one-t...
f1eq3 6011 Equality theorem for one-t...
nff1 6012 Bound-variable hypothesis ...
dff12 6013 Alternate definition of a ...
f1f 6014 A one-to-one mapping is a ...
f1fn 6015 A one-to-one mapping is a ...
f1fun 6016 A one-to-one mapping is a ...
f1rel 6017 A one-to-one onto mapping ...
f1dm 6018 The domain of a one-to-one...
f1ss 6019 A function that is one-to-...
f1ssr 6020 A function that is one-to-...
f1ssres 6021 A function that is one-to-...
f1cnvcnv 6022 Two ways to express that a...
f1co 6023 Composition of one-to-one ...
foeq1 6024 Equality theorem for onto ...
foeq2 6025 Equality theorem for onto ...
foeq3 6026 Equality theorem for onto ...
nffo 6027 Bound-variable hypothesis ...
fof 6028 An onto mapping is a mappi...
fofun 6029 An onto mapping is a funct...
fofn 6030 An onto mapping is a funct...
forn 6031 The codomain of an onto fu...
dffo2 6032 Alternate definition of an...
foima 6033 The image of the domain of...
dffn4 6034 A function maps onto its r...
funforn 6035 A function maps its domain...
fodmrnu 6036 An onto function has uniqu...
fores 6037 Restriction of an onto fun...
foco 6038 Composition of onto functi...
foconst 6039 A nonzero constant functio...
f1oeq1 6040 Equality theorem for one-t...
f1oeq2 6041 Equality theorem for one-t...
f1oeq3 6042 Equality theorem for one-t...
f1oeq23 6043 Equality theorem for one-t...
f1eq123d 6044 Equality deduction for one...
foeq123d 6045 Equality deduction for ont...
f1oeq123d 6046 Equality deduction for one...
f1oeq3d 6047 Equality deduction for one...
nff1o 6048 Bound-variable hypothesis ...
f1of1 6049 A one-to-one onto mapping ...
f1of 6050 A one-to-one onto mapping ...
f1ofn 6051 A one-to-one onto mapping ...
f1ofun 6052 A one-to-one onto mapping ...
f1orel 6053 A one-to-one onto mapping ...
f1odm 6054 The domain of a one-to-one...
dff1o2 6055 Alternate definition of on...
dff1o3 6056 Alternate definition of on...
f1ofo 6057 A one-to-one onto function...
dff1o4 6058 Alternate definition of on...
dff1o5 6059 Alternate definition of on...
f1orn 6060 A one-to-one function maps...
f1f1orn 6061 A one-to-one function maps...
f1ocnv 6062 The converse of a one-to-o...
f1ocnvb 6063 A relation is a one-to-one...
f1ores 6064 The restriction of a one-t...
f1orescnv 6065 The converse of a one-to-o...
f1imacnv 6066 Preimage of an image. (Co...
foimacnv 6067 A reverse version of ~ f1i...
foun 6068 The union of two onto func...
f1oun 6069 The union of two one-to-on...
resdif 6070 The restriction of a one-t...
resin 6071 The restriction of a one-t...
f1oco 6072 Composition of one-to-one ...
f1cnv 6073 The converse of an injecti...
funcocnv2 6074 Composition with the conve...
fococnv2 6075 The composition of an onto...
f1ococnv2 6076 The composition of a one-t...
f1cocnv2 6077 Composition of an injectiv...
f1ococnv1 6078 The composition of a one-t...
f1cocnv1 6079 Composition of an injectiv...
funcoeqres 6080 Re-express a constraint on...
f10 6081 The empty set maps one-to-...
f10d 6082 The empty set maps one-to-...
f1o00 6083 One-to-one onto mapping of...
fo00 6084 Onto mapping of the empty ...
f1o0 6085 One-to-one onto mapping of...
f1oi 6086 A restriction of the ident...
f1ovi 6087 The identity relation is a...
f1osn 6088 A singleton of an ordered ...
f1osng 6089 A singleton of an ordered ...
f1sng 6090 A singleton of an ordered ...
fsnd 6091 A singleton of an ordered ...
f1oprswap 6092 A two-element swap is a bi...
f1oprg 6093 An unordered pair of order...
tz6.12-2 6094 Function value when ` F ` ...
fveu 6095 The value of a function at...
brprcneu 6096 If ` A ` is a proper class...
fvprc 6097 A function's value at a pr...
fv2 6098 Alternate definition of fu...
dffv3 6099 A definition of function v...
dffv4 6100 The previous definition of...
elfv 6101 Membership in a function v...
fveq1 6102 Equality theorem for funct...
fveq2 6103 Equality theorem for funct...
fveq1i 6104 Equality inference for fun...
fveq1d 6105 Equality deduction for fun...
fveq2i 6106 Equality inference for fun...
fveq2d 6107 Equality deduction for fun...
fveq12i 6108 Equality deduction for fun...
fveq12d 6109 Equality deduction for fun...
nffv 6110 Bound-variable hypothesis ...
nffvmpt1 6111 Bound-variable hypothesis ...
nffvd 6112 Deduction version of bound...
fvex 6113 The value of a class exist...
fvif 6114 Move a conditional outside...
iffv 6115 Move a conditional outside...
fv3 6116 Alternate definition of th...
fvres 6117 The value of a restricted ...
fvresd 6118 The value of a restricted ...
funssfv 6119 The value of a member of t...
tz6.12-1 6120 Function value. Theorem 6...
tz6.12 6121 Function value. Theorem 6...
tz6.12f 6122 Function value, using boun...
tz6.12c 6123 Corollary of Theorem 6.12(...
tz6.12i 6124 Corollary of Theorem 6.12(...
fvbr0 6125 Two possibilities for the ...
fvrn0 6126 A function value is a memb...
fvssunirn 6127 The result of a function v...
ndmfv 6128 The value of a class outsi...
ndmfvrcl 6129 Reverse closure law for fu...
elfvdm 6130 If a function value has a ...
elfvex 6131 If a function value has a ...
elfvexd 6132 If a function value is non...
eliman0 6133 A non-nul function value i...
nfvres 6134 The value of a non-member ...
nfunsn 6135 If the restriction of a cl...
fvfundmfvn0 6136 If a class' value at an ar...
0fv 6137 Function value of the empt...
fv2prc 6138 A function's value at a fu...
elfv2ex 6139 If a function value of a f...
fveqres 6140 Equal values imply equal v...
csbfv12 6141 Move class substitution in...
csbfv2g 6142 Move class substitution in...
csbfv 6143 Substitution for a functio...
funbrfv 6144 The second argument of a b...
funopfv 6145 The second element in an o...
fnbrfvb 6146 Equivalence of function va...
fnopfvb 6147 Equivalence of function va...
funbrfvb 6148 Equivalence of function va...
funopfvb 6149 Equivalence of function va...
funbrfv2b 6150 Function value in terms of...
dffn5 6151 Representation of a functi...
fnrnfv 6152 The range of a function ex...
fvelrnb 6153 A member of a function's r...
foelrni 6154 A member of a surjective f...
dfimafn 6155 Alternate definition of th...
dfimafn2 6156 Alternate definition of th...
funimass4 6157 Membership relation for th...
fvelima 6158 Function value in an image...
feqmptd 6159 Deduction form of ~ dffn5 ...
feqresmpt 6160 Express a restricted funct...
feqmptdf 6161 Deduction form of ~ dffn5f...
dffn5f 6162 Representation of a functi...
fvelimab 6163 Function value in an image...
fvelimabd 6164 Deduction form of ~ fvelim...
fvi 6165 The value of the identity ...
fviss 6166 The value of the identity ...
fniinfv 6167 The indexed intersection o...
fnsnfv 6168 Singleton of function valu...
opabiotafun 6169 Define a function whose va...
opabiotadm 6170 Define a function whose va...
opabiota 6171 Define a function whose va...
fnimapr 6172 The image of a pair under ...
ssimaex 6173 The existence of a subimag...
ssimaexg 6174 The existence of a subimag...
funfv 6175 A simplified expression fo...
funfv2 6176 The value of a function. ...
funfv2f 6177 The value of a function. ...
fvun 6178 Value of the union of two ...
fvun1 6179 The value of a union when ...
fvun2 6180 The value of a union when ...
dffv2 6181 Alternate definition of fu...
dmfco 6182 Domains of a function comp...
fvco2 6183 Value of a function compos...
fvco 6184 Value of a function compos...
fvco3 6185 Value of a function compos...
fvco4i 6186 Conditions for a compositi...
fvopab3g 6187 Value of a function given ...
fvopab3ig 6188 Value of a function given ...
fvmptg 6189 Value of a function given ...
fvmpti 6190 Value of a function given ...
fvmpt 6191 Value of a function given ...
fvmpt2f 6192 Value of a function given ...
fvtresfn 6193 Functionality of a tuple-r...
fvmpts 6194 Value of a function given ...
fvmpt3 6195 Value of a function given ...
fvmpt3i 6196 Value of a function given ...
fvmptd 6197 Deduction version of ~ fvm...
mptrcl 6198 Reverse closure for a mapp...
fvmpt2i 6199 Value of a function given ...
fvmpt2 6200 Value of a function given ...
fvmptss 6201 If all the values of the m...
fvmpt2d 6202 Deduction version of ~ fvm...
fvmptex 6203 Express a function ` F ` w...
fvmptdf 6204 Alternate deduction versio...
fvmptdv 6205 Alternate deduction versio...
fvmptdv2 6206 Alternate deduction versio...
mpteqb 6207 Bidirectional equality the...
fvmptt 6208 Closed theorem form of ~ f...
fvmptf 6209 Value of a function given ...
fvmptnf 6210 The value of a function gi...
fvmptn 6211 This somewhat non-intuitiv...
fvmptss2 6212 A mapping always evaluates...
elfvmptrab1 6213 Implications for the value...
elfvmptrab 6214 Implications for the value...
fvopab4ndm 6215 Value of a function given ...
fvmptndm 6216 Value of a function given ...
fvopab5 6217 The value of a function th...
fvopab6 6218 Value of a function given ...
eqfnfv 6219 Equality of functions is d...
eqfnfv2 6220 Equality of functions is d...
eqfnfv3 6221 Derive equality of functio...
eqfnfvd 6222 Deduction for equality of ...
eqfnfv2f 6223 Equality of functions is d...
eqfunfv 6224 Equality of functions is d...
fvreseq0 6225 Equality of restricted fun...
fvreseq1 6226 Equality of a function res...
fvreseq 6227 Equality of restricted fun...
fnmptfvd 6228 A function with a given do...
fndmdif 6229 Two ways to express the lo...
fndmdifcom 6230 The difference set between...
fndmdifeq0 6231 The difference set of two ...
fndmin 6232 Two ways to express the lo...
fneqeql 6233 Two functions are equal if...
fneqeql2 6234 Two functions are equal if...
fnreseql 6235 Two functions are equal on...
chfnrn 6236 The range of a choice func...
funfvop 6237 Ordered pair with function...
funfvbrb 6238 Two ways to say that ` A `...
fvimacnvi 6239 A member of a preimage is ...
fvimacnv 6240 The argument of a function...
funimass3 6241 A kind of contraposition l...
funimass5 6242 A subclass of a preimage i...
funconstss 6243 Two ways of specifying tha...
fvimacnvALT 6244 Alternate proof of ~ fvima...
elpreima 6245 Membership in the preimage...
fniniseg 6246 Membership in the preimage...
fncnvima2 6247 Inverse images under funct...
fniniseg2 6248 Inverse point images under...
unpreima 6249 Preimage of a union. (Con...
inpreima 6250 Preimage of an intersectio...
difpreima 6251 Preimage of a difference. ...
respreima 6252 The preimage of a restrict...
iinpreima 6253 Preimage of an intersectio...
intpreima 6254 Preimage of an intersectio...
fimacnv 6255 The preimage of the codoma...
fimacnvinrn 6256 Taking the converse image ...
fimacnvinrn2 6257 Taking the converse image ...
fvn0ssdmfun 6258 If a class' function value...
fnopfv 6259 Ordered pair with function...
fvelrn 6260 A function's value belongs...
nelrnfvne 6261 A function value cannot be...
fveqdmss 6262 If the empty set is not co...
fveqressseq 6263 If the empty set is not co...
fnfvelrn 6264 A function's value belongs...
ffvelrn 6265 A function's value belongs...
ffvelrni 6266 A function's value belongs...
ffvelrnda 6267 A function's value belongs...
ffvelrnd 6268 A function's value belongs...
rexrn 6269 Restricted existential qua...
ralrn 6270 Restricted universal quant...
elrnrexdm 6271 For any element in the ran...
elrnrexdmb 6272 For any element in the ran...
eldmrexrn 6273 For any element in the dom...
eldmrexrnb 6274 For any element in the dom...
fvcofneq 6275 The values of two function...
ralrnmpt 6276 A restricted quantifier ov...
rexrnmpt 6277 A restricted quantifier ov...
f0cli 6278 Unconditional closure of a...
dff2 6279 Alternate definition of a ...
dff3 6280 Alternate definition of a ...
dff4 6281 Alternate definition of a ...
dffo3 6282 An onto mapping expressed ...
dffo4 6283 Alternate definition of an...
dffo5 6284 Alternate definition of an...
exfo 6285 A relation equivalent to t...
foelrn 6286 Property of a surjective f...
foco2 6287 If a composition of two fu...
foco2OLD 6288 Obsolete proof of ~ foco2 ...
fmpt 6289 Functionality of the mappi...
f1ompt 6290 Express bijection for a ma...
fmpti 6291 Functionality of the mappi...
fmptd 6292 Domain and codomain of the...
fmpt3d 6293 Domain and co-domain of th...
fmptdf 6294 A version of ~ fmptd using...
ffnfv 6295 A function maps to a class...
ffnfvf 6296 A function maps to a class...
fnfvrnss 6297 An upper bound for range d...
frnssb 6298 A function is a function i...
rnmptss 6299 The range of an operation ...
fmpt2d 6300 Domain and codomain of the...
ffvresb 6301 A necessary and sufficient...
f1oresrab 6302 Build a bijection between ...
fmptco 6303 Composition of two functio...
fmptcof 6304 Version of ~ fmptco where ...
fmptcos 6305 Composition of two functio...
fcompt 6306 Express composition of two...
fcoconst 6307 Composition with a constan...
fsn 6308 A function maps a singleto...
fsn2 6309 A function that maps a sin...
fsng 6310 A function maps a singleto...
fsn2g 6311 A function that maps a sin...
xpsng 6312 The Cartesian product of t...
xpsn 6313 The Cartesian product of t...
f1o2sn 6314 A singleton with a nested ...
residpr 6315 Restriction of the identit...
dfmpt 6316 Alternate definition for t...
fnasrn 6317 A function expressed as th...
funiun 6318 A function is a union of s...
funopsn 6319 If a function is an ordere...
funop 6320 An ordered pair is a funct...
funsndifnop 6321 A singleton of an ordered ...
funsneqopsn 6322 A singleton of an ordered ...
funsneqop 6323 A singleton of an ordered ...
funsneqopb 6324 A singleton of an ordered ...
ressnop0 6325 If ` A ` is not in ` C ` ,...
fpr 6326 A function with a domain o...
fprg 6327 A function with a domain o...
ftpg 6328 A function with a domain o...
ftp 6329 A function with a domain o...
fnressn 6330 A function restricted to a...
funressn 6331 A function restricted to a...
fressnfv 6332 The value of a function re...
fvrnressn 6333 If the value of a function...
fvressn 6334 The value of a function re...
fvn0fvelrn 6335 If the value of a function...
fvconst 6336 The value of a constant fu...
fnsnb 6337 A function whose domain is...
fmptsn 6338 Express a singleton functi...
fmptsng 6339 Express a singleton functi...
fmptsnd 6340 Express a singleton functi...
fmptap 6341 Append an additional value...
fmptapd 6342 Append an additional value...
fmptpr 6343 Express a pair function in...
fvresi 6344 The value of a restricted ...
fninfp 6345 Express the class of fixed...
fnelfp 6346 Property of a fixed point ...
fndifnfp 6347 Express the class of non-f...
fnelnfp 6348 Property of a non-fixed po...
fnnfpeq0 6349 A function is the identity...
fvunsn 6350 Remove an ordered pair not...
fvsn 6351 The value of a singleton o...
fvsng 6352 The value of a singleton o...
fvsnun1 6353 The value of a function wi...
fvsnun2 6354 The value of a function wi...
fnsnsplit 6355 Split a function into a si...
fsnunf 6356 Adjoining a point to a fun...
fsnunf2 6357 Adjoining a point to a pun...
fsnunfv 6358 Recover the added point fr...
fsnunres 6359 Recover the original funct...
funresdfunsn 6360 Restricting a function to ...
fvpr1 6361 The value of a function wi...
fvpr2 6362 The value of a function wi...
fvpr1g 6363 The value of a function wi...
fvpr2g 6364 The value of a function wi...
fvtp1 6365 The first value of a funct...
fvtp2 6366 The second value of a func...
fvtp3 6367 The third value of a funct...
fvtp1g 6368 The value of a function wi...
fvtp2g 6369 The value of a function wi...
fvtp3g 6370 The value of a function wi...
tpres 6371 An unordered triple of ord...
fvconst2g 6372 The value of a constant fu...
fconst2g 6373 A constant function expres...
fvconst2 6374 The value of a constant fu...
fconst2 6375 A constant function expres...
fconst5 6376 Two ways to express that a...
fnprb 6377 A function whose domain ha...
fntpb 6378 A function whose domain ha...
fnpr2g 6379 A function whose domain ha...
fpr2g 6380 A function that maps a pai...
fconstfv 6381 A constant function expres...
fconst3 6382 Two ways to express a cons...
fconst4 6383 Two ways to express a cons...
resfunexg 6384 The restriction of a funct...
resiexd 6385 The restriction of the ide...
fnex 6386 If the domain of a functio...
funex 6387 If the domain of a functio...
opabex 6388 Existence of a function ex...
mptexg 6389 If the domain of a functio...
mptex 6390 If the domain of a functio...
mptexd 6391 If the domain of a functio...
mptrabex 6392 If the domain of a functio...
mptrabexOLD 6393 Obsolete version of ~ mptr...
fex 6394 If the domain of a mapping...
eufnfv 6395 A function is uniquely det...
funfvima 6396 A function's value in a pr...
funfvima2 6397 A function's value in an i...
resfvresima 6398 The value of the function ...
funfvima3 6399 A class including a functi...
fnfvima 6400 The function value of an o...
rexima 6401 Existential quantification...
ralima 6402 Universal quantification u...
idref 6403 TODO: This is the same as...
fvclss 6404 Upper bound for the class ...
elabrex 6405 Elementhood in an image se...
abrexco 6406 Composition of two image m...
imaiun 6407 The image of an indexed un...
imauni 6408 The image of a union is th...
fniunfv 6409 The indexed union of a fun...
funiunfv 6410 The indexed union of a fun...
funiunfvf 6411 The indexed union of a fun...
eluniima 6412 Membership in the union of...
elunirn 6413 Membership in the union of...
elunirnALT 6414 Alternate proof of ~ eluni...
fnunirn 6415 Membership in a union of s...
dff13 6416 A one-to-one function in t...
dff13f 6417 A one-to-one function in t...
f1veqaeq 6418 If the values of a one-to-...
f1mpt 6419 Express injection for a ma...
f1fveq 6420 Equality of function value...
f1elima 6421 Membership in the image of...
f1imass 6422 Taking images under a one-...
f1imaeq 6423 Taking images under a one-...
f1imapss 6424 Taking images under a one-...
f1dom3fv3dif 6425 The function values for a ...
f1dom3el3dif 6426 The range of a 1-1 functio...
dff14a 6427 A one-to-one function in t...
dff14b 6428 A one-to-one function in t...
f12dfv 6429 A one-to-one function with...
f13dfv 6430 A one-to-one function with...
dff1o6 6431 A one-to-one onto function...
f1ocnvfv1 6432 The converse value of the ...
f1ocnvfv2 6433 The value of the converse ...
f1ocnvfv 6434 Relationship between the v...
f1ocnvfvb 6435 Relationship between the v...
nvof1o 6436 An involution is a bijecti...
nvocnv 6437 The converse of an involut...
fsnex 6438 Relate a function with a s...
f1prex 6439 Relate a one-to-one functi...
f1ocnvdm 6440 The value of the converse ...
f1ocnvfvrneq 6441 If the values of a one-to-...
fcof1 6442 An application is injectiv...
fcofo 6443 An application is surjecti...
cbvfo 6444 Change bound variable betw...
cbvexfo 6445 Change bound variable betw...
cocan1 6446 An injection is left-cance...
cocan2 6447 A surjection is right-canc...
fcof1oinvd 6448 Show that a function is th...
fcof1od 6449 A function is bijective if...
2fcoidinvd 6450 Show that a function is th...
fcof1o 6451 Show that two functions ar...
2fvcoidd 6452 Show that the composition ...
2fvidf1od 6453 A function is bijective if...
2fvidinvd 6454 Show that two functions ar...
foeqcnvco 6455 Condition for function equ...
f1eqcocnv 6456 Condition for function equ...
fveqf1o 6457 Given a bijection ` F ` , ...
fliftrel 6458 ` F ` , a function lift, i...
fliftel 6459 Elementhood in the relatio...
fliftel1 6460 Elementhood in the relatio...
fliftcnv 6461 Converse of the relation `...
fliftfun 6462 The function ` F ` is the ...
fliftfund 6463 The function ` F ` is the ...
fliftfuns 6464 The function ` F ` is the ...
fliftf 6465 The domain and range of th...
fliftval 6466 The value of the function ...
isoeq1 6467 Equality theorem for isomo...
isoeq2 6468 Equality theorem for isomo...
isoeq3 6469 Equality theorem for isomo...
isoeq4 6470 Equality theorem for isomo...
isoeq5 6471 Equality theorem for isomo...
nfiso 6472 Bound-variable hypothesis ...
isof1o 6473 An isomorphism is a one-to...
isof1oidb 6474 A function is a bijection ...
isof1oopb 6475 A function is a bijection ...
isorel 6476 An isomorphism connects bi...
soisores 6477 Express the condition of i...
soisoi 6478 Infer isomorphism from one...
isoid 6479 Identity law for isomorphi...
isocnv 6480 Converse law for isomorphi...
isocnv2 6481 Converse law for isomorphi...
isocnv3 6482 Complementation law for is...
isores2 6483 An isomorphism from one we...
isores1 6484 An isomorphism from one we...
isores3 6485 Induced isomorphism on a s...
isotr 6486 Composition (transitive) l...
isomin 6487 Isomorphisms preserve mini...
isoini 6488 Isomorphisms preserve init...
isoini2 6489 Isomorphisms are isomorphi...
isofrlem 6490 Lemma for ~ isofr . (Cont...
isoselem 6491 Lemma for ~ isose . (Cont...
isofr 6492 An isomorphism preserves w...
isose 6493 An isomorphism preserves s...
isofr2 6494 A weak form of ~ isofr tha...
isopolem 6495 Lemma for ~ isopo . (Cont...
isopo 6496 An isomorphism preserves p...
isosolem 6497 Lemma for ~ isoso . (Cont...
isoso 6498 An isomorphism preserves s...
isowe 6499 An isomorphism preserves w...
isowe2 6500 A weak form of ~ isowe tha...
f1oiso 6501 Any one-to-one onto functi...
f1oiso2 6502 Any one-to-one onto functi...
f1owe 6503 Well-ordering of isomorphi...
weniso 6504 A set-like well-ordering h...
weisoeq 6505 Thus, there is at most one...
weisoeq2 6506 Thus, there is at most one...
knatar 6507 The Knaster-Tarski theorem...
canth 6508 No set ` A ` is equinumero...
ncanth 6509 Cantor's theorem fails for...
riotaeqdv 6512 Formula-building deduction...
riotabidv 6513 Formula-building deduction...
riotaeqbidv 6514 Equality deduction for res...
riotaex 6515 Restricted iota is a set. ...
riotav 6516 An iota restricted to the ...
riotauni 6517 Restricted iota in terms o...
nfriota1 6518 The abstraction variable i...
nfriotad 6519 Deduction version of ~ nfr...
nfriota 6520 A variable not free in a w...
cbvriota 6521 Change bound variable in a...
cbvriotav 6522 Change bound variable in a...
csbriota 6523 Interchange class substitu...
riotacl2 6524 Membership law for "the un...
riotacl 6525 Closure of restricted iota...
riotasbc 6526 Substitution law for descr...
riotabidva 6527 Equivalent wff's yield equ...
riotabiia 6528 Equivalent wff's yield equ...
riota1 6529 Property of restricted iot...
riota1a 6530 Property of iota. (Contri...
riota2df 6531 A deduction version of ~ r...
riota2f 6532 This theorem shows a condi...
riota2 6533 This theorem shows a condi...
riotaprop 6534 Properties of a restricted...
riota5f 6535 A method for computing res...
riota5 6536 A method for computing res...
riotass2 6537 Restriction of a unique el...
riotass 6538 Restriction of a unique el...
moriotass 6539 Restriction of a unique el...
snriota 6540 A restricted class abstrac...
riotaxfrd 6541 Change the variable ` x ` ...
eusvobj2 6542 Specify the same property ...
eusvobj1 6543 Specify the same object in...
f1ofveu 6544 There is one domain elemen...
f1ocnvfv3 6545 Value of the converse of a...
riotaund 6546 Restricted iota equals the...
riotassuni 6547 The restricted iota class ...
riotaclb 6548 Bidirectional closure of r...
oveq 6555 Equality theorem for opera...
oveq1 6556 Equality theorem for opera...
oveq2 6557 Equality theorem for opera...
oveq12 6558 Equality theorem for opera...
oveq1i 6559 Equality inference for ope...
oveq2i 6560 Equality inference for ope...
oveq12i 6561 Equality inference for ope...
oveqi 6562 Equality inference for ope...
oveq123i 6563 Equality inference for ope...
oveq1d 6564 Equality deduction for ope...
oveq2d 6565 Equality deduction for ope...
oveqd 6566 Equality deduction for ope...
oveq12d 6567 Equality deduction for ope...
oveqan12d 6568 Equality deduction for ope...
oveqan12rd 6569 Equality deduction for ope...
oveq123d 6570 Equality deduction for ope...
ovrspc2v 6571 If an operation value is e...
oveqrspc2v 6572 Restricted specialization ...
oveqdr 6573 Equality of two operations...
nfovd 6574 Deduction version of bound...
nfov 6575 Bound-variable hypothesis ...
oprabid 6576 The law of concretion. Sp...
ovex 6577 The result of an operation...
ovexi 6578 The result of an operation...
ovssunirn 6579 The result of an operation...
0ov 6580 Operation value of the emp...
ovprc 6581 The value of an operation ...
ovprc1 6582 The value of an operation ...
ovprc2 6583 The value of an operation ...
ovrcl 6584 Reverse closure for an ope...
csbov123 6585 Move class substitution in...
csbov 6586 Move class substitution in...
csbov12g 6587 Move class substitution in...
csbov1g 6588 Move class substitution in...
csbov2g 6589 Move class substitution in...
rspceov 6590 A frequently used special ...
elovimad 6591 Elementhood of the image s...
fnotovb 6592 Equivalence of operation v...
opabbrex 6593 A collection of ordered pa...
fvmptopab1 6594 The function value of a ma...
fvmptopab2 6595 The function value of a ma...
0neqopab 6596 The empty set is never an ...
brabv 6597 If two classes are in a re...
brfvopab 6598 The classes involved in a ...
dfoprab2 6599 Class abstraction for oper...
reloprab 6600 An operation class abstrac...
oprabv 6601 If a pair and a class are ...
nfoprab1 6602 The abstraction variables ...
nfoprab2 6603 The abstraction variables ...
nfoprab3 6604 The abstraction variables ...
nfoprab 6605 Bound-variable hypothesis ...
oprabbid 6606 Equivalent wff's yield equ...
oprabbidv 6607 Equivalent wff's yield equ...
oprabbii 6608 Equivalent wff's yield equ...
ssoprab2 6609 Equivalence of ordered pai...
ssoprab2b 6610 Equivalence of ordered pai...
eqoprab2b 6611 Equivalence of ordered pai...
mpt2eq123 6612 An equality theorem for th...
mpt2eq12 6613 An equality theorem for th...
mpt2eq123dva 6614 An equality deduction for ...
mpt2eq123dv 6615 An equality deduction for ...
mpt2eq123i 6616 An equality inference for ...
mpt2eq3dva 6617 Slightly more general equa...
mpt2eq3ia 6618 An equality inference for ...
mpt2eq3dv 6619 An equality deduction for ...
nfmpt21 6620 Bound-variable hypothesis ...
nfmpt22 6621 Bound-variable hypothesis ...
nfmpt2 6622 Bound-variable hypothesis ...
mpt20 6623 A mapping operation with e...
oprab4 6624 Two ways to state the doma...
cbvoprab1 6625 Rule used to change first ...
cbvoprab2 6626 Change the second bound va...
cbvoprab12 6627 Rule used to change first ...
cbvoprab12v 6628 Rule used to change first ...
cbvoprab3 6629 Rule used to change the th...
cbvoprab3v 6630 Rule used to change the th...
cbvmpt2x 6631 Rule to change the bound v...
cbvmpt2 6632 Rule to change the bound v...
cbvmpt2v 6633 Rule to change the bound v...
elimdelov 6634 Eliminate a hypothesis whi...
ovif 6635 Move a conditional outside...
ovif2 6636 Move a conditional outside...
ovif12 6637 Move a conditional outside...
ifov 6638 Move a conditional outside...
dmoprab 6639 The domain of an operation...
dmoprabss 6640 The domain of an operation...
rnoprab 6641 The range of an operation ...
rnoprab2 6642 The range of a restricted ...
reldmoprab 6643 The domain of an operation...
oprabss 6644 Structure of an operation ...
eloprabga 6645 The law of concretion for ...
eloprabg 6646 The law of concretion for ...
ssoprab2i 6647 Inference of operation cla...
mpt2v 6648 Operation with universal d...
mpt2mptx 6649 Express a two-argument fun...
mpt2mpt 6650 Express a two-argument fun...
mpt2difsnif 6651 A mapping with two argumen...
mpt2snif 6652 A mapping with two argumen...
fconstmpt2 6653 Representation of a consta...
resoprab 6654 Restriction of an operatio...
resoprab2 6655 Restriction of an operator...
resmpt2 6656 Restriction of the mapping...
funoprabg 6657 "At most one" is a suffici...
funoprab 6658 "At most one" is a suffici...
fnoprabg 6659 Functionality and domain o...
mpt2fun 6660 The maps-to notation for a...
fnoprab 6661 Functionality and domain o...
ffnov 6662 An operation maps to a cla...
fovcl 6663 Closure law for an operati...
eqfnov 6664 Equality of two operations...
eqfnov2 6665 Two operators with the sam...
fnov 6666 Representation of a functi...
mpt22eqb 6667 Bidirectional equality the...
rnmpt2 6668 The range of an operation ...
reldmmpt2 6669 The domain of an operation...
elrnmpt2g 6670 Membership in the range of...
elrnmpt2 6671 Membership in the range of...
elrnmpt2res 6672 Membership in the range of...
ralrnmpt2 6673 A restricted quantifier ov...
rexrnmpt2 6674 A restricted quantifier ov...
ovid 6675 The value of an operation ...
ovidig 6676 The value of an operation ...
ovidi 6677 The value of an operation ...
ov 6678 The value of an operation ...
ovigg 6679 The value of an operation ...
ovig 6680 The value of an operation ...
ovmpt4g 6681 Value of a function given ...
ovmpt2s 6682 Value of a function given ...
ov2gf 6683 The value of an operation ...
ovmpt2dxf 6684 Value of an operation give...
ovmpt2dx 6685 Value of an operation give...
ovmpt2d 6686 Value of an operation give...
ovmpt2x 6687 The value of an operation ...
ovmpt2ga 6688 Value of an operation give...
ovmpt2a 6689 Value of an operation give...
ovmpt2df 6690 Alternate deduction versio...
ovmpt2dv 6691 Alternate deduction versio...
ovmpt2dv2 6692 Alternate deduction versio...
ovmpt2g 6693 Value of an operation give...
ovmpt2 6694 Value of an operation give...
ov3 6695 The value of an operation ...
ov6g 6696 The value of an operation ...
ovg 6697 The value of an operation ...
ovres 6698 The value of a restricted ...
ovresd 6699 Lemma for converting metri...
oprres 6700 The restriction of an oper...
oprssov 6701 The value of a member of t...
fovrn 6702 An operation's value belon...
fovrnda 6703 An operation's value belon...
fovrnd 6704 An operation's value belon...
fnrnov 6705 The range of an operation ...
foov 6706 An onto mapping of an oper...
fnovrn 6707 An operation's value belon...
ovelrn 6708 A member of an operation's...
funimassov 6709 Membership relation for th...
ovelimab 6710 Operation value in an imag...
ovima0 6711 An operation value is a me...
ovconst2 6712 The value of a constant op...
oprssdm 6713 Domain of closure of an op...
nssdmovg 6714 The value of an operation ...
ndmovg 6715 The value of an operation ...
ndmov 6716 The value of an operation ...
ndmovcl 6717 The closure of an operatio...
ndmovrcl 6718 Reverse closure law, when ...
ndmovcom 6719 Any operation is commutati...
ndmovass 6720 Any operation is associati...
ndmovdistr 6721 Any operation is distribut...
ndmovord 6722 Elimination of redundant a...
ndmovordi 6723 Elimination of redundant a...
caovclg 6724 Convert an operation closu...
caovcld 6725 Convert an operation closu...
caovcl 6726 Convert an operation closu...
caovcomg 6727 Convert an operation commu...
caovcomd 6728 Convert an operation commu...
caovcom 6729 Convert an operation commu...
caovassg 6730 Convert an operation assoc...
caovassd 6731 Convert an operation assoc...
caovass 6732 Convert an operation assoc...
caovcang 6733 Convert an operation cance...
caovcand 6734 Convert an operation cance...
caovcanrd 6735 Commute the arguments of a...
caovcan 6736 Convert an operation cance...
caovordig 6737 Convert an operation order...
caovordid 6738 Convert an operation order...
caovordg 6739 Convert an operation order...
caovordd 6740 Convert an operation order...
caovord2d 6741 Operation ordering law wit...
caovord3d 6742 Ordering law. (Contribute...
caovord 6743 Convert an operation order...
caovord2 6744 Operation ordering law wit...
caovord3 6745 Ordering law. (Contribute...
caovdig 6746 Convert an operation distr...
caovdid 6747 Convert an operation distr...
caovdir2d 6748 Convert an operation distr...
caovdirg 6749 Convert an operation rever...
caovdird 6750 Convert an operation distr...
caovdi 6751 Convert an operation distr...
caov32d 6752 Rearrange arguments in a c...
caov12d 6753 Rearrange arguments in a c...
caov31d 6754 Rearrange arguments in a c...
caov13d 6755 Rearrange arguments in a c...
caov4d 6756 Rearrange arguments in a c...
caov411d 6757 Rearrange arguments in a c...
caov42d 6758 Rearrange arguments in a c...
caov32 6759 Rearrange arguments in a c...
caov12 6760 Rearrange arguments in a c...
caov31 6761 Rearrange arguments in a c...
caov13 6762 Rearrange arguments in a c...
caov4 6763 Rearrange arguments in a c...
caov411 6764 Rearrange arguments in a c...
caov42 6765 Rearrange arguments in a c...
caovdir 6766 Reverse distributive law. ...
caovdilem 6767 Lemma used by real number ...
caovlem2 6768 Lemma used in real number ...
caovmo 6769 Uniqueness of inverse elem...
grprinvlem 6770 Lemma for ~ grprinvd . (C...
grprinvd 6771 Deduce right inverse from ...
grpridd 6772 Deduce right identity from...
mpt2ndm0 6773 The value of an operation ...
elmpt2cl 6774 If a two-parameter class i...
elmpt2cl1 6775 If a two-parameter class i...
elmpt2cl2 6776 If a two-parameter class i...
elovmpt2 6777 Utility lemma for two-para...
elovmpt2rab 6778 Implications for the value...
elovmpt2rab1 6779 Implications for the value...
2mpt20 6780 If the operation value of ...
relmptopab 6781 Any function to sets of or...
f1ocnvd 6782 Describe an implicit one-t...
f1od 6783 Describe an implicit one-t...
f1ocnv2d 6784 Describe an implicit one-t...
f1o2d 6785 Describe an implicit one-t...
f1opw2 6786 A one-to-one mapping induc...
f1opw 6787 A one-to-one mapping induc...
elovmpt3imp 6788 If the value of a function...
ovmpt3rab1 6789 The value of an operation ...
ovmpt3rabdm 6790 If the value of a function...
elovmpt3rab1 6791 Implications for the value...
elovmpt3rab 6792 Implications for the value...
ofeq 6797 Equality theorem for funct...
ofreq 6798 Equality theorem for funct...
ofexg 6799 A function operation restr...
nfof 6800 Hypothesis builder for fun...
nfofr 6801 Hypothesis builder for fun...
offval 6802 Value of an operation appl...
ofrfval 6803 Value of a relation applie...
ofval 6804 Evaluate a function operat...
ofrval 6805 Exhibit a function relatio...
offn 6806 The function operation pro...
offval2f 6807 The function operation exp...
ofmresval 6808 Value of a restriction of ...
fnfvof 6809 Function value of a pointw...
off 6810 The function operation pro...
ofres 6811 Restrict the operands of a...
offval2 6812 The function operation exp...
ofrfval2 6813 The function relation acti...
ofmpteq 6814 Value of a pointwise opera...
ofco 6815 The composition of a funct...
offveq 6816 Convert an identity of the...
offveqb 6817 Equivalent expressions for...
ofc1 6818 Left operation by a consta...
ofc2 6819 Right operation by a const...
ofc12 6820 Function operation on two ...
caofref 6821 Transfer a reflexive law t...
caofinvl 6822 Transfer a left inverse la...
caofid0l 6823 Transfer a left identity l...
caofid0r 6824 Transfer a right identity ...
caofid1 6825 Transfer a right absorptio...
caofid2 6826 Transfer a right absorptio...
caofcom 6827 Transfer a commutative law...
caofrss 6828 Transfer a relation subset...
caofass 6829 Transfer an associative la...
caoftrn 6830 Transfer a transitivity la...
caofdi 6831 Transfer a distributive la...
caofdir 6832 Transfer a reverse distrib...
caonncan 6833 Transfer ~ nncan -shaped l...
relrpss 6836 The proper subset relation...
brrpssg 6837 The proper subset relation...
brrpss 6838 The proper subset relation...
porpss 6839 Every class is partially o...
sorpss 6840 Express strict ordering un...
sorpssi 6841 Property of a chain of set...
sorpssun 6842 A chain of sets is closed ...
sorpssin 6843 A chain of sets is closed ...
sorpssuni 6844 In a chain of sets, a maxi...
sorpssint 6845 In a chain of sets, a mini...
sorpsscmpl 6846 The componentwise compleme...
zfun 6848 Axiom of Union expressed w...
axun2 6849 A variant of the Axiom of ...
uniex2 6850 The Axiom of Union using t...
uniex 6851 The Axiom of Union in clas...
vuniex 6852 The union of a setvar is a...
uniexg 6853 The ZF Axiom of Union in c...
unex 6854 The union of two sets is a...
tpex 6855 An unordered triple of cla...
unexb 6856 Existence of union is equi...
unexg 6857 A union of two sets is a s...
xpexg 6858 The Cartesian product of t...
3xpexg 6859 The Cartesian product of t...
xpex 6860 The Cartesian product of t...
sqxpexg 6861 The Cartesian square of a ...
snnex 6862 The class of all singleton...
difex2 6863 If the subtrahend of a cla...
difsnexi 6864 If the difference of a cla...
uniuni 6865 Expression for double unio...
uniexb 6866 The Axiom of Union and its...
pwexb 6867 The Axiom of Power Sets an...
eldifpw 6868 Membership in a power clas...
elpwun 6869 Membership in the power cl...
iunpw 6870 An indexed union of a powe...
fr3nr 6871 A well-founded relation ha...
epne3 6872 A set well-founded by epsi...
dfwe2 6873 Alternate definition of we...
ordon 6874 The class of all ordinal n...
epweon 6875 The epsilon relation well-...
onprc 6876 No set contains all ordina...
ssorduni 6877 The union of a class of or...
ssonuni 6878 The union of a set of ordi...
ssonunii 6879 The union of a set of ordi...
ordeleqon 6880 A way to express the ordin...
ordsson 6881 Any ordinal class is a sub...
onss 6882 An ordinal number is a sub...
predon 6883 For an ordinal, the predec...
ssonprc 6884 Two ways of saying a class...
onuni 6885 The union of an ordinal nu...
orduni 6886 The union of an ordinal cl...
onint 6887 The intersection (infimum)...
onint0 6888 The intersection of a clas...
onssmin 6889 A nonempty class of ordina...
onminesb 6890 If a property is true for ...
onminsb 6891 If a property is true for ...
oninton 6892 The intersection of a none...
onintrab 6893 The intersection of a clas...
onintrab2 6894 An existence condition equ...
onnmin 6895 No member of a set of ordi...
onnminsb 6896 An ordinal number smaller ...
oneqmin 6897 A way to show that an ordi...
bm2.5ii 6898 Problem 2.5(ii) of [BellMa...
onminex 6899 If a wff is true for an or...
sucon 6900 The class of all ordinal n...
sucexb 6901 A successor exists iff its...
sucexg 6902 The successor of a set is ...
sucex 6903 The successor of a set is ...
onmindif2 6904 The minimum of a class of ...
suceloni 6905 The successor of an ordina...
ordsuc 6906 The successor of an ordina...
ordpwsuc 6907 The collection of ordinals...
onpwsuc 6908 The collection of ordinal ...
sucelon 6909 The successor of an ordina...
ordsucss 6910 The successor of an elemen...
onpsssuc 6911 An ordinal number is a pro...
ordelsuc 6912 A set belongs to an ordina...
onsucmin 6913 The successor of an ordina...
ordsucelsuc 6914 Membership is inherited by...
ordsucsssuc 6915 The subclass relationship ...
ordsucuniel 6916 Given an element ` A ` of ...
ordsucun 6917 The successor of the maxim...
ordunpr 6918 The maximum of two ordinal...
ordunel 6919 The maximum of two ordinal...
onsucuni 6920 A class of ordinal numbers...
ordsucuni 6921 An ordinal class is a subc...
orduniorsuc 6922 An ordinal class is either...
unon 6923 The class of all ordinal n...
ordunisuc 6924 An ordinal class is equal ...
orduniss2 6925 The union of the ordinal s...
onsucuni2 6926 A successor ordinal is the...
0elsuc 6927 The successor of an ordina...
limon 6928 The class of ordinal numbe...
onssi 6929 An ordinal number is a sub...
onsuci 6930 The successor of an ordina...
onuniorsuci 6931 An ordinal number is eithe...
onuninsuci 6932 A limit ordinal is not a s...
onsucssi 6933 A set belongs to an ordina...
nlimsucg 6934 A successor is not a limit...
orduninsuc 6935 An ordinal equal to its un...
ordunisuc2 6936 An ordinal equal to its un...
ordzsl 6937 An ordinal is zero, a succ...
onzsl 6938 An ordinal number is zero,...
dflim3 6939 An alternate definition of...
dflim4 6940 An alternate definition of...
limsuc 6941 The successor of a member ...
limsssuc 6942 A class includes a limit o...
nlimon 6943 Two ways to express the cl...
limuni3 6944 The union of a nonempty cl...
tfi 6945 The Principle of Transfini...
tfis 6946 Transfinite Induction Sche...
tfis2f 6947 Transfinite Induction Sche...
tfis2 6948 Transfinite Induction Sche...
tfis3 6949 Transfinite Induction Sche...
tfisi 6950 A transfinite induction sc...
tfinds 6951 Principle of Transfinite I...
tfindsg 6952 Transfinite Induction (inf...
tfindsg2 6953 Transfinite Induction (inf...
tfindes 6954 Transfinite Induction with...
tfinds2 6955 Transfinite Induction (inf...
tfinds3 6956 Principle of Transfinite I...
dfom2 6959 An alternate definition of...
elom 6960 Membership in omega. The ...
omsson 6961 Omega is a subset of ` On ...
limomss 6962 The class of natural numbe...
nnon 6963 A natural number is an ord...
nnoni 6964 A natural number is an ord...
nnord 6965 A natural number is ordina...
ordom 6966 Omega is ordinal. Theorem...
elnn 6967 A member of a natural numb...
omon 6968 The class of natural numbe...
omelon2 6969 Omega is an ordinal number...
nnlim 6970 A natural number is not a ...
omssnlim 6971 The class of natural numbe...
limom 6972 Omega is a limit ordinal. ...
peano2b 6973 A class belongs to omega i...
nnsuc 6974 A nonzero natural number i...
ssnlim 6975 An ordinal subclass of non...
omsinds 6976 Strong (or "total") induct...
peano1 6977 Zero is a natural number. ...
peano2 6978 The successor of any natur...
peano3 6979 The successor of any natur...
peano4 6980 Two natural numbers are eq...
peano5 6981 The induction postulate: a...
nn0suc 6982 A natural number is either...
find 6983 The Principle of Finite In...
finds 6984 Principle of Finite Induct...
findsg 6985 Principle of Finite Induct...
finds2 6986 Principle of Finite Induct...
finds1 6987 Principle of Finite Induct...
findes 6988 Finite induction with expl...
dmexg 6989 The domain of a set is a s...
rnexg 6990 The range of a set is a se...
dmex 6991 The domain of a set is a s...
rnex 6992 The range of a set is a se...
iprc 6993 The identity function is a...
resiexg 6994 The existence of a restric...
imaexg 6995 The image of a set is a se...
imaex 6996 The image of a set is a se...
opabex2 6997 Condition for an operation...
exse2 6998 Any set relation is set-li...
xpexr 6999 If a Cartesian product is ...
xpexr2 7000 If a nonempty Cartesian pr...
xpexcnv 7001 A condition where the conv...
soex 7002 If the relation in a stric...
elxp4 7003 Membership in a Cartesian ...
elxp5 7004 Membership in a Cartesian ...
cnvexg 7005 The converse of a set is a...
cnvex 7006 The converse of a set is a...
relcnvexb 7007 A relation is a set iff it...
f1oexrnex 7008 If the range of a 1-1 onto...
f1oexbi 7009 There is a one-to-one onto...
coexg 7010 The composition of two set...
coex 7011 The composition of two set...
funcnvuni 7012 The union of a chain (with...
fun11uni 7013 The union of a chain (with...
fex2 7014 A function with bounded do...
fabexg 7015 Existence of a set of func...
fabex 7016 Existence of a set of func...
dmfex 7017 If a mapping is a set, its...
f1oabexg 7018 The class of all 1-1-onto ...
fun11iun 7019 The union of a chain (with...
ffoss 7020 Relationship between a map...
f11o 7021 Relationship between one-t...
resfunexgALT 7022 Alternate proof of ~ resfu...
cofunexg 7023 Existence of a composition...
cofunex2g 7024 Existence of a composition...
fnexALT 7025 Alternate proof of ~ fnex ...
funrnex 7026 If the domain of a functio...
zfrep6 7027 A version of the Axiom of ...
fornex 7028 If the domain of an onto f...
f1dmex 7029 If the codomain of a one-t...
f1ovv 7030 The range of a 1-1 onto fu...
fvclex 7031 Existence of the class of ...
fvresex 7032 Existence of the class of ...
abrexex 7033 Existence of a class abstr...
abrexexg 7034 Existence of a class abstr...
iunexg 7035 The existence of an indexe...
abrexex2g 7036 Existence of an existentia...
opabex3d 7037 Existence of an ordered pa...
opabex3 7038 Existence of an ordered pa...
iunex 7039 The existence of an indexe...
abrexex2 7040 Existence of an existentia...
abexssex 7041 Existence of a class abstr...
abexex 7042 A condition where a class ...
f1oweALT 7043 Alternate proof of ~ f1owe...
wemoiso 7044 Thus, there is at most one...
wemoiso2 7045 Thus, there is at most one...
oprabexd 7046 Existence of an operator a...
oprabex 7047 Existence of an operation ...
oprabex3 7048 Existence of an operation ...
oprabrexex2 7049 Existence of an existentia...
ab2rexex 7050 Existence of a class abstr...
ab2rexex2 7051 Existence of an existentia...
xpexgALT 7052 Alternate proof of ~ xpexg...
offval3 7053 General value of ` ( F oF ...
offres 7054 Pointwise combination comm...
ofmres 7055 Equivalent expressions for...
ofmresex 7056 Existence of a restriction...
1stval 7061 The value of the function ...
2ndval 7062 The value of the function ...
1stnpr 7063 Value of the first-member ...
2ndnpr 7064 Value of the second-member...
1st0 7065 The value of the first-mem...
2nd0 7066 The value of the second-me...
op1st 7067 Extract the first member o...
op2nd 7068 Extract the second member ...
op1std 7069 Extract the first member o...
op2ndd 7070 Extract the second member ...
op1stg 7071 Extract the first member o...
op2ndg 7072 Extract the second member ...
ot1stg 7073 Extract the first member o...
ot2ndg 7074 Extract the second member ...
ot3rdg 7075 Extract the third member o...
1stval2 7076 Alternate value of the fun...
2ndval2 7077 Alternate value of the fun...
oteqimp 7078 The components of an order...
fo1st 7079 The ` 1st ` function maps ...
fo2nd 7080 The ` 2nd ` function maps ...
f1stres 7081 Mapping of a restriction o...
f2ndres 7082 Mapping of a restriction o...
fo1stres 7083 Onto mapping of a restrict...
fo2ndres 7084 Onto mapping of a restrict...
1st2val 7085 Value of an alternate defi...
2nd2val 7086 Value of an alternate defi...
1stcof 7087 Composition of the first m...
2ndcof 7088 Composition of the second ...
xp1st 7089 Location of the first elem...
xp2nd 7090 Location of the second ele...
elxp6 7091 Membership in a Cartesian ...
elxp7 7092 Membership in a Cartesian ...
eqopi 7093 Equality with an ordered p...
xp2 7094 Representation of Cartesia...
unielxp 7095 The membership relation fo...
1st2nd2 7096 Reconstruction of a member...
1st2ndb 7097 Reconstruction of an order...
xpopth 7098 An ordered pair theorem fo...
eqop 7099 Two ways to express equali...
eqop2 7100 Two ways to express equali...
op1steq 7101 Two ways of expressing tha...
el2xptp 7102 A member of a nested Carte...
el2xptp0 7103 A member of a nested Carte...
2nd1st 7104 Swap the members of an ord...
1st2nd 7105 Reconstruction of a member...
1stdm 7106 The first ordered pair com...
2ndrn 7107 The second ordered pair co...
1st2ndbr 7108 Express an element of a re...
releldm2 7109 Two ways of expressing mem...
reldm 7110 An expression for the doma...
sbcopeq1a 7111 Equality theorem for subst...
csbopeq1a 7112 Equality theorem for subst...
dfopab2 7113 A way to define an ordered...
dfoprab3s 7114 A way to define an operati...
dfoprab3 7115 Operation class abstractio...
dfoprab4 7116 Operation class abstractio...
dfoprab4f 7117 Operation class abstractio...
opiota 7118 The property of a uniquely...
dfxp3 7119 Define the Cartesian produ...
elopabi 7120 A consequence of membershi...
eloprabi 7121 A consequence of membershi...
mpt2mptsx 7122 Express a two-argument fun...
mpt2mpts 7123 Express a two-argument fun...
dmmpt2ssx 7124 The domain of a mapping is...
fmpt2x 7125 Functionality, domain and ...
fmpt2 7126 Functionality, domain and ...
fnmpt2 7127 Functionality and domain o...
fnmpt2i 7128 Functionality and domain o...
dmmpt2 7129 Domain of a class given by...
ovmpt2elrn 7130 An operation's value belon...
dmmpt2ga 7131 Domain of an operation giv...
dmmpt2g 7132 Domain of an operation giv...
mpt2exxg 7133 Existence of an operation ...
mpt2exg 7134 Existence of an operation ...
mpt2exga 7135 If the domain of a functio...
mpt2ex 7136 If the domain of a functio...
el2mpt2csbcl 7137 If the operation value of ...
el2mpt2cl 7138 If the operation value of ...
fnmpt2ovd 7139 A function with a Cartesia...
offval22 7140 The function operation exp...
brovpreldm 7141 If a binary relation holds...
bropopvvv 7142 If a binary relation holds...
bropfvvvvlem 7143 Lemma for ~ bropfvvvv . (...
bropfvvvv 7144 If a binary relation holds...
ovmptss 7145 If all the values of the m...
relmpt2opab 7146 Any function to sets of or...
fmpt2co 7147 Composition of two functio...
oprabco 7148 Composition of a function ...
oprab2co 7149 Composition of operator ab...
df1st2 7150 An alternate possible defi...
df2nd2 7151 An alternate possible defi...
1stconst 7152 The mapping of a restricti...
2ndconst 7153 The mapping of a restricti...
dfmpt2 7154 Alternate definition for t...
mpt2sn 7155 An operation (in maps-to n...
curry1 7156 Composition with ` ``' ( 2...
curry1val 7157 The value of a curried fun...
curry1f 7158 Functionality of a curried...
curry2 7159 Composition with ` ``' ( 1...
curry2f 7160 Functionality of a curried...
curry2val 7161 The value of a curried fun...
cnvf1olem 7162 Lemma for ~ cnvf1o . (Con...
cnvf1o 7163 Describe a function that m...
fparlem1 7164 Lemma for ~ fpar . (Contr...
fparlem2 7165 Lemma for ~ fpar . (Contr...
fparlem3 7166 Lemma for ~ fpar . (Contr...
fparlem4 7167 Lemma for ~ fpar . (Contr...
fpar 7168 Merge two functions in par...
fsplit 7169 A function that can be use...
f2ndf 7170 The ` 2nd ` (second member...
fo2ndf 7171 The ` 2nd ` (second member...
f1o2ndf1 7172 The ` 2nd ` (second member...
algrflem 7173 Lemma for ~ algrf and rela...
frxp 7174 A lexicographical ordering...
xporderlem 7175 Lemma for lexicographical ...
poxp 7176 A lexicographical ordering...
soxp 7177 A lexicographical ordering...
wexp 7178 A lexicographical ordering...
fnwelem 7179 Lemma for ~ fnwe . (Contr...
fnwe 7180 A variant on lexicographic...
fnse 7181 Condition for the well-ord...
suppval 7184 The value of the operation...
supp0prc 7185 The support of a class is ...
suppvalbr 7186 The value of the operation...
supp0 7187 The support of the empty s...
suppval1 7188 The value of the operation...
suppvalfn 7189 The value of the operation...
elsuppfn 7190 An element of the support ...
cnvimadfsn 7191 The support of functions "...
suppimacnvss 7192 The support of functions "...
suppimacnv 7193 Support sets of functions ...
frnsuppeq 7194 Two ways of writing the su...
suppssdm 7195 The support of a function ...
suppsnop 7196 The support of a singleton...
snopsuppss 7197 The support of a singleton...
fvn0elsupp 7198 If the function value for ...
fvn0elsuppb 7199 The function value for a g...
rexsupp 7200 Existential quantification...
ressuppss 7201 The support of the restric...
suppun 7202 The support of a class/fun...
ressuppssdif 7203 The support of the restric...
mptsuppdifd 7204 The support of a function ...
mptsuppd 7205 The support of a function ...
extmptsuppeq 7206 The support of an extended...
suppfnss 7207 The support of a function ...
funsssuppss 7208 The support of a function ...
fnsuppres 7209 Two ways to express restri...
fnsuppeq0 7210 The support of a function ...
fczsupp0 7211 The support of a constant ...
suppss 7212 Show that the support of a...
suppssr 7213 A function is zero outside...
suppssov1 7214 Formula building theorem f...
suppssof1 7215 Formula building theorem f...
suppss2 7216 Show that the support of a...
suppsssn 7217 Show that the support of a...
suppssfv 7218 Formula building theorem f...
suppofss1d 7219 Condition for the support ...
suppofss2d 7220 Condition for the support ...
supp0cosupp0 7221 The support of the composi...
imacosupp 7222 The image of the support o...
opeliunxp2f 7223 Membership in a union of C...
mpt2xeldm 7224 If there is an element of ...
mpt2xneldm 7225 If the first argument of a...
mpt2xopn0yelv 7226 If there is an element of ...
mpt2xopynvov0g 7227 If the second argument of ...
mpt2xopxnop0 7228 If the first argument of a...
mpt2xopx0ov0 7229 If the first argument of a...
mpt2xopxprcov0 7230 If the components of the f...
mpt2xopynvov0 7231 If the second argument of ...
mpt2xopoveq 7232 Value of an operation give...
mpt2xopovel 7233 Element of the value of an...
mpt2xopoveqd 7234 Value of an operation give...
brovex 7235 A binary relation of the v...
brovmpt2ex 7236 A binary relation of the v...
sprmpt2d 7237 The extension of a binary ...
tposss 7240 Subset theorem for transpo...
tposeq 7241 Equality theorem for trans...
tposeqd 7242 Equality theorem for trans...
tposssxp 7243 The transposition is a sub...
reltpos 7244 The transposition is a rel...
brtpos2 7245 Value of the transposition...
brtpos0 7246 The behavior of ` tpos ` w...
reldmtpos 7247 Necessary and sufficient c...
brtpos 7248 The transposition swaps ar...
ottpos 7249 The transposition swaps th...
relbrtpos 7250 The transposition swaps ar...
dmtpos 7251 The domain of ` tpos F ` w...
rntpos 7252 The range of ` tpos F ` wh...
tposexg 7253 The transposition of a set...
ovtpos 7254 The transposition swaps th...
tposfun 7255 The transposition of a fun...
dftpos2 7256 Alternate definition of ` ...
dftpos3 7257 Alternate definition of ` ...
dftpos4 7258 Alternate definition of ` ...
tpostpos 7259 Value of the double transp...
tpostpos2 7260 Value of the double transp...
tposfn2 7261 The domain of a transposit...
tposfo2 7262 Condition for a surjective...
tposf2 7263 The domain and range of a ...
tposf12 7264 Condition for an injective...
tposf1o2 7265 Condition of a bijective t...
tposfo 7266 The domain and range of a ...
tposf 7267 The domain and range of a ...
tposfn 7268 Functionality of a transpo...
tpos0 7269 Transposition of the empty...
tposco 7270 Transposition of a composi...
tpossym 7271 Two ways to say a function...
tposeqi 7272 Equality theorem for trans...
tposex 7273 A transposition is a set. ...
nftpos 7274 Hypothesis builder for tra...
tposoprab 7275 Transposition of a class o...
tposmpt2 7276 Transposition of a two-arg...
tposconst 7277 The transposition of a con...
mpt2curryd 7282 The currying of an operati...
mpt2curryvald 7283 The value of a curried ope...
fvmpt2curryd 7284 The value of the value of ...
pwuninel2 7287 Direct proof of ~ pwuninel...
pwuninel 7288 The power set of the union...
undefval 7289 Value of the undefined val...
undefnel2 7290 The undefined value genera...
undefnel 7291 The undefined value genera...
undefne0 7292 The undefined value genera...
wrecseq123 7295 General equality theorem f...
nfwrecs 7296 Bound-variable hypothesis ...
wrecseq1 7297 Equality theorem for the w...
wrecseq2 7298 Equality theorem for the w...
wrecseq3 7299 Equality theorem for the w...
wfr3g 7300 Functions defined by well-...
wfrlem1 7301 Lemma for well-founded rec...
wfrlem2 7302 Lemma for well-founded rec...
wfrlem3 7303 Lemma for well-founded rec...
wfrlem3a 7304 Lemma for well-founded rec...
wfrlem4 7305 Lemma for well-founded rec...
wfrlem5 7306 Lemma for well-founded rec...
wfrrel 7307 The well-founded recursion...
wfrdmss 7308 The domain of the well-fou...
wfrlem8 7309 Lemma for well-founded rec...
wfrdmcl 7310 Given ` F = wrecs ( R , A ...
wfrlem10 7311 Lemma for well-founded rec...
wfrfun 7312 The well-founded function ...
wfrlem12 7313 Lemma for well-founded rec...
wfrlem13 7314 Lemma for well-founded rec...
wfrlem14 7315 Lemma for well-founded rec...
wfrlem15 7316 Lemma for well-founded rec...
wfrlem16 7317 Lemma for well-founded rec...
wfrlem17 7318 Without using ~ ax-rep , s...
wfr2a 7319 A weak version of ~ wfr2 w...
wfr1 7320 The Principle of Well-Foun...
wfr2 7321 The Principle of Well-Foun...
wfr3 7322 The principle of Well-Foun...
iunon 7323 The indexed union of a set...
iinon 7324 The nonempty indexed inter...
onfununi 7325 A property of functions on...
onovuni 7326 A variant of ~ onfununi fo...
onoviun 7327 A variant of ~ onovuni wit...
onnseq 7328 There are no length ` _om ...
dfsmo2 7331 Alternate definition of a ...
issmo 7332 Conditions for which ` A `...
issmo2 7333 Alternate definition of a ...
smoeq 7334 Equality theorem for stric...
smodm 7335 The domain of a strictly m...
smores 7336 A strictly monotone functi...
smores3 7337 A strictly monotone functi...
smores2 7338 A strictly monotone ordina...
smodm2 7339 The domain of a strictly m...
smofvon2 7340 The function values of a s...
iordsmo 7341 The identity relation rest...
smo0 7342 The null set is a strictly...
smofvon 7343 If ` B ` is a strictly mon...
smoel 7344 If ` x ` is less than ` y ...
smoiun 7345 The value of a strictly mo...
smoiso 7346 If ` F ` is an isomorphism...
smoel2 7347 A strictly monotone ordina...
smo11 7348 A strictly monotone ordina...
smoord 7349 A strictly monotone ordina...
smoword 7350 A strictly monotone ordina...
smogt 7351 A strictly monotone ordina...
smorndom 7352 The range of a strictly mo...
smoiso2 7353 The strictly monotone ordi...
dfrecs3 7356 The old definition of tran...
recseq 7357 Equality theorem for ` rec...
nfrecs 7358 Bound-variable hypothesis ...
tfrlem1 7359 A technical lemma for tran...
tfrlem3a 7360 Lemma for transfinite recu...
tfrlem3 7361 Lemma for transfinite recu...
tfrlem4 7362 Lemma for transfinite recu...
tfrlem5 7363 Lemma for transfinite recu...
recsfval 7364 Lemma for transfinite recu...
tfrlem6 7365 Lemma for transfinite recu...
tfrlem7 7366 Lemma for transfinite recu...
tfrlem8 7367 Lemma for transfinite recu...
tfrlem9 7368 Lemma for transfinite recu...
tfrlem9a 7369 Lemma for transfinite recu...
tfrlem10 7370 Lemma for transfinite recu...
tfrlem11 7371 Lemma for transfinite recu...
tfrlem12 7372 Lemma for transfinite recu...
tfrlem13 7373 Lemma for transfinite recu...
tfrlem14 7374 Lemma for transfinite recu...
tfrlem15 7375 Lemma for transfinite recu...
tfrlem16 7376 Lemma for finite recursion...
tfr1a 7377 A weak version of ~ tfr1 w...
tfr2a 7378 A weak version of ~ tfr2 w...
tfr2b 7379 Without assuming ~ ax-rep ...
tfr1 7380 Principle of Transfinite R...
tfr2 7381 Principle of Transfinite R...
tfr3 7382 Principle of Transfinite R...
tfr1ALT 7383 Alternate proof of ~ tfr1 ...
tfr2ALT 7384 Alternate proof of ~ tfr2 ...
tfr3ALT 7385 Alternate proof of ~ tfr3 ...
recsfnon 7386 Strong transfinite recursi...
recsval 7387 Strong transfinite recursi...
tz7.44lem1 7388 ` G ` is a function. Lemm...
tz7.44-1 7389 The value of ` F ` at ` (/...
tz7.44-2 7390 The value of ` F ` at a su...
tz7.44-3 7391 The value of ` F ` at a li...
rdgeq1 7394 Equality theorem for the r...
rdgeq2 7395 Equality theorem for the r...
rdgeq12 7396 Equality theorem for the r...
nfrdg 7397 Bound-variable hypothesis ...
rdglem1 7398 Lemma used with the recurs...
rdgfun 7399 The recursive definition g...
rdgdmlim 7400 The domain of the recursiv...
rdgfnon 7401 The recursive definition g...
rdgvalg 7402 Value of the recursive def...
rdgval 7403 Value of the recursive def...
rdg0 7404 The initial value of the r...
rdgseg 7405 The initial segments of th...
rdgsucg 7406 The value of the recursive...
rdgsuc 7407 The value of the recursive...
rdglimg 7408 The value of the recursive...
rdglim 7409 The value of the recursive...
rdg0g 7410 The initial value of the r...
rdgsucmptf 7411 The value of the recursive...
rdgsucmptnf 7412 The value of the recursive...
rdgsucmpt2 7413 This version of ~ rdgsucmp...
rdgsucmpt 7414 The value of the recursive...
rdglim2 7415 The value of the recursive...
rdglim2a 7416 The value of the recursive...
frfnom 7417 The function generated by ...
fr0g 7418 The initial value resultin...
frsuc 7419 The successor value result...
frsucmpt 7420 The successor value result...
frsucmptn 7421 The value of the finite re...
frsucmpt2 7422 The successor value result...
tz7.48lem 7423 A way of showing an ordina...
tz7.48-2 7424 Proposition 7.48(2) of [Ta...
tz7.48-1 7425 Proposition 7.48(1) of [Ta...
tz7.48-3 7426 Proposition 7.48(3) of [Ta...
tz7.49 7427 Proposition 7.49 of [Takeu...
tz7.49c 7428 Corollary of Proposition 7...
seqomlem0 7431 Lemma for ` seqom ` . Cha...
seqomlem1 7432 Lemma for ` seqom ` . The...
seqomlem2 7433 Lemma for ` seqom ` . (Co...
seqomlem3 7434 Lemma for ` seqom ` . (Co...
seqomlem4 7435 Lemma for ` seqom ` . (Co...
seqomeq12 7436 Equality theorem for ` seq...
fnseqom 7437 An index-aware recursive d...
seqom0g 7438 Value of an index-aware re...
seqomsuc 7439 Value of an index-aware re...
1on 7454 Ordinal 1 is an ordinal nu...
2on 7455 Ordinal 2 is an ordinal nu...
2on0 7456 Ordinal two is not zero. ...
3on 7457 Ordinal 3 is an ordinal nu...
4on 7458 Ordinal 3 is an ordinal nu...
df1o2 7459 Expanded value of the ordi...
df2o3 7460 Expanded value of the ordi...
df2o2 7461 Expanded value of the ordi...
1n0 7462 Ordinal one is not equal t...
xp01disj 7463 Cartesian products with th...
ordgt0ge1 7464 Two ways to express that a...
ordge1n0 7465 An ordinal greater than or...
el1o 7466 Membership in ordinal one....
dif1o 7467 Two ways to say that ` A `...
ondif1 7468 Two ways to say that ` A `...
ondif2 7469 Two ways to say that ` A `...
2oconcl 7470 Closure of the pair swappi...
0lt1o 7471 Ordinal zero is less than ...
dif20el 7472 An ordinal greater than on...
0we1 7473 The empty set is a well-or...
brwitnlem 7474 Lemma for relations which ...
fnoa 7475 Functionality and domain o...
fnom 7476 Functionality and domain o...
fnoe 7477 Functionality and domain o...
oav 7478 Value of ordinal addition....
omv 7479 Value of ordinal multiplic...
oe0lem 7480 A helper lemma for ~ oe0 a...
oev 7481 Value of ordinal exponenti...
oevn0 7482 Value of ordinal exponenti...
oa0 7483 Addition with zero. Propo...
om0 7484 Ordinal multiplication wit...
oe0m 7485 Ordinal exponentiation wit...
om0x 7486 Ordinal multiplication wit...
oe0m0 7487 Ordinal exponentiation wit...
oe0m1 7488 Ordinal exponentiation wit...
oe0 7489 Ordinal exponentiation wit...
oev2 7490 Alternate value of ordinal...
oasuc 7491 Addition with successor. ...
oesuclem 7492 Lemma for ~ oesuc . (Cont...
omsuc 7493 Multiplication with succes...
oesuc 7494 Ordinal exponentiation wit...
onasuc 7495 Addition with successor. ...
onmsuc 7496 Multiplication with succes...
onesuc 7497 Exponentiation with a succ...
oa1suc 7498 Addition with 1 is same as...
oalim 7499 Ordinal addition with a li...
omlim 7500 Ordinal multiplication wit...
oelim 7501 Ordinal exponentiation wit...
oacl 7502 Closure law for ordinal ad...
omcl 7503 Closure law for ordinal mu...
oecl 7504 Closure law for ordinal ex...
oa0r 7505 Ordinal addition with zero...
om0r 7506 Ordinal multiplication wit...
o1p1e2 7507 1 + 1 = 2 for ordinal numb...
o2p2e4 7508 2 + 2 = 4 for ordinal numb...
om1 7509 Ordinal multiplication wit...
om1r 7510 Ordinal multiplication wit...
oe1 7511 Ordinal exponentiation wit...
oe1m 7512 Ordinal exponentiation wit...
oaordi 7513 Ordering property of ordin...
oaord 7514 Ordering property of ordin...
oacan 7515 Left cancellation law for ...
oaword 7516 Weak ordering property of ...
oawordri 7517 Weak ordering property of ...
oaord1 7518 An ordinal is less than it...
oaword1 7519 An ordinal is less than or...
oaword2 7520 An ordinal is less than or...
oawordeulem 7521 Lemma for ~ oawordex . (C...
oawordeu 7522 Existence theorem for weak...
oawordexr 7523 Existence theorem for weak...
oawordex 7524 Existence theorem for weak...
oaordex 7525 Existence theorem for orde...
oa00 7526 An ordinal sum is zero iff...
oalimcl 7527 The ordinal sum with a lim...
oaass 7528 Ordinal addition is associ...
oarec 7529 Recursive definition of or...
oaf1o 7530 Left addition by a constan...
oacomf1olem 7531 Lemma for ~ oacomf1o . (C...
oacomf1o 7532 Define a bijection from ` ...
omordi 7533 Ordering property of ordin...
omord2 7534 Ordering property of ordin...
omord 7535 Ordering property of ordin...
omcan 7536 Left cancellation law for ...
omword 7537 Weak ordering property of ...
omwordi 7538 Weak ordering property of ...
omwordri 7539 Weak ordering property of ...
omword1 7540 An ordinal is less than or...
omword2 7541 An ordinal is less than or...
om00 7542 The product of two ordinal...
om00el 7543 The product of two nonzero...
omordlim 7544 Ordering involving the pro...
omlimcl 7545 The product of any nonzero...
odi 7546 Distributive law for ordin...
omass 7547 Multiplication of ordinal ...
oneo 7548 If an ordinal number is ev...
omeulem1 7549 Lemma for ~ omeu : existen...
omeulem2 7550 Lemma for ~ omeu : uniquen...
omopth2 7551 An ordered pair-like theor...
omeu 7552 The division algorithm for...
oen0 7553 Ordinal exponentiation wit...
oeordi 7554 Ordering law for ordinal e...
oeord 7555 Ordering property of ordin...
oecan 7556 Left cancellation law for ...
oeword 7557 Weak ordering property of ...
oewordi 7558 Weak ordering property of ...
oewordri 7559 Weak ordering property of ...
oeworde 7560 Ordinal exponentiation com...
oeordsuc 7561 Ordering property of ordin...
oelim2 7562 Ordinal exponentiation wit...
oeoalem 7563 Lemma for ~ oeoa . (Contr...
oeoa 7564 Sum of exponents law for o...
oeoelem 7565 Lemma for ~ oeoe . (Contr...
oeoe 7566 Product of exponents law f...
oelimcl 7567 The ordinal exponential wi...
oeeulem 7568 Lemma for ~ oeeu . (Contr...
oeeui 7569 The division algorithm for...
oeeu 7570 The division algorithm for...
nna0 7571 Addition with zero. Theor...
nnm0 7572 Multiplication with zero. ...
nnasuc 7573 Addition with successor. ...
nnmsuc 7574 Multiplication with succes...
nnesuc 7575 Exponentiation with a succ...
nna0r 7576 Addition to zero. Remark ...
nnm0r 7577 Multiplication with zero. ...
nnacl 7578 Closure of addition of nat...
nnmcl 7579 Closure of multiplication ...
nnecl 7580 Closure of exponentiation ...
nnacli 7581 ` _om ` is closed under ad...
nnmcli 7582 ` _om ` is closed under mu...
nnarcl 7583 Reverse closure law for ad...
nnacom 7584 Addition of natural number...
nnaordi 7585 Ordering property of addit...
nnaord 7586 Ordering property of addit...
nnaordr 7587 Ordering property of addit...
nnawordi 7588 Adding to both sides of an...
nnaass 7589 Addition of natural number...
nndi 7590 Distributive law for natur...
nnmass 7591 Multiplication of natural ...
nnmsucr 7592 Multiplication with succes...
nnmcom 7593 Multiplication of natural ...
nnaword 7594 Weak ordering property of ...
nnacan 7595 Cancellation law for addit...
nnaword1 7596 Weak ordering property of ...
nnaword2 7597 Weak ordering property of ...
nnmordi 7598 Ordering property of multi...
nnmord 7599 Ordering property of multi...
nnmword 7600 Weak ordering property of ...
nnmcan 7601 Cancellation law for multi...
nnmwordi 7602 Weak ordering property of ...
nnmwordri 7603 Weak ordering property of ...
nnawordex 7604 Equivalence for weak order...
nnaordex 7605 Equivalence for ordering. ...
1onn 7606 One is a natural number. ...
2onn 7607 The ordinal 2 is a natural...
3onn 7608 The ordinal 3 is a natural...
4onn 7609 The ordinal 4 is a natural...
oaabslem 7610 Lemma for ~ oaabs . (Cont...
oaabs 7611 Ordinal addition absorbs a...
oaabs2 7612 The absorption law ~ oaabs...
omabslem 7613 Lemma for ~ omabs . (Cont...
omabs 7614 Ordinal multiplication is ...
nnm1 7615 Multiply an element of ` _...
nnm2 7616 Multiply an element of ` _...
nn2m 7617 Multiply an element of ` _...
nnneo 7618 If a natural number is eve...
nneob 7619 A natural number is even i...
omsmolem 7620 Lemma for ~ omsmo . (Cont...
omsmo 7621 A strictly monotonic ordin...
omopthlem1 7622 Lemma for ~ omopthi . (Co...
omopthlem2 7623 Lemma for ~ omopthi . (Co...
omopthi 7624 An ordered pair theorem fo...
omopth 7625 An ordered pair theorem fo...
dfer2 7630 Alternate definition of eq...
dfec2 7632 Alternate definition of ` ...
ecexg 7633 An equivalence class modul...
ecexr 7634 A nonempty equivalence cla...
ereq1 7636 Equality theorem for equiv...
ereq2 7637 Equality theorem for equiv...
errel 7638 An equivalence relation is...
erdm 7639 The domain of an equivalen...
ercl 7640 Elementhood in the field o...
ersym 7641 An equivalence relation is...
ercl2 7642 Elementhood in the field o...
ersymb 7643 An equivalence relation is...
ertr 7644 An equivalence relation is...
ertrd 7645 A transitivity relation fo...
ertr2d 7646 A transitivity relation fo...
ertr3d 7647 A transitivity relation fo...
ertr4d 7648 A transitivity relation fo...
erref 7649 An equivalence relation is...
ercnv 7650 The converse of an equival...
errn 7651 The range and domain of an...
erssxp 7652 An equivalence relation is...
erex 7653 An equivalence relation is...
erexb 7654 An equivalence relation is...
iserd 7655 A reflexive, symmetric, tr...
iseri 7656 A reflexive, symmetric, tr...
iseriALT 7657 Alternate proof of ~ iseri...
brdifun 7658 Evaluate the incomparabili...
swoer 7659 Incomparability under a st...
swoord1 7660 The incomparability equiva...
swoord2 7661 The incomparability equiva...
swoso 7662 If the incomparability rel...
eqerlem 7663 Lemma for ~ eqer . (Contr...
eqer 7664 Equivalence relation invol...
eqerOLD 7665 Obsolete proof of ~ eqer a...
ider 7666 The identity relation is a...
0er 7667 The empty set is an equiva...
0erOLD 7668 Obsolete proof of ~ 0er as...
eceq1 7669 Equality theorem for equiv...
eceq1d 7670 Equality theorem for equiv...
eceq2 7671 Equality theorem for equiv...
elecg 7672 Membership in an equivalen...
elec 7673 Membership in an equivalen...
relelec 7674 Membership in an equivalen...
ecss 7675 An equivalence class is a ...
ecdmn0 7676 A representative of a none...
ereldm 7677 Equality of equivalence cl...
erth 7678 Basic property of equivale...
erth2 7679 Basic property of equivale...
erthi 7680 Basic property of equivale...
erdisj 7681 Equivalence classes do not...
ecidsn 7682 An equivalence class modul...
qseq1 7683 Equality theorem for quoti...
qseq2 7684 Equality theorem for quoti...
elqsg 7685 Closed form of ~ elqs . (...
elqs 7686 Membership in a quotient s...
elqsi 7687 Membership in a quotient s...
elqsecl 7688 Membership in a quotient s...
ecelqsg 7689 Membership of an equivalen...
ecelqsi 7690 Membership of an equivalen...
ecopqsi 7691 "Closure" law for equivale...
qsexg 7692 A quotient set exists. (C...
qsex 7693 A quotient set exists. (C...
uniqs 7694 The union of a quotient se...
qsss 7695 A quotient set is a set of...
uniqs2 7696 The union of a quotient se...
snec 7697 The singleton of an equiva...
ecqs 7698 Equivalence class in terms...
ecid 7699 A set is equal to its conv...
qsid 7700 A set is equal to its quot...
ectocld 7701 Implicit substitution of c...
ectocl 7702 Implicit substitution of c...
elqsn0 7703 A quotient set doesn't con...
ecelqsdm 7704 Membership of an equivalen...
xpider 7705 A square Cartesian product...
iiner 7706 The intersection of a none...
riiner 7707 The relative intersection ...
erinxp 7708 A restricted equivalence r...
ecinxp 7709 Restrict the relation in a...
qsinxp 7710 Restrict the equivalence r...
qsdisj 7711 Members of a quotient set ...
qsdisj2 7712 A quotient set is a disjoi...
qsel 7713 If an element of a quotien...
uniinqs 7714 Class union distributes ov...
qliftlem 7715 ` F ` , a function lift, i...
qliftrel 7716 ` F ` , a function lift, i...
qliftel 7717 Elementhood in the relatio...
qliftel1 7718 Elementhood in the relatio...
qliftfun 7719 The function ` F ` is the ...
qliftfund 7720 The function ` F ` is the ...
qliftfuns 7721 The function ` F ` is the ...
qliftf 7722 The domain and range of th...
qliftval 7723 The value of the function ...
ecoptocl 7724 Implicit substitution of c...
2ecoptocl 7725 Implicit substitution of c...
3ecoptocl 7726 Implicit substitution of c...
brecop 7727 Binary relation on a quoti...
brecop2 7728 Binary relation on a quoti...
eroveu 7729 Lemma for ~ erov and ~ ero...
erovlem 7730 Lemma for ~ erov and ~ ero...
erov 7731 The value of an operation ...
eroprf 7732 Functionality of an operat...
erov2 7733 The value of an operation ...
eroprf2 7734 Functionality of an operat...
ecopoveq 7735 This is the first of sever...
ecopovsym 7736 Assuming the operation ` F...
ecopovtrn 7737 Assuming that operation ` ...
ecopover 7738 Assuming that operation ` ...
ecopoverOLD 7739 Obsolete proof of ~ ecopov...
eceqoveq 7740 Equality of equivalence re...
ecovcom 7741 Lemma used to transfer a c...
ecovass 7742 Lemma used to transfer an ...
ecovdi 7743 Lemma used to transfer a d...
mapprc 7748 When ` A ` is a proper cla...
pmex 7749 The class of all partial f...
mapex 7750 The class of all functions...
fnmap 7751 Set exponentiation has a u...
fnpm 7752 Partial function exponenti...
reldmmap 7753 Set exponentiation is a we...
mapvalg 7754 The value of set exponenti...
pmvalg 7755 The value of the partial m...
mapval 7756 The value of set exponenti...
elmapg 7757 Membership relation for se...
elmapd 7758 Deduction form of ~ elmapg...
elpmg 7759 The predicate "is a partia...
elpm2g 7760 The predicate "is a partia...
elpm2r 7761 Sufficient condition for b...
elpmi 7762 A partial function is a fu...
pmfun 7763 A partial function is a fu...
elmapex 7764 Eliminate antecedent for m...
elmapi 7765 A mapping is a function, f...
elmapfn 7766 A mapping is a function wi...
elmapfun 7767 A mapping is always a func...
elmapssres 7768 A restricted mapping is a ...
fpmg 7769 A total function is a part...
pmss12g 7770 Subset relation for the se...
pmresg 7771 Elementhood of a restricte...
elmap 7772 Membership relation for se...
mapval2 7773 Alternate expression for t...
elpm 7774 The predicate "is a partia...
elpm2 7775 The predicate "is a partia...
fpm 7776 A total function is a part...
mapsspm 7777 Set exponentiation is a su...
pmsspw 7778 Partial maps are a subset ...
mapsspw 7779 Set exponentiation is a su...
fvmptmap 7780 Special case of ~ fvmpt fo...
map0e 7781 Set exponentiation with an...
map0b 7782 Set exponentiation with an...
map0g 7783 Set exponentiation is empt...
map0 7784 Set exponentiation is empt...
mapsn 7785 The value of set exponenti...
mapss 7786 Subset inheritance for set...
fdiagfn 7787 Functionality of the diago...
fvdiagfn 7788 Functionality of the diago...
mapsnconst 7789 Every singleton map is a c...
mapsncnv 7790 Expression for the inverse...
mapsnf1o2 7791 Explicit bijection between...
mapsnf1o3 7792 Explicit bijection in the ...
ralxpmap 7793 Quantification over functi...
dfixp 7796 Eliminate the expression `...
ixpsnval 7797 The value of an infinite C...
elixp2 7798 Membership in an infinite ...
fvixp 7799 Projection of a factor of ...
ixpfn 7800 A nuple is a function. (C...
elixp 7801 Membership in an infinite ...
elixpconst 7802 Membership in an infinite ...
ixpconstg 7803 Infinite Cartesian product...
ixpconst 7804 Infinite Cartesian product...
ixpeq1 7805 Equality theorem for infin...
ixpeq1d 7806 Equality theorem for infin...
ss2ixp 7807 Subclass theorem for infin...
ixpeq2 7808 Equality theorem for infin...
ixpeq2dva 7809 Equality theorem for infin...
ixpeq2dv 7810 Equality theorem for infin...
cbvixp 7811 Change bound variable in a...
cbvixpv 7812 Change bound variable in a...
nfixp 7813 Bound-variable hypothesis ...
nfixp1 7814 The index variable in an i...
ixpprc 7815 A cartesian product of pro...
ixpf 7816 A member of an infinite Ca...
uniixp 7817 The union of an infinite C...
ixpexg 7818 The existence of an infini...
ixpin 7819 The intersection of two in...
ixpiin 7820 The indexed intersection o...
ixpint 7821 The intersection of a coll...
ixp0x 7822 An infinite Cartesian prod...
ixpssmap2g 7823 An infinite Cartesian prod...
ixpssmapg 7824 An infinite Cartesian prod...
0elixp 7825 Membership of the empty se...
ixpn0 7826 The infinite Cartesian pro...
ixp0 7827 The infinite Cartesian pro...
ixpssmap 7828 An infinite Cartesian prod...
resixp 7829 Restriction of an element ...
undifixp 7830 Union of two projections o...
mptelixpg 7831 Condition for an explicit ...
resixpfo 7832 Restriction of elements of...
elixpsn 7833 Membership in a class of s...
ixpsnf1o 7834 A bijection between a clas...
mapsnf1o 7835 A bijection between a set ...
boxriin 7836 A rectangular subset of a ...
boxcutc 7837 The relative complement of...
relen 7846 Equinumerosity is a relati...
reldom 7847 Dominance is a relation. ...
relsdom 7848 Strict dominance is a rela...
encv 7849 If two classes are equinum...
bren 7850 Equinumerosity relation. ...
brdomg 7851 Dominance relation. (Cont...
brdomi 7852 Dominance relation. (Cont...
brdom 7853 Dominance relation. (Cont...
domen 7854 Dominance in terms of equi...
domeng 7855 Dominance in terms of equi...
ctex 7856 A countable set is a set. ...
f1oen3g 7857 The domain and range of a ...
f1oen2g 7858 The domain and range of a ...
f1dom2g 7859 The domain of a one-to-one...
f1oeng 7860 The domain and range of a ...
f1domg 7861 The domain of a one-to-one...
f1oen 7862 The domain and range of a ...
f1dom 7863 The domain of a one-to-one...
brsdom 7864 Strict dominance relation,...
isfi 7865 Express " ` A ` is finite....
enssdom 7866 Equinumerosity implies dom...
dfdom2 7867 Alternate definition of do...
endom 7868 Equinumerosity implies dom...
sdomdom 7869 Strict dominance implies d...
sdomnen 7870 Strict dominance implies n...
brdom2 7871 Dominance in terms of stri...
bren2 7872 Equinumerosity expressed i...
enrefg 7873 Equinumerosity is reflexiv...
enref 7874 Equinumerosity is reflexiv...
eqeng 7875 Equality implies equinumer...
domrefg 7876 Dominance is reflexive. (...
en2d 7877 Equinumerosity inference f...
en3d 7878 Equinumerosity inference f...
en2i 7879 Equinumerosity inference f...
en3i 7880 Equinumerosity inference f...
dom2lem 7881 A mapping (first hypothesi...
dom2d 7882 A mapping (first hypothesi...
dom3d 7883 A mapping (first hypothesi...
dom2 7884 A mapping (first hypothesi...
dom3 7885 A mapping (first hypothesi...
idssen 7886 Equality implies equinumer...
ssdomg 7887 A set dominates its subset...
ener 7888 Equinumerosity is an equiv...
enerOLD 7889 Obsolete proof of ~ ener a...
ensymb 7890 Symmetry of equinumerosity...
ensym 7891 Symmetry of equinumerosity...
ensymi 7892 Symmetry of equinumerosity...
ensymd 7893 Symmetry of equinumerosity...
entr 7894 Transitivity of equinumero...
domtr 7895 Transitivity of dominance ...
entri 7896 A chained equinumerosity i...
entr2i 7897 A chained equinumerosity i...
entr3i 7898 A chained equinumerosity i...
entr4i 7899 A chained equinumerosity i...
endomtr 7900 Transitivity of equinumero...
domentr 7901 Transitivity of dominance ...
f1imaeng 7902 A one-to-one function's im...
f1imaen2g 7903 A one-to-one function's im...
f1imaen 7904 A one-to-one function's im...
en0 7905 The empty set is equinumer...
ensn1 7906 A singleton is equinumerou...
ensn1g 7907 A singleton is equinumerou...
enpr1g 7908 ` { A , A } ` has only one...
en1 7909 A set is equinumerous to o...
en1b 7910 A set is equinumerous to o...
reuen1 7911 Two ways to express "exact...
euen1 7912 Two ways to express "exact...
euen1b 7913 Two ways to express " ` A ...
en1uniel 7914 A singleton contains its s...
2dom 7915 A set that dominates ordin...
fundmen 7916 A function is equinumerous...
fundmeng 7917 A function is equinumerous...
cnven 7918 A relational set is equinu...
fndmeng 7919 A function is equinumerate...
mapsnen 7920 Set exponentiation to a si...
map1 7921 Set exponentiation: ordina...
en2sn 7922 Two singletons are equinum...
snfi 7923 A singleton is finite. (C...
fiprc 7924 The class of finite sets i...
unen 7925 Equinumerosity of union of...
ssct 7926 Any subset of a countable ...
difsnen 7927 All decrements of a set ar...
domdifsn 7928 Dominance over a set with ...
xpsnen 7929 A set is equinumerous to i...
xpsneng 7930 A set is equinumerous to i...
xp1en 7931 One times a cardinal numbe...
endisj 7932 Any two sets are equinumer...
undom 7933 Dominance law for union. ...
xpcomf1o 7934 The canonical bijection fr...
xpcomco 7935 Composition with the bijec...
xpcomen 7936 Commutative law for equinu...
xpcomeng 7937 Commutative law for equinu...
xpsnen2g 7938 A set is equinumerous to i...
xpassen 7939 Associative law for equinu...
xpdom2 7940 Dominance law for Cartesia...
xpdom2g 7941 Dominance law for Cartesia...
xpdom1g 7942 Dominance law for Cartesia...
xpdom3 7943 A set is dominated by its ...
xpdom1 7944 Dominance law for Cartesia...
domunsncan 7945 A singleton cancellation l...
omxpenlem 7946 Lemma for ~ omxpen . (Con...
omxpen 7947 The cardinal and ordinal p...
omf1o 7948 Construct an explicit bije...
pw2f1olem 7949 Lemma for ~ pw2f1o . (Con...
pw2f1o 7950 The power set of a set is ...
pw2eng 7951 The power set of a set is ...
pw2en 7952 The power set of a set is ...
fopwdom 7953 Covering implies injection...
enfixsn 7954 Given two equipollent sets...
sbthlem1 7955 Lemma for ~ sbth . (Contr...
sbthlem2 7956 Lemma for ~ sbth . (Contr...
sbthlem3 7957 Lemma for ~ sbth . (Contr...
sbthlem4 7958 Lemma for ~ sbth . (Contr...
sbthlem5 7959 Lemma for ~ sbth . (Contr...
sbthlem6 7960 Lemma for ~ sbth . (Contr...
sbthlem7 7961 Lemma for ~ sbth . (Contr...
sbthlem8 7962 Lemma for ~ sbth . (Contr...
sbthlem9 7963 Lemma for ~ sbth . (Contr...
sbthlem10 7964 Lemma for ~ sbth . (Contr...
sbth 7965 Schroeder-Bernstein Theore...
sbthb 7966 Schroeder-Bernstein Theore...
sbthcl 7967 Schroeder-Bernstein Theore...
dfsdom2 7968 Alternate definition of st...
brsdom2 7969 Alternate definition of st...
sdomnsym 7970 Strict dominance is asymme...
domnsym 7971 Theorem 22(i) of [Suppes] ...
0domg 7972 Any set dominates the empt...
dom0 7973 A set dominated by the emp...
0sdomg 7974 A set strictly dominates t...
0dom 7975 Any set dominates the empt...
0sdom 7976 A set strictly dominates t...
sdom0 7977 The empty set does not str...
sdomdomtr 7978 Transitivity of strict dom...
sdomentr 7979 Transitivity of strict dom...
domsdomtr 7980 Transitivity of dominance ...
ensdomtr 7981 Transitivity of equinumero...
sdomirr 7982 Strict dominance is irrefl...
sdomtr 7983 Strict dominance is transi...
sdomn2lp 7984 Strict dominance has no 2-...
enen1 7985 Equality-like theorem for ...
enen2 7986 Equality-like theorem for ...
domen1 7987 Equality-like theorem for ...
domen2 7988 Equality-like theorem for ...
sdomen1 7989 Equality-like theorem for ...
sdomen2 7990 Equality-like theorem for ...
domtriord 7991 Dominance is trichotomous ...
sdomel 7992 Strict dominance implies o...
sdomdif 7993 The difference of a set fr...
onsdominel 7994 An ordinal with more eleme...
domunsn 7995 Dominance over a set with ...
fodomr 7996 There exists a mapping fro...
pwdom 7997 Injection of sets implies ...
canth2 7998 Cantor's Theorem. No set ...
canth2g 7999 Cantor's theorem with the ...
2pwuninel 8000 The power set of the power...
2pwne 8001 No set equals the power se...
disjen 8002 A stronger form of ~ pwuni...
disjenex 8003 Existence version of ~ dis...
domss2 8004 A corollary of ~ disjenex ...
domssex2 8005 A corollary of ~ disjenex ...
domssex 8006 Weakening of ~ domssex to ...
xpf1o 8007 Construct a bijection on a...
xpen 8008 Equinumerosity law for Car...
mapen 8009 Two set exponentiations ar...
mapdom1 8010 Order-preserving property ...
mapxpen 8011 Equinumerosity law for dou...
xpmapenlem 8012 Lemma for ~ xpmapen . (Co...
xpmapen 8013 Equinumerosity law for set...
mapunen 8014 Equinumerosity law for set...
map2xp 8015 A cardinal power with expo...
mapdom2 8016 Order-preserving property ...
mapdom3 8017 Set exponentiation dominat...
pwen 8018 If two sets are equinumero...
ssenen 8019 Equinumerosity of equinume...
limenpsi 8020 A limit ordinal is equinum...
limensuci 8021 A limit ordinal is equinum...
limensuc 8022 A limit ordinal is equinum...
infensuc 8023 Any infinite ordinal is eq...
phplem1 8024 Lemma for Pigeonhole Princ...
phplem2 8025 Lemma for Pigeonhole Princ...
phplem3 8026 Lemma for Pigeonhole Princ...
phplem4 8027 Lemma for Pigeonhole Princ...
nneneq 8028 Two equinumerous natural n...
php 8029 Pigeonhole Principle. A n...
php2 8030 Corollary of Pigeonhole Pr...
php3 8031 Corollary of Pigeonhole Pr...
php4 8032 Corollary of the Pigeonhol...
php5 8033 Corollary of the Pigeonhol...
snnen2o 8034 A singleton ` { A } ` is n...
onomeneq 8035 An ordinal number equinume...
onfin 8036 An ordinal number is finit...
onfin2 8037 A set is a natural number ...
nnfi 8038 Natural numbers are finite...
nndomo 8039 Cardinal ordering agrees w...
nnsdomo 8040 Cardinal ordering agrees w...
sucdom2 8041 Strict dominance of a set ...
sucdom 8042 Strict dominance of a set ...
0sdom1dom 8043 Strict dominance over zero...
1sdom2 8044 Ordinal 1 is strictly domi...
sdom1 8045 A set has less than one me...
modom 8046 Two ways to express "at mo...
modom2 8047 Two ways to express "at mo...
1sdom 8048 A set that strictly domina...
unxpdomlem1 8049 Lemma for ~ unxpdom . (Tr...
unxpdomlem2 8050 Lemma for ~ unxpdom . (Co...
unxpdomlem3 8051 Lemma for ~ unxpdom . (Co...
unxpdom 8052 Cartesian product dominate...
unxpdom2 8053 Corollary of ~ unxpdom . ...
sucxpdom 8054 Cartesian product dominate...
pssinf 8055 A set equinumerous to a pr...
fisseneq 8056 A finite set is equal to i...
ominf 8057 The set of natural numbers...
isinf 8058 Any set that is not finite...
fineqvlem 8059 Lemma for ~ fineqv . (Con...
fineqv 8060 If the Axiom of Infinity i...
enfi 8061 Equinumerous sets have the...
enfii 8062 A set equinumerous to a fi...
pssnn 8063 A proper subset of a natur...
ssnnfi 8064 A subset of a natural numb...
ssfi 8065 A subset of a finite set i...
domfi 8066 A set dominated by a finit...
xpfir 8067 The components of a nonemp...
ssfid 8068 A subset of a finite set i...
infi 8069 The intersection of two se...
rabfi 8070 A restricted class built f...
finresfin 8071 The restriction of a finit...
f1finf1o 8072 Any injection from one fin...
0fin 8073 The empty set is finite. ...
nfielex 8074 If a class is not finite, ...
en1eqsn 8075 A set with one element is ...
en1eqsnbi 8076 A set containing an elemen...
diffi 8077 If ` A ` is finite, ` ( A ...
dif1en 8078 If a set ` A ` is equinume...
enp1ilem 8079 Lemma for uses of ~ enp1i ...
enp1i 8080 Proof induction for ~ en2i...
en2 8081 A set equinumerous to ordi...
en3 8082 A set equinumerous to ordi...
en4 8083 A set equinumerous to ordi...
findcard 8084 Schema for induction on th...
findcard2 8085 Schema for induction on th...
findcard2s 8086 Variation of ~ findcard2 r...
findcard2d 8087 Deduction version of ~ fin...
findcard3 8088 Schema for strong inductio...
ac6sfi 8089 A version of ~ ac6s for fi...
frfi 8090 A partial order is well-fo...
fimax2g 8091 A finite set has a maximum...
fimaxg 8092 A finite set has a maximum...
fisupg 8093 Lemma showing existence an...
wofi 8094 A total order on a finite ...
ordunifi 8095 The maximum of a finite co...
nnunifi 8096 The union (supremum) of a ...
unblem1 8097 Lemma for ~ unbnn . After...
unblem2 8098 Lemma for ~ unbnn . The v...
unblem3 8099 Lemma for ~ unbnn . The v...
unblem4 8100 Lemma for ~ unbnn . The f...
unbnn 8101 Any unbounded subset of na...
unbnn2 8102 Version of ~ unbnn that do...
isfinite2 8103 Any set strictly dominated...
nnsdomg 8104 Omega strictly dominates a...
isfiniteg 8105 A set is finite iff it is ...
infsdomnn 8106 An infinite set strictly d...
infn0 8107 An infinite set is not emp...
fin2inf 8108 This (useless) theorem, wh...
unfilem1 8109 Lemma for proving that the...
unfilem2 8110 Lemma for proving that the...
unfilem3 8111 Lemma for proving that the...
unfi 8112 The union of two finite se...
unfir 8113 If a union is finite, the ...
unfi2 8114 The union of two finite se...
difinf 8115 An infinite set ` A ` minu...
xpfi 8116 The Cartesian product of t...
3xpfi 8117 The Cartesian product of t...
domunfican 8118 A finite set union cancell...
infcntss 8119 Every infinite set has a d...
prfi 8120 An unordered pair is finit...
tpfi 8121 An unordered triple is fin...
fiint 8122 Equivalent ways of stating...
fnfi 8123 A version of ~ fnex for fi...
fodomfi 8124 An onto function implies d...
fodomfib 8125 Equivalence of an onto map...
fofinf1o 8126 Any surjection from one fi...
rneqdmfinf1o 8127 Any function from a finite...
fidomdm 8128 Any finite set dominates i...
dmfi 8129 The domain of a finite set...
fundmfibi 8130 A function (set) is finite...
cnvfi 8131 If a set is finite, its co...
rnfi 8132 The range of a finite set ...
f1dmvrnfibi 8133 A 1-1 function (class) wit...
f1vrnfibi 8134 A 1-1 function (set) is fi...
fofi 8135 If a function has a finite...
f1fi 8136 If a 1-to-1 function has a...
iunfi 8137 The finite union of finite...
unifi 8138 The finite union of finite...
unifi2 8139 The finite union of finite...
infssuni 8140 If an infinite set ` A ` i...
unirnffid 8141 The union of the range of ...
imafi 8142 Images of finite sets are ...
pwfilem 8143 Lemma for ~ pwfi . (Contr...
pwfi 8144 The power set of a finite ...
mapfi 8145 Set exponentiation of fini...
ixpfi 8146 A Cartesian product of fin...
ixpfi2 8147 A Cartesian product of fin...
mptfi 8148 A finite mapping set is fi...
abrexfi 8149 An image set from a finite...
cnvimamptfin 8150 A preimage of a mapping wi...
elfpw 8151 Membership in a class of f...
unifpw 8152 A set is the union of its ...
f1opwfi 8153 A one-to-one mapping induc...
fissuni 8154 A finite subset of a union...
fipreima 8155 Given a finite subset ` A ...
finsschain 8156 A finite subset of the uni...
indexfi 8157 If for every element of a ...
relfsupp 8160 The property of a function...
relprcnfsupp 8161 A proper class is never fi...
isfsupp 8162 The property of a class to...
funisfsupp 8163 The property of a function...
fsuppimp 8164 Implications of a class be...
fsuppimpd 8165 A finitely supported funct...
fisuppfi 8166 A function on a finite set...
fdmfisuppfi 8167 The support of a function ...
fdmfifsupp 8168 A function with a finite d...
fsuppmptdm 8169 A mapping with a finite do...
fndmfisuppfi 8170 The support of a function ...
fndmfifsupp 8171 A function with a finite d...
suppeqfsuppbi 8172 If two functions have the ...
suppssfifsupp 8173 If the support of a functi...
fsuppsssupp 8174 If the support of a functi...
fsuppxpfi 8175 The cartesian product of t...
fczfsuppd 8176 A constant function with v...
fsuppun 8177 The union of two finitely ...
fsuppunfi 8178 The union of the support o...
fsuppunbi 8179 If the union of two classe...
0fsupp 8180 The empty set is a finitel...
snopfsupp 8181 A singleton containing an ...
funsnfsupp 8182 Finite support for a funct...
fsuppres 8183 The restriction of a finit...
ressuppfi 8184 If the support of the rest...
resfsupp 8185 If the restriction of a fu...
resfifsupp 8186 The restriction of a funct...
frnfsuppbi 8187 Two ways of saying that a ...
fsuppmptif 8188 A function mapping an argu...
fsuppcolem 8189 Lemma for ~ fsuppco . For...
fsuppco 8190 The composition of a 1-1 f...
fsuppco2 8191 The composition of a funct...
fsuppcor 8192 The composition of a funct...
mapfienlem1 8193 Lemma 1 for ~ mapfien . (...
mapfienlem2 8194 Lemma 2 for ~ mapfien . (...
mapfienlem3 8195 Lemma 3 for ~ mapfien . (...
mapfien 8196 A bijection of the base se...
mapfien2 8197 Equinumerousity relation f...
sniffsupp 8198 A function mapping all but...
fival 8201 The set of all the finite ...
elfi 8202 Specific properties of an ...
elfi2 8203 The empty intersection nee...
elfir 8204 Sufficient condition for a...
intrnfi 8205 Sufficient condition for t...
iinfi 8206 An indexed intersection of...
inelfi 8207 The intersection of two se...
ssfii 8208 Any element of a set ` A `...
fi0 8209 The set of finite intersec...
fieq0 8210 If ` A ` is not empty, the...
fiin 8211 The elements of ` ( fi `` ...
dffi2 8212 The set of finite intersec...
fiss 8213 Subset relationship for fu...
inficl 8214 A set which is closed unde...
fipwuni 8215 The set of finite intersec...
fisn 8216 A singleton is closed unde...
fiuni 8217 The union of the finite in...
fipwss 8218 If a set is a family of su...
elfiun 8219 A finite intersection of e...
dffi3 8220 The set of finite intersec...
fifo 8221 Describe a surjection from...
marypha1lem 8222 Core induction for Philip ...
marypha1 8223 (Philip) Hall's marriage t...
marypha2lem1 8224 Lemma for ~ marypha2 . Pr...
marypha2lem2 8225 Lemma for ~ marypha2 . Pr...
marypha2lem3 8226 Lemma for ~ marypha2 . Pr...
marypha2lem4 8227 Lemma for ~ marypha2 . Pr...
marypha2 8228 Version of ~ marypha1 usin...
dfsup2 8233 Quantifier free definition...
supeq1 8234 Equality theorem for supre...
supeq1d 8235 Equality deduction for sup...
supeq1i 8236 Equality inference for sup...
supeq2 8237 Equality theorem for supre...
supeq3 8238 Equality theorem for supre...
supeq123d 8239 Equality deduction for sup...
nfsup 8240 Hypothesis builder for sup...
supmo 8241 Any class ` B ` has at mos...
supexd 8242 A supremum is a set. (Con...
supeu 8243 A supremum is unique. Sim...
supval2 8244 Alternate expression for t...
eqsup 8245 Sufficient condition for a...
eqsupd 8246 Sufficient condition for a...
supcl 8247 A supremum belongs to its ...
supub 8248 A supremum is an upper bou...
suplub 8249 A supremum is the least up...
suplub2 8250 Bidirectional form of ~ su...
supnub 8251 An upper bound is not less...
supex 8252 A supremum is a set. (Con...
sup00 8253 The supremum under an empt...
sup0riota 8254 The supremum of an empty s...
sup0 8255 The supremum of an empty s...
supmax 8256 The greatest element of a ...
fisup2g 8257 A finite set satisfies the...
fisupcl 8258 A nonempty finite set cont...
supgtoreq 8259 The supremum of a finite s...
suppr 8260 The supremum of a pair. (...
supsn 8261 The supremum of a singleto...
supisolem 8262 Lemma for ~ supiso . (Con...
supisoex 8263 Lemma for ~ supiso . (Con...
supiso 8264 Image of a supremum under ...
infeq1 8265 Equality theorem for infim...
infeq1d 8266 Equality deduction for inf...
infeq1i 8267 Equality inference for inf...
infeq2 8268 Equality theorem for infim...
infeq3 8269 Equality theorem for infim...
infeq123d 8270 Equality deduction for inf...
nfinf 8271 Hypothesis builder for inf...
infexd 8272 An infimum is a set. (Con...
eqinf 8273 Sufficient condition for a...
eqinfd 8274 Sufficient condition for a...
infval 8275 Alternate expression for t...
infcllem 8276 Lemma for ~ infcl , ~ infl...
infcl 8277 An infimum belongs to its ...
inflb 8278 An infimum is a lower boun...
infglb 8279 An infimum is the greatest...
infglbb 8280 Bidirectional form of ~ in...
infnlb 8281 A lower bound is not great...
infex 8282 An infimum is a set. (Con...
infmin 8283 The smallest element of a ...
infmo 8284 Any class ` B ` has at mos...
infeu 8285 An infimum is unique. (Co...
fimin2g 8286 A finite set has a minimum...
fiming 8287 A finite set has a minimum...
fiinfg 8288 Lemma showing existence an...
fiinf2g 8289 A finite set satisfies the...
fiinfcl 8290 A nonempty finite set cont...
infltoreq 8291 The infimum of a finite se...
infpr 8292 The infimum of a pair. (C...
infsn 8293 The infimum of a singleton...
inf00 8294 The infimum regarding an e...
infempty 8295 The infimum of an empty se...
infiso 8296 Image of an infimum under ...
dfoi 8299 Rewrite ~ df-oi with abbre...
oieq1 8300 Equality theorem for ordin...
oieq2 8301 Equality theorem for ordin...
nfoi 8302 Hypothesis builder for ord...
ordiso2 8303 Generalize ~ ordiso to pro...
ordiso 8304 Order-isomorphic ordinal n...
ordtypecbv 8305 Lemma for ~ ordtype . (Co...
ordtypelem1 8306 Lemma for ~ ordtype . (Co...
ordtypelem2 8307 Lemma for ~ ordtype . (Co...
ordtypelem3 8308 Lemma for ~ ordtype . (Co...
ordtypelem4 8309 Lemma for ~ ordtype . (Co...
ordtypelem5 8310 Lemma for ~ ordtype . (Co...
ordtypelem6 8311 Lemma for ~ ordtype . (Co...
ordtypelem7 8312 Lemma for ~ ordtype . ` ra...
ordtypelem8 8313 Lemma for ~ ordtype . (Co...
ordtypelem9 8314 Lemma for ~ ordtype . Eit...
ordtypelem10 8315 Lemma for ~ ordtype . Usi...
oi0 8316 Definition of the ordinal ...
oicl 8317 The order type of the well...
oif 8318 The order isomorphism of t...
oiiso2 8319 The order isomorphism of t...
ordtype 8320 For any set-like well-orde...
oiiniseg 8321 ` ran F ` is an initial se...
ordtype2 8322 For any set-like well-orde...
oiexg 8323 The order isomorphism on a...
oion 8324 The order type of the well...
oiiso 8325 The order isomorphism of t...
oien 8326 The order type of a well-o...
oieu 8327 Uniqueness of the unique o...
oismo 8328 When ` A ` is a subclass o...
oiid 8329 The order type of an ordin...
hartogslem1 8330 Lemma for ~ hartogs . (Co...
hartogslem2 8331 Lemma for ~ hartogs . (Co...
hartogs 8332 Given any set, the Hartogs...
wofib 8333 The only sets which are we...
wemaplem1 8334 Value of the lexicographic...
wemaplem2 8335 Lemma for ~ wemapso . Tra...
wemaplem3 8336 Lemma for ~ wemapso . Tra...
wemappo 8337 Construct lexicographic or...
wemapsolem 8338 Lemma for ~ wemapso . (Co...
wemapso 8339 Construct lexicographic or...
wemapso2lem 8340 Lemma for ~ wemapso2 . (C...
wemapso2 8341 An alternative to having a...
card2on 8342 Proof that the alternate d...
card2inf 8343 The definition ~ cardval2 ...
harf 8348 Functionality of the Harto...
harcl 8349 Closure of the Hartogs fun...
harval 8350 Function value of the Hart...
elharval 8351 The Hartogs number of a se...
harndom 8352 The Hartogs number of a se...
harword 8353 Weak ordering property of ...
relwdom 8354 Weak dominance is a relati...
brwdom 8355 Property of weak dominance...
brwdomi 8356 Property of weak dominance...
brwdomn0 8357 Weak dominance over nonemp...
0wdom 8358 Any set weakly dominates t...
fowdom 8359 An onto function implies w...
wdomref 8360 Reflexivity of weak domina...
brwdom2 8361 Alternate characterization...
domwdom 8362 Weak dominance is implied ...
wdomtr 8363 Transitivity of weak domin...
wdomen1 8364 Equality-like theorem for ...
wdomen2 8365 Equality-like theorem for ...
wdompwdom 8366 Weak dominance strengthens...
canthwdom 8367 Cantor's Theorem, stated u...
wdom2d 8368 Deduce weak dominance from...
wdomd 8369 Deduce weak dominance from...
brwdom3 8370 Condition for weak dominan...
brwdom3i 8371 Weak dominance implies exi...
unwdomg 8372 Weak dominance of a (disjo...
xpwdomg 8373 Weak dominance of a Cartes...
wdomima2g 8374 A set is weakly dominant o...
wdomimag 8375 A set is weakly dominant o...
unxpwdom2 8376 Lemma for ~ unxpwdom . (C...
unxpwdom 8377 If a Cartesian product is ...
harwdom 8378 The Hartogs function is we...
ixpiunwdom 8379 Describe an onto function ...
axreg2 8381 Axiom of Regularity expres...
zfregcl 8382 The Axiom of Regularity wi...
zfreg 8383 The Axiom of Regularity us...
zfregclOLD 8384 Obsolete version of ~ zfre...
zfregOLD 8385 Obsolete version of ~ zfre...
zfreg2OLD 8386 Alternate version of ~ zfr...
elirrv 8387 The membership relation is...
elirr 8388 No class is a member of it...
sucprcreg 8389 A class is equal to its su...
ruv 8390 The Russell class is equal...
ruALT 8391 Alternate proof of ~ ru , ...
zfregfr 8392 The epsilon relation is we...
en2lp 8393 No class has 2-cycle membe...
en3lplem1 8394 Lemma for ~ en3lp . (Cont...
en3lplem2 8395 Lemma for ~ en3lp . (Cont...
en3lp 8396 No class has 3-cycle membe...
preleq 8397 Equality of two unordered ...
opthreg 8398 Theorem for alternate repr...
suc11reg 8399 The successor operation be...
dford2 8400 Assuming ~ ax-reg , an ord...
inf0 8401 Our Axiom of Infinity deri...
inf1 8402 Variation of Axiom of Infi...
inf2 8403 Variation of Axiom of Infi...
inf3lema 8404 Lemma for our Axiom of Inf...
inf3lemb 8405 Lemma for our Axiom of Inf...
inf3lemc 8406 Lemma for our Axiom of Inf...
inf3lemd 8407 Lemma for our Axiom of Inf...
inf3lem1 8408 Lemma for our Axiom of Inf...
inf3lem2 8409 Lemma for our Axiom of Inf...
inf3lem3 8410 Lemma for our Axiom of Inf...
inf3lem4 8411 Lemma for our Axiom of Inf...
inf3lem5 8412 Lemma for our Axiom of Inf...
inf3lem6 8413 Lemma for our Axiom of Inf...
inf3lem7 8414 Lemma for our Axiom of Inf...
inf3 8415 Our Axiom of Infinity ~ ax...
infeq5i 8416 Half of ~ infeq5 . (Contr...
infeq5 8417 The statement "there exist...
zfinf 8419 Axiom of Infinity expresse...
axinf2 8420 A standard version of Axio...
zfinf2 8422 A standard version of the ...
omex 8423 The existence of omega (th...
axinf 8424 The first version of the A...
inf5 8425 The statement "there exist...
omelon 8426 Omega is an ordinal number...
dfom3 8427 The class of natural numbe...
elom3 8428 A simplification of ~ elom...
dfom4 8429 A simplification of ~ df-o...
dfom5 8430 ` _om ` is the smallest li...
oancom 8431 Ordinal addition is not co...
isfinite 8432 A set is finite iff it is ...
fict 8433 A finite set is countable ...
nnsdom 8434 A natural number is strict...
omenps 8435 Omega is equinumerous to a...
omensuc 8436 The set of natural numbers...
infdifsn 8437 Removing a singleton from ...
infdiffi 8438 Removing a finite set from...
unbnn3 8439 Any unbounded subset of na...
noinfep 8440 Using the Axiom of Regular...
cantnffval 8443 The value of the Cantor no...
cantnfdm 8444 The domain of the Cantor n...
cantnfvalf 8445 Lemma for ~ cantnf . The ...
cantnfs 8446 Elementhood in the set of ...
cantnfcl 8447 Basic properties of the or...
cantnfval 8448 The value of the Cantor no...
cantnfval2 8449 Alternate expression for t...
cantnfsuc 8450 The value of the recursive...
cantnfle 8451 A lower bound on the ` CNF...
cantnflt 8452 An upper bound on the part...
cantnflt2 8453 An upper bound on the ` CN...
cantnff 8454 The ` CNF ` function is a ...
cantnf0 8455 The value of the zero func...
cantnfrescl 8456 A function is finitely sup...
cantnfres 8457 The ` CNF ` function respe...
cantnfp1lem1 8458 Lemma for ~ cantnfp1 . (C...
cantnfp1lem2 8459 Lemma for ~ cantnfp1 . (C...
cantnfp1lem3 8460 Lemma for ~ cantnfp1 . (C...
cantnfp1 8461 If ` F ` is created by add...
oemapso 8462 The relation ` T ` is a st...
oemapval 8463 Value of the relation ` T ...
oemapvali 8464 If ` F < G ` , then there ...
cantnflem1a 8465 Lemma for ~ cantnf . (Con...
cantnflem1b 8466 Lemma for ~ cantnf . (Con...
cantnflem1c 8467 Lemma for ~ cantnf . (Con...
cantnflem1d 8468 Lemma for ~ cantnf . (Con...
cantnflem1 8469 Lemma for ~ cantnf . This...
cantnflem2 8470 Lemma for ~ cantnf . (Con...
cantnflem3 8471 Lemma for ~ cantnf . Here...
cantnflem4 8472 Lemma for ~ cantnf . Comp...
cantnf 8473 The Cantor Normal Form the...
oemapwe 8474 The lexicographic order on...
cantnffval2 8475 An alternate definition of...
cantnff1o 8476 Simplify the isomorphism o...
wemapwe 8477 Construct lexicographic or...
oef1o 8478 A bijection of the base se...
cnfcomlem 8479 Lemma for ~ cnfcom . (Con...
cnfcom 8480 Any ordinal ` B ` is equin...
cnfcom2lem 8481 Lemma for ~ cnfcom2 . (Co...
cnfcom2 8482 Any nonzero ordinal ` B ` ...
cnfcom3lem 8483 Lemma for ~ cnfcom3 . (Co...
cnfcom3 8484 Any infinite ordinal ` B `...
cnfcom3clem 8485 Lemma for ~ cnfcom3c . (C...
cnfcom3c 8486 Wrap the construction of ~...
trcl 8487 For any set ` A ` , show t...
tz9.1 8488 Every set has a transitive...
tz9.1c 8489 Alternate expression for t...
epfrs 8490 The strong form of the Axi...
zfregs 8491 The strong form of the Axi...
zfregs2 8492 Alternate strong form of t...
setind 8493 Set (epsilon) induction. ...
setind2 8494 Set (epsilon) induction, s...
tcvalg 8497 Value of the transitive cl...
tcid 8498 Defining property of the t...
tctr 8499 Defining property of the t...
tcmin 8500 Defining property of the t...
tc2 8501 A variant of the definitio...
tcsni 8502 The transitive closure of ...
tcss 8503 The transitive closure fun...
tcel 8504 The transitive closure fun...
tcidm 8505 The transitive closure fun...
tc0 8506 The transitive closure of ...
tc00 8507 The transitive closure is ...
r1funlim 8512 The cumulative hierarchy o...
r1fnon 8513 The cumulative hierarchy o...
r10 8514 Value of the cumulative hi...
r1sucg 8515 Value of the cumulative hi...
r1suc 8516 Value of the cumulative hi...
r1limg 8517 Value of the cumulative hi...
r1lim 8518 Value of the cumulative hi...
r1fin 8519 The first ` _om ` levels o...
r1sdom 8520 Each stage in the cumulati...
r111 8521 The cumulative hierarchy i...
r1tr 8522 The cumulative hierarchy o...
r1tr2 8523 The union of a cumulative ...
r1ordg 8524 Ordering relation for the ...
r1ord3g 8525 Ordering relation for the ...
r1ord 8526 Ordering relation for the ...
r1ord2 8527 Ordering relation for the ...
r1ord3 8528 Ordering relation for the ...
r1sssuc 8529 The value of the cumulativ...
r1pwss 8530 Each set of the cumulative...
r1sscl 8531 Each set of the cumulative...
r1val1 8532 The value of the cumulativ...
tz9.12lem1 8533 Lemma for ~ tz9.12 . (Con...
tz9.12lem2 8534 Lemma for ~ tz9.12 . (Con...
tz9.12lem3 8535 Lemma for ~ tz9.12 . (Con...
tz9.12 8536 A set is well-founded if a...
tz9.13 8537 Every set is well-founded,...
tz9.13g 8538 Every set is well-founded,...
rankwflemb 8539 Two ways of saying a set i...
rankf 8540 The domain and range of th...
rankon 8541 The rank of a set is an or...
r1elwf 8542 Any member of the cumulati...
rankvalb 8543 Value of the rank function...
rankr1ai 8544 One direction of ~ rankr1a...
rankvaln 8545 Value of the rank function...
rankidb 8546 Identity law for the rank ...
rankdmr1 8547 A rank is a member of the ...
rankr1ag 8548 A version of ~ rankr1a tha...
rankr1bg 8549 A relationship between ran...
r1rankidb 8550 Any set is a subset of the...
r1elssi 8551 The range of the ` R1 ` fu...
r1elss 8552 The range of the ` R1 ` fu...
pwwf 8553 A power set is well-founde...
sswf 8554 A subset of a well-founded...
snwf 8555 A singleton is well-founde...
unwf 8556 A binary union is well-fou...
prwf 8557 An unordered pair is well-...
opwf 8558 An ordered pair is well-fo...
unir1 8559 The cumulative hierarchy o...
jech9.3 8560 Every set belongs to some ...
rankwflem 8561 Every set is well-founded,...
rankval 8562 Value of the rank function...
rankvalg 8563 Value of the rank function...
rankval2 8564 Value of an alternate defi...
uniwf 8565 A union is well-founded if...
rankr1clem 8566 Lemma for ~ rankr1c . (Co...
rankr1c 8567 A relationship between the...
rankidn 8568 A relationship between the...
rankpwi 8569 The rank of a power set. ...
rankelb 8570 The membership relation is...
wfelirr 8571 A well-founded set is not ...
rankval3b 8572 The value of the rank func...
ranksnb 8573 The rank of a singleton. ...
rankonidlem 8574 Lemma for ~ rankonid . (C...
rankonid 8575 The rank of an ordinal num...
onwf 8576 The ordinals are all well-...
onssr1 8577 Initial segments of the or...
rankr1g 8578 A relationship between the...
rankid 8579 Identity law for the rank ...
rankr1 8580 A relationship between the...
ssrankr1 8581 A relationship between an ...
rankr1a 8582 A relationship between ran...
r1val2 8583 The value of the cumulativ...
r1val3 8584 The value of the cumulativ...
rankel 8585 The membership relation is...
rankval3 8586 The value of the rank func...
bndrank 8587 Any class whose elements h...
unbndrank 8588 The elements of a proper c...
rankpw 8589 The rank of a power set. ...
ranklim 8590 The rank of a set belongs ...
r1pw 8591 A stronger property of ` R...
r1pwALT 8592 Alternate shorter proof of...
r1pwcl 8593 The cumulative hierarchy o...
rankssb 8594 The subset relation is inh...
rankss 8595 The subset relation is inh...
rankunb 8596 The rank of the union of t...
rankprb 8597 The rank of an unordered p...
rankopb 8598 The rank of an ordered pai...
rankuni2b 8599 The value of the rank func...
ranksn 8600 The rank of a singleton. ...
rankuni2 8601 The rank of a union. Part...
rankun 8602 The rank of the union of t...
rankpr 8603 The rank of an unordered p...
rankop 8604 The rank of an ordered pai...
r1rankid 8605 Any set is a subset of the...
rankeq0b 8606 A set is empty iff its ran...
rankeq0 8607 A set is empty iff its ran...
rankr1id 8608 The rank of the hierarchy ...
rankuni 8609 The rank of a union. Part...
rankr1b 8610 A relationship between ran...
ranksuc 8611 The rank of a successor. ...
rankuniss 8612 Upper bound of the rank of...
rankval4 8613 The rank of a set is the s...
rankbnd 8614 The rank of a set is bound...
rankbnd2 8615 The rank of a set is bound...
rankc1 8616 A relationship that can be...
rankc2 8617 A relationship that can be...
rankelun 8618 Rank membership is inherit...
rankelpr 8619 Rank membership is inherit...
rankelop 8620 Rank membership is inherit...
rankxpl 8621 A lower bound on the rank ...
rankxpu 8622 An upper bound on the rank...
rankfu 8623 An upper bound on the rank...
rankmapu 8624 An upper bound on the rank...
rankxplim 8625 The rank of a Cartesian pr...
rankxplim2 8626 If the rank of a Cartesian...
rankxplim3 8627 The rank of a Cartesian pr...
rankxpsuc 8628 The rank of a Cartesian pr...
tcwf 8629 The transitive closure fun...
tcrank 8630 This theorem expresses two...
scottex 8631 Scott's trick collects all...
scott0 8632 Scott's trick collects all...
scottexs 8633 Theorem scheme version of ...
scott0s 8634 Theorem scheme version of ...
cplem1 8635 Lemma for the Collection P...
cplem2 8636 -Lemma for the Collection ...
cp 8637 Collection Principle. Thi...
bnd 8638 A very strong generalizati...
bnd2 8639 A variant of the Boundedne...
kardex 8640 The collection of all sets...
karden 8641 If we allow the Axiom of R...
htalem 8642 Lemma for defining an emul...
hta 8643 A ZFC emulation of Hilbert...
cardf2 8652 The cardinality function i...
cardon 8653 The cardinal number of a s...
isnum2 8654 A way to express well-orde...
isnumi 8655 A set equinumerous to an o...
ennum 8656 Equinumerous sets are equi...
finnum 8657 Every finite set is numera...
onenon 8658 Every ordinal number is nu...
tskwe 8659 A Tarski set is well-order...
xpnum 8660 The cartesian product of n...
cardval3 8661 An alternate definition of...
cardid2 8662 Any numerable set is equin...
isnum3 8663 A set is numerable iff it ...
oncardval 8664 The value of the cardinal ...
oncardid 8665 Any ordinal number is equi...
cardonle 8666 The cardinal of an ordinal...
card0 8667 The cardinality of the emp...
cardidm 8668 The cardinality function i...
oncard 8669 A set is a cardinal number...
ficardom 8670 The cardinal number of a f...
ficardid 8671 A finite set is equinumero...
cardnn 8672 The cardinality of a natur...
cardnueq0 8673 The empty set is the only ...
cardne 8674 No member of a cardinal nu...
carden2a 8675 If two sets have equal non...
carden2b 8676 If two sets are equinumero...
card1 8677 A set has cardinality one ...
cardsn 8678 A singleton has cardinalit...
carddomi2 8679 Two sets have the dominanc...
sdomsdomcardi 8680 A set strictly dominates i...
cardlim 8681 An infinite cardinal is a ...
cardsdomelir 8682 A cardinal strictly domina...
cardsdomel 8683 A cardinal strictly domina...
iscard 8684 Two ways to express the pr...
iscard2 8685 Two ways to express the pr...
carddom2 8686 Two numerable sets have th...
harcard 8687 The class of ordinal numbe...
cardprclem 8688 Lemma for ~ cardprc . (Co...
cardprc 8689 The class of all cardinal ...
carduni 8690 The union of a set of card...
cardiun 8691 The indexed union of a set...
cardennn 8692 If ` A ` is equinumerous t...
cardsucinf 8693 The cardinality of the suc...
cardsucnn 8694 The cardinality of the suc...
cardom 8695 The set of natural numbers...
carden2 8696 Two numerable sets are equ...
cardsdom2 8697 A numerable set is strictl...
domtri2 8698 Trichotomy of dominance fo...
nnsdomel 8699 Strict dominance and eleme...
cardval2 8700 An alternate version of th...
isinffi 8701 An infinite set contains s...
fidomtri 8702 Trichotomy of dominance wi...
fidomtri2 8703 Trichotomy of dominance wi...
harsdom 8704 The Hartogs number of a we...
onsdom 8705 Any well-orderable set is ...
harval2 8706 An alternate expression fo...
cardmin2 8707 The smallest ordinal that ...
pm54.43lem 8708 In Theorem *54.43 of [Whit...
pm54.43 8709 Theorem *54.43 of [Whitehe...
pr2nelem 8710 Lemma for ~ pr2ne . (Cont...
pr2ne 8711 If an unordered pair has t...
prdom2 8712 An unordered pair has at m...
en2eqpr 8713 Building a set with two el...
en2eleq 8714 Express a set of pair card...
en2other2 8715 Taking the other element t...
dif1card 8716 The cardinality of a nonem...
leweon 8717 Lexicographical order is a...
r0weon 8718 A set-like well-ordering o...
infxpenlem 8719 Lemma for ~ infxpen . (Co...
infxpen 8720 Every infinite ordinal is ...
xpomen 8721 The Cartesian product of o...
xpct 8722 The cartesian product of t...
infxpidm2 8723 The Cartesian product of a...
infxpenc 8724 A canonical version of ~ i...
infxpenc2lem1 8725 Lemma for ~ infxpenc2 . (...
infxpenc2lem2 8726 Lemma for ~ infxpenc2 . (...
infxpenc2lem3 8727 Lemma for ~ infxpenc2 . (...
infxpenc2 8728 Existence form of ~ infxpe...
iunmapdisj 8729 The union ` U_ n e. C ( A ...
fseqenlem1 8730 Lemma for ~ fseqen . (Con...
fseqenlem2 8731 Lemma for ~ fseqen . (Con...
fseqdom 8732 One half of ~ fseqen . (C...
fseqen 8733 A set that is equinumerous...
infpwfidom 8734 The collection of finite s...
dfac8alem 8735 Lemma for ~ dfac8a . If t...
dfac8a 8736 Numeration theorem: every ...
dfac8b 8737 The well-ordering theorem:...
dfac8clem 8738 Lemma for ~ dfac8c . (Con...
dfac8c 8739 If the union of a set is w...
ac10ct 8740 A proof of the Well orderi...
ween 8741 A set is numerable iff it ...
ac5num 8742 A version of ~ ac5b with t...
ondomen 8743 If a set is dominated by a...
numdom 8744 A set dominated by a numer...
ssnum 8745 A subset of a numerable se...
onssnum 8746 All subsets of the ordinal...
indcardi 8747 Indirect strong induction ...
acnrcl 8748 Reverse closure for the ch...
acneq 8749 Equality theorem for the c...
isacn 8750 The property of being a ch...
acni 8751 The property of being a ch...
acni2 8752 The property of being a ch...
acni3 8753 The property of being a ch...
acnlem 8754 Construct a mapping satisf...
numacn 8755 A well-orderable set has c...
finacn 8756 Every set has finite choic...
acndom 8757 A set with long choice seq...
acnnum 8758 A set ` X ` which has choi...
acnen 8759 The class of choice sets o...
acndom2 8760 A set smaller than one wit...
acnen2 8761 The class of sets with cho...
fodomacn 8762 A version of ~ fodom that ...
fodomnum 8763 A version of ~ fodom that ...
fonum 8764 A surjection maps numerabl...
numwdom 8765 A surjection maps numerabl...
fodomfi2 8766 Onto functions define domi...
wdomfil 8767 Weak dominance agrees with...
infpwfien 8768 Any infinite well-orderabl...
inffien 8769 The set of finite intersec...
wdomnumr 8770 Weak dominance agrees with...
alephfnon 8771 The aleph function is a fu...
aleph0 8772 The first infinite cardina...
alephlim 8773 Value of the aleph functio...
alephsuc 8774 Value of the aleph functio...
alephon 8775 An aleph is an ordinal num...
alephcard 8776 Every aleph is a cardinal ...
alephnbtwn 8777 No cardinal can be sandwic...
alephnbtwn2 8778 No set has equinumerosity ...
alephordilem1 8779 Lemma for ~ alephordi . (...
alephordi 8780 Strict ordering property o...
alephord 8781 Ordering property of the a...
alephord2 8782 Ordering property of the a...
alephord2i 8783 Ordering property of the a...
alephord3 8784 Ordering property of the a...
alephsucdom 8785 A set dominated by an alep...
alephsuc2 8786 An alternate representatio...
alephdom 8787 Relationship between inclu...
alephgeom 8788 Every aleph is greater tha...
alephislim 8789 Every aleph is a limit ord...
aleph11 8790 The aleph function is one-...
alephf1 8791 The aleph function is a on...
alephsdom 8792 If an ordinal is smaller t...
alephdom2 8793 A dominated initial ordina...
alephle 8794 The argument of the aleph ...
cardaleph 8795 Given any transfinite card...
cardalephex 8796 Every transfinite cardinal...
infenaleph 8797 An infinite numerable set ...
isinfcard 8798 Two ways to express the pr...
iscard3 8799 Two ways to express the pr...
cardnum 8800 Two ways to express the cl...
alephinit 8801 An infinite initial ordina...
carduniima 8802 The union of the image of ...
cardinfima 8803 If a mapping to cardinals ...
alephiso 8804 Aleph is an order isomorph...
alephprc 8805 The class of all transfini...
alephsson 8806 The class of transfinite c...
unialeph 8807 The union of the class of ...
alephsmo 8808 The aleph function is stri...
alephf1ALT 8809 Alternate proof of ~ aleph...
alephfplem1 8810 Lemma for ~ alephfp . (Co...
alephfplem2 8811 Lemma for ~ alephfp . (Co...
alephfplem3 8812 Lemma for ~ alephfp . (Co...
alephfplem4 8813 Lemma for ~ alephfp . (Co...
alephfp 8814 The aleph function has a f...
alephfp2 8815 The aleph function has at ...
alephval3 8816 An alternate way to expres...
alephsucpw2 8817 The power set of an aleph ...
mappwen 8818 Power rule for cardinal ar...
finnisoeu 8819 A finite totally ordered s...
iunfictbso 8820 Countability of a countabl...
aceq1 8823 Equivalence of two version...
aceq0 8824 Equivalence of two version...
aceq2 8825 Equivalence of two version...
aceq3lem 8826 Lemma for ~ dfac3 . (Cont...
dfac3 8827 Equivalence of two version...
dfac4 8828 Equivalence of two version...
dfac5lem1 8829 Lemma for ~ dfac5 . (Cont...
dfac5lem2 8830 Lemma for ~ dfac5 . (Cont...
dfac5lem3 8831 Lemma for ~ dfac5 . (Cont...
dfac5lem4 8832 Lemma for ~ dfac5 . (Cont...
dfac5lem5 8833 Lemma for ~ dfac5 . (Cont...
dfac5 8834 Equivalence of two version...
dfac2a 8835 Our Axiom of Choice (in th...
dfac2 8836 Axiom of Choice (first for...
dfac7 8837 Equivalence of the Axiom o...
dfac0 8838 Equivalence of two version...
dfac1 8839 Equivalence of two version...
dfac8 8840 A proof of the equivalency...
dfac9 8841 Equivalence of the axiom o...
dfac10 8842 Axiom of Choice equivalent...
dfac10c 8843 Axiom of Choice equivalent...
dfac10b 8844 Axiom of Choice equivalent...
acacni 8845 A choice equivalent: every...
dfacacn 8846 A choice equivalent: every...
dfac13 8847 The axiom of choice holds ...
dfac12lem1 8848 Lemma for ~ dfac12 . (Con...
dfac12lem2 8849 Lemma for ~ dfac12 . (Con...
dfac12lem3 8850 Lemma for ~ dfac12 . (Con...
dfac12r 8851 The axiom of choice holds ...
dfac12k 8852 Equivalence of ~ dfac12 an...
dfac12a 8853 The axiom of choice holds ...
dfac12 8854 The axiom of choice holds ...
kmlem1 8855 Lemma for 5-quantifier AC ...
kmlem2 8856 Lemma for 5-quantifier AC ...
kmlem3 8857 Lemma for 5-quantifier AC ...
kmlem4 8858 Lemma for 5-quantifier AC ...
kmlem5 8859 Lemma for 5-quantifier AC ...
kmlem6 8860 Lemma for 5-quantifier AC ...
kmlem7 8861 Lemma for 5-quantifier AC ...
kmlem8 8862 Lemma for 5-quantifier AC ...
kmlem9 8863 Lemma for 5-quantifier AC ...
kmlem10 8864 Lemma for 5-quantifier AC ...
kmlem11 8865 Lemma for 5-quantifier AC ...
kmlem12 8866 Lemma for 5-quantifier AC ...
kmlem13 8867 Lemma for 5-quantifier AC ...
kmlem14 8868 Lemma for 5-quantifier AC ...
kmlem15 8869 Lemma for 5-quantifier AC ...
kmlem16 8870 Lemma for 5-quantifier AC ...
dfackm 8871 Equivalence of the Axiom o...
cdafn 8874 Cardinal number addition i...
cdaval 8875 Value of cardinal addition...
uncdadom 8876 Cardinal addition dominate...
cdaun 8877 Cardinal addition is equin...
cdaen 8878 Cardinal addition of equin...
cdaenun 8879 Cardinal addition is equin...
cda1en 8880 Cardinal addition with car...
cda1dif 8881 Adding and subtracting one...
pm110.643 8882 1+1=2 for cardinal number ...
pm110.643ALT 8883 Alternate proof of ~ pm110...
cda0en 8884 Cardinal addition with car...
xp2cda 8885 Two times a cardinal numbe...
cdacomen 8886 Commutative law for cardin...
cdaassen 8887 Associative law for cardin...
xpcdaen 8888 Cardinal multiplication di...
mapcdaen 8889 Sum of exponents law for c...
pwcdaen 8890 Sum of exponents law for c...
cdadom1 8891 Ordering law for cardinal ...
cdadom2 8892 Ordering law for cardinal ...
cdadom3 8893 A set is dominated by its ...
cdaxpdom 8894 Cartesian product dominate...
cdafi 8895 The cardinal sum of two fi...
cdainflem 8896 Any partition of omega int...
cdainf 8897 A set is infinite iff the ...
infcda1 8898 An infinite set is equinum...
pwcda1 8899 The sum of a powerset with...
pwcdaidm 8900 If the natural numbers inj...
cdalepw 8901 If ` A ` is idempotent und...
onacda 8902 The cardinal and ordinal s...
cardacda 8903 The cardinal sum is equinu...
cdanum 8904 The cardinal sum of two nu...
unnum 8905 The union of two numerable...
nnacda 8906 The cardinal and ordinal s...
ficardun 8907 The cardinality of the uni...
ficardun2 8908 The cardinality of the uni...
pwsdompw 8909 Lemma for ~ domtriom . Th...
unctb 8910 The union of two countable...
infcdaabs 8911 Absorption law for additio...
infunabs 8912 An infinite set is equinum...
infcda 8913 The sum of two cardinal nu...
infdif 8914 The cardinality of an infi...
infdif2 8915 Cardinality ordering for a...
infxpdom 8916 Dominance law for multipli...
infxpabs 8917 Absorption law for multipl...
infunsdom1 8918 The union of two sets that...
infunsdom 8919 The union of two sets that...
infxp 8920 Absorption law for multipl...
pwcdadom 8921 A property of dominance ov...
infpss 8922 Every infinite set has an ...
infmap2 8923 An exponentiation law for ...
ackbij2lem1 8924 Lemma for ~ ackbij2 . (Co...
ackbij1lem1 8925 Lemma for ~ ackbij2 . (Co...
ackbij1lem2 8926 Lemma for ~ ackbij2 . (Co...
ackbij1lem3 8927 Lemma for ~ ackbij2 . (Co...
ackbij1lem4 8928 Lemma for ~ ackbij2 . (Co...
ackbij1lem5 8929 Lemma for ~ ackbij2 . (Co...
ackbij1lem6 8930 Lemma for ~ ackbij2 . (Co...
ackbij1lem7 8931 Lemma for ~ ackbij1 . (Co...
ackbij1lem8 8932 Lemma for ~ ackbij1 . (Co...
ackbij1lem9 8933 Lemma for ~ ackbij1 . (Co...
ackbij1lem10 8934 Lemma for ~ ackbij1 . (Co...
ackbij1lem11 8935 Lemma for ~ ackbij1 . (Co...
ackbij1lem12 8936 Lemma for ~ ackbij1 . (Co...
ackbij1lem13 8937 Lemma for ~ ackbij1 . (Co...
ackbij1lem14 8938 Lemma for ~ ackbij1 . (Co...
ackbij1lem15 8939 Lemma for ~ ackbij1 . (Co...
ackbij1lem16 8940 Lemma for ~ ackbij1 . (Co...
ackbij1lem17 8941 Lemma for ~ ackbij1 . (Co...
ackbij1lem18 8942 Lemma for ~ ackbij1 . (Co...
ackbij1 8943 The Ackermann bijection, p...
ackbij1b 8944 The Ackermann bijection, p...
ackbij2lem2 8945 Lemma for ~ ackbij2 . (Co...
ackbij2lem3 8946 Lemma for ~ ackbij2 . (Co...
ackbij2lem4 8947 Lemma for ~ ackbij2 . (Co...
ackbij2 8948 The Ackermann bijection, p...
r1om 8949 The set of hereditarily fi...
fictb 8950 A set is countable iff its...
cflem 8951 A lemma used to simplify c...
cfval 8952 Value of the cofinality fu...
cff 8953 Cofinality is a function o...
cfub 8954 An upper bound on cofinali...
cflm 8955 Value of the cofinality fu...
cf0 8956 Value of the cofinality fu...
cardcf 8957 Cofinality is a cardinal n...
cflecard 8958 Cofinality is bounded by t...
cfle 8959 Cofinality is bounded by i...
cfon 8960 The cofinality of any set ...
cfeq0 8961 Only the ordinal zero has ...
cfsuc 8962 Value of the cofinality fu...
cff1 8963 There is always a map from...
cfflb 8964 If there is a cofinal map ...
cfval2 8965 Another expression for the...
coflim 8966 A simpler expression for t...
cflim3 8967 Another expression for the...
cflim2 8968 The cofinality function is...
cfom 8969 Value of the cofinality fu...
cfss 8970 There is a cofinal subset ...
cfslb 8971 Any cofinal subset of ` A ...
cfslbn 8972 Any subset of ` A ` smalle...
cfslb2n 8973 Any small collection of sm...
cofsmo 8974 Any cofinal map implies th...
cfsmolem 8975 Lemma for ~ cfsmo . (Cont...
cfsmo 8976 The map in ~ cff1 can be a...
cfcoflem 8977 Lemma for ~ cfcof , showin...
coftr 8978 If there is a cofinal map ...
cfcof 8979 If there is a cofinal map ...
cfidm 8980 The cofinality function is...
alephsing 8981 The cofinality of a limit ...
sornom 8982 The range of a single-step...
isfin1a 8997 Definition of a Ia-finite ...
fin1ai 8998 Property of a Ia-finite se...
isfin2 8999 Definition of a II-finite ...
fin2i 9000 Property of a II-finite se...
isfin3 9001 Definition of a III-finite...
isfin4 9002 Definition of a IV-finite ...
fin4i 9003 Infer that a set is IV-inf...
isfin5 9004 Definition of a V-finite s...
isfin6 9005 Definition of a VI-finite ...
isfin7 9006 Definition of a VII-finite...
sdom2en01 9007 A set with less than two e...
infpssrlem1 9008 Lemma for ~ infpssr . (Co...
infpssrlem2 9009 Lemma for ~ infpssr . (Co...
infpssrlem3 9010 Lemma for ~ infpssr . (Co...
infpssrlem4 9011 Lemma for ~ infpssr . (Co...
infpssrlem5 9012 Lemma for ~ infpssr . (Co...
infpssr 9013 Dedekind infinity implies ...
fin4en1 9014 Dedekind finite is a cardi...
ssfin4 9015 Dedekind finite sets have ...
domfin4 9016 A set dominated by a Dedek...
ominf4 9017 ` _om ` is Dedekind infini...
infpssALT 9018 Alternate proof of ~ infps...
isfin4-2 9019 Alternate definition of IV...
isfin4-3 9020 Alternate definition of IV...
fin23lem7 9021 Lemma for ~ isfin2-2 . Th...
fin23lem11 9022 Lemma for ~ isfin2-2 . (C...
fin2i2 9023 A II-finite set contains m...
isfin2-2 9024 ` Fin2 ` expressed in term...
ssfin2 9025 A subset of a II-finite se...
enfin2i 9026 II-finiteness is a cardina...
fin23lem24 9027 Lemma for ~ fin23 . In a ...
fincssdom 9028 In a chain of finite sets,...
fin23lem25 9029 Lemma for ~ fin23 . In a ...
fin23lem26 9030 Lemma for ~ fin23lem22 . ...
fin23lem23 9031 Lemma for ~ fin23lem22 . ...
fin23lem22 9032 Lemma for ~ fin23 but coul...
fin23lem27 9033 The mapping constructed in...
isfin3ds 9034 Property of a III-finite s...
ssfin3ds 9035 A subset of a III-finite s...
fin23lem12 9036 The beginning of the proof...
fin23lem13 9037 Lemma for ~ fin23 . Each ...
fin23lem14 9038 Lemma for ~ fin23 . ` U ` ...
fin23lem15 9039 Lemma for ~ fin23 . ` U ` ...
fin23lem16 9040 Lemma for ~ fin23 . ` U ` ...
fin23lem19 9041 Lemma for ~ fin23 . The f...
fin23lem20 9042 Lemma for ~ fin23 . ` X ` ...
fin23lem17 9043 Lemma for ~ fin23 . By ? ...
fin23lem21 9044 Lemma for ~ fin23 . ` X ` ...
fin23lem28 9045 Lemma for ~ fin23 . The r...
fin23lem29 9046 Lemma for ~ fin23 . The r...
fin23lem30 9047 Lemma for ~ fin23 . The r...
fin23lem31 9048 Lemma for ~ fin23 . The r...
fin23lem32 9049 Lemma for ~ fin23 . Wrap ...
fin23lem33 9050 Lemma for ~ fin23 . Disch...
fin23lem34 9051 Lemma for ~ fin23 . Estab...
fin23lem35 9052 Lemma for ~ fin23 . Stric...
fin23lem36 9053 Lemma for ~ fin23 . Weak ...
fin23lem38 9054 Lemma for ~ fin23 . The c...
fin23lem39 9055 Lemma for ~ fin23 . Thus,...
fin23lem40 9056 Lemma for ~ fin23 . ` Fin2...
fin23lem41 9057 Lemma for ~ fin23 . A set...
isf32lem1 9058 Lemma for ~ isfin3-2 . De...
isf32lem2 9059 Lemma for ~ isfin3-2 . No...
isf32lem3 9060 Lemma for ~ isfin3-2 . Be...
isf32lem4 9061 Lemma for ~ isfin3-2 . Be...
isf32lem5 9062 Lemma for ~ isfin3-2 . Th...
isf32lem6 9063 Lemma for ~ isfin3-2 . Ea...
isf32lem7 9064 Lemma for ~ isfin3-2 . Di...
isf32lem8 9065 Lemma for ~ isfin3-2 . K ...
isf32lem9 9066 Lemma for ~ isfin3-2 . Co...
isf32lem10 9067 Lemma for isfin3-2 . Writ...
isf32lem11 9068 Lemma for ~ isfin3-2 . Re...
isf32lem12 9069 Lemma for ~ isfin3-2 . (C...
isfin32i 9070 One half of ~ isfin3-2 . ...
isf33lem 9071 Lemma for ~ isfin3-3 . (C...
isfin3-2 9072 Weakly Dedekind-infinite s...
isfin3-3 9073 Weakly Dedekind-infinite s...
fin33i 9074 Inference from ~ isfin3-3 ...
compsscnvlem 9075 Lemma for ~ compsscnv . (...
compsscnv 9076 Complementation on a power...
isf34lem1 9077 Lemma for ~ isfin3-4 . (C...
isf34lem2 9078 Lemma for ~ isfin3-4 . (C...
compssiso 9079 Complementation is an anti...
isf34lem3 9080 Lemma for ~ isfin3-4 . (C...
compss 9081 Express image under of the...
isf34lem4 9082 Lemma for ~ isfin3-4 . (C...
isf34lem5 9083 Lemma for ~ isfin3-4 . (C...
isf34lem7 9084 Lemma for ~ isfin3-4 . (C...
isf34lem6 9085 Lemma for ~ isfin3-4 . (C...
fin34i 9086 Inference from ~ isfin3-4 ...
isfin3-4 9087 Weakly Dedekind-infinite s...
fin11a 9088 Every I-finite set is Ia-f...
enfin1ai 9089 Ia-finiteness is a cardina...
isfin1-2 9090 A set is finite in the usu...
isfin1-3 9091 A set is I-finite iff ever...
isfin1-4 9092 A set is I-finite iff ever...
dffin1-5 9093 Compact quantifier-free ve...
fin23 9094 Every II-finite set (every...
fin34 9095 Every III-finite set is IV...
isfin5-2 9096 Alternate definition of V-...
fin45 9097 Every IV-finite set is V-f...
fin56 9098 Every V-finite set is VI-f...
fin17 9099 Every I-finite set is VII-...
fin67 9100 Every VI-finite set is VII...
isfin7-2 9101 A set is VII-finite iff it...
fin71num 9102 A well-orderable set is VI...
dffin7-2 9103 Class form of ~ isfin7-2 ....
dfacfin7 9104 Axiom of Choice equivalent...
fin1a2lem1 9105 Lemma for ~ fin1a2 . (Con...
fin1a2lem2 9106 Lemma for ~ fin1a2 . (Con...
fin1a2lem3 9107 Lemma for ~ fin1a2 . (Con...
fin1a2lem4 9108 Lemma for ~ fin1a2 . (Con...
fin1a2lem5 9109 Lemma for ~ fin1a2 . (Con...
fin1a2lem6 9110 Lemma for ~ fin1a2 . Esta...
fin1a2lem7 9111 Lemma for ~ fin1a2 . Spli...
fin1a2lem8 9112 Lemma for ~ fin1a2 . Spli...
fin1a2lem9 9113 Lemma for ~ fin1a2 . In a...
fin1a2lem10 9114 Lemma for ~ fin1a2 . A no...
fin1a2lem11 9115 Lemma for ~ fin1a2 . (Con...
fin1a2lem12 9116 Lemma for ~ fin1a2 . (Con...
fin1a2lem13 9117 Lemma for ~ fin1a2 . (Con...
fin12 9118 Weak theorem which skips I...
fin1a2s 9119 An II-infinite set can hav...
fin1a2 9120 Every Ia-finite set is II-...
itunifval 9121 Function value of iterated...
itunifn 9122 Functionality of the itera...
ituni0 9123 A zero-fold iterated union...
itunisuc 9124 Successor iterated union. ...
itunitc1 9125 Each union iterate is a me...
itunitc 9126 The union of all union ite...
ituniiun 9127 Unwrap an iterated union f...
hsmexlem7 9128 Lemma for ~ hsmex . Prope...
hsmexlem8 9129 Lemma for ~ hsmex . Prope...
hsmexlem9 9130 Lemma for ~ hsmex . Prope...
hsmexlem1 9131 Lemma for ~ hsmex . Bound...
hsmexlem2 9132 Lemma for ~ hsmex . Bound...
hsmexlem3 9133 Lemma for ~ hsmex . Clear...
hsmexlem4 9134 Lemma for ~ hsmex . The c...
hsmexlem5 9135 Lemma for ~ hsmex . Combi...
hsmexlem6 9136 Lemma for ~ hsmex . (Cont...
hsmex 9137 The collection of heredita...
hsmex2 9138 The set of hereditary size...
hsmex3 9139 The set of hereditary size...
axcc2lem 9141 Lemma for ~ axcc2 . (Cont...
axcc2 9142 A possibly more useful ver...
axcc3 9143 A possibly more useful ver...
axcc4 9144 A version of ~ axcc3 that ...
acncc 9145 An ~ ax-cc equivalent: eve...
axcc4dom 9146 Relax the constraint on ~ ...
domtriomlem 9147 Lemma for ~ domtriom . (C...
domtriom 9148 Trichotomy of equinumerosi...
fin41 9149 Under countable choice, th...
dominf 9150 A nonempty set that is a s...
dcomex 9152 The Axiom of Dependent Cho...
axdc2lem 9153 Lemma for ~ axdc2 . We co...
axdc2 9154 An apparent strengthening ...
axdc3lem 9155 The class ` S ` of finite ...
axdc3lem2 9156 Lemma for ~ axdc3 . We ha...
axdc3lem3 9157 Simple substitution lemma ...
axdc3lem4 9158 Lemma for ~ axdc3 . We ha...
axdc3 9159 Dependent Choice. Axiom D...
axdc4lem 9160 Lemma for ~ axdc4 . (Cont...
axdc4 9161 A more general version of ...
axcclem 9162 Lemma for ~ axcc . (Contr...
axcc 9163 Although CC can be proven ...
zfac 9165 Axiom of Choice expressed ...
ac2 9166 Axiom of Choice equivalent...
ac3 9167 Axiom of Choice using abbr...
axac3 9169 This theorem asserts that ...
ackm 9170 A remarkable equivalent to...
axac2 9171 Derive ~ ax-ac2 from ~ ax-...
axac 9172 Derive ~ ax-ac from ~ ax-a...
axaci 9173 Apply a choice equivalent....
cardeqv 9174 All sets are well-orderabl...
numth3 9175 All sets are well-orderabl...
numth2 9176 Numeration theorem: any se...
numth 9177 Numeration theorem: every ...
ac7 9178 An Axiom of Choice equival...
ac7g 9179 An Axiom of Choice equival...
ac4 9180 Equivalent of Axiom of Cho...
ac4c 9181 Equivalent of Axiom of Cho...
ac5 9182 An Axiom of Choice equival...
ac5b 9183 Equivalent of Axiom of Cho...
ac6num 9184 A version of ~ ac6 which t...
ac6 9185 Equivalent of Axiom of Cho...
ac6c4 9186 Equivalent of Axiom of Cho...
ac6c5 9187 Equivalent of Axiom of Cho...
ac9 9188 An Axiom of Choice equival...
ac6s 9189 Equivalent of Axiom of Cho...
ac6n 9190 Equivalent of Axiom of Cho...
ac6s2 9191 Generalization of the Axio...
ac6s3 9192 Generalization of the Axio...
ac6sg 9193 ~ ac6s with sethood as ant...
ac6sf 9194 Version of ~ ac6 with boun...
ac6s4 9195 Generalization of the Axio...
ac6s5 9196 Generalization of the Axio...
ac8 9197 An Axiom of Choice equival...
ac9s 9198 An Axiom of Choice equival...
numthcor 9199 Any set is strictly domina...
weth 9200 Well-ordering theorem: any...
zorn2lem1 9201 Lemma for ~ zorn2 . (Cont...
zorn2lem2 9202 Lemma for ~ zorn2 . (Cont...
zorn2lem3 9203 Lemma for ~ zorn2 . (Cont...
zorn2lem4 9204 Lemma for ~ zorn2 . (Cont...
zorn2lem5 9205 Lemma for ~ zorn2 . (Cont...
zorn2lem6 9206 Lemma for ~ zorn2 . (Cont...
zorn2lem7 9207 Lemma for ~ zorn2 . (Cont...
zorn2g 9208 Zorn's Lemma of [Monk1] p....
zorng 9209 Zorn's Lemma. If the unio...
zornn0g 9210 Variant of Zorn's lemma ~ ...
zorn2 9211 Zorn's Lemma of [Monk1] p....
zorn 9212 Zorn's Lemma. If the unio...
zornn0 9213 Variant of Zorn's lemma ~ ...
ttukeylem1 9214 Lemma for ~ ttukey . Expa...
ttukeylem2 9215 Lemma for ~ ttukey . A pr...
ttukeylem3 9216 Lemma for ~ ttukey . (Con...
ttukeylem4 9217 Lemma for ~ ttukey . (Con...
ttukeylem5 9218 Lemma for ~ ttukey . The ...
ttukeylem6 9219 Lemma for ~ ttukey . (Con...
ttukeylem7 9220 Lemma for ~ ttukey . (Con...
ttukey2g 9221 The Teichmüller-Tukey...
ttukeyg 9222 The Teichmüller-Tukey...
ttukey 9223 The Teichmüller-Tukey...
axdclem 9224 Lemma for ~ axdc . (Contr...
axdclem2 9225 Lemma for ~ axdc . Using ...
axdc 9226 This theorem derives ~ ax-...
fodom 9227 An onto function implies d...
fodomg 9228 An onto function implies d...
fodomb 9229 Equivalence of an onto map...
wdomac 9230 When assuming AC, weak and...
brdom3 9231 Equivalence to a dominance...
brdom5 9232 An equivalence to a domina...
brdom4 9233 An equivalence to a domina...
brdom7disj 9234 An equivalence to a domina...
brdom6disj 9235 An equivalence to a domina...
fin71ac 9236 Once we allow AC, the "str...
imadomg 9237 An image of a function und...
fimact 9238 The image by a function of...
fnrndomg 9239 The range of a function is...
iunfo 9240 Existence of an onto funct...
iundom2g 9241 An upper bound for the car...
iundomg 9242 An upper bound for the car...
iundom 9243 An upper bound for the car...
unidom 9244 An upper bound for the car...
uniimadom 9245 An upper bound for the car...
uniimadomf 9246 An upper bound for the car...
cardval 9247 The value of the cardinal ...
cardid 9248 Any set is equinumerous to...
cardidg 9249 Any set is equinumerous to...
cardidd 9250 Any set is equinumerous to...
cardf 9251 The cardinality function i...
carden 9252 Two sets are equinumerous ...
cardeq0 9253 Only the empty set has car...
unsnen 9254 Equinumerosity of a set wi...
carddom 9255 Two sets have the dominanc...
cardsdom 9256 Two sets have the strict d...
domtri 9257 Trichotomy law for dominan...
entric 9258 Trichotomy of equinumerosi...
entri2 9259 Trichotomy of dominance an...
entri3 9260 Trichotomy of dominance. ...
sdomsdomcard 9261 A set strictly dominates i...
canth3 9262 Cantor's theorem in terms ...
infxpidm 9263 The Cartesian product of a...
ondomon 9264 The collection of ordinal ...
cardmin 9265 The smallest ordinal that ...
ficard 9266 A set is finite iff its ca...
infinf 9267 Equivalence between two in...
unirnfdomd 9268 The union of the range of ...
konigthlem 9269 Lemma for ~ konigth . (Co...
konigth 9270 Konig's Theorem. If ` m (...
alephsucpw 9271 The power set of an aleph ...
aleph1 9272 The set exponentiation of ...
alephval2 9273 An alternate way to expres...
dominfac 9274 A nonempty set that is a s...
iunctb 9275 The countable union of cou...
unictb 9276 The countable union of cou...
infmap 9277 An exponentiation law for ...
alephadd 9278 The sum of two alephs is t...
alephmul 9279 The product of two alephs ...
alephexp1 9280 An exponentiation law for ...
alephsuc3 9281 An alternate representatio...
alephexp2 9282 An expression equinumerous...
alephreg 9283 A successor aleph is regul...
pwcfsdom 9284 A corollary of Konig's The...
cfpwsdom 9285 A corollary of Konig's The...
alephom 9286 From ~ canth2 , we know th...
smobeth 9287 The beth function is stric...
nd1 9288 A lemma for proving condit...
nd2 9289 A lemma for proving condit...
nd3 9290 A lemma for proving condit...
nd4 9291 A lemma for proving condit...
axextnd 9292 A version of the Axiom of ...
axrepndlem1 9293 Lemma for the Axiom of Rep...
axrepndlem2 9294 Lemma for the Axiom of Rep...
axrepnd 9295 A version of the Axiom of ...
axunndlem1 9296 Lemma for the Axiom of Uni...
axunnd 9297 A version of the Axiom of ...
axpowndlem1 9298 Lemma for the Axiom of Pow...
axpowndlem2 9299 Lemma for the Axiom of Pow...
axpowndlem3 9300 Lemma for the Axiom of Pow...
axpowndlem4 9301 Lemma for the Axiom of Pow...
axpownd 9302 A version of the Axiom of ...
axregndlem1 9303 Lemma for the Axiom of Reg...
axregndlem2 9304 Lemma for the Axiom of Reg...
axregnd 9305 A version of the Axiom of ...
axinfndlem1 9306 Lemma for the Axiom of Inf...
axinfnd 9307 A version of the Axiom of ...
axacndlem1 9308 Lemma for the Axiom of Cho...
axacndlem2 9309 Lemma for the Axiom of Cho...
axacndlem3 9310 Lemma for the Axiom of Cho...
axacndlem4 9311 Lemma for the Axiom of Cho...
axacndlem5 9312 Lemma for the Axiom of Cho...
axacnd 9313 A version of the Axiom of ...
zfcndext 9314 Axiom of Extensionality ~ ...
zfcndrep 9315 Axiom of Replacement ~ ax-...
zfcndun 9316 Axiom of Union ~ ax-un , r...
zfcndpow 9317 Axiom of Power Sets ~ ax-p...
zfcndreg 9318 Axiom of Regularity ~ ax-r...
zfcndinf 9319 Axiom of Infinity ~ ax-inf...
zfcndac 9320 Axiom of Choice ~ ax-ac , ...
elgch 9323 Elementhood in the collect...
fingch 9324 A finite set is a GCH-set....
gchi 9325 The only GCH-sets which ha...
gchen1 9326 If ` A <_ B < ~P A ` , and...
gchen2 9327 If ` A < B <_ ~P A ` , and...
gchor 9328 If ` A <_ B <_ ~P A ` , an...
engch 9329 The property of being a GC...
gchdomtri 9330 Under certain conditions, ...
fpwwe2cbv 9331 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem1 9332 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem2 9333 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem3 9334 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem5 9335 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem6 9336 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem7 9337 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem8 9338 Lemma for ~ fpwwe2 . Show...
fpwwe2lem9 9339 Lemma for ~ fpwwe2 . Give...
fpwwe2lem10 9340 Lemma for ~ fpwwe2 . Give...
fpwwe2lem11 9341 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem12 9342 Lemma for ~ fpwwe2 . (Con...
fpwwe2lem13 9343 Lemma for ~ fpwwe2 . (Con...
fpwwe2 9344 Given any function ` F ` f...
fpwwecbv 9345 Lemma for ~ fpwwe . (Cont...
fpwwelem 9346 Lemma for ~ fpwwe . (Cont...
fpwwe 9347 Given any function ` F ` f...
canth4 9348 An "effective" form of Can...
canthnumlem 9349 Lemma for ~ canthnum . (C...
canthnum 9350 The set of well-orderable ...
canthwelem 9351 Lemma for ~ canthnum . (C...
canthwe 9352 The set of well-orders of ...
canthp1lem1 9353 Lemma for ~ canthp1 . (Co...
canthp1lem2 9354 Lemma for ~ canthp1 . (Co...
canthp1 9355 A slightly stronger form o...
finngch 9356 The exclusion of finite se...
gchcda1 9357 An infinite GCH-set is ide...
gchinf 9358 An infinite GCH-set is Ded...
pwfseqlem1 9359 Lemma for ~ pwfseq . Deri...
pwfseqlem2 9360 Lemma for ~ pwfseq . (Con...
pwfseqlem3 9361 Lemma for ~ pwfseq . Usin...
pwfseqlem4a 9362 Lemma for ~ pwfseqlem4 . ...
pwfseqlem4 9363 Lemma for ~ pwfseq . Deri...
pwfseqlem5 9364 Lemma for ~ pwfseq . Alth...
pwfseq 9365 The powerset of a Dedekind...
pwxpndom2 9366 The powerset of a Dedekind...
pwxpndom 9367 The powerset of a Dedekind...
pwcdandom 9368 The powerset of a Dedekind...
gchcdaidm 9369 An infinite GCH-set is ide...
gchxpidm 9370 An infinite GCH-set is ide...
gchpwdom 9371 A relationship between dom...
gchaleph 9372 If ` ( aleph `` A ) ` is a...
gchaleph2 9373 If ` ( aleph `` A ) ` and ...
hargch 9374 If ` A + ~~ ~P A ` , then ...
alephgch 9375 If ` ( aleph `` suc A ) ` ...
gch2 9376 It is sufficient to requir...
gch3 9377 An equivalent formulation ...
gch-kn 9378 The equivalence of two ver...
gchaclem 9379 Lemma for ~ gchac (obsolet...
gchhar 9380 A "local" form of ~ gchac ...
gchacg 9381 A "local" form of ~ gchac ...
gchac 9382 The Generalized Continuum ...
elwina 9387 Conditions of weak inacces...
elina 9388 Conditions of strong inacc...
winaon 9389 A weakly inaccessible card...
inawinalem 9390 Lemma for ~ inawina . (Co...
inawina 9391 Every strongly inaccessibl...
omina 9392 ` _om ` is a strongly inac...
winacard 9393 A weakly inaccessible card...
winainflem 9394 A weakly inaccessible card...
winainf 9395 A weakly inaccessible card...
winalim 9396 A weakly inaccessible card...
winalim2 9397 A nontrivial weakly inacce...
winafp 9398 A nontrivial weakly inacce...
winafpi 9399 This theorem, which states...
gchina 9400 Assuming the GCH, weakly a...
iswun 9405 Properties of a weak unive...
wuntr 9406 A weak universe is transit...
wununi 9407 A weak universe is closed ...
wunpw 9408 A weak universe is closed ...
wunelss 9409 The elements of a weak uni...
wunpr 9410 A weak universe is closed ...
wunun 9411 A weak universe is closed ...
wuntp 9412 A weak universe is closed ...
wunss 9413 A weak universe is closed ...
wunin 9414 A weak universe is closed ...
wundif 9415 A weak universe is closed ...
wunint 9416 A weak universe is closed ...
wunsn 9417 A weak universe is closed ...
wunsuc 9418 A weak universe is closed ...
wun0 9419 A weak universe contains t...
wunr1om 9420 A weak universe is infinit...
wunom 9421 A weak universe contains a...
wunfi 9422 A weak universe contains a...
wunop 9423 A weak universe is closed ...
wunot 9424 A weak universe is closed ...
wunxp 9425 A weak universe is closed ...
wunpm 9426 A weak universe is closed ...
wunmap 9427 A weak universe is closed ...
wunf 9428 A weak universe is closed ...
wundm 9429 A weak universe is closed ...
wunrn 9430 A weak universe is closed ...
wuncnv 9431 A weak universe is closed ...
wunres 9432 A weak universe is closed ...
wunfv 9433 A weak universe is closed ...
wunco 9434 A weak universe is closed ...
wuntpos 9435 A weak universe is closed ...
intwun 9436 The intersection of a coll...
r1limwun 9437 Each limit stage in the cu...
r1wunlim 9438 The weak universes in the ...
wunex2 9439 Construct a weak universe ...
wunex 9440 Construct a weak universe ...
uniwun 9441 Every set is contained in ...
wunex3 9442 Construct a weak universe ...
wuncval 9443 Value of the weak universe...
wuncid 9444 The weak universe closure ...
wunccl 9445 The weak universe closure ...
wuncss 9446 The weak universe closure ...
wuncidm 9447 The weak universe closure ...
wuncval2 9448 Our earlier expression for...
eltskg 9451 Properties of a Tarski cla...
eltsk2g 9452 Properties of a Tarski cla...
tskpwss 9453 First axiom of a Tarski cl...
tskpw 9454 Second axiom of a Tarski c...
tsken 9455 Third axiom of a Tarski cl...
0tsk 9456 The empty set is a (transi...
tsksdom 9457 An element of a Tarski cla...
tskssel 9458 A part of a Tarski class s...
tskss 9459 The subsets of an element ...
tskin 9460 The intersection of two el...
tsksn 9461 A singleton of an element ...
tsktrss 9462 A transitive element of a ...
tsksuc 9463 If an element of a Tarski ...
tsk0 9464 A nonempty Tarski class co...
tsk1 9465 One is an element of a non...
tsk2 9466 Two is an element of a non...
2domtsk 9467 If a Tarski class is not e...
tskr1om 9468 A nonempty Tarski class is...
tskr1om2 9469 A nonempty Tarski class co...
tskinf 9470 A nonempty Tarski class is...
tskpr 9471 If ` A ` and ` B ` are mem...
tskop 9472 If ` A ` and ` B ` are mem...
tskxpss 9473 A Cartesian product of two...
tskwe2 9474 A Tarski class is well-ord...
inttsk 9475 The intersection of a coll...
inar1 9476 ` ( R1 `` A ) ` for ` A ` ...
r1omALT 9477 Alternate proof of ~ r1om ...
rankcf 9478 Any set must be at least a...
inatsk 9479 ` ( R1 `` A ) ` for ` A ` ...
r1omtsk 9480 The set of hereditarily fi...
tskord 9481 A Tarski class contains al...
tskcard 9482 An even more direct relati...
r1tskina 9483 There is a direct relation...
tskuni 9484 The union of an element of...
tskwun 9485 A nonempty transitive Tars...
tskint 9486 The intersection of an ele...
tskun 9487 The union of two elements ...
tskxp 9488 The Cartesian product of t...
tskmap 9489 Set exponentiation is an e...
tskurn 9490 A transitive Tarski class ...
elgrug 9493 Properties of a Grothendie...
grutr 9494 A Grothendieck universe is...
gruelss 9495 A Grothendieck universe is...
grupw 9496 A Grothendieck universe co...
gruss 9497 Any subset of an element o...
grupr 9498 A Grothendieck universe co...
gruurn 9499 A Grothendieck universe co...
gruiun 9500 If ` B ( x ) ` is a family...
gruuni 9501 A Grothendieck universe co...
grurn 9502 A Grothendieck universe co...
gruima 9503 A Grothendieck universe co...
gruel 9504 Any element of an element ...
grusn 9505 A Grothendieck universe co...
gruop 9506 A Grothendieck universe co...
gruun 9507 A Grothendieck universe co...
gruxp 9508 A Grothendieck universe co...
grumap 9509 A Grothendieck universe co...
gruixp 9510 A Grothendieck universe co...
gruiin 9511 A Grothendieck universe co...
gruf 9512 A Grothendieck universe co...
gruen 9513 A Grothendieck universe co...
gruwun 9514 A nonempty Grothendieck un...
intgru 9515 The intersection of a fami...
ingru 9516 The intersection of a univ...
wfgru 9517 The wellfounded part of a ...
grudomon 9518 Each ordinal that is compa...
gruina 9519 If a Grothendieck universe...
grur1a 9520 A characterization of Grot...
grur1 9521 A characterization of Grot...
grutsk1 9522 Grothendieck universes are...
grutsk 9523 Grothendieck universes are...
axgroth5 9525 The Tarski-Grothendieck ax...
axgroth2 9526 Alternate version of the T...
grothpw 9527 Derive the Axiom of Power ...
grothpwex 9528 Derive the Axiom of Power ...
axgroth6 9529 The Tarski-Grothendieck ax...
grothomex 9530 The Tarski-Grothendieck Ax...
grothac 9531 The Tarski-Grothendieck Ax...
axgroth3 9532 Alternate version of the T...
axgroth4 9533 Alternate version of the T...
grothprimlem 9534 Lemma for ~ grothprim . E...
grothprim 9535 The Tarski-Grothendieck Ax...
grothtsk 9536 The Tarski-Grothendieck Ax...
inaprc 9537 An equivalent to the Tarsk...
tskmval 9540 Value of our tarski map. ...
tskmid 9541 The set ` A ` is an elemen...
tskmcl 9542 A Tarski class that contai...
sstskm 9543 Being a part of ` ( tarski...
eltskm 9544 Belonging to ` ( tarskiMap...
elni 9577 Membership in the class of...
elni2 9578 Membership in the class of...
pinn 9579 A positive integer is a na...
pion 9580 A positive integer is an o...
piord 9581 A positive integer is ordi...
niex 9582 The class of positive inte...
0npi 9583 The empty set is not a pos...
1pi 9584 Ordinal 'one' is a positiv...
addpiord 9585 Positive integer addition ...
mulpiord 9586 Positive integer multiplic...
mulidpi 9587 1 is an identity element f...
ltpiord 9588 Positive integer 'less tha...
ltsopi 9589 Positive integer 'less tha...
ltrelpi 9590 Positive integer 'less tha...
dmaddpi 9591 Domain of addition on posi...
dmmulpi 9592 Domain of multiplication o...
addclpi 9593 Closure of addition of pos...
mulclpi 9594 Closure of multiplication ...
addcompi 9595 Addition of positive integ...
addasspi 9596 Addition of positive integ...
mulcompi 9597 Multiplication of positive...
mulasspi 9598 Multiplication of positive...
distrpi 9599 Multiplication of positive...
addcanpi 9600 Addition cancellation law ...
mulcanpi 9601 Multiplication cancellatio...
addnidpi 9602 There is no identity eleme...
ltexpi 9603 Ordering on positive integ...
ltapi 9604 Ordering property of addit...
ltmpi 9605 Ordering property of multi...
1lt2pi 9606 One is less than two (one ...
nlt1pi 9607 No positive integer is les...
indpi 9608 Principle of Finite Induct...
enqbreq 9620 Equivalence relation for p...
enqbreq2 9621 Equivalence relation for p...
enqer 9622 The equivalence relation f...
enqex 9623 The equivalence relation f...
nqex 9624 The class of positive frac...
0nnq 9625 The empty set is not a pos...
elpqn 9626 Each positive fraction is ...
ltrelnq 9627 Positive fraction 'less th...
pinq 9628 The representatives of pos...
1nq 9629 The positive fraction 'one...
nqereu 9630 There is a unique element ...
nqerf 9631 Corollary of ~ nqereu : th...
nqercl 9632 Corollary of ~ nqereu : cl...
nqerrel 9633 Any member of ` ( N. X. N....
nqerid 9634 Corollary of ~ nqereu : th...
enqeq 9635 Corollary of ~ nqereu : if...
nqereq 9636 The function ` /Q ` acts a...
addpipq2 9637 Addition of positive fract...
addpipq 9638 Addition of positive fract...
addpqnq 9639 Addition of positive fract...
mulpipq2 9640 Multiplication of positive...
mulpipq 9641 Multiplication of positive...
mulpqnq 9642 Multiplication of positive...
ordpipq 9643 Ordering of positive fract...
ordpinq 9644 Ordering of positive fract...
addpqf 9645 Closure of addition on pos...
addclnq 9646 Closure of addition on pos...
mulpqf 9647 Closure of multiplication ...
mulclnq 9648 Closure of multiplication ...
addnqf 9649 Domain of addition on posi...
mulnqf 9650 Domain of multiplication o...
addcompq 9651 Addition of positive fract...
addcomnq 9652 Addition of positive fract...
mulcompq 9653 Multiplication of positive...
mulcomnq 9654 Multiplication of positive...
adderpqlem 9655 Lemma for ~ adderpq . (Co...
mulerpqlem 9656 Lemma for ~ mulerpq . (Co...
adderpq 9657 Addition is compatible wit...
mulerpq 9658 Multiplication is compatib...
addassnq 9659 Addition of positive fract...
mulassnq 9660 Multiplication of positive...
mulcanenq 9661 Lemma for distributive law...
distrnq 9662 Multiplication of positive...
1nqenq 9663 The equivalence class of r...
mulidnq 9664 Multiplication identity el...
recmulnq 9665 Relationship between recip...
recidnq 9666 A positive fraction times ...
recclnq 9667 Closure law for positive f...
recrecnq 9668 Reciprocal of reciprocal o...
dmrecnq 9669 Domain of reciprocal on po...
ltsonq 9670 'Less than' is a strict or...
lterpq 9671 Compatibility of ordering ...
ltanq 9672 Ordering property of addit...
ltmnq 9673 Ordering property of multi...
1lt2nq 9674 One is less than two (one ...
ltaddnq 9675 The sum of two fractions i...
ltexnq 9676 Ordering on positive fract...
halfnq 9677 One-half of any positive f...
nsmallnq 9678 The is no smallest positiv...
ltbtwnnq 9679 There exists a number betw...
ltrnq 9680 Ordering property of recip...
archnq 9681 For any fraction, there is...
npex 9687 The class of positive real...
elnp 9688 Membership in positive rea...
elnpi 9689 Membership in positive rea...
prn0 9690 A positive real is not emp...
prpssnq 9691 A positive real is a subse...
elprnq 9692 A positive real is a set o...
0npr 9693 The empty set is not a pos...
prcdnq 9694 A positive real is closed ...
prub 9695 A positive fraction not in...
prnmax 9696 A positive real has no lar...
npomex 9697 A simplifying observation,...
prnmadd 9698 A positive real has no lar...
ltrelpr 9699 Positive real 'less than' ...
genpv 9700 Value of general operation...
genpelv 9701 Membership in value of gen...
genpprecl 9702 Pre-closure law for genera...
genpdm 9703 Domain of general operatio...
genpn0 9704 The result of an operation...
genpss 9705 The result of an operation...
genpnnp 9706 The result of an operation...
genpcd 9707 Downward closure of an ope...
genpnmax 9708 An operation on positive r...
genpcl 9709 Closure of an operation on...
genpass 9710 Associativity of an operat...
plpv 9711 Value of addition on posit...
mpv 9712 Value of multiplication on...
dmplp 9713 Domain of addition on posi...
dmmp 9714 Domain of multiplication o...
nqpr 9715 The canonical embedding of...
1pr 9716 The positive real number '...
addclprlem1 9717 Lemma to prove downward cl...
addclprlem2 9718 Lemma to prove downward cl...
addclpr 9719 Closure of addition on pos...
mulclprlem 9720 Lemma to prove downward cl...
mulclpr 9721 Closure of multiplication ...
addcompr 9722 Addition of positive reals...
addasspr 9723 Addition of positive reals...
mulcompr 9724 Multiplication of positive...
mulasspr 9725 Multiplication of positive...
distrlem1pr 9726 Lemma for distributive law...
distrlem4pr 9727 Lemma for distributive law...
distrlem5pr 9728 Lemma for distributive law...
distrpr 9729 Multiplication of positive...
1idpr 9730 1 is an identity element f...
ltprord 9731 Positive real 'less than' ...
psslinpr 9732 Proper subset is a linear ...
ltsopr 9733 Positive real 'less than' ...
prlem934 9734 Lemma 9-3.4 of [Gleason] p...
ltaddpr 9735 The sum of two positive re...
ltaddpr2 9736 The sum of two positive re...
ltexprlem1 9737 Lemma for Proposition 9-3....
ltexprlem2 9738 Lemma for Proposition 9-3....
ltexprlem3 9739 Lemma for Proposition 9-3....
ltexprlem4 9740 Lemma for Proposition 9-3....
ltexprlem5 9741 Lemma for Proposition 9-3....
ltexprlem6 9742 Lemma for Proposition 9-3....
ltexprlem7 9743 Lemma for Proposition 9-3....
ltexpri 9744 Proposition 9-3.5(iv) of [...
ltaprlem 9745 Lemma for Proposition 9-3....
ltapr 9746 Ordering property of addit...
addcanpr 9747 Addition cancellation law ...
prlem936 9748 Lemma 9-3.6 of [Gleason] p...
reclem2pr 9749 Lemma for Proposition 9-3....
reclem3pr 9750 Lemma for Proposition 9-3....
reclem4pr 9751 Lemma for Proposition 9-3....
recexpr 9752 The reciprocal of a positi...
suplem1pr 9753 The union of a nonempty, b...
suplem2pr 9754 The union of a set of posi...
supexpr 9755 The union of a nonempty, b...
enrbreq 9764 Equivalence relation for s...
enrer 9765 The equivalence relation f...
enreceq 9766 Equivalence class equality...
enrex 9767 The equivalence relation f...
ltrelsr 9768 Signed real 'less than' is...
addcmpblnr 9769 Lemma showing compatibilit...
mulcmpblnrlem 9770 Lemma used in lemma showin...
mulcmpblnr 9771 Lemma showing compatibilit...
prsrlem1 9772 Decomposing signed reals i...
addsrmo 9773 There is at most one resul...
mulsrmo 9774 There is at most one resul...
addsrpr 9775 Addition of signed reals i...
mulsrpr 9776 Multiplication of signed r...
ltsrpr 9777 Ordering of signed reals i...
gt0srpr 9778 Greater than zero in terms...
0nsr 9779 The empty set is not a sig...
0r 9780 The constant ` 0R ` is a s...
1sr 9781 The constant ` 1R ` is a s...
m1r 9782 The constant ` -1R ` is a ...
addclsr 9783 Closure of addition on sig...
mulclsr 9784 Closure of multiplication ...
dmaddsr 9785 Domain of addition on sign...
dmmulsr 9786 Domain of multiplication o...
addcomsr 9787 Addition of signed reals i...
addasssr 9788 Addition of signed reals i...
mulcomsr 9789 Multiplication of signed r...
mulasssr 9790 Multiplication of signed r...
distrsr 9791 Multiplication of signed r...
m1p1sr 9792 Minus one plus one is zero...
m1m1sr 9793 Minus one times minus one ...
ltsosr 9794 Signed real 'less than' is...
0lt1sr 9795 0 is less than 1 for signe...
1ne0sr 9796 1 and 0 are distinct for s...
0idsr 9797 The signed real number 0 i...
1idsr 9798 1 is an identity element f...
00sr 9799 A signed real times 0 is 0...
ltasr 9800 Ordering property of addit...
pn0sr 9801 A signed real plus its neg...
negexsr 9802 Existence of negative sign...
recexsrlem 9803 The reciprocal of a positi...
addgt0sr 9804 The sum of two positive si...
mulgt0sr 9805 The product of two positiv...
sqgt0sr 9806 The square of a nonzero si...
recexsr 9807 The reciprocal of a nonzer...
mappsrpr 9808 Mapping from positive sign...
ltpsrpr 9809 Mapping of order from posi...
map2psrpr 9810 Equivalence for positive s...
supsrlem 9811 Lemma for supremum theorem...
supsr 9812 A nonempty, bounded set of...
opelcn 9829 Ordered pair membership in...
opelreal 9830 Ordered pair membership in...
elreal 9831 Membership in class of rea...
elreal2 9832 Ordered pair membership in...
0ncn 9833 The empty set is not a com...
ltrelre 9834 'Less than' is a relation ...
addcnsr 9835 Addition of complex number...
mulcnsr 9836 Multiplication of complex ...
eqresr 9837 Equality of real numbers i...
addresr 9838 Addition of real numbers i...
mulresr 9839 Multiplication of real num...
ltresr 9840 Ordering of real subset of...
ltresr2 9841 Ordering of real subset of...
dfcnqs 9842 Technical trick to permit ...
addcnsrec 9843 Technical trick to permit ...
mulcnsrec 9844 Technical trick to permit ...
axaddf 9845 Addition is an operation o...
axmulf 9846 Multiplication is an opera...
axcnex 9847 The complex numbers form a...
axresscn 9848 The real numbers are a sub...
ax1cn 9849 1 is a complex number. Ax...
axicn 9850 ` _i ` is a complex number...
axaddcl 9851 Closure law for addition o...
axaddrcl 9852 Closure law for addition i...
axmulcl 9853 Closure law for multiplica...
axmulrcl 9854 Closure law for multiplica...
axmulcom 9855 Multiplication of complex ...
axaddass 9856 Addition of complex number...
axmulass 9857 Multiplication of complex ...
axdistr 9858 Distributive law for compl...
axi2m1 9859 i-squared equals -1 (expre...
ax1ne0 9860 1 and 0 are distinct. Axi...
ax1rid 9861 ` 1 ` is an identity eleme...
axrnegex 9862 Existence of negative of r...
axrrecex 9863 Existence of reciprocal of...
axcnre 9864 A complex number can be ex...
axpre-lttri 9865 Ordering on reals satisfie...
axpre-lttrn 9866 Ordering on reals is trans...
axpre-ltadd 9867 Ordering property of addit...
axpre-mulgt0 9868 The product of two positiv...
axpre-sup 9869 A nonempty, bounded-above ...
wuncn 9870 A weak universe containing...
cnex 9896 Alias for ~ ax-cnex . See...
addcl 9897 Alias for ~ ax-addcl , for...
readdcl 9898 Alias for ~ ax-addrcl , fo...
mulcl 9899 Alias for ~ ax-mulcl , for...
remulcl 9900 Alias for ~ ax-mulrcl , fo...
mulcom 9901 Alias for ~ ax-mulcom , fo...
addass 9902 Alias for ~ ax-addass , fo...
mulass 9903 Alias for ~ ax-mulass , fo...
adddi 9904 Alias for ~ ax-distr , for...
recn 9905 A real number is a complex...
reex 9906 The real numbers form a se...
reelprrecn 9907 Reals are a subset of the ...
cnelprrecn 9908 Complex numbers are a subs...
elimne0 9909 Hypothesis for weak deduct...
adddir 9910 Distributive law for compl...
0cn 9911 0 is a complex number. Se...
0cnd 9912 0 is a complex number, ded...
c0ex 9913 0 is a set (common case). ...
1ex 9914 1 is a set. Common specia...
cnre 9915 Alias for ~ ax-cnre , for ...
mulid1 9916 ` 1 ` is an identity eleme...
mulid2 9917 Identity law for multiplic...
1re 9918 ` 1 ` is a real number. T...
0re 9919 ` 0 ` is a real number. S...
0red 9920 ` 0 ` is a real number, de...
mulid1i 9921 Identity law for multiplic...
mulid2i 9922 Identity law for multiplic...
addcli 9923 Closure law for addition. ...
mulcli 9924 Closure law for multiplica...
mulcomi 9925 Commutative law for multip...
mulcomli 9926 Commutative law for multip...
addassi 9927 Associative law for additi...
mulassi 9928 Associative law for multip...
adddii 9929 Distributive law (left-dis...
adddiri 9930 Distributive law (right-di...
recni 9931 A real number is a complex...
readdcli 9932 Closure law for addition o...
remulcli 9933 Closure law for multiplica...
1red 9934 1 is an real number, deduc...
1cnd 9935 1 is a complex number, ded...
mulid1d 9936 Identity law for multiplic...
mulid2d 9937 Identity law for multiplic...
addcld 9938 Closure law for addition. ...
mulcld 9939 Closure law for multiplica...
mulcomd 9940 Commutative law for multip...
addassd 9941 Associative law for additi...
mulassd 9942 Associative law for multip...
adddid 9943 Distributive law (left-dis...
adddird 9944 Distributive law (right-di...
adddirp1d 9945 Distributive law, plus 1 v...
joinlmuladdmuld 9946 Join AB+CB into (A+C) on L...
recnd 9947 Deduction from real number...
readdcld 9948 Closure law for addition o...
remulcld 9949 Closure law for multiplica...
pnfnre 9960 Plus infinity is not a rea...
mnfnre 9961 Minus infinity is not a re...
ressxr 9962 The standard reals are a s...
rexpssxrxp 9963 The Cartesian product of s...
rexr 9964 A standard real is an exte...
0xr 9965 Zero is an extended real. ...
renepnf 9966 No (finite) real equals pl...
renemnf 9967 No real equals minus infin...
rexrd 9968 A standard real is an exte...
renepnfd 9969 No (finite) real equals pl...
renemnfd 9970 No real equals minus infin...
pnfxr 9971 Plus infinity belongs to t...
pnfex 9972 Plus infinity exists (comm...
pnfnemnf 9973 Plus and minus infinity ar...
mnfnepnf 9974 Minus and plus infinity ar...
mnfxr 9975 Minus infinity belongs to ...
rexri 9976 A standard real is an exte...
renfdisj 9977 The reals and the infiniti...
ltrelxr 9978 'Less than' is a relation ...
ltrel 9979 'Less than' is a relation....
lerelxr 9980 'Less than or equal' is a ...
lerel 9981 'Less or equal to' is a re...
xrlenlt 9982 'Less than or equal to' ex...
xrlenltd 9983 'Less than or equal to' ex...
xrltnle 9984 'Less than' expressed in t...
xrnltled 9985 'Not less than ' implies '...
ssxr 9986 The three (non-exclusive) ...
ltxrlt 9987 The standard less-than ` <...
axlttri 9988 Ordering on reals satisfie...
axlttrn 9989 Ordering on reals is trans...
axltadd 9990 Ordering property of addit...
axmulgt0 9991 The product of two positiv...
axsup 9992 A nonempty, bounded-above ...
lttr 9993 Alias for ~ axlttrn , for ...
mulgt0 9994 The product of two positiv...
lenlt 9995 'Less than or equal to' ex...
ltnle 9996 'Less than' expressed in t...
ltso 9997 'Less than' is a strict or...
gtso 9998 'Greater than' is a strict...
lttri2 9999 Consequence of trichotomy....
lttri3 10000 Trichotomy law for 'less t...
lttri4 10001 Trichotomy law for 'less t...
letri3 10002 Trichotomy law. (Contribu...
leloe 10003 'Less than or equal to' ex...
eqlelt 10004 Equality in terms of 'less...
ltle 10005 'Less than' implies 'less ...
leltne 10006 'Less than or equal to' im...
lelttr 10007 Transitive law. (Contribu...
ltletr 10008 Transitive law. (Contribu...
ltleletr 10009 Transitive law, weaker for...
letr 10010 Transitive law. (Contribu...
ltnr 10011 'Less than' is irreflexive...
leid 10012 'Less than or equal to' is...
ltne 10013 'Less than' implies not eq...
ltnsym 10014 'Less than' is not symmetr...
ltnsym2 10015 'Less than' is antisymmetr...
letric 10016 Trichotomy law. (Contribu...
ltlen 10017 'Less than' expressed in t...
eqle 10018 Equality implies 'less tha...
eqled 10019 Equality implies 'less tha...
ltadd2 10020 Addition to both sides of ...
ne0gt0 10021 A nonzero nonnegative numb...
lecasei 10022 Ordering elimination by ca...
lelttric 10023 Trichotomy law. (Contribu...
ltlecasei 10024 Ordering elimination by ca...
ltnri 10025 'Less than' is irreflexive...
eqlei 10026 Equality implies 'less tha...
eqlei2 10027 Equality implies 'less tha...
gtneii 10028 'Less than' implies not eq...
ltneii 10029 'Greater than' implies not...
lttri2i 10030 Consequence of trichotomy....
lttri3i 10031 Consequence of trichotomy....
letri3i 10032 Consequence of trichotomy....
leloei 10033 'Less than or equal to' in...
ltleni 10034 'Less than' expressed in t...
ltnsymi 10035 'Less than' is not symmetr...
lenlti 10036 'Less than or equal to' in...
ltnlei 10037 'Less than' in terms of 'l...
ltlei 10038 'Less than' implies 'less ...
ltleii 10039 'Less than' implies 'less ...
ltnei 10040 'Less than' implies not eq...
letrii 10041 Trichotomy law for 'less t...
lttri 10042 'Less than' is transitive....
lelttri 10043 'Less than or equal to', '...
ltletri 10044 'Less than', 'less than or...
letri 10045 'Less than or equal to' is...
le2tri3i 10046 Extended trichotomy law fo...
ltadd2i 10047 Addition to both sides of ...
mulgt0i 10048 The product of two positiv...
mulgt0ii 10049 The product of two positiv...
ltnrd 10050 'Less than' is irreflexive...
gtned 10051 'Less than' implies not eq...
ltned 10052 'Greater than' implies not...
ne0gt0d 10053 A nonzero nonnegative numb...
lttrid 10054 Ordering on reals satisfie...
lttri2d 10055 Consequence of trichotomy....
lttri3d 10056 Consequence of trichotomy....
lttri4d 10057 Trichotomy law for 'less t...
letri3d 10058 Consequence of trichotomy....
leloed 10059 'Less than or equal to' in...
eqleltd 10060 Equality in terms of 'less...
ltlend 10061 'Less than' expressed in t...
lenltd 10062 'Less than or equal to' in...
ltnled 10063 'Less than' in terms of 'l...
ltled 10064 'Less than' implies 'less ...
ltnsymd 10065 'Less than' implies 'less ...
nltled 10066 'Not less than ' implies '...
lensymd 10067 'Less than or equal to' im...
letrid 10068 Trichotomy law for 'less t...
leltned 10069 'Less than or equal to' im...
leneltd 10070 'Less than or equal to' an...
mulgt0d 10071 The product of two positiv...
ltadd2d 10072 Addition to both sides of ...
letrd 10073 Transitive law deduction f...
lelttrd 10074 Transitive law deduction f...
ltadd2dd 10075 Addition to both sides of ...
ltletrd 10076 Transitive law deduction f...
lttrd 10077 Transitive law deduction f...
lelttrdi 10078 If a number is less than a...
dedekind 10079 The Dedekind cut theorem. ...
dedekindle 10080 The Dedekind cut theorem, ...
mul12 10081 Commutative/associative la...
mul32 10082 Commutative/associative la...
mul31 10083 Commutative/associative la...
mul4 10084 Rearrangement of 4 factors...
muladd11 10085 A simple product of sums e...
1p1times 10086 Two times a number. (Cont...
peano2cn 10087 A theorem for complex numb...
peano2re 10088 A theorem for reals analog...
readdcan 10089 Cancellation law for addit...
00id 10090 ` 0 ` is its own additive ...
mul02lem1 10091 Lemma for ~ mul02 . If an...
mul02lem2 10092 Lemma for ~ mul02 . Zero ...
mul02 10093 Multiplication by ` 0 ` . ...
mul01 10094 Multiplication by ` 0 ` . ...
addid1 10095 ` 0 ` is an additive ident...
cnegex 10096 Existence of the negative ...
cnegex2 10097 Existence of a left invers...
addid2 10098 ` 0 ` is a left identity f...
addcan 10099 Cancellation law for addit...
addcan2 10100 Cancellation law for addit...
addcom 10101 Addition commutes. This u...
addid1i 10102 ` 0 ` is an additive ident...
addid2i 10103 ` 0 ` is a left identity f...
mul02i 10104 Multiplication by 0. Theo...
mul01i 10105 Multiplication by ` 0 ` . ...
addcomi 10106 Addition commutes. Based ...
addcomli 10107 Addition commutes. (Contr...
addcani 10108 Cancellation law for addit...
addcan2i 10109 Cancellation law for addit...
mul12i 10110 Commutative/associative la...
mul32i 10111 Commutative/associative la...
mul4i 10112 Rearrangement of 4 factors...
mul02d 10113 Multiplication by 0. Theo...
mul01d 10114 Multiplication by ` 0 ` . ...
addid1d 10115 ` 0 ` is an additive ident...
addid2d 10116 ` 0 ` is a left identity f...
addcomd 10117 Addition commutes. Based ...
addcand 10118 Cancellation law for addit...
addcan2d 10119 Cancellation law for addit...
addcanad 10120 Cancelling a term on the l...
addcan2ad 10121 Cancelling a term on the r...
addneintrd 10122 Introducing a term on the ...
addneintr2d 10123 Introducing a term on the ...
mul12d 10124 Commutative/associative la...
mul32d 10125 Commutative/associative la...
mul31d 10126 Commutative/associative la...
mul4d 10127 Rearrangement of 4 factors...
muladd11r 10128 A simple product of sums e...
comraddd 10129 Commute RHS addition, in d...
ltaddneg 10130 Adding a negative number t...
ltaddnegr 10131 Adding a negative number t...
add12 10132 Commutative/associative la...
add32 10133 Commutative/associative la...
add32r 10134 Commutative/associative la...
add4 10135 Rearrangement of 4 terms i...
add42 10136 Rearrangement of 4 terms i...
add12i 10137 Commutative/associative la...
add32i 10138 Commutative/associative la...
add4i 10139 Rearrangement of 4 terms i...
add42i 10140 Rearrangement of 4 terms i...
add12d 10141 Commutative/associative la...
add32d 10142 Commutative/associative la...
add4d 10143 Rearrangement of 4 terms i...
add42d 10144 Rearrangement of 4 terms i...
0cnALT 10149 Alternate proof of ~ 0cn w...
negeu 10150 Existential uniqueness of ...
subval 10151 Value of subtraction, whic...
negeq 10152 Equality theorem for negat...
negeqi 10153 Equality inference for neg...
negeqd 10154 Equality deduction for neg...
nfnegd 10155 Deduction version of ~ nfn...
nfneg 10156 Bound-variable hypothesis ...
csbnegg 10157 Move class substitution in...
negex 10158 A negative is a set. (Con...
subcl 10159 Closure law for subtractio...
negcl 10160 Closure law for negative. ...
negicn 10161 ` -u _i ` is a complex num...
subf 10162 Subtraction is an operatio...
subadd 10163 Relationship between subtr...
subadd2 10164 Relationship between subtr...
subsub23 10165 Swap subtrahend and result...
pncan 10166 Cancellation law for subtr...
pncan2 10167 Cancellation law for subtr...
pncan3 10168 Subtraction and addition o...
npcan 10169 Cancellation law for subtr...
addsubass 10170 Associative-type law for a...
addsub 10171 Law for addition and subtr...
subadd23 10172 Commutative/associative la...
addsub12 10173 Commutative/associative la...
2addsub 10174 Law for subtraction and ad...
addsubeq4 10175 Relation between sums and ...
pncan3oi 10176 Subtraction and addition o...
mvrraddi 10177 Move RHS right addition to...
mvlladdi 10178 Move LHS left addition to ...
subid 10179 Subtraction of a number fr...
subid1 10180 Identity law for subtracti...
npncan 10181 Cancellation law for subtr...
nppcan 10182 Cancellation law for subtr...
nnpcan 10183 Cancellation law for subtr...
nppcan3 10184 Cancellation law for subtr...
subcan2 10185 Cancellation law for subtr...
subeq0 10186 If the difference between ...
npncan2 10187 Cancellation law for subtr...
subsub2 10188 Law for double subtraction...
nncan 10189 Cancellation law for subtr...
subsub 10190 Law for double subtraction...
nppcan2 10191 Cancellation law for subtr...
subsub3 10192 Law for double subtraction...
subsub4 10193 Law for double subtraction...
sub32 10194 Swap the second and third ...
nnncan 10195 Cancellation law for subtr...
nnncan1 10196 Cancellation law for subtr...
nnncan2 10197 Cancellation law for subtr...
npncan3 10198 Cancellation law for subtr...
pnpcan 10199 Cancellation law for mixed...
pnpcan2 10200 Cancellation law for mixed...
pnncan 10201 Cancellation law for mixed...
ppncan 10202 Cancellation law for mixed...
addsub4 10203 Rearrangement of 4 terms i...
subadd4 10204 Rearrangement of 4 terms i...
sub4 10205 Rearrangement of 4 terms i...
neg0 10206 Minus 0 equals 0. (Contri...
negid 10207 Addition of a number and i...
negsub 10208 Relationship between subtr...
subneg 10209 Relationship between subtr...
negneg 10210 A number is equal to the n...
neg11 10211 Negative is one-to-one. (...
negcon1 10212 Negative contraposition la...
negcon2 10213 Negative contraposition la...
negeq0 10214 A number is zero iff its n...
subcan 10215 Cancellation law for subtr...
negsubdi 10216 Distribution of negative o...
negdi 10217 Distribution of negative o...
negdi2 10218 Distribution of negative o...
negsubdi2 10219 Distribution of negative o...
neg2sub 10220 Relationship between subtr...
renegcli 10221 Closure law for negative o...
resubcli 10222 Closure law for subtractio...
renegcl 10223 Closure law for negative o...
resubcl 10224 Closure law for subtractio...
negreb 10225 The negative of a real is ...
peano2cnm 10226 "Reverse" second Peano pos...
peano2rem 10227 "Reverse" second Peano pos...
negcli 10228 Closure law for negative. ...
negidi 10229 Addition of a number and i...
negnegi 10230 A number is equal to the n...
subidi 10231 Subtraction of a number fr...
subid1i 10232 Identity law for subtracti...
negne0bi 10233 A number is nonzero iff it...
negrebi 10234 The negative of a real is ...
negne0i 10235 The negative of a nonzero ...
subcli 10236 Closure law for subtractio...
pncan3i 10237 Subtraction and addition o...
negsubi 10238 Relationship between subtr...
subnegi 10239 Relationship between subtr...
subeq0i 10240 If the difference between ...
neg11i 10241 Negative is one-to-one. (...
negcon1i 10242 Negative contraposition la...
negcon2i 10243 Negative contraposition la...
negdii 10244 Distribution of negative o...
negsubdii 10245 Distribution of negative o...
negsubdi2i 10246 Distribution of negative o...
subaddi 10247 Relationship between subtr...
subadd2i 10248 Relationship between subtr...
subaddrii 10249 Relationship between subtr...
subsub23i 10250 Swap subtrahend and result...
addsubassi 10251 Associative-type law for s...
addsubi 10252 Law for subtraction and ad...
subcani 10253 Cancellation law for subtr...
subcan2i 10254 Cancellation law for subtr...
pnncani 10255 Cancellation law for mixed...
addsub4i 10256 Rearrangement of 4 terms i...
0reALT 10257 Alternate proof of ~ 0re ....
negcld 10258 Closure law for negative. ...
subidd 10259 Subtraction of a number fr...
subid1d 10260 Identity law for subtracti...
negidd 10261 Addition of a number and i...
negnegd 10262 A number is equal to the n...
negeq0d 10263 A number is zero iff its n...
negne0bd 10264 A number is nonzero iff it...
negcon1d 10265 Contraposition law for una...
negcon1ad 10266 Contraposition law for una...
neg11ad 10267 The negatives of two compl...
negned 10268 If two complex numbers are...
negne0d 10269 The negative of a nonzero ...
negrebd 10270 The negative of a real is ...
subcld 10271 Closure law for subtractio...
pncand 10272 Cancellation law for subtr...
pncan2d 10273 Cancellation law for subtr...
pncan3d 10274 Subtraction and addition o...
npcand 10275 Cancellation law for subtr...
nncand 10276 Cancellation law for subtr...
negsubd 10277 Relationship between subtr...
subnegd 10278 Relationship between subtr...
subeq0d 10279 If the difference between ...
subne0d 10280 Two unequal numbers have n...
subeq0ad 10281 The difference of two comp...
subne0ad 10282 If the difference of two c...
neg11d 10283 If the difference between ...
negdid 10284 Distribution of negative o...
negdi2d 10285 Distribution of negative o...
negsubdid 10286 Distribution of negative o...
negsubdi2d 10287 Distribution of negative o...
neg2subd 10288 Relationship between subtr...
subaddd 10289 Relationship between subtr...
subadd2d 10290 Relationship between subtr...
addsubassd 10291 Associative-type law for s...
addsubd 10292 Law for subtraction and ad...
subadd23d 10293 Commutative/associative la...
addsub12d 10294 Commutative/associative la...
npncand 10295 Cancellation law for subtr...
nppcand 10296 Cancellation law for subtr...
nppcan2d 10297 Cancellation law for subtr...
nppcan3d 10298 Cancellation law for subtr...
subsubd 10299 Law for double subtraction...
subsub2d 10300 Law for double subtraction...
subsub3d 10301 Law for double subtraction...
subsub4d 10302 Law for double subtraction...
sub32d 10303 Swap the second and third ...
nnncand 10304 Cancellation law for subtr...
nnncan1d 10305 Cancellation law for subtr...
nnncan2d 10306 Cancellation law for subtr...
npncan3d 10307 Cancellation law for subtr...
pnpcand 10308 Cancellation law for mixed...
pnpcan2d 10309 Cancellation law for mixed...
pnncand 10310 Cancellation law for mixed...
ppncand 10311 Cancellation law for mixed...
subcand 10312 Cancellation law for subtr...
subcan2d 10313 Cancellation law for subtr...
subcanad 10314 Cancellation law for subtr...
subneintrd 10315 Introducing subtraction on...
subcan2ad 10316 Cancellation law for subtr...
subneintr2d 10317 Introducing subtraction on...
addsub4d 10318 Rearrangement of 4 terms i...
subadd4d 10319 Rearrangement of 4 terms i...
sub4d 10320 Rearrangement of 4 terms i...
2addsubd 10321 Law for subtraction and ad...
addsubeq4d 10322 Relation between sums and ...
mvlraddd 10323 Move LHS right addition to...
mvrraddd 10324 Move RHS right addition to...
subaddeqd 10325 Transfer two terms of a su...
addlsub 10326 Left-subtraction: Subtrac...
addrsub 10327 Right-subtraction: Subtra...
subexsub 10328 A subtraction law: Exchan...
addid0 10329 If adding a number to a an...
addn0nid 10330 Adding a nonzero number to...
pnpncand 10331 Addition/subtraction cance...
subeqrev 10332 Reverse the order of subtr...
pncan1 10333 Cancellation law for addit...
npcan1 10334 Cancellation law for subtr...
subeq0bd 10335 If two complex numbers are...
renegcld 10336 Closure law for negative o...
resubcld 10337 Closure law for subtractio...
negn0 10338 The image under negation o...
negf1o 10339 Negation is an isomorphism...
kcnktkm1cn 10340 k times k minus 1 is a com...
muladd 10341 Product of two sums. (Con...
subdi 10342 Distribution of multiplica...
subdir 10343 Distribution of multiplica...
ine0 10344 The imaginary unit ` _i ` ...
mulneg1 10345 Product with negative is n...
mulneg2 10346 The product with a negativ...
mulneg12 10347 Swap the negative sign in ...
mul2neg 10348 Product of two negatives. ...
submul2 10349 Convert a subtraction to a...
mulm1 10350 Product with minus one is ...
addneg1mul 10351 Addition with product with...
mulsub 10352 Product of two differences...
mulsub2 10353 Swap the order of subtract...
mulm1i 10354 Product with minus one is ...
mulneg1i 10355 Product with negative is n...
mulneg2i 10356 Product with negative is n...
mul2negi 10357 Product of two negatives. ...
subdii 10358 Distribution of multiplica...
subdiri 10359 Distribution of multiplica...
muladdi 10360 Product of two sums. (Con...
mulm1d 10361 Product with minus one is ...
mulneg1d 10362 Product with negative is n...
mulneg2d 10363 Product with negative is n...
mul2negd 10364 Product of two negatives. ...
subdid 10365 Distribution of multiplica...
subdird 10366 Distribution of multiplica...
subdir2d 10367 Distribution of multiplica...
muladdd 10368 Product of two sums. (Con...
mulsubd 10369 Product of two differences...
muls1d 10370 Multiplication by one minu...
mulsubfacd 10371 Multiplication followed by...
gt0ne0 10372 Positive implies nonzero. ...
lt0ne0 10373 A number which is less tha...
ltadd1 10374 Addition to both sides of ...
leadd1 10375 Addition to both sides of ...
leadd2 10376 Addition to both sides of ...
ltsubadd 10377 'Less than' relationship b...
ltsubadd2 10378 'Less than' relationship b...
lesubadd 10379 'Less than or equal to' re...
lesubadd2 10380 'Less than or equal to' re...
ltaddsub 10381 'Less than' relationship b...
ltaddsub2 10382 'Less than' relationship b...
leaddsub 10383 'Less than or equal to' re...
leaddsub2 10384 'Less than or equal to' re...
suble 10385 Swap subtrahends in an ine...
lesub 10386 Swap subtrahends in an ine...
ltsub23 10387 'Less than' relationship b...
ltsub13 10388 'Less than' relationship b...
le2add 10389 Adding both sides of two '...
ltleadd 10390 Adding both sides of two o...
leltadd 10391 Adding both sides of two o...
lt2add 10392 Adding both sides of two '...
addgt0 10393 The sum of 2 positive numb...
addgegt0 10394 The sum of nonnegative and...
addgtge0 10395 The sum of nonnegative and...
addge0 10396 The sum of 2 nonnegative n...
ltaddpos 10397 Adding a positive number t...
ltaddpos2 10398 Adding a positive number t...
ltsubpos 10399 Subtracting a positive num...
posdif 10400 Comparison of two numbers ...
lesub1 10401 Subtraction from both side...
lesub2 10402 Subtraction of both sides ...
ltsub1 10403 Subtraction from both side...
ltsub2 10404 Subtraction of both sides ...
lt2sub 10405 Subtracting both sides of ...
le2sub 10406 Subtracting both sides of ...
ltneg 10407 Negative of both sides of ...
ltnegcon1 10408 Contraposition of negative...
ltnegcon2 10409 Contraposition of negative...
leneg 10410 Negative of both sides of ...
lenegcon1 10411 Contraposition of negative...
lenegcon2 10412 Contraposition of negative...
lt0neg1 10413 Comparison of a number and...
lt0neg2 10414 Comparison of a number and...
le0neg1 10415 Comparison of a number and...
le0neg2 10416 Comparison of a number and...
addge01 10417 A number is less than or e...
addge02 10418 A number is less than or e...
add20 10419 Two nonnegative numbers ar...
subge0 10420 Nonnegative subtraction. ...
suble0 10421 Nonpositive subtraction. ...
leaddle0 10422 The sum of a real number a...
subge02 10423 Nonnegative subtraction. ...
lesub0 10424 Lemma to show a nonnegativ...
mulge0 10425 The product of two nonnega...
mullt0 10426 The product of two negativ...
msqgt0 10427 A nonzero square is positi...
msqge0 10428 A square is nonnegative. ...
0lt1 10429 0 is less than 1. Theorem...
0le1 10430 0 is less than or equal to...
relin01 10431 An interval law for less t...
ltordlem 10432 Lemma for ~ ltord1 . (Con...
ltord1 10433 Infer an ordering relation...
leord1 10434 Infer an ordering relation...
eqord1 10435 Infer an ordering relation...
ltord2 10436 Infer an ordering relation...
leord2 10437 Infer an ordering relation...
eqord2 10438 Infer an ordering relation...
wloglei 10439 Form of ~ wlogle where bot...
wlogle 10440 If the predicate ` ch ( x ...
leidi 10441 'Less than or equal to' is...
gt0ne0i 10442 Positive means nonzero (us...
gt0ne0ii 10443 Positive implies nonzero. ...
msqgt0i 10444 A nonzero square is positi...
msqge0i 10445 A square is nonnegative. ...
addgt0i 10446 Addition of 2 positive num...
addge0i 10447 Addition of 2 nonnegative ...
addgegt0i 10448 Addition of nonnegative an...
addgt0ii 10449 Addition of 2 positive num...
add20i 10450 Two nonnegative numbers ar...
ltnegi 10451 Negative of both sides of ...
lenegi 10452 Negative of both sides of ...
ltnegcon2i 10453 Contraposition of negative...
mulge0i 10454 The product of two nonnega...
lesub0i 10455 Lemma to show a nonnegativ...
ltaddposi 10456 Adding a positive number t...
posdifi 10457 Comparison of two numbers ...
ltnegcon1i 10458 Contraposition of negative...
lenegcon1i 10459 Contraposition of negative...
subge0i 10460 Nonnegative subtraction. ...
ltadd1i 10461 Addition to both sides of ...
leadd1i 10462 Addition to both sides of ...
leadd2i 10463 Addition to both sides of ...
ltsubaddi 10464 'Less than' relationship b...
lesubaddi 10465 'Less than or equal to' re...
ltsubadd2i 10466 'Less than' relationship b...
lesubadd2i 10467 'Less than or equal to' re...
ltaddsubi 10468 'Less than' relationship b...
lt2addi 10469 Adding both side of two in...
le2addi 10470 Adding both side of two in...
gt0ne0d 10471 Positive implies nonzero. ...
lt0ne0d 10472 Something less than zero i...
leidd 10473 'Less than or equal to' is...
msqgt0d 10474 A nonzero square is positi...
msqge0d 10475 A square is nonnegative. ...
lt0neg1d 10476 Comparison of a number and...
lt0neg2d 10477 Comparison of a number and...
le0neg1d 10478 Comparison of a number and...
le0neg2d 10479 Comparison of a number and...
addgegt0d 10480 Addition of nonnegative an...
addgt0d 10481 Addition of 2 positive num...
addge0d 10482 Addition of 2 nonnegative ...
mulge0d 10483 The product of two nonnega...
ltnegd 10484 Negative of both sides of ...
lenegd 10485 Negative of both sides of ...
ltnegcon1d 10486 Contraposition of negative...
ltnegcon2d 10487 Contraposition of negative...
lenegcon1d 10488 Contraposition of negative...
lenegcon2d 10489 Contraposition of negative...
ltaddposd 10490 Adding a positive number t...
ltaddpos2d 10491 Adding a positive number t...
ltsubposd 10492 Subtracting a positive num...
posdifd 10493 Comparison of two numbers ...
addge01d 10494 A number is less than or e...
addge02d 10495 A number is less than or e...
subge0d 10496 Nonnegative subtraction. ...
suble0d 10497 Nonpositive subtraction. ...
subge02d 10498 Nonnegative subtraction. ...
ltadd1d 10499 Addition to both sides of ...
leadd1d 10500 Addition to both sides of ...
leadd2d 10501 Addition to both sides of ...
ltsubaddd 10502 'Less than' relationship b...
lesubaddd 10503 'Less than or equal to' re...
ltsubadd2d 10504 'Less than' relationship b...
lesubadd2d 10505 'Less than or equal to' re...
ltaddsubd 10506 'Less than' relationship b...
ltaddsub2d 10507 'Less than' relationship b...
leaddsub2d 10508 'Less than or equal to' re...
subled 10509 Swap subtrahends in an ine...
lesubd 10510 Swap subtrahends in an ine...
ltsub23d 10511 'Less than' relationship b...
ltsub13d 10512 'Less than' relationship b...
lesub1d 10513 Subtraction from both side...
lesub2d 10514 Subtraction of both sides ...
ltsub1d 10515 Subtraction from both side...
ltsub2d 10516 Subtraction of both sides ...
ltadd1dd 10517 Addition to both sides of ...
ltsub1dd 10518 Subtraction from both side...
ltsub2dd 10519 Subtraction of both sides ...
leadd1dd 10520 Addition to both sides of ...
leadd2dd 10521 Addition to both sides of ...
lesub1dd 10522 Subtraction from both side...
lesub2dd 10523 Subtraction of both sides ...
lesub3d 10524 The result of subtracting ...
le2addd 10525 Adding both side of two in...
le2subd 10526 Subtracting both sides of ...
ltleaddd 10527 Adding both sides of two o...
leltaddd 10528 Adding both sides of two o...
lt2addd 10529 Adding both side of two in...
lt2subd 10530 Subtracting both sides of ...
possumd 10531 Condition for a positive s...
sublt0d 10532 When a subtraction gives a...
ltaddsublt 10533 Addition and subtraction o...
1le1 10534 ` 1 <_ 1 ` . Common speci...
ixi 10535 ` _i ` times itself is min...
recextlem1 10536 Lemma for ~ recex . (Cont...
recextlem2 10537 Lemma for ~ recex . (Cont...
recex 10538 Existence of reciprocal of...
mulcand 10539 Cancellation law for multi...
mulcan2d 10540 Cancellation law for multi...
mulcanad 10541 Cancellation of a nonzero ...
mulcan2ad 10542 Cancellation of a nonzero ...
mulcan 10543 Cancellation law for multi...
mulcan2 10544 Cancellation law for multi...
mulcani 10545 Cancellation law for multi...
mul0or 10546 If a product is zero, one ...
mulne0b 10547 The product of two nonzero...
mulne0 10548 The product of two nonzero...
mulne0i 10549 The product of two nonzero...
muleqadd 10550 Property of numbers whose ...
receu 10551 Existential uniqueness of ...
mulnzcnopr 10552 Multiplication maps nonzer...
msq0i 10553 A number is zero iff its s...
mul0ori 10554 If a product is zero, one ...
msq0d 10555 A number is zero iff its s...
mul0ord 10556 If a product is zero, one ...
mulne0bd 10557 The product of two nonzero...
mulne0d 10558 The product of two nonzero...
mulcan1g 10559 A generalized form of the ...
mulcan2g 10560 A generalized form of the ...
mulne0bad 10561 A factor of a nonzero comp...
mulne0bbd 10562 A factor of a nonzero comp...
1div0 10565 You can't divide by zero, ...
divval 10566 Value of division: if ` A ...
divmul 10567 Relationship between divis...
divmul2 10568 Relationship between divis...
divmul3 10569 Relationship between divis...
divcl 10570 Closure law for division. ...
reccl 10571 Closure law for reciprocal...
divcan2 10572 A cancellation law for div...
divcan1 10573 A cancellation law for div...
diveq0 10574 A ratio is zero iff the nu...
divne0b 10575 The ratio of nonzero numbe...
divne0 10576 The ratio of nonzero numbe...
recne0 10577 The reciprocal of a nonzer...
recid 10578 Multiplication of a number...
recid2 10579 Multiplication of a number...
divrec 10580 Relationship between divis...
divrec2 10581 Relationship between divis...
divass 10582 An associative law for div...
div23 10583 A commutative/associative ...
div32 10584 A commutative/associative ...
div13 10585 A commutative/associative ...
div12 10586 A commutative/associative ...
divmulass 10587 An associative law for div...
divmulasscom 10588 An associative/commutative...
divdir 10589 Distribution of division o...
divcan3 10590 A cancellation law for div...
divcan4 10591 A cancellation law for div...
div11 10592 One-to-one relationship fo...
divid 10593 A number divided by itself...
div0 10594 Division into zero is zero...
div1 10595 A number divided by 1 is i...
1div1e1 10596 1 divided by 1 is 1 (commo...
diveq1 10597 Equality in terms of unit ...
divneg 10598 Move negative sign inside ...
muldivdir 10599 Distribution of division o...
divsubdir 10600 Distribution of division o...
recrec 10601 A number is equal to the r...
rec11 10602 Reciprocal is one-to-one. ...
rec11r 10603 Mutual reciprocals. (Cont...
divmuldiv 10604 Multiplication of two rati...
divdivdiv 10605 Division of two ratios. T...
divcan5 10606 Cancellation of common fac...
divmul13 10607 Swap the denominators in t...
divmul24 10608 Swap the numerators in the...
divmuleq 10609 Cross-multiply in an equal...
recdiv 10610 The reciprocal of a ratio....
divcan6 10611 Cancellation of inverted f...
divdiv32 10612 Swap denominators in a div...
divcan7 10613 Cancel equal divisors in a...
dmdcan 10614 Cancellation law for divis...
divdiv1 10615 Division into a fraction. ...
divdiv2 10616 Division by a fraction. (...
recdiv2 10617 Division into a reciprocal...
ddcan 10618 Cancellation in a double d...
divadddiv 10619 Addition of two ratios. T...
divsubdiv 10620 Subtraction of two ratios....
conjmul 10621 Two numbers whose reciproc...
rereccl 10622 Closure law for reciprocal...
redivcl 10623 Closure law for division o...
eqneg 10624 A number equal to its nega...
eqnegd 10625 A complex number equals it...
eqnegad 10626 If a complex number equals...
div2neg 10627 Quotient of two negatives....
divneg2 10628 Move negative sign inside ...
recclzi 10629 Closure law for reciprocal...
recne0zi 10630 The reciprocal of a nonzer...
recidzi 10631 Multiplication of a number...
div1i 10632 A number divided by 1 is i...
eqnegi 10633 A number equal to its nega...
reccli 10634 Closure law for reciprocal...
recidi 10635 Multiplication of a number...
recreci 10636 A number is equal to the r...
dividi 10637 A number divided by itself...
div0i 10638 Division into zero is zero...
divclzi 10639 Closure law for division. ...
divcan1zi 10640 A cancellation law for div...
divcan2zi 10641 A cancellation law for div...
divreczi 10642 Relationship between divis...
divcan3zi 10643 A cancellation law for div...
divcan4zi 10644 A cancellation law for div...
rec11i 10645 Reciprocal is one-to-one. ...
divcli 10646 Closure law for division. ...
divcan2i 10647 A cancellation law for div...
divcan1i 10648 A cancellation law for div...
divreci 10649 Relationship between divis...
divcan3i 10650 A cancellation law for div...
divcan4i 10651 A cancellation law for div...
divne0i 10652 The ratio of nonzero numbe...
rec11ii 10653 Reciprocal is one-to-one. ...
divasszi 10654 An associative law for div...
divmulzi 10655 Relationship between divis...
divdirzi 10656 Distribution of division o...
divdiv23zi 10657 Swap denominators in a div...
divmuli 10658 Relationship between divis...
divdiv32i 10659 Swap denominators in a div...
divassi 10660 An associative law for div...
divdiri 10661 Distribution of division o...
div23i 10662 A commutative/associative ...
div11i 10663 One-to-one relationship fo...
divmuldivi 10664 Multiplication of two rati...
divmul13i 10665 Swap denominators of two r...
divadddivi 10666 Addition of two ratios. T...
divdivdivi 10667 Division of two ratios. T...
rerecclzi 10668 Closure law for reciprocal...
rereccli 10669 Closure law for reciprocal...
redivclzi 10670 Closure law for division o...
redivcli 10671 Closure law for division o...
div1d 10672 A number divided by 1 is i...
reccld 10673 Closure law for reciprocal...
recne0d 10674 The reciprocal of a nonzer...
recidd 10675 Multiplication of a number...
recid2d 10676 Multiplication of a number...
recrecd 10677 A number is equal to the r...
dividd 10678 A number divided by itself...
div0d 10679 Division into zero is zero...
divcld 10680 Closure law for division. ...
divcan1d 10681 A cancellation law for div...
divcan2d 10682 A cancellation law for div...
divrecd 10683 Relationship between divis...
divrec2d 10684 Relationship between divis...
divcan3d 10685 A cancellation law for div...
divcan4d 10686 A cancellation law for div...
diveq0d 10687 A ratio is zero iff the nu...
diveq1d 10688 Equality in terms of unit ...
diveq1ad 10689 The quotient of two comple...
diveq0ad 10690 A fraction of complex numb...
divne1d 10691 If two complex numbers are...
divne0bd 10692 A ratio is zero iff the nu...
divnegd 10693 Move negative sign inside ...
divneg2d 10694 Move negative sign inside ...
div2negd 10695 Quotient of two negatives....
divne0d 10696 The ratio of nonzero numbe...
recdivd 10697 The reciprocal of a ratio....
recdiv2d 10698 Division into a reciprocal...
divcan6d 10699 Cancellation of inverted f...
ddcand 10700 Cancellation in a double d...
rec11d 10701 Reciprocal is one-to-one. ...
divmuld 10702 Relationship between divis...
div32d 10703 A commutative/associative ...
div13d 10704 A commutative/associative ...
divdiv32d 10705 Swap denominators in a div...
divcan5d 10706 Cancellation of common fac...
divcan5rd 10707 Cancellation of common fac...
divcan7d 10708 Cancel equal divisors in a...
dmdcand 10709 Cancellation law for divis...
dmdcan2d 10710 Cancellation law for divis...
divdiv1d 10711 Division into a fraction. ...
divdiv2d 10712 Division by a fraction. (...
divmul2d 10713 Relationship between divis...
divmul3d 10714 Relationship between divis...
divassd 10715 An associative law for div...
div12d 10716 A commutative/associative ...
div23d 10717 A commutative/associative ...
divdird 10718 Distribution of division o...
divsubdird 10719 Distribution of division o...
div11d 10720 One-to-one relationship fo...
divmuldivd 10721 Multiplication of two rati...
divmul13d 10722 Swap denominators of two r...
divmul24d 10723 Swap the numerators in the...
divadddivd 10724 Addition of two ratios. T...
divsubdivd 10725 Subtraction of two ratios....
divmuleqd 10726 Cross-multiply in an equal...
divdivdivd 10727 Division of two ratios. T...
diveq1bd 10728 If two complex numbers are...
div2sub 10729 Swap the order of subtract...
div2subd 10730 Swap subtrahend and minuen...
rereccld 10731 Closure law for reciprocal...
redivcld 10732 Closure law for division o...
subrec 10733 Subtraction of reciprocals...
subreci 10734 Subtraction of reciprocals...
subrecd 10735 Subtraction of reciprocals...
mvllmuld 10736 Move LHS left multiplicati...
mvllmuli 10737 Move LHS left multiplicati...
elimgt0 10738 Hypothesis for weak deduct...
elimge0 10739 Hypothesis for weak deduct...
ltp1 10740 A number is less than itse...
lep1 10741 A number is less than or e...
ltm1 10742 A number minus 1 is less t...
lem1 10743 A number minus 1 is less t...
letrp1 10744 A transitive property of '...
p1le 10745 A transitive property of p...
recgt0 10746 The reciprocal of a positi...
prodgt0 10747 Infer that a multiplicand ...
prodgt02 10748 Infer that a multiplier is...
prodge0 10749 Infer that a multiplicand ...
prodge02 10750 Infer that a multiplier is...
ltmul1a 10751 Lemma for ~ ltmul1 . Mult...
ltmul1 10752 Multiplication of both sid...
ltmul2 10753 Multiplication of both sid...
lemul1 10754 Multiplication of both sid...
lemul2 10755 Multiplication of both sid...
lemul1a 10756 Multiplication of both sid...
lemul2a 10757 Multiplication of both sid...
ltmul12a 10758 Comparison of product of t...
lemul12b 10759 Comparison of product of t...
lemul12a 10760 Comparison of product of t...
mulgt1 10761 The product of two numbers...
ltmulgt11 10762 Multiplication by a number...
ltmulgt12 10763 Multiplication by a number...
lemulge11 10764 Multiplication by a number...
lemulge12 10765 Multiplication by a number...
ltdiv1 10766 Division of both sides of ...
lediv1 10767 Division of both sides of ...
gt0div 10768 Division of a positive num...
ge0div 10769 Division of a nonnegative ...
divgt0 10770 The ratio of two positive ...
divge0 10771 The ratio of nonnegative a...
mulge0b 10772 A condition for multiplica...
mulle0b 10773 A condition for multiplica...
mulsuble0b 10774 A condition for multiplica...
ltmuldiv 10775 'Less than' relationship b...
ltmuldiv2 10776 'Less than' relationship b...
ltdivmul 10777 'Less than' relationship b...
ledivmul 10778 'Less than or equal to' re...
ltdivmul2 10779 'Less than' relationship b...
lt2mul2div 10780 'Less than' relationship b...
ledivmul2 10781 'Less than or equal to' re...
lemuldiv 10782 'Less than or equal' relat...
lemuldiv2 10783 'Less than or equal' relat...
ltrec 10784 The reciprocal of both sid...
lerec 10785 The reciprocal of both sid...
lt2msq1 10786 Lemma for ~ lt2msq . (Con...
lt2msq 10787 Two nonnegative numbers co...
ltdiv2 10788 Division of a positive num...
ltrec1 10789 Reciprocal swap in a 'less...
lerec2 10790 Reciprocal swap in a 'less...
ledivdiv 10791 Invert ratios of positive ...
lediv2 10792 Division of a positive num...
ltdiv23 10793 Swap denominator with othe...
lediv23 10794 Swap denominator with othe...
lediv12a 10795 Comparison of ratio of two...
lediv2a 10796 Division of both sides of ...
reclt1 10797 The reciprocal of a positi...
recgt1 10798 The reciprocal of a positi...
recgt1i 10799 The reciprocal of a number...
recp1lt1 10800 Construct a number less th...
recreclt 10801 Given a positive number ` ...
le2msq 10802 The square function on non...
msq11 10803 The square of a nonnegativ...
ledivp1 10804 Less-than-or-equal-to and ...
squeeze0 10805 If a nonnegative number is...
ltp1i 10806 A number is less than itse...
recgt0i 10807 The reciprocal of a positi...
recgt0ii 10808 The reciprocal of a positi...
prodgt0i 10809 Infer that a multiplicand ...
prodge0i 10810 Infer that a multiplicand ...
divgt0i 10811 The ratio of two positive ...
divge0i 10812 The ratio of nonnegative a...
ltreci 10813 The reciprocal of both sid...
lereci 10814 The reciprocal of both sid...
lt2msqi 10815 The square function on non...
le2msqi 10816 The square function on non...
msq11i 10817 The square of a nonnegativ...
divgt0i2i 10818 The ratio of two positive ...
ltrecii 10819 The reciprocal of both sid...
divgt0ii 10820 The ratio of two positive ...
ltmul1i 10821 Multiplication of both sid...
ltdiv1i 10822 Division of both sides of ...
ltmuldivi 10823 'Less than' relationship b...
ltmul2i 10824 Multiplication of both sid...
lemul1i 10825 Multiplication of both sid...
lemul2i 10826 Multiplication of both sid...
ltdiv23i 10827 Swap denominator with othe...
ledivp1i 10828 Less-than-or-equal-to and ...
ltdivp1i 10829 Less-than and division rel...
ltdiv23ii 10830 Swap denominator with othe...
ltmul1ii 10831 Multiplication of both sid...
ltdiv1ii 10832 Division of both sides of ...
ltp1d 10833 A number is less than itse...
lep1d 10834 A number is less than or e...
ltm1d 10835 A number minus 1 is less t...
lem1d 10836 A number minus 1 is less t...
recgt0d 10837 The reciprocal of a positi...
divgt0d 10838 The ratio of two positive ...
mulgt1d 10839 The product of two numbers...
lemulge11d 10840 Multiplication by a number...
lemulge12d 10841 Multiplication by a number...
lemul1ad 10842 Multiplication of both sid...
lemul2ad 10843 Multiplication of both sid...
ltmul12ad 10844 Comparison of product of t...
lemul12ad 10845 Comparison of product of t...
lemul12bd 10846 Comparison of product of t...
fimaxre 10847 A finite set of real numbe...
fimaxre2 10848 A nonempty finite set of r...
fimaxre3 10849 A nonempty finite set of r...
negfi 10850 The negation of a finite s...
fiminre 10851 A nonempty finite set of r...
lbreu 10852 If a set of reals contains...
lbcl 10853 If a set of reals contains...
lble 10854 If a set of reals contains...
lbinf 10855 If a set of reals contains...
lbinfcl 10856 If a set of reals contains...
lbinfle 10857 If a set of reals contains...
sup2 10858 A nonempty, bounded-above ...
sup3 10859 A version of the completen...
infm3lem 10860 Lemma for ~ infm3 . (Cont...
infm3 10861 The completeness axiom for...
suprcl 10862 Closure of supremum of a n...
suprub 10863 A member of a nonempty bou...
suprlub 10864 The supremum of a nonempty...
suprnub 10865 An upper bound is not less...
suprleub 10866 The supremum of a nonempty...
supaddc 10867 The supremum function dist...
supadd 10868 The supremum function dist...
supmul1 10869 The supremum function dist...
supmullem1 10870 Lemma for ~ supmul . (Con...
supmullem2 10871 Lemma for ~ supmul . (Con...
supmul 10872 The supremum function dist...
sup3ii 10873 A version of the completen...
suprclii 10874 Closure of supremum of a n...
suprubii 10875 A member of a nonempty bou...
suprlubii 10876 The supremum of a nonempty...
suprnubii 10877 An upper bound is not less...
suprleubii 10878 The supremum of a nonempty...
riotaneg 10879 The negative of the unique...
negiso 10880 Negation is an order anti-...
dfinfre 10881 The infimum of a set of re...
infrecl 10882 Closure of infimum of a no...
infrenegsup 10883 The infimum of a set of re...
infregelb 10884 Any lower bound of a nonem...
infrelb 10885 If a nonempty set of real ...
supfirege 10886 The supremum of a finite s...
inelr 10887 The imaginary unit ` _i ` ...
rimul 10888 A real number times the im...
cru 10889 The representation of comp...
crne0 10890 The real representation of...
creur 10891 The real part of a complex...
creui 10892 The imaginary part of a co...
cju 10893 The complex conjugate of a...
ofsubeq0 10894 Function analogue of ~ sub...
ofnegsub 10895 Function analogue of ~ neg...
ofsubge0 10896 Function analogue of ~ sub...
nnexALT 10899 Alternate proof of ~ nnex ...
peano5nni 10900 Peano's inductive postulat...
nnssre 10901 The positive integers are ...
nnsscn 10902 The positive integers are ...
nnex 10903 The set of positive intege...
nnre 10904 A positive integer is a re...
nncn 10905 A positive integer is a co...
nnrei 10906 A positive integer is a re...
nncni 10907 A positive integer is a co...
1nn 10908 Peano postulate: 1 is a po...
peano2nn 10909 Peano postulate: a success...
dfnn2 10910 Alternate definition of th...
dfnn3 10911 Alternate definition of th...
nnred 10912 A positive integer is a re...
nncnd 10913 A positive integer is a co...
peano2nnd 10914 Peano postulate: a success...
nnind 10915 Principle of Mathematical ...
nnindALT 10916 Principle of Mathematical ...
nn1m1nn 10917 Every positive integer is ...
nn1suc 10918 If a statement holds for 1...
nnaddcl 10919 Closure of addition of pos...
nnmulcl 10920 Closure of multiplication ...
nnmulcli 10921 Closure of multiplication ...
nn2ge 10922 There exists a positive in...
nnge1 10923 A positive integer is one ...
nngt1ne1 10924 A positive integer is grea...
nnle1eq1 10925 A positive integer is less...
nngt0 10926 A positive integer is posi...
nnnlt1 10927 A positive integer is not ...
nnnle0 10928 A positive integer is not ...
0nnn 10929 Zero is not a positive int...
nnne0 10930 A positive integer is nonz...
nngt0i 10931 A positive integer is posi...
nnne0i 10932 A positive integer is nonz...
nndivre 10933 The quotient of a real and...
nnrecre 10934 The reciprocal of a positi...
nnrecgt0 10935 The reciprocal of a positi...
nnsub 10936 Subtraction of positive in...
nnsubi 10937 Subtraction of positive in...
nndiv 10938 Two ways to express " ` A ...
nndivtr 10939 Transitive property of div...
nnge1d 10940 A positive integer is one ...
nngt0d 10941 A positive integer is posi...
nnne0d 10942 A positive integer is nonz...
nnrecred 10943 The reciprocal of a positi...
nnaddcld 10944 Closure of addition of pos...
nnmulcld 10945 Closure of multiplication ...
nndivred 10946 A positive integer is one ...
0ne1 10965 ` 0 =/= 1 ` (common case);...
1m1e0 10966 ` ( 1 - 1 ) = 0 ` (common ...
2re 10967 The number 2 is real. (Co...
2cn 10968 The number 2 is a complex ...
2ex 10969 2 is a set (common case). ...
2cnd 10970 2 is a complex number, ded...
3re 10971 The number 3 is real. (Co...
3cn 10972 The number 3 is a complex ...
3ex 10973 3 is a set (common case). ...
4re 10974 The number 4 is real. (Co...
4cn 10975 The number 4 is a complex ...
5re 10976 The number 5 is real. (Co...
5cn 10977 The number 5 is complex. ...
6re 10978 The number 6 is real. (Co...
6cn 10979 The number 6 is complex. ...
7re 10980 The number 7 is real. (Co...
7cn 10981 The number 7 is complex. ...
8re 10982 The number 8 is real. (Co...
8cn 10983 The number 8 is complex. ...
9re 10984 The number 9 is real. (Co...
9cn 10985 The number 9 is complex. ...
10reOLD 10986 Obsolete version of ~ 10re...
0le0 10987 Zero is nonnegative. (Con...
0le2 10988 0 is less than or equal to...
2pos 10989 The number 2 is positive. ...
2ne0 10990 The number 2 is nonzero. ...
3pos 10991 The number 3 is positive. ...
3ne0 10992 The number 3 is nonzero. ...
4pos 10993 The number 4 is positive. ...
4ne0 10994 The number 4 is nonzero. ...
5pos 10995 The number 5 is positive. ...
6pos 10996 The number 6 is positive. ...
7pos 10997 The number 7 is positive. ...
8pos 10998 The number 8 is positive. ...
9pos 10999 The number 9 is positive. ...
10posOLD 11000 The number 10 is positive....
neg1cn 11001 -1 is a complex number (co...
neg1rr 11002 -1 is a real number (commo...
neg1ne0 11003 -1 is nonzero (common case...
neg1lt0 11004 -1 is less than 0 (common ...
negneg1e1 11005 ` -u -u 1 ` is 1 (common c...
1pneg1e0 11006 ` 1 + -u 1 ` is 0 (common ...
0m0e0 11007 0 minus 0 equals 0 (common...
1m0e1 11008 1 - 0 = 1 (common case). ...
0p1e1 11009 0 + 1 = 1. (Contributed b...
1p0e1 11010 1 + 0 = 1. (Contributed b...
1p1e2 11011 1 + 1 = 2. (Contributed b...
2m1e1 11012 2 - 1 = 1. The result is ...
1e2m1 11013 1 = 2 - 1 (common case). ...
3m1e2 11014 3 - 1 = 2. (Contributed b...
4m1e3 11015 4 - 1 = 3. (Contributed b...
5m1e4 11016 5 - 1 = 4. (Contributed b...
6m1e5 11017 6 - 1 = 5. (Contributed b...
7m1e6 11018 7 - 1 = 6. (Contributed b...
8m1e7 11019 8 - 1 = 7. (Contributed b...
9m1e8 11020 9 - 1 = 8. (Contributed b...
2p2e4 11021 Two plus two equals four. ...
2times 11022 Two times a number. (Cont...
times2 11023 A number times 2. (Contri...
2timesi 11024 Two times a number. (Cont...
times2i 11025 A number times 2. (Contri...
2txmxeqx 11026 Two times a complex number...
2div2e1 11027 2 divided by 2 is 1 (commo...
2p1e3 11028 2 + 1 = 3. (Contributed b...
1p2e3 11029 1 + 2 = 3 (common case). ...
3p1e4 11030 3 + 1 = 4. (Contributed b...
4p1e5 11031 4 + 1 = 5. (Contributed b...
5p1e6 11032 5 + 1 = 6. (Contributed b...
6p1e7 11033 6 + 1 = 7. (Contributed b...
7p1e8 11034 7 + 1 = 8. (Contributed b...
8p1e9 11035 8 + 1 = 9. (Contributed b...
9p1e10OLD 11036 9 + 1 = 10. (Contributed ...
3p2e5 11037 3 + 2 = 5. (Contributed b...
3p3e6 11038 3 + 3 = 6. (Contributed b...
4p2e6 11039 4 + 2 = 6. (Contributed b...
4p3e7 11040 4 + 3 = 7. (Contributed b...
4p4e8 11041 4 + 4 = 8. (Contributed b...
5p2e7 11042 5 + 2 = 7. (Contributed b...
5p3e8 11043 5 + 3 = 8. (Contributed b...
5p4e9 11044 5 + 4 = 9. (Contributed b...
5p5e10OLD 11045 5 + 5 = 10. (Contributed ...
6p2e8 11046 6 + 2 = 8. (Contributed b...
6p3e9 11047 6 + 3 = 9. (Contributed b...
6p4e10OLD 11048 6 + 4 = 10. (Contributed ...
7p2e9 11049 7 + 2 = 9. (Contributed b...
7p3e10OLD 11050 7 + 3 = 10. (Contributed ...
8p2e10OLD 11051 8 + 2 = 10. (Contributed ...
1t1e1 11052 1 times 1 equals 1. (Cont...
2t1e2 11053 2 times 1 equals 2. (Cont...
2t2e4 11054 2 times 2 equals 4. (Cont...
3t1e3 11055 3 times 1 equals 3. (Cont...
3t2e6 11056 3 times 2 equals 6. (Cont...
3t3e9 11057 3 times 3 equals 9. (Cont...
4t2e8 11058 4 times 2 equals 8. (Cont...
5t2e10OLD 11059 5 times 2 equals 10. (Con...
2t0e0 11060 2 times 0 equals 0. (Cont...
4d2e2 11061 One half of four is two. ...
2nn 11062 2 is a positive integer. ...
3nn 11063 3 is a positive integer. ...
4nn 11064 4 is a positive integer. ...
5nn 11065 5 is a positive integer. ...
6nn 11066 6 is a positive integer. ...
7nn 11067 7 is a positive integer. ...
8nn 11068 8 is a positive integer. ...
9nn 11069 9 is a positive integer. ...
10nnOLD 11070 Obsolete version of ~ 10nn...
1lt2 11071 1 is less than 2. (Contri...
2lt3 11072 2 is less than 3. (Contri...
1lt3 11073 1 is less than 3. (Contri...
3lt4 11074 3 is less than 4. (Contri...
2lt4 11075 2 is less than 4. (Contri...
1lt4 11076 1 is less than 4. (Contri...
4lt5 11077 4 is less than 5. (Contri...
3lt5 11078 3 is less than 5. (Contri...
2lt5 11079 2 is less than 5. (Contri...
1lt5 11080 1 is less than 5. (Contri...
5lt6 11081 5 is less than 6. (Contri...
4lt6 11082 4 is less than 6. (Contri...
3lt6 11083 3 is less than 6. (Contri...
2lt6 11084 2 is less than 6. (Contri...
1lt6 11085 1 is less than 6. (Contri...
6lt7 11086 6 is less than 7. (Contri...
5lt7 11087 5 is less than 7. (Contri...
4lt7 11088 4 is less than 7. (Contri...
3lt7 11089 3 is less than 7. (Contri...
2lt7 11090 2 is less than 7. (Contri...
1lt7 11091 1 is less than 7. (Contri...
7lt8 11092 7 is less than 8. (Contri...
6lt8 11093 6 is less than 8. (Contri...
5lt8 11094 5 is less than 8. (Contri...
4lt8 11095 4 is less than 8. (Contri...
3lt8 11096 3 is less than 8. (Contri...
2lt8 11097 2 is less than 8. (Contri...
1lt8 11098 1 is less than 8. (Contri...
8lt9 11099 8 is less than 9. (Contri...
7lt9 11100 7 is less than 9. (Contri...
6lt9 11101 6 is less than 9. (Contri...
5lt9 11102 5 is less than 9. (Contri...
4lt9 11103 4 is less than 9. (Contri...
3lt9 11104 3 is less than 9. (Contri...
2lt9 11105 2 is less than 9. (Contri...
1lt9 11106 1 is less than 9. (Contri...
9lt10OLD 11107 9 is less than 10. (Contr...
8lt10OLD 11108 8 is less than 10. (Contr...
7lt10OLD 11109 7 is less than 10. (Contr...
6lt10OLD 11110 6 is less than 10. (Contr...
5lt10OLD 11111 5 is less than 10. (Contr...
4lt10OLD 11112 4 is less than 10. (Contr...
3lt10OLD 11113 3 is less than 10. (Contr...
2lt10OLD 11114 2 is less than 10. (Contr...
1lt10OLD 11115 1 is less than 10. (Contr...
0ne2 11116 0 is not equal to 2. (Con...
1ne2 11117 1 is not equal to 2. (Con...
1le2 11118 1 is less than or equal to...
2cnne0 11119 2 is a nonzero complex num...
2rene0 11120 2 is a nonzero real number...
1le3 11121 1 is less than or equal to...
neg1mulneg1e1 11122 ` -u 1 x. -u 1 ` is 1 (com...
halfre 11123 One-half is real. (Contri...
halfcn 11124 One-half is complex. (Con...
halfgt0 11125 One-half is greater than z...
halfge0 11126 One-half is not negative. ...
halflt1 11127 One-half is less than one....
1mhlfehlf 11128 Prove that 1 - 1/2 = 1/2. ...
8th4div3 11129 An eighth of four thirds i...
halfpm6th 11130 One half plus or minus one...
it0e0 11131 i times 0 equals 0 (common...
2mulicn 11132 ` ( 2 x. _i ) e. CC ` (com...
2muline0 11133 ` ( 2 x. _i ) =/= 0 ` (com...
halfcl 11134 Closure of half of a numbe...
rehalfcl 11135 Real closure of half. (Co...
half0 11136 Half of a number is zero i...
2halves 11137 Two halves make a whole. ...
halfpos2 11138 A number is positive iff i...
halfpos 11139 A positive number is great...
halfnneg2 11140 A number is nonnegative if...
halfaddsubcl 11141 Closure of half-sum and ha...
halfaddsub 11142 Sum and difference of half...
subhalfhalf 11143 Subtracting the half of a ...
lt2halves 11144 A sum is less than the who...
addltmul 11145 Sum is less than product f...
nominpos 11146 There is no smallest posit...
avglt1 11147 Ordering property for aver...
avglt2 11148 Ordering property for aver...
avgle1 11149 Ordering property for aver...
avgle2 11150 Ordering property for aver...
avgle 11151 The average of two numbers...
2timesd 11152 Two times a number. (Cont...
times2d 11153 A number times 2. (Contri...
halfcld 11154 Closure of half of a numbe...
2halvesd 11155 Two halves make a whole. ...
rehalfcld 11156 Real closure of half. (Co...
lt2halvesd 11157 A sum is less than the who...
rehalfcli 11158 Half a real number is real...
lt2addmuld 11159 If two real numbers are le...
add1p1 11160 Adding two times 1 to a nu...
sub1m1 11161 Subtracting two times 1 fr...
cnm2m1cnm3 11162 Subtracting 2 and afterwar...
xp1d2m1eqxm1d2 11163 A complex number increased...
div4p1lem1div2 11164 An integer greater than 5,...
nnunb 11165 The set of positive intege...
arch 11166 Archimedean property of re...
nnrecl 11167 There exists a positive in...
bndndx 11168 A bounded real sequence ` ...
elnn0 11171 Nonnegative integers expre...
nnssnn0 11172 Positive naturals are a su...
nn0ssre 11173 Nonnegative integers are a...
nn0sscn 11174 Nonnegative integers are a...
nn0ex 11175 The set of nonnegative int...
nnnn0 11176 A positive integer is a no...
nnnn0i 11177 A positive integer is a no...
nn0re 11178 A nonnegative integer is a...
nn0cn 11179 A nonnegative integer is a...
nn0rei 11180 A nonnegative integer is a...
nn0cni 11181 A nonnegative integer is a...
dfn2 11182 The set of positive intege...
elnnne0 11183 The positive integer prope...
0nn0 11184 0 is a nonnegative integer...
1nn0 11185 1 is a nonnegative integer...
2nn0 11186 2 is a nonnegative integer...
3nn0 11187 3 is a nonnegative integer...
4nn0 11188 4 is a nonnegative integer...
5nn0 11189 5 is a nonnegative integer...
6nn0 11190 6 is a nonnegative integer...
7nn0 11191 7 is a nonnegative integer...
8nn0 11192 8 is a nonnegative integer...
9nn0 11193 9 is a nonnegative integer...
10nn0OLD 11194 Obsolete version of ~ 10nn...
nn0ge0 11195 A nonnegative integer is g...
nn0nlt0 11196 A nonnegative integer is n...
nn0ge0i 11197 Nonnegative integers are n...
nn0le0eq0 11198 A nonnegative integer is l...
nn0p1gt0 11199 A nonnegative integer incr...
nnnn0addcl 11200 A positive integer plus a ...
nn0nnaddcl 11201 A nonnegative integer plus...
0mnnnnn0 11202 The result of subtracting ...
un0addcl 11203 If ` S ` is closed under a...
un0mulcl 11204 If ` S ` is closed under m...
nn0addcl 11205 Closure of addition of non...
nn0mulcl 11206 Closure of multiplication ...
nn0addcli 11207 Closure of addition of non...
nn0mulcli 11208 Closure of multiplication ...
nn0p1nn 11209 A nonnegative integer plus...
peano2nn0 11210 Second Peano postulate for...
nnm1nn0 11211 A positive integer minus 1...
elnn0nn 11212 The nonnegative integer pr...
elnnnn0 11213 The positive integer prope...
elnnnn0b 11214 The positive integer prope...
elnnnn0c 11215 The positive integer prope...
nn0addge1 11216 A number is less than or e...
nn0addge2 11217 A number is less than or e...
nn0addge1i 11218 A number is less than or e...
nn0addge2i 11219 A number is less than or e...
nn0sub 11220 Subtraction of nonnegative...
ltsubnn0 11221 Subtracting a nonnegative ...
nn0negleid 11222 A nonnegative integer is g...
difgtsumgt 11223 If the difference of a rea...
nn0le2xi 11224 A nonnegative integer is l...
nn0lele2xi 11225 'Less than or equal to' im...
frnnn0supp 11226 Two ways to write the supp...
frnnn0fsupp 11227 A function on ` NN0 ` is f...
nnnn0d 11228 A positive integer is a no...
nn0red 11229 A nonnegative integer is a...
nn0cnd 11230 A nonnegative integer is a...
nn0ge0d 11231 A nonnegative integer is g...
nn0addcld 11232 Closure of addition of non...
nn0mulcld 11233 Closure of multiplication ...
nn0readdcl 11234 Closure law for addition o...
nn0n0n1ge2 11235 A nonnegative integer whic...
nn0n0n1ge2b 11236 A nonnegative integer is n...
nn0ge2m1nn 11237 If a nonnegative integer i...
nn0ge2m1nn0 11238 If a nonnegative integer i...
nn0nndivcl 11239 Closure law for dividing o...
elxnn0 11242 An extended nonnegative in...
nn0ssxnn0 11243 The standard nonnegative i...
nn0xnn0 11244 A standard nonnegative int...
xnn0xr 11245 An extended nonnegative in...
0xnn0 11246 Zero is an extended nonneg...
pnf0xnn0 11247 Positive infinity is an ex...
nn0nepnf 11248 No standard nonnegative in...
nn0xnn0d 11249 A standard nonnegative int...
nn0nepnfd 11250 No standard nonnegative in...
xnn0nemnf 11251 No extended nonnegative in...
xnn0xrnemnf 11252 The extended nonnegative i...
xnn0nnn0pnf 11253 An extended nonnegative in...
elz 11256 Membership in the set of i...
nnnegz 11257 The negative of a positive...
zre 11258 An integer is a real. (Co...
zcn 11259 An integer is a complex nu...
zrei 11260 An integer is a real numbe...
zssre 11261 The integers are a subset ...
zsscn 11262 The integers are a subset ...
zex 11263 The set of integers exists...
elnnz 11264 Positive integer property ...
0z 11265 Zero is an integer. (Cont...
0zd 11266 Zero is an integer, deduct...
elnn0z 11267 Nonnegative integer proper...
elznn0nn 11268 Integer property expressed...
elznn0 11269 Integer property expressed...
elznn 11270 Integer property expressed...
elz2 11271 Membership in the set of i...
dfz2 11272 Alternative definition of ...
zexALT 11273 Alternate proof of ~ zex ....
nnssz 11274 Positive integers are a su...
nn0ssz 11275 Nonnegative integers are a...
nnz 11276 A positive integer is an i...
nn0z 11277 A nonnegative integer is a...
nnzi 11278 A positive integer is an i...
nn0zi 11279 A nonnegative integer is a...
elnnz1 11280 Positive integer property ...
znnnlt1 11281 An integer is not a positi...
nnzrab 11282 Positive integers expresse...
nn0zrab 11283 Nonnegative integers expre...
1z 11284 One is an integer. (Contr...
1zzd 11285 1 is an integer, deductive...
2z 11286 2 is an integer. (Contrib...
3z 11287 3 is an integer. (Contrib...
4z 11288 4 is an integer. (Contrib...
znegcl 11289 Closure law for negative i...
neg1z 11290 -1 is an integer (common c...
znegclb 11291 A complex number is an int...
nn0negz 11292 The negative of a nonnegat...
nn0negzi 11293 The negative of a nonnegat...
zaddcl 11294 Closure of addition of int...
peano2z 11295 Second Peano postulate gen...
zsubcl 11296 Closure of subtraction of ...
peano2zm 11297 "Reverse" second Peano pos...
zletr 11298 Transitive law of ordering...
zrevaddcl 11299 Reverse closure law for ad...
znnsub 11300 The positive difference of...
znn0sub 11301 The nonnegative difference...
nzadd 11302 The sum of a real number n...
zmulcl 11303 Closure of multiplication ...
zltp1le 11304 Integer ordering relation....
zleltp1 11305 Integer ordering relation....
zlem1lt 11306 Integer ordering relation....
zltlem1 11307 Integer ordering relation....
zgt0ge1 11308 An integer greater than ` ...
nnleltp1 11309 Positive integer ordering ...
nnltp1le 11310 Positive integer ordering ...
nnaddm1cl 11311 Closure of addition of pos...
nn0ltp1le 11312 Nonnegative integer orderi...
nn0leltp1 11313 Nonnegative integer orderi...
nn0ltlem1 11314 Nonnegative integer orderi...
nn0sub2 11315 Subtraction of nonnegative...
nn0lt10b 11316 A nonnegative integer less...
nn0lt2 11317 A nonnegative integer less...
nn0lem1lt 11318 Nonnegative integer orderi...
nnlem1lt 11319 Positive integer ordering ...
nnltlem1 11320 Positive integer ordering ...
nnm1ge0 11321 A positive integer decreas...
nn0ge0div 11322 Division of a nonnegative ...
zdiv 11323 Two ways to express " ` M ...
zdivadd 11324 Property of divisibility: ...
zdivmul 11325 Property of divisibility: ...
zextle 11326 An extensionality-like pro...
zextlt 11327 An extensionality-like pro...
recnz 11328 The reciprocal of a number...
btwnnz 11329 A number between an intege...
gtndiv 11330 A larger number does not d...
halfnz 11331 One-half is not an integer...
3halfnz 11332 Three halves is not an int...
suprzcl 11333 The supremum of a bounded-...
prime 11334 Two ways to express " ` A ...
msqznn 11335 The square of a nonzero in...
zneo 11336 No even integer equals an ...
nneo 11337 A positive integer is even...
nneoi 11338 A positive integer is even...
zeo 11339 An integer is even or odd....
zeo2 11340 An integer is even or odd ...
peano2uz2 11341 Second Peano postulate for...
peano5uzi 11342 Peano's inductive postulat...
peano5uzti 11343 Peano's inductive postulat...
dfuzi 11344 An expression for the uppe...
uzind 11345 Induction on the upper int...
uzind2 11346 Induction on the upper int...
uzind3 11347 Induction on the upper int...
nn0ind 11348 Principle of Mathematical ...
nn0indALT 11349 Principle of Mathematical ...
nn0indd 11350 Principle of Mathematical ...
fzind 11351 Induction on the integers ...
fnn0ind 11352 Induction on the integers ...
nn0ind-raph 11353 Principle of Mathematical ...
zindd 11354 Principle of Mathematical ...
btwnz 11355 Any real number can be san...
nn0zd 11356 A positive integer is an i...
nnzd 11357 A nonnegative integer is a...
zred 11358 An integer is a real numbe...
zcnd 11359 An integer is a complex nu...
znegcld 11360 Closure law for negative i...
peano2zd 11361 Deduction from second Pean...
zaddcld 11362 Closure of addition of int...
zsubcld 11363 Closure of subtraction of ...
zmulcld 11364 Closure of multiplication ...
znnn0nn 11365 The negative of a negative...
zadd2cl 11366 Increasing an integer by 2...
zriotaneg 11367 The negative of the unique...
suprfinzcl 11368 The supremum of a nonempty...
dfdecOLD 11371 Define the "decimal constr...
9p1e10 11372 9 + 1 = 10. (Contributed ...
dfdec10 11373 Version of the definition ...
decex 11374 A decimal number is a set....
decexOLD 11375 Obsolete proof of ~ decex ...
deceq1 11376 Equality theorem for the d...
deceq1OLD 11377 Obsolete proof of ~ deceq1...
deceq2 11378 Equality theorem for the d...
deceq2OLD 11379 Obsolete proof of ~ deceq1...
deceq1i 11380 Equality theorem for the d...
deceq2i 11381 Equality theorem for the d...
deceq12i 11382 Equality theorem for the d...
numnncl 11383 Closure for a numeral (wit...
num0u 11384 Add a zero in the units pl...
num0h 11385 Add a zero in the higher p...
numcl 11386 Closure for a decimal inte...
numsuc 11387 The successor of a decimal...
deccl 11388 Closure for a numeral. (C...
decclOLD 11389 Obsolete proof of ~ deccl ...
10nn 11390 10 is a positive integer. ...
10pos 11391 The number 10 is positive....
10nn0 11392 10 is a nonnegative intege...
10re 11393 The number 10 is real. (C...
decnncl 11394 Closure for a numeral. (C...
decnnclOLD 11395 Obsolete proof of ~ decnnc...
dec0u 11396 Add a zero in the units pl...
dec0uOLD 11397 Obsolete version of ~ dec0...
dec0h 11398 Add a zero in the higher p...
dec0hOLD 11399 Obsolete proof of ~ dec0h ...
numnncl2 11400 Closure for a decimal inte...
decnncl2 11401 Closure for a decimal inte...
decnncl2OLD 11402 Obsolete proof of ~ decnnc...
numlt 11403 Comparing two decimal inte...
numltc 11404 Comparing two decimal inte...
le9lt10 11405 A "decimal digit" (i.e. a ...
declt 11406 Comparing two decimal inte...
decltOLD 11407 Obsolete proof of ~ declt ...
decltc 11408 Comparing two decimal inte...
decltcOLD 11409 Obsolete version of ~ decl...
declth 11410 Comparing two decimal inte...
decsuc 11411 The successor of a decimal...
decsucOLD 11412 Obsolete proof of ~ decsuc...
3declth 11413 Comparing two decimal inte...
3decltc 11414 Comparing two decimal inte...
3decltcOLD 11415 Obsolete version of ~ 3dec...
decle 11416 Comparing two decimal inte...
decleh 11417 Comparing two decimal inte...
declei 11418 Comparing a digit to a dec...
decleOLD 11419 Obsolete version of ~ decl...
declecOLD 11420 Obsolete version of ~ decl...
numlti 11421 Comparing a digit to a dec...
declti 11422 Comparing a digit to a dec...
decltdi 11423 Comparing a digit to a dec...
decltiOLD 11424 Obsolete version of ~ decl...
numsucc 11425 The successor of a decimal...
decsucc 11426 The successor of a decimal...
decsuccOLD 11427 Obsolete version of ~ decs...
1e0p1 11428 The successor of zero. (C...
dec10p 11429 Ten plus an integer. (Con...
dec10pOLD 11430 Obsolete version of ~ dec1...
dec10OLD 11431 The decimal form of 10. N...
9p1e10bOLD 11432 Obsolete proof of ~ 9p1e10...
numma 11433 Perform a multiply-add of ...
nummac 11434 Perform a multiply-add of ...
numma2c 11435 Perform a multiply-add of ...
numadd 11436 Add two decimal integers `...
numaddc 11437 Add two decimal integers `...
nummul1c 11438 The product of a decimal i...
nummul2c 11439 The product of a decimal i...
decma 11440 Perform a multiply-add of ...
decmaOLD 11441 Obsolete proof of ~ decma ...
decmac 11442 Perform a multiply-add of ...
decmacOLD 11443 Obsolete proof of ~ decmac...
decma2c 11444 Perform a multiply-add of ...
decma2cOLD 11445 Obsolete proof of ~ decma2...
decadd 11446 Add two numerals ` M ` and...
decaddOLD 11447 Obsolete proof of ~ decadd...
decaddc 11448 Add two numerals ` M ` and...
decaddcOLD 11449 Obsolete proof of ~ decadd...
decaddc2OLD 11450 Obsolete version of ~ deca...
decaddc2 11451 Add two numerals ` M ` and...
decrmanc 11452 Perform a multiply-add of ...
decrmac 11453 Perform a multiply-add of ...
decaddm10 11454 The sum of two multiples o...
decaddi 11455 Add two numerals ` M ` and...
decaddci 11456 Add two numerals ` M ` and...
decaddci2 11457 Add two numerals ` M ` and...
decaddci2OLD 11458 Obsolete version of ~ deca...
decsubi 11459 Difference between a numer...
decsubiOLD 11460 Obsolete proof of ~ decsub...
decmul1 11461 The product of a numeral w...
decmul1OLD 11462 Obsolete proof of ~ decmul...
decmul1c 11463 The product of a numeral w...
decmul1cOLD 11464 Obsolete proof of ~ decmul...
decmul2c 11465 The product of a numeral w...
decmul2cOLD 11466 Obsolete proof of ~ decmul...
decmulnc 11467 The product of a numeral w...
11multnc 11468 The product of 11 (as nume...
decmul10add 11469 A multiplication of a numb...
decmul10addOLD 11470 Obsolete proof of ~ decmul...
6p5lem 11471 Lemma for ~ 6p5e11 and rel...
5p5e10 11472 5 + 5 = 10. (Contributed ...
5p5e10bOLD 11473 Obsolete proof of ~ 5p5e10...
6p4e10 11474 6 + 4 = 10. (Contributed ...
6p4e10bOLD 11475 Obsolete proof of ~ 6p4e10...
6p5e11 11476 6 + 5 = 11. (Contributed ...
6p5e11OLD 11477 Obsolete proof of ~ 6p5e11...
6p6e12 11478 6 + 6 = 12. (Contributed ...
7p3e10 11479 7 + 3 = 10. (Contributed ...
7p3e10bOLD 11480 Obsolete proof of ~ 7p3e10...
7p4e11 11481 7 + 4 = 11. (Contributed ...
7p4e11OLD 11482 Obsolete proof of ~ 7p4e11...
7p5e12 11483 7 + 5 = 12. (Contributed ...
7p6e13 11484 7 + 6 = 13. (Contributed ...
7p7e14 11485 7 + 7 = 14. (Contributed ...
8p2e10 11486 8 + 2 = 10. (Contributed ...
8p2e10bOLD 11487 Obsolete proof of ~ 8p2e10...
8p3e11 11488 8 + 3 = 11. (Contributed ...
8p3e11OLD 11489 Obsolete proof of ~ 8p3e11...
8p4e12 11490 8 + 4 = 12. (Contributed ...
8p5e13 11491 8 + 5 = 13. (Contributed ...
8p6e14 11492 8 + 6 = 14. (Contributed ...
8p7e15 11493 8 + 7 = 15. (Contributed ...
8p8e16 11494 8 + 8 = 16. (Contributed ...
9p2e11 11495 9 + 2 = 11. (Contributed ...
9p2e11OLD 11496 Obsolete proof of ~ 9p2e11...
9p3e12 11497 9 + 3 = 12. (Contributed ...
9p4e13 11498 9 + 4 = 13. (Contributed ...
9p5e14 11499 9 + 5 = 14. (Contributed ...
9p6e15 11500 9 + 6 = 15. (Contributed ...
9p7e16 11501 9 + 7 = 16. (Contributed ...
9p8e17 11502 9 + 8 = 17. (Contributed ...
9p9e18 11503 9 + 9 = 18. (Contributed ...
10p10e20 11504 10 + 10 = 20. (Contribute...
10p10e20OLD 11505 Obsolete version of ~ 10p1...
10m1e9 11506 10 - 1 = 9. (Contributed ...
4t3lem 11507 Lemma for ~ 4t3e12 and rel...
4t3e12 11508 4 times 3 equals 12. (Con...
4t4e16 11509 4 times 4 equals 16. (Con...
5t2e10 11510 5 times 2 equals 10. (Con...
5t3e15 11511 5 times 3 equals 15. (Con...
5t3e15OLD 11512 Obsolete proof of ~ 5t3e15...
5t4e20 11513 5 times 4 equals 20. (Con...
5t4e20OLD 11514 Obsolete proof of ~ 5t4e20...
5t5e25 11515 5 times 5 equals 25. (Con...
5t5e25OLD 11516 Obsolete proof of ~ 5t5e25...
6t2e12 11517 6 times 2 equals 12. (Con...
6t3e18 11518 6 times 3 equals 18. (Con...
6t4e24 11519 6 times 4 equals 24. (Con...
6t5e30 11520 6 times 5 equals 30. (Con...
6t5e30OLD 11521 Obsolete proof of ~ 6t5e30...
6t6e36 11522 6 times 6 equals 36. (Con...
6t6e36OLD 11523 Obsolete proof of ~ 6t6e36...
7t2e14 11524 7 times 2 equals 14. (Con...
7t3e21 11525 7 times 3 equals 21. (Con...
7t4e28 11526 7 times 4 equals 28. (Con...
7t5e35 11527 7 times 5 equals 35. (Con...
7t6e42 11528 7 times 6 equals 42. (Con...
7t7e49 11529 7 times 7 equals 49. (Con...
8t2e16 11530 8 times 2 equals 16. (Con...
8t3e24 11531 8 times 3 equals 24. (Con...
8t4e32 11532 8 times 4 equals 32. (Con...
8t5e40 11533 8 times 5 equals 40. (Con...
8t5e40OLD 11534 Obsolete proof of ~ 8t5e40...
8t6e48 11535 8 times 6 equals 48. (Con...
8t6e48OLD 11536 Obsolete proof of ~ 8t6e48...
8t7e56 11537 8 times 7 equals 56. (Con...
8t8e64 11538 8 times 8 equals 64. (Con...
9t2e18 11539 9 times 2 equals 18. (Con...
9t3e27 11540 9 times 3 equals 27. (Con...
9t4e36 11541 9 times 4 equals 36. (Con...
9t5e45 11542 9 times 5 equals 45. (Con...
9t6e54 11543 9 times 6 equals 54. (Con...
9t7e63 11544 9 times 7 equals 63. (Con...
9t8e72 11545 9 times 8 equals 72. (Con...
9t9e81 11546 9 times 9 equals 81. (Con...
9t11e99 11547 9 times 11 equals 99. (Co...
9t11e99OLD 11548 Obsolete proof of ~ 9t11e9...
9lt10 11549 9 is less than 10. (Contr...
8lt10 11550 8 is less than 10. (Contr...
7lt10 11551 7 is less than 10. (Contr...
6lt10 11552 6 is less than 10. (Contr...
5lt10 11553 5 is less than 10. (Contr...
4lt10 11554 4 is less than 10. (Contr...
3lt10 11555 3 is less than 10. (Contr...
2lt10 11556 2 is less than 10. (Contr...
1lt10 11557 1 is less than 10. (Contr...
decbin0 11558 Decompose base 4 into base...
decbin2 11559 Decompose base 4 into base...
decbin3 11560 Decompose base 4 into base...
halfthird 11561 Half minus a third. (Cont...
5recm6rec 11562 One fifth minus one sixth....
uzval 11565 The value of the upper int...
uzf 11566 The domain and range of th...
eluz1 11567 Membership in the upper se...
eluzel2 11568 Implication of membership ...
eluz2 11569 Membership in an upper set...
eluzmn 11570 Membership in an earlier u...
eluz1i 11571 Membership in an upper set...
eluzuzle 11572 An integer in an upper set...
eluzelz 11573 A member of an upper set o...
eluzelre 11574 A member of an upper set o...
eluzelcn 11575 A member of an upper set o...
eluzle 11576 Implication of membership ...
eluz 11577 Membership in an upper set...
uzid 11578 Membership of the least me...
uzn0 11579 The upper integers are all...
uztrn 11580 Transitive law for sets of...
uztrn2 11581 Transitive law for sets of...
uzneg 11582 Contraposition law for upp...
uzssz 11583 An upper set of integers i...
uzss 11584 Subset relationship for tw...
uztric 11585 Totality of the ordering r...
uz11 11586 The upper integers functio...
eluzp1m1 11587 Membership in the next upp...
eluzp1l 11588 Strict ordering implied by...
eluzp1p1 11589 Membership in the next upp...
eluzaddi 11590 Membership in a later uppe...
eluzsubi 11591 Membership in an earlier u...
eluzadd 11592 Membership in a later uppe...
eluzsub 11593 Membership in an earlier u...
uzm1 11594 Choices for an element of ...
uznn0sub 11595 The nonnegative difference...
uzin 11596 Intersection of two upper ...
uzp1 11597 Choices for an element of ...
nn0uz 11598 Nonnegative integers expre...
nnuz 11599 Positive integers expresse...
elnnuz 11600 A positive integer express...
elnn0uz 11601 A nonnegative integer expr...
eluz2nn 11602 An integer is greater than...
eluzge2nn0 11603 If an integer is greater t...
eluz2n0 11604 An integer greater than or...
uzuzle23 11605 An integer in the upper se...
eluzge3nn 11606 If an integer is greater t...
uz3m2nn 11607 An integer greater than or...
1eluzge0 11608 1 is an integer greater th...
2eluzge0 11609 2 is an integer greater th...
2eluzge1 11610 2 is an integer greater th...
uznnssnn 11611 The upper integers startin...
raluz 11612 Restricted universal quant...
raluz2 11613 Restricted universal quant...
rexuz 11614 Restricted existential qua...
rexuz2 11615 Restricted existential qua...
2rexuz 11616 Double existential quantif...
peano2uz 11617 Second Peano postulate for...
peano2uzs 11618 Second Peano postulate for...
peano2uzr 11619 Reversed second Peano axio...
uzaddcl 11620 Addition closure law for a...
nn0pzuz 11621 The sum of a nonnegative i...
uzind4 11622 Induction on the upper set...
uzind4ALT 11623 Induction on the upper set...
uzind4s 11624 Induction on the upper set...
uzind4s2 11625 Induction on the upper set...
uzind4i 11626 Induction on the upper int...
uzwo 11627 Well-ordering principle: a...
uzwo2 11628 Well-ordering principle: a...
nnwo 11629 Well-ordering principle: a...
nnwof 11630 Well-ordering principle: a...
nnwos 11631 Well-ordering principle: a...
indstr 11632 Strong Mathematical Induct...
eluznn0 11633 Membership in a nonnegativ...
eluznn 11634 Membership in a positive u...
eluz2b1 11635 Two ways to say "an intege...
eluz2gt1 11636 An integer greater than or...
eluz2b2 11637 Two ways to say "an intege...
eluz2b3 11638 Two ways to say "an intege...
uz2m1nn 11639 One less than an integer g...
1nuz2 11640 1 is not in ` ( ZZ>= `` 2 ...
elnn1uz2 11641 A positive integer is eith...
uz2mulcl 11642 Closure of multiplication ...
indstr2 11643 Strong Mathematical Induct...
uzinfi 11644 Extract the lower bound of...
nninf 11645 The infimum of the set of ...
nn0inf 11646 The infimum of the set of ...
infssuzle 11647 The infimum of a subset of...
infssuzcl 11648 The infimum of a subset of...
ublbneg 11649 The image under negation o...
eqreznegel 11650 Two ways to express the im...
supminf 11651 The supremum of a bounded-...
lbzbi 11652 If a set of reals is bound...
zsupss 11653 Any nonempty bounded subse...
suprzcl2 11654 The supremum of a bounded-...
suprzub 11655 The supremum of a bounded-...
uzsupss 11656 Any bounded subset of an u...
nn01to3 11657 A (nonnegative) integer be...
nn0ge2m1nnALT 11658 Alternate proof of ~ nn0ge...
uzwo3 11659 Well-ordering principle: a...
zmin 11660 There is a unique smallest...
zmax 11661 There is a unique largest ...
zbtwnre 11662 There is a unique integer ...
rebtwnz 11663 There is a unique greatest...
elq 11666 Membership in the set of r...
qmulz 11667 If ` A ` is rational, then...
znq 11668 The ratio of an integer an...
qre 11669 A rational number is a rea...
zq 11670 An integer is a rational n...
zssq 11671 The integers are a subset ...
nn0ssq 11672 The nonnegative integers a...
nnssq 11673 The positive integers are ...
qssre 11674 The rationals are a subset...
qsscn 11675 The rationals are a subset...
qex 11676 The set of rational number...
nnq 11677 A positive integer is rati...
qcn 11678 A rational number is a com...
qexALT 11679 Alternate proof of ~ qex ....
qaddcl 11680 Closure of addition of rat...
qnegcl 11681 Closure law for the negati...
qmulcl 11682 Closure of multiplication ...
qsubcl 11683 Closure of subtraction of ...
qreccl 11684 Closure of reciprocal of r...
qdivcl 11685 Closure of division of rat...
qrevaddcl 11686 Reverse closure law for ad...
nnrecq 11687 The reciprocal of a positi...
irradd 11688 The sum of an irrational n...
irrmul 11689 The product of an irration...
rpnnen1lem2 11690 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem1 11691 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem3 11692 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem4 11693 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem5 11694 Lemma for ~ rpnnen1 . (Co...
rpnnen1lem6 11695 Lemma for ~ rpnnen1 . (Co...
rpnnen1 11696 One half of ~ rpnnen , whe...
rpnnen1lem1OLD 11697 Lemma for ~ rpnnen1OLD . ...
rpnnen1lem3OLD 11698 Lemma for ~ rpnnen1OLD . ...
rpnnen1lem4OLD 11699 Lemma for ~ rpnnen1OLD . ...
rpnnen1lem5OLD 11700 Lemma for ~ rpnnen1OLD . ...
rpnnen1OLD 11701 One half of ~ rpnnen , whe...
reexALT 11702 Alternate proof of ~ reex ...
cnref1o 11703 There is a natural one-to-...
cnexALT 11704 The set of complex numbers...
xrex 11705 The set of extended reals ...
addex 11706 The addition operation is ...
mulex 11707 The multiplication operati...
elrp 11710 Membership in the set of p...
elrpii 11711 Membership in the set of p...
1rp 11712 1 is a positive real. (Co...
2rp 11713 2 is a positive real. (Co...
3rp 11714 3 is a positive real. (Co...
rpre 11715 A positive real is a real....
rpxr 11716 A positive real is an exte...
rpcn 11717 A positive real is a compl...
nnrp 11718 A positive integer is a po...
rpssre 11719 The positive reals are a s...
rpgt0 11720 A positive real is greater...
rpge0 11721 A positive real is greater...
rpregt0 11722 A positive real is a posit...
rprege0 11723 A positive real is a nonne...
rpne0 11724 A positive real is nonzero...
rprene0 11725 A positive real is a nonze...
rpcnne0 11726 A positive real is a nonze...
rpcndif0 11727 A positive real number is ...
ralrp 11728 Quantification over positi...
rexrp 11729 Quantification over positi...
rpaddcl 11730 Closure law for addition o...
rpmulcl 11731 Closure law for multiplica...
rpdivcl 11732 Closure law for division o...
rpreccl 11733 Closure law for reciprocat...
rphalfcl 11734 Closure law for half of a ...
rpgecl 11735 A number greater or equal ...
rphalflt 11736 Half of a positive real is...
rerpdivcl 11737 Closure law for division o...
ge0p1rp 11738 A nonnegative number plus ...
rpneg 11739 Either a nonzero real or i...
negelrp 11740 Elementhood of a negation ...
0nrp 11741 Zero is not a positive rea...
ltsubrp 11742 Subtracting a positive rea...
ltaddrp 11743 Adding a positive number t...
difrp 11744 Two ways to say one number...
elrpd 11745 Membership in the set of p...
nnrpd 11746 A positive integer is a po...
zgt1rpn0n1 11747 An integer greater than 1 ...
rpred 11748 A positive real is a real....
rpxrd 11749 A positive real is an exte...
rpcnd 11750 A positive real is a compl...
rpgt0d 11751 A positive real is greater...
rpge0d 11752 A positive real is greater...
rpne0d 11753 A positive real is nonzero...
rpregt0d 11754 A positive real is real an...
rprege0d 11755 A positive real is real an...
rprene0d 11756 A positive real is a nonze...
rpcnne0d 11757 A positive real is a nonze...
rpreccld 11758 Closure law for reciprocat...
rprecred 11759 Closure law for reciprocat...
rphalfcld 11760 Closure law for half of a ...
reclt1d 11761 The reciprocal of a positi...
recgt1d 11762 The reciprocal of a positi...
rpaddcld 11763 Closure law for addition o...
rpmulcld 11764 Closure law for multiplica...
rpdivcld 11765 Closure law for division o...
ltrecd 11766 The reciprocal of both sid...
lerecd 11767 The reciprocal of both sid...
ltrec1d 11768 Reciprocal swap in a 'less...
lerec2d 11769 Reciprocal swap in a 'less...
lediv2ad 11770 Division of both sides of ...
ltdiv2d 11771 Division of a positive num...
lediv2d 11772 Division of a positive num...
ledivdivd 11773 Invert ratios of positive ...
divge1 11774 The ratio of a number over...
divlt1lt 11775 A real number divided by a...
divle1le 11776 A real number divided by a...
ledivge1le 11777 If a number is less than o...
ge0p1rpd 11778 A nonnegative number plus ...
rerpdivcld 11779 Closure law for division o...
ltsubrpd 11780 Subtracting a positive rea...
ltaddrpd 11781 Adding a positive number t...
ltaddrp2d 11782 Adding a positive number t...
ltmulgt11d 11783 Multiplication by a number...
ltmulgt12d 11784 Multiplication by a number...
gt0divd 11785 Division of a positive num...
ge0divd 11786 Division of a nonnegative ...
rpgecld 11787 A number greater or equal ...
divge0d 11788 The ratio of nonnegative a...
ltmul1d 11789 The ratio of nonnegative a...
ltmul2d 11790 Multiplication of both sid...
lemul1d 11791 Multiplication of both sid...
lemul2d 11792 Multiplication of both sid...
ltdiv1d 11793 Division of both sides of ...
lediv1d 11794 Division of both sides of ...
ltmuldivd 11795 'Less than' relationship b...
ltmuldiv2d 11796 'Less than' relationship b...
lemuldivd 11797 'Less than or equal to' re...
lemuldiv2d 11798 'Less than or equal to' re...
ltdivmuld 11799 'Less than' relationship b...
ltdivmul2d 11800 'Less than' relationship b...
ledivmuld 11801 'Less than or equal to' re...
ledivmul2d 11802 'Less than or equal to' re...
ltmul1dd 11803 The ratio of nonnegative a...
ltmul2dd 11804 Multiplication of both sid...
ltdiv1dd 11805 Division of both sides of ...
lediv1dd 11806 Division of both sides of ...
lediv12ad 11807 Comparison of ratio of two...
mul2lt0rlt0 11808 If the result of a multipl...
mul2lt0rgt0 11809 If the result of a multipl...
mul2lt0llt0 11810 If the result of a multipl...
mul2lt0lgt0 11811 If the result of a multipl...
mul2lt0bi 11812 If the result of a multipl...
ltdiv23d 11813 Swap denominator with othe...
lediv23d 11814 Swap denominator with othe...
lt2mul2divd 11815 The ratio of nonnegative a...
nnledivrp 11816 Division of a positive int...
nn0ledivnn 11817 Division of a nonnegative ...
addlelt 11818 If the sum of a real numbe...
ltxr 11825 The 'less than' binary rel...
elxr 11826 Membership in the set of e...
xrnemnf 11827 An extended real other tha...
xrnepnf 11828 An extended real other tha...
xrltnr 11829 The extended real 'less th...
ltpnf 11830 Any (finite) real is less ...
ltpnfd 11831 Any (finite) real is less ...
0ltpnf 11832 Zero is less than plus inf...
mnflt 11833 Minus infinity is less tha...
mnfltd 11834 Minus infinity is less tha...
mnflt0 11835 Minus infinity is less tha...
mnfltpnf 11836 Minus infinity is less tha...
mnfltxr 11837 Minus infinity is less tha...
pnfnlt 11838 No extended real is greate...
nltmnf 11839 No extended real is less t...
pnfge 11840 Plus infinity is an upper ...
xnn0n0n1ge2b 11841 An extended nonnegative in...
0lepnf 11842 0 less than or equal to po...
xnn0ge0 11843 An extended nonnegative in...
nn0pnfge0OLD 11844 Obsolete version of ~ xnn0...
mnfle 11845 Minus infinity is less tha...
xrltnsym 11846 Ordering on the extended r...
xrltnsym2 11847 'Less than' is antisymmetr...
xrlttri 11848 Ordering on the extended r...
xrlttr 11849 Ordering on the extended r...
xrltso 11850 'Less than' is a strict or...
xrlttri2 11851 Trichotomy law for 'less t...
xrlttri3 11852 Trichotomy law for 'less t...
xrleloe 11853 'Less than or equal' expre...
xrleltne 11854 'Less than or equal to' im...
xrltlen 11855 'Less than' expressed in t...
dfle2 11856 Alternative definition of ...
dflt2 11857 Alternative definition of ...
xrltle 11858 'Less than' implies 'less ...
xrleid 11859 'Less than or equal to' is...
xrletri 11860 Trichotomy law for extende...
xrletri3 11861 Trichotomy law for extende...
xrletrid 11862 Trichotomy law for extende...
xrlelttr 11863 Transitive law for orderin...
xrltletr 11864 Transitive law for orderin...
xrletr 11865 Transitive law for orderin...
xrlttrd 11866 Transitive law for orderin...
xrlelttrd 11867 Transitive law for orderin...
xrltletrd 11868 Transitive law for orderin...
xrletrd 11869 Transitive law for orderin...
xrltne 11870 'Less than' implies not eq...
nltpnft 11871 An extended real is not le...
ngtmnft 11872 An extended real is not gr...
xrrebnd 11873 An extended real is real i...
xrre 11874 A way of proving that an e...
xrre2 11875 An extended real between t...
xrre3 11876 A way of proving that an e...
ge0gtmnf 11877 A nonnegative extended rea...
ge0nemnf 11878 A nonnegative extended rea...
xrrege0 11879 A nonnegative extended rea...
xrmax1 11880 An extended real is less t...
xrmax2 11881 An extended real is less t...
xrmin1 11882 The minimum of two extende...
xrmin2 11883 The minimum of two extende...
xrmaxeq 11884 The maximum of two extende...
xrmineq 11885 The minimum of two extende...
xrmaxlt 11886 Two ways of saying the max...
xrltmin 11887 Two ways of saying an exte...
xrmaxle 11888 Two ways of saying the max...
xrlemin 11889 Two ways of saying a numbe...
max1 11890 A number is less than or e...
max1ALT 11891 A number is less than or e...
max2 11892 A number is less than or e...
2resupmax 11893 The supremum of two real n...
min1 11894 The minimum of two numbers...
min2 11895 The minimum of two numbers...
maxle 11896 Two ways of saying the max...
lemin 11897 Two ways of saying a numbe...
maxlt 11898 Two ways of saying the max...
ltmin 11899 Two ways of saying a numbe...
lemaxle 11900 A real number which is les...
max0sub 11901 Decompose a real number in...
ifle 11902 An if statement transforms...
z2ge 11903 There exists an integer gr...
qbtwnre 11904 The rational numbers are d...
qbtwnxr 11905 The rational numbers are d...
qsqueeze 11906 If a nonnegative real is l...
qextltlem 11907 Lemma for ~ qextlt and qex...
qextlt 11908 An extensionality-like pro...
qextle 11909 An extensionality-like pro...
xralrple 11910 Show that ` A ` is less th...
alrple 11911 Show that ` A ` is less th...
xnegeq 11912 Equality of two extended n...
xnegex 11913 A negative extended real e...
xnegpnf 11914 Minus ` +oo ` . Remark of...
xnegmnf 11915 Minus ` -oo ` . Remark of...
rexneg 11916 Minus a real number. Rema...
xneg0 11917 The negative of zero. (Co...
xnegcl 11918 Closure of extended real n...
xnegneg 11919 Extended real version of ~...
xneg11 11920 Extended real version of ~...
xltnegi 11921 Forward direction of ~ xlt...
xltneg 11922 Extended real version of ~...
xleneg 11923 Extended real version of ~...
xlt0neg1 11924 Extended real version of ~...
xlt0neg2 11925 Extended real version of ~...
xle0neg1 11926 Extended real version of ~...
xle0neg2 11927 Extended real version of ~...
xaddval 11928 Value of the extended real...
xaddf 11929 The extended real addition...
xmulval 11930 Value of the extended real...
xaddpnf1 11931 Addition of positive infin...
xaddpnf2 11932 Addition of positive infin...
xaddmnf1 11933 Addition of negative infin...
xaddmnf2 11934 Addition of negative infin...
pnfaddmnf 11935 Addition of positive and n...
mnfaddpnf 11936 Addition of negative and p...
rexadd 11937 The extended real addition...
rexsub 11938 Extended real subtraction ...
rexaddd 11939 The extended real addition...
xnn0xaddcl 11940 The extended nonnegative i...
xaddnemnf 11941 Closure of extended real a...
xaddnepnf 11942 Closure of extended real a...
xnegid 11943 Extended real version of ~...
xaddcl 11944 The extended real addition...
xaddcom 11945 The extended real addition...
xaddid1 11946 Extended real version of ~...
xaddid2 11947 Extended real version of ~...
xaddid1d 11948 ` 0 ` is a right identity ...
xnn0xadd0 11949 The sum of two extended no...
xnegdi 11950 Extended real version of ~...
xaddass 11951 Associativity of extended ...
xaddass2 11952 Associativity of extended ...
xpncan 11953 Extended real version of ~...
xnpcan 11954 Extended real version of ~...
xleadd1a 11955 Extended real version of ~...
xleadd2a 11956 Commuted form of ~ xleadd1...
xleadd1 11957 Weakened version of ~ xlea...
xltadd1 11958 Extended real version of ~...
xltadd2 11959 Extended real version of ~...
xaddge0 11960 The sum of nonnegative ext...
xle2add 11961 Extended real version of ~...
xlt2add 11962 Extended real version of ~...
xsubge0 11963 Extended real version of ~...
xposdif 11964 Extended real version of ~...
xlesubadd 11965 Under certain conditions, ...
xmullem 11966 Lemma for ~ rexmul . (Con...
xmullem2 11967 Lemma for ~ xmulneg1 . (C...
xmulcom 11968 Extended real multiplicati...
xmul01 11969 Extended real version of ~...
xmul02 11970 Extended real version of ~...
xmulneg1 11971 Extended real version of ~...
xmulneg2 11972 Extended real version of ~...
rexmul 11973 The extended real multipli...
xmulf 11974 The extended real multipli...
xmulcl 11975 Closure of extended real m...
xmulpnf1 11976 Multiplication by plus inf...
xmulpnf2 11977 Multiplication by plus inf...
xmulmnf1 11978 Multiplication by minus in...
xmulmnf2 11979 Multiplication by minus in...
xmulpnf1n 11980 Multiplication by plus inf...
xmulid1 11981 Extended real version of ~...
xmulid2 11982 Extended real version of ~...
xmulm1 11983 Extended real version of ~...
xmulasslem2 11984 Lemma for ~ xmulass . (Co...
xmulgt0 11985 Extended real version of ~...
xmulge0 11986 Extended real version of ~...
xmulasslem 11987 Lemma for ~ xmulass . (Co...
xmulasslem3 11988 Lemma for ~ xmulass . (Co...
xmulass 11989 Associativity of the exten...
xlemul1a 11990 Extended real version of ~...
xlemul2a 11991 Extended real version of ~...
xlemul1 11992 Extended real version of ~...
xlemul2 11993 Extended real version of ~...
xltmul1 11994 Extended real version of ~...
xltmul2 11995 Extended real version of ~...
xadddilem 11996 Lemma for ~ xadddi . (Con...
xadddi 11997 Distributive property for ...
xadddir 11998 Commuted version of ~ xadd...
xadddi2 11999 The assumption that the mu...
xadddi2r 12000 Commuted version of ~ xadd...
x2times 12001 Extended real version of ~...
xnegcld 12002 Closure of extended real n...
xaddcld 12003 The extended real addition...
xmulcld 12004 Closure of extended real m...
xadd4d 12005 Rearrangement of 4 terms i...
xnn0add4d 12006 Rearrangement of 4 terms i...
xrsupexmnf 12007 Adding minus infinity to a...
xrinfmexpnf 12008 Adding plus infinity to a ...
xrsupsslem 12009 Lemma for ~ xrsupss . (Co...
xrinfmsslem 12010 Lemma for ~ xrinfmss . (C...
xrsupss 12011 Any subset of extended rea...
xrinfmss 12012 Any subset of extended rea...
xrinfmss2 12013 Any subset of extended rea...
xrub 12014 By quantifying only over r...
supxr 12015 The supremum of a set of e...
supxr2 12016 The supremum of a set of e...
supxrcl 12017 The supremum of an arbitra...
supxrun 12018 The supremum of the union ...
supxrmnf 12019 Adding minus infinity to a...
supxrpnf 12020 The supremum of a set of e...
supxrunb1 12021 The supremum of an unbound...
supxrunb2 12022 The supremum of an unbound...
supxrbnd1 12023 The supremum of a bounded-...
supxrbnd2 12024 The supremum of a bounded-...
xrsup0 12025 The supremum of an empty s...
supxrub 12026 A member of a set of exten...
supxrlub 12027 The supremum of a set of e...
supxrleub 12028 The supremum of a set of e...
supxrre 12029 The real and extended real...
supxrbnd 12030 The supremum of a bounded-...
supxrgtmnf 12031 The supremum of a nonempty...
supxrre1 12032 The supremum of a nonempty...
supxrre2 12033 The supremum of a nonempty...
supxrss 12034 Smaller sets of extended r...
infxrcl 12035 The infimum of an arbitrar...
infxrlb 12036 A member of a set of exten...
infxrgelb 12037 The infimum of a set of ex...
infxrre 12038 The real and extended real...
xrinf0 12039 The infimum of the empty s...
infxrss 12040 Larger sets of extended re...
reltre 12041 For all real numbers there...
rpltrp 12042 For all positive real numb...
reltxrnmnf 12043 For all extended real numb...
infmremnf 12044 The infimum of the reals i...
infmrp1 12045 The infimum of the positiv...
ixxval 12054 Value of the interval func...
elixx1 12055 Membership in an interval ...
ixxf 12056 The set of intervals of ex...
ixxex 12057 The set of intervals of ex...
ixxssxr 12058 The set of intervals of ex...
elixx3g 12059 Membership in a set of ope...
ixxssixx 12060 An interval is a subset of...
ixxdisj 12061 Split an interval into dis...
ixxun 12062 Split an interval into two...
ixxin 12063 Intersection of two interv...
ixxss1 12064 Subset relationship for in...
ixxss2 12065 Subset relationship for in...
ixxss12 12066 Subset relationship for in...
ixxub 12067 Extract the upper bound of...
ixxlb 12068 Extract the lower bound of...
iooex 12069 The set of open intervals ...
iooval 12070 Value of the open interval...
ioo0 12071 An empty open interval of ...
ioon0 12072 An open interval of extend...
ndmioo 12073 The open interval function...
iooid 12074 An open interval with iden...
elioo3g 12075 Membership in a set of ope...
elioore 12076 A member of an open interv...
lbioo 12077 An open interval does not ...
ubioo 12078 An open interval does not ...
iooval2 12079 Value of the open interval...
iooin 12080 Intersection of two open i...
iooss1 12081 Subset relationship for op...
iooss2 12082 Subset relationship for op...
iocval 12083 Value of the open-below, c...
icoval 12084 Value of the closed-below,...
iccval 12085 Value of the closed interv...
elioo1 12086 Membership in an open inte...
elioo2 12087 Membership in an open inte...
elioc1 12088 Membership in an open-belo...
elico1 12089 Membership in a closed-bel...
elicc1 12090 Membership in a closed int...
iccid 12091 A closed interval with ide...
ico0 12092 An empty open interval of ...
ioc0 12093 An empty open interval of ...
icc0 12094 An empty closed interval o...
elicod 12095 Membership in a left close...
icogelb 12096 An element of a left close...
elicore 12097 A member of a left closed,...
ubioc1 12098 The upper bound belongs to...
lbico1 12099 The lower bound belongs to...
iccleub 12100 An element of a closed int...
iccgelb 12101 An element of a closed int...
elioo5 12102 Membership in an open inte...
eliooxr 12103 A nonempty open interval s...
eliooord 12104 Ordering implied by a memb...
elioo4g 12105 Membership in an open inte...
ioossre 12106 An open interval is a set ...
elioc2 12107 Membership in an open-belo...
elico2 12108 Membership in a closed-bel...
elicc2 12109 Membership in a closed rea...
elicc2i 12110 Inference for membership i...
elicc4 12111 Membership in a closed rea...
iccss 12112 Condition for a closed int...
iccssioo 12113 Condition for a closed int...
icossico 12114 Condition for a closed-bel...
iccss2 12115 Condition for a closed int...
iccssico 12116 Condition for a closed int...
iccssioo2 12117 Condition for a closed int...
iccssico2 12118 Condition for a closed int...
ioomax 12119 The open interval from min...
iccmax 12120 The closed interval from m...
ioopos 12121 The set of positive reals ...
ioorp 12122 The set of positive reals ...
iooshf 12123 Shift the arguments of the...
iocssre 12124 A closed-above interval wi...
icossre 12125 A closed-below interval wi...
iccssre 12126 A closed real interval is ...
iccssxr 12127 A closed interval is a set...
iocssxr 12128 An open-below, closed-abov...
icossxr 12129 A closed-below, open-above...
ioossicc 12130 An open interval is a subs...
icossicc 12131 A closed-below, open-above...
iocssicc 12132 A closed-above, open-below...
ioossico 12133 An open interval is a subs...
iocssioo 12134 Condition for a closed int...
icossioo 12135 Condition for a closed int...
ioossioo 12136 Condition for an open inte...
iccsupr 12137 A nonempty subset of a clo...
elioopnf 12138 Membership in an unbounded...
elioomnf 12139 Membership in an unbounded...
elicopnf 12140 Membership in a closed unb...
repos 12141 Two ways of saying that a ...
ioof 12142 The set of open intervals ...
iccf 12143 The set of closed interval...
unirnioo 12144 The union of the range of ...
dfioo2 12145 Alternate definition of th...
ioorebas 12146 Open intervals are element...
xrge0neqmnf 12147 An extended nonnegative re...
xrge0nre 12148 An extended real which is ...
elrege0 12149 The predicate "is a nonneg...
nn0rp0 12150 A nonnegative integer is a...
rge0ssre 12151 Nonnegative real numbers a...
elxrge0 12152 Elementhood in the set of ...
0e0icopnf 12153 0 is a member of ` ( 0 [,)...
0e0iccpnf 12154 0 is a member of ` ( 0 [,]...
ge0addcl 12155 The nonnegative reals are ...
ge0mulcl 12156 The nonnegative reals are ...
ge0xaddcl 12157 The nonnegative reals are ...
ge0xmulcl 12158 The nonnegative extended r...
lbicc2 12159 The lower bound of a close...
ubicc2 12160 The upper bound of a close...
0elunit 12161 Zero is an element of the ...
1elunit 12162 One is an element of the c...
iooneg 12163 Membership in a negated op...
iccneg 12164 Membership in a negated cl...
icoshft 12165 A shifted real is a member...
icoshftf1o 12166 Shifting a closed-below, o...
icoun 12167 The union of end-to-end cl...
icodisj 12168 End-to-end closed-below, o...
snunioo 12169 The closure of one end of ...
snunico 12170 The closure of the open en...
snunioc 12171 The closure of the open en...
prunioo 12172 The closure of an open rea...
ioodisj 12173 If the upper bound of one ...
ioojoin 12174 Join two open intervals to...
difreicc 12175 The class difference of ` ...
iccsplit 12176 Split a closed interval in...
iccshftr 12177 Membership in a shifted in...
iccshftri 12178 Membership in a shifted in...
iccshftl 12179 Membership in a shifted in...
iccshftli 12180 Membership in a shifted in...
iccdil 12181 Membership in a dilated in...
iccdili 12182 Membership in a dilated in...
icccntr 12183 Membership in a contracted...
icccntri 12184 Membership in a contracted...
divelunit 12185 A condition for a ratio to...
lincmb01cmp 12186 A linear combination of tw...
iccf1o 12187 Describe a bijection from ...
iccen 12188 Any nontrivial closed inte...
xov1plusxeqvd 12189 A complex number ` X ` is ...
unitssre 12190 ` ( 0 [,] 1 ) ` is a subse...
supicc 12191 Supremum of a bounded set ...
supiccub 12192 The supremum of a bounded ...
supicclub 12193 The supremum of a bounded ...
supicclub2 12194 The supremum of a bounded ...
zltaddlt1le 12195 The sum of an integer and ...
xnn0xrge0 12196 An extended nonnegative in...
fzval 12199 The value of a finite set ...
fzval2 12200 An alternative way of expr...
fzf 12201 Establish the domain and c...
elfz1 12202 Membership in a finite set...
elfz 12203 Membership in a finite set...
elfz2 12204 Membership in a finite set...
elfz5 12205 Membership in a finite set...
elfz4 12206 Membership in a finite set...
elfzuzb 12207 Membership in a finite set...
eluzfz 12208 Membership in a finite set...
elfzuz 12209 A member of a finite set o...
elfzuz3 12210 Membership in a finite set...
elfzel2 12211 Membership in a finite set...
elfzel1 12212 Membership in a finite set...
elfzelz 12213 A member of a finite set o...
fzssz 12214 A finite sequence of integ...
elfzle1 12215 A member of a finite set o...
elfzle2 12216 A member of a finite set o...
elfzuz2 12217 Implication of membership ...
elfzle3 12218 Membership in a finite set...
eluzfz1 12219 Membership in a finite set...
eluzfz2 12220 Membership in a finite set...
eluzfz2b 12221 Membership in a finite set...
elfz3 12222 Membership in a finite set...
elfz1eq 12223 Membership in a finite set...
elfzubelfz 12224 If there is a member in a ...
peano2fzr 12225 A Peano-postulate-like the...
fzn0 12226 Properties of a finite int...
fz0 12227 A finite set of sequential...
fzn 12228 A finite set of sequential...
fzen 12229 A shifted finite set of se...
fz1n 12230 A 1-based finite set of se...
0nelfz1 12231 0 is not an element of a f...
0fz1 12232 Two ways to say a finite 1...
fz10 12233 There are no integers betw...
uzsubsubfz 12234 Membership of an integer g...
uzsubsubfz1 12235 Membership of an integer g...
ige3m2fz 12236 Membership of an integer g...
fzsplit2 12237 Split a finite interval of...
fzsplit 12238 Split a finite interval of...
fzdisj 12239 Condition for two finite i...
fz01en 12240 0-based and 1-based finite...
elfznn 12241 A member of a finite set o...
elfz1end 12242 A nonempty finite range of...
fz1ssnn 12243 A finite set of positive i...
fznn0sub 12244 Subtraction closure for a ...
fzmmmeqm 12245 Subtracting the difference...
fzaddel 12246 Membership of a sum in a f...
fzadd2 12247 Membership of a sum in a f...
fzsubel 12248 Membership of a difference...
fzopth 12249 A finite set of sequential...
fzass4 12250 Two ways to express a nond...
fzss1 12251 Subset relationship for fi...
fzss2 12252 Subset relationship for fi...
fzssuz 12253 A finite set of sequential...
fzsn 12254 A finite interval of integ...
fzssp1 12255 Subset relationship for fi...
fzssnn 12256 Finite sets of sequential ...
ssfzunsn 12257 A subset of a finite seque...
fzsuc 12258 Join a successor to the en...
fzpred 12259 Join a predecessor to the ...
fzpreddisj 12260 A finite set of sequential...
elfzp1 12261 Append an element to a fin...
fzp1ss 12262 Subset relationship for fi...
fzelp1 12263 Membership in a set of seq...
fzp1elp1 12264 Add one to an element of a...
fznatpl1 12265 Shift membership in a fini...
fzpr 12266 A finite interval of integ...
fztp 12267 A finite interval of integ...
fzsuc2 12268 Join a successor to the en...
fzp1disj 12269 ` ( M ... ( N + 1 ) ) ` is...
fzdifsuc 12270 Remove a successor from th...
fzprval 12271 Two ways of defining the f...
fztpval 12272 Two ways of defining the f...
fzrev 12273 Reversal of start and end ...
fzrev2 12274 Reversal of start and end ...
fzrev2i 12275 Reversal of start and end ...
fzrev3 12276 The "complement" of a memb...
fzrev3i 12277 The "complement" of a memb...
fznn 12278 Finite set of sequential i...
elfz1b 12279 Membership in a 1 based fi...
elfzm11 12280 Membership in a finite set...
uzsplit 12281 Express an upper integer s...
uzdisj 12282 The first ` N ` elements o...
fseq1p1m1 12283 Add/remove an item to/from...
fseq1m1p1 12284 Add/remove an item to/from...
fz1sbc 12285 Quantification over a one-...
elfzp1b 12286 An integer is a member of ...
elfzm1b 12287 An integer is a member of ...
elfzp12 12288 Options for membership in ...
fzm1 12289 Choices for an element of ...
fzneuz 12290 No finite set of sequentia...
fznuz 12291 Disjointness of the upper ...
uznfz 12292 Disjointness of the upper ...
fzp1nel 12293 One plus the upper bound o...
fzrevral 12294 Reversal of scanning order...
fzrevral2 12295 Reversal of scanning order...
fzrevral3 12296 Reversal of scanning order...
fzshftral 12297 Shift the scanning order i...
ige2m1fz1 12298 Membership of an integer g...
ige2m1fz 12299 Membership in a 0 based fi...
elfz2nn0 12300 Membership in a finite set...
fznn0 12301 Characterization of a fini...
elfznn0 12302 A member of a finite set o...
elfz3nn0 12303 The upper bound of a nonem...
fz0ssnn0 12304 Finite sets of sequential ...
0elfz 12305 0 is an element of a finit...
nn0fz0 12306 A nonnegative integer is a...
elfz0add 12307 An element of a finite set...
fz0sn 12308 An integer range from 0 to...
fz0tp 12309 An integer range from 0 to...
fz0to3un2pr 12310 An integer range from 0 to...
fz0to4untppr 12311 An integer range from 0 to...
elfz0ubfz0 12312 An element of a finite set...
elfz0fzfz0 12313 A member of a finite set o...
fz0fzelfz0 12314 If a member of a finite se...
fznn0sub2 12315 Subtraction closure for a ...
uzsubfz0 12316 Membership of an integer g...
fz0fzdiffz0 12317 The difference of an integ...
elfzmlbm 12318 Subtracting the lower boun...
elfzmlbp 12319 Subtracting the lower boun...
fzctr 12320 Lemma for theorems about t...
difelfzle 12321 The difference of two inte...
difelfznle 12322 The difference of two inte...
nn0split 12323 Express the set of nonnega...
nn0disj 12324 The first ` N + 1 ` elemen...
fz0sn0fz1 12325 A finite set of sequential...
fvffz0 12326 The function value of a fu...
1fv 12327 A one value function. (Co...
4fvwrd4 12328 The first four function va...
2ffzeq 12329 Two functions over 0 based...
preduz 12330 The value of the predecess...
prednn 12331 The value of the predecess...
prednn0 12332 The value of the predecess...
predfz 12333 Calculate the predecessor ...
fzof 12336 Functionality of the half-...
elfzoel1 12337 Reverse closure for half-o...
elfzoel2 12338 Reverse closure for half-o...
elfzoelz 12339 Reverse closure for half-o...
fzoval 12340 Value of the half-open int...
elfzo 12341 Membership in a half-open ...
elfzo2 12342 Membership in a half-open ...
elfzouz 12343 Membership in a half-open ...
nelfzo 12344 An integer not being a mem...
fzolb 12345 The left endpoint of a hal...
fzolb2 12346 The left endpoint of a hal...
elfzole1 12347 A member in a half-open in...
elfzolt2 12348 A member in a half-open in...
elfzolt3 12349 Membership in a half-open ...
elfzolt2b 12350 A member in a half-open in...
elfzolt3b 12351 Membership in a half-open ...
fzonel 12352 A half-open range does not...
elfzouz2 12353 The upper bound of a half-...
elfzofz 12354 A half-open range is conta...
elfzo3 12355 Express membership in a ha...
fzon0 12356 A half-open integer interv...
fzossfz 12357 A half-open range is conta...
fzon 12358 A half-open set of sequent...
fzo0n 12359 A half-open range of nonne...
fzonlt0 12360 A half-open integer range ...
fzo0 12361 Half-open sets with equal ...
fzonnsub 12362 If ` K < N ` then ` N - K ...
fzonnsub2 12363 If ` M < N ` then ` N - M ...
fzoss1 12364 Subset relationship for ha...
fzoss2 12365 Subset relationship for ha...
fzossrbm1 12366 Subset of a half open rang...
fzo0ss1 12367 Subset relationship for ha...
fzossnn0 12368 A half-open integer range ...
fzospliti 12369 One direction of splitting...
fzosplit 12370 Split a half-open integer ...
fzodisj 12371 Abutting half-open integer...
fzouzsplit 12372 Split an upper integer set...
fzouzdisj 12373 A half-open integer range ...
fzodisjsn 12374 A half-open integer range ...
lbfzo0 12375 An integer is strictly gre...
elfzo0 12376 Membership in a half-open ...
elfzo0z 12377 Membership in a half-open ...
nn0p1elfzo 12378 A nonnegative integer incr...
elfzo0le 12379 A member in a half-open ra...
elfzonn0 12380 A member of a half-open ra...
fzonmapblen 12381 The result of subtracting ...
fzofzim 12382 If a nonnegative integer i...
fz1fzo0m1 12383 Translation of one between...
fzossnn 12384 Half-open integer ranges s...
elfzo1 12385 Membership in a half-open ...
fzo1fzo0n0 12386 An integer between 1 and a...
fzo0n0 12387 A half-open integer range ...
fzoaddel 12388 Translate membership in a ...
fzo0addel 12389 Translate membership in a ...
fzo0addelr 12390 Translate membership in a ...
fzoaddel2 12391 Translate membership in a ...
elfzoext 12392 Membership of an integer i...
elincfzoext 12393 Membership of an increased...
fzosubel 12394 Translate membership in a ...
fzosubel2 12395 Membership in a translated...
fzosubel3 12396 Membership in a translated...
eluzgtdifelfzo 12397 Membership of the differen...
ige2m2fzo 12398 Membership of an integer g...
fzocatel 12399 Translate membership in a ...
ubmelfzo 12400 If an integer in a 1 based...
elfzodifsumelfzo 12401 If an integer is in a half...
elfzom1elp1fzo 12402 Membership of an integer i...
elfzom1elfzo 12403 Membership in a half-open ...
fzval3 12404 Expressing a closed intege...
fzosn 12405 Expressing a singleton as ...
elfzomin 12406 Membership of an integer i...
zpnn0elfzo 12407 Membership of an integer i...
zpnn0elfzo1 12408 Membership of an integer i...
fzosplitsnm1 12409 Removing a singleton from ...
elfzonlteqm1 12410 If an element of a half-op...
fzonn0p1 12411 A nonnegative integer is e...
fzossfzop1 12412 A half-open range of nonne...
fzonn0p1p1 12413 If a nonnegative integer i...
elfzom1p1elfzo 12414 Increasing an element of a...
fzo0ssnn0 12415 Half-open integer ranges s...
fzo0ssnn0OLD 12416 Obsolete proof of ~ fzo0ss...
fzo01 12417 Expressing the singleton o...
fzo12sn 12418 A 1-based half-open intege...
fzo13pr 12419 A 1-based half-open intege...
fzo0to2pr 12420 A half-open integer range ...
fzo0to3tp 12421 A half-open integer range ...
fzo0to42pr 12422 A half-open integer range ...
fzo1to4tp 12423 A half-open integer range ...
fzo0sn0fzo1 12424 A half-open range of nonne...
fzoend 12425 The endpoint of a half-ope...
fzo0end 12426 The endpoint of a zero-bas...
ssfzo12 12427 Subset relationship for ha...
ssfzoulel 12428 If a half-open integer ran...
ssfzo12bi 12429 Subset relationship for ha...
ubmelm1fzo 12430 The result of subtracting ...
fzofzp1 12431 If a point is in a half-op...
fzofzp1b 12432 If a point is in a half-op...
elfzom1b 12433 An integer is a member of ...
elfzom1elp1fzo1 12434 Membership of a nonnegativ...
elfzo1elm1fzo0 12435 Membership of a positive i...
elfzonelfzo 12436 If an element of a half-op...
fzonfzoufzol 12437 If an element of a half-op...
elfzomelpfzo 12438 An integer increased by an...
elfznelfzo 12439 A value in a finite set of...
elfznelfzob 12440 A value in a finite set of...
peano2fzor 12441 A Peano-postulate-like the...
fzosplitsn 12442 Extending a half-open rang...
fzosplitprm1 12443 Extending a half-open inte...
fzosplitsni 12444 Membership in a half-open ...
fzisfzounsn 12445 A finite interval of integ...
fzostep1 12446 Two possibilities for a nu...
fzoshftral 12447 Shift the scanning order i...
fzind2 12448 Induction on the integers ...
fvinim0ffz 12449 The function values for th...
injresinjlem 12450 Lemma for ~ injresinj . (...
injresinj 12451 A function whose restricti...
subfzo0 12452 The difference between two...
flval 12457 Value of the floor (greate...
flcl 12458 The floor (greatest intege...
reflcl 12459 The floor (greatest intege...
fllelt 12460 A basic property of the fl...
flcld 12461 The floor (greatest intege...
flle 12462 A basic property of the fl...
flltp1 12463 A basic property of the fl...
fllep1 12464 A basic property of the fl...
fraclt1 12465 The fractional part of a r...
fracle1 12466 The fractional part of a r...
fracge0 12467 The fractional part of a r...
flge 12468 The floor function value i...
fllt 12469 The floor function value i...
flflp1 12470 Move floor function betwee...
flid 12471 An integer is its own floo...
flidm 12472 The floor function is idem...
flidz 12473 A real number equals its f...
flltnz 12474 If A is not an integer, th...
flwordi 12475 Ordering relationship for ...
flword2 12476 Ordering relationship for ...
flval2 12477 An alternate way to define...
flval3 12478 An alternate way to define...
flbi 12479 A condition equivalent to ...
flbi2 12480 A condition equivalent to ...
adddivflid 12481 The floor of a sum of an i...
ico01fl0 12482 The floor of a real number...
flge0nn0 12483 The floor of a number grea...
flge1nn 12484 The floor of a number grea...
fldivnn0 12485 The floor function of a di...
refldivcl 12486 The floor function of a di...
divfl0 12487 The floor of a fraction is...
fladdz 12488 An integer can be moved in...
flzadd 12489 An integer can be moved in...
flmulnn0 12490 Move a nonnegative integer...
btwnzge0 12491 A real bounded between an ...
2tnp1ge0ge0 12492 Two times an integer plus ...
flhalf 12493 Ordering relation for the ...
fldivle 12494 The floor function of a di...
fldivnn0le 12495 The floor function of a di...
flltdivnn0lt 12496 The floor function of a di...
ltdifltdiv 12497 If the dividend of a divis...
fldiv4p1lem1div2 12498 The floor of an integer eq...
fldiv4lem1div2uz2 12499 The floor of an integer gr...
fldiv4lem1div2 12500 The floor of a positive in...
ceilval 12501 The value of the ceiling f...
dfceil2 12502 Alternative definition of ...
ceilval2 12503 The value of the ceiling f...
ceicl 12504 The ceiling function retur...
ceilcl 12505 Closure of the ceiling fun...
ceige 12506 The ceiling of a real numb...
ceilge 12507 The ceiling of a real numb...
ceim1l 12508 One less than the ceiling ...
ceilm1lt 12509 One less than the ceiling ...
ceile 12510 The ceiling of a real numb...
ceille 12511 The ceiling of a real numb...
ceilid 12512 An integer is its own ceil...
ceilidz 12513 A real number equals its c...
flleceil 12514 The floor of a real number...
fleqceilz 12515 A real number is an intege...
quoremz 12516 Quotient and remainder of ...
quoremnn0 12517 Quotient and remainder of ...
quoremnn0ALT 12518 Alternate proof of ~ quore...
intfrac2 12519 Decompose a real into inte...
intfracq 12520 Decompose a rational numbe...
fldiv 12521 Cancellation of the embedd...
fldiv2 12522 Cancellation of an embedde...
fznnfl 12523 Finite set of sequential i...
uzsup 12524 An upper set of integers i...
ioopnfsup 12525 An upper set of reals is u...
icopnfsup 12526 An upper set of reals is u...
rpsup 12527 The positive reals are unb...
resup 12528 The real numbers are unbou...
xrsup 12529 The extended real numbers ...
modval 12532 The value of the modulo op...
modvalr 12533 The value of the modulo op...
modcl 12534 Closure law for the modulo...
flpmodeq 12535 Partition of a division in...
modcld 12536 Closure law for the modulo...
mod0 12537 ` A mod B ` is zero iff ` ...
mulmod0 12538 The product of an integer ...
negmod0 12539 ` A ` is divisible by ` B ...
modge0 12540 The modulo operation is no...
modlt 12541 The modulo operation is le...
modelico 12542 Modular reduction produces...
moddiffl 12543 The modulo operation diffe...
moddifz 12544 The modulo operation diffe...
modfrac 12545 The fractional part of a n...
flmod 12546 The floor function express...
intfrac 12547 Break a number into its in...
zmod10 12548 An integer modulo 1 is 0. ...
zmod1congr 12549 Two arbitrary integers are...
modmulnn 12550 Move a positive integer in...
modvalp1 12551 The value of the modulo op...
zmodcl 12552 Closure law for the modulo...
zmodcld 12553 Closure law for the modulo...
zmodfz 12554 An integer mod ` B ` lies ...
zmodfzo 12555 An integer mod ` B ` lies ...
zmodfzp1 12556 An integer mod ` B ` lies ...
modid 12557 Identity law for modulo. ...
modid0 12558 A positive real number mod...
modid2 12559 Identity law for modulo. ...
zmodid2 12560 Identity law for modulo re...
zmodidfzo 12561 Identity law for modulo re...
zmodidfzoimp 12562 Identity law for modulo re...
0mod 12563 Special case: 0 modulo a p...
1mod 12564 Special case: 1 modulo a r...
modabs 12565 Absorption law for modulo....
modabs2 12566 Absorption law for modulo....
modcyc 12567 The modulo operation is pe...
modcyc2 12568 The modulo operation is pe...
modadd1 12569 Addition property of the m...
modaddabs 12570 Absorption law for modulo....
modaddmod 12571 The sum of a real number m...
muladdmodid 12572 The sum of a positive real...
mulp1mod1 12573 The product of an integer ...
modmuladd 12574 Decomposition of an intege...
modmuladdim 12575 Implication of a decomposi...
modmuladdnn0 12576 Implication of a decomposi...
negmod 12577 The negation of a number m...
m1modnnsub1 12578 Minus one modulo a positiv...
m1modge3gt1 12579 Minus one modulo an intege...
addmodid 12580 The sum of a positive inte...
addmodidr 12581 The sum of a positive inte...
modadd2mod 12582 The sum of a real number m...
modm1p1mod0 12583 If an real number modulo a...
modltm1p1mod 12584 If a real number modulo a ...
modmul1 12585 Multiplication property of...
modmul12d 12586 Multiplication property of...
modnegd 12587 Negation property of the m...
modadd12d 12588 Additive property of the m...
modsub12d 12589 Subtraction property of th...
modsubmod 12590 The difference of a real n...
modsubmodmod 12591 The difference of a real n...
2txmodxeq0 12592 Two times a positive real ...
2submod 12593 If a real number is betwee...
modifeq2int 12594 If a nonnegative integer i...
modaddmodup 12595 The sum of an integer modu...
modaddmodlo 12596 The sum of an integer modu...
modmulmod 12597 The product of a real numb...
modmulmodr 12598 The product of an integer ...
modaddmulmod 12599 The sum of a real number a...
moddi 12600 Distribute multiplication ...
modsubdir 12601 Distribute the modulo oper...
modeqmodmin 12602 A real number equals the d...
modirr 12603 A number modulo an irratio...
modfzo0difsn 12604 For a number within a half...
modsumfzodifsn 12605 The sum of a number within...
modlteq 12606 Two nonnegative integers l...
addmodlteq 12607 Two nonnegative integers l...
om2uz0i 12608 The mapping ` G ` is a one...
om2uzsuci 12609 The value of ` G ` (see ~ ...
om2uzuzi 12610 The value ` G ` (see ~ om2...
om2uzlti 12611 Less-than relation for ` G...
om2uzlt2i 12612 The mapping ` G ` (see ~ o...
om2uzrani 12613 Range of ` G ` (see ~ om2u...
om2uzf1oi 12614 ` G ` (see ~ om2uz0i ) is ...
om2uzisoi 12615 ` G ` (see ~ om2uz0i ) is ...
om2uzoi 12616 An alternative definition ...
om2uzrdg 12617 A helper lemma for the val...
uzrdglem 12618 A helper lemma for the val...
uzrdgfni 12619 The recursive definition g...
uzrdg0i 12620 Initial value of a recursi...
uzrdgsuci 12621 Successor value of a recur...
ltweuz 12622 ` < ` is a well-founded re...
ltwenn 12623 Less than well-orders the ...
ltwefz 12624 Less than well-orders a se...
uzenom 12625 An upper integer set is de...
uzinf 12626 An upper integer set is in...
nnnfi 12627 The set of positive intege...
uzrdgxfr 12628 Transfer the value of the ...
fzennn 12629 The cardinality of a finit...
fzen2 12630 The cardinality of a finit...
cardfz 12631 The cardinality of a finit...
hashgf1o 12632 ` G ` maps ` _om ` one-to-...
fzfi 12633 A finite interval of integ...
fzfid 12634 Commonly used special case...
fzofi 12635 Half-open integer sets are...
fsequb 12636 The values of a finite rea...
fsequb2 12637 The values of a finite rea...
fseqsupcl 12638 The values of a finite rea...
fseqsupubi 12639 The values of a finite rea...
nn0ennn 12640 The nonnegative integers a...
nnenom 12641 The set of positive intege...
nnct 12642 ` NN ` is countable. (Con...
uzindi 12643 Indirect strong induction ...
axdc4uzlem 12644 Lemma for ~ axdc4uz . (Co...
axdc4uz 12645 A version of ~ axdc4 that ...
ssnn0fi 12646 A subset of the nonnegativ...
rabssnn0fi 12647 A subset of the nonnegativ...
uzsinds 12648 Strong (or "total") induct...
nnsinds 12649 Strong (or "total") induct...
nn0sinds 12650 Strong (or "total") induct...
fsuppmapnn0fiublem 12651 Lemma for ~ fsuppmapnn0fiu...
fsuppmapnn0fiub 12652 If all functions of a fini...
fsuppmapnn0fiubOLD 12653 Obsolete proof of ~ fsuppm...
fsuppmapnn0fiubex 12654 If all functions of a fini...
fsuppmapnn0fiub0 12655 If all functions of a fini...
suppssfz 12656 Condition for a function o...
fsuppmapnn0ub 12657 If a function over the non...
fsuppmapnn0fz 12658 If a function over the non...
mptnn0fsupp 12659 A mapping from the nonnega...
mptnn0fsuppd 12660 A mapping from the nonnega...
mptnn0fsuppr 12661 A finitely supported mappi...
f13idfv 12662 A one-to-one function with...
seqex 12665 Existence of the sequence ...
seqeq1 12666 Equality theorem for the s...
seqeq2 12667 Equality theorem for the s...
seqeq3 12668 Equality theorem for the s...
seqeq1d 12669 Equality deduction for the...
seqeq2d 12670 Equality deduction for the...
seqeq3d 12671 Equality deduction for the...
seqeq123d 12672 Equality deduction for the...
nfseq 12673 Hypothesis builder for the...
seqval 12674 Value of the sequence buil...
seqfn 12675 The sequence builder funct...
seq1 12676 Value of the sequence buil...
seq1i 12677 Value of the sequence buil...
seqp1 12678 Value of the sequence buil...
seqp1i 12679 Value of the sequence buil...
seqm1 12680 Value of the sequence buil...
seqcl2 12681 Closure properties of the ...
seqf2 12682 Range of the recursive seq...
seqcl 12683 Closure properties of the ...
seqf 12684 Range of the recursive seq...
seqfveq2 12685 Equality of sequences. (C...
seqfeq2 12686 Equality of sequences. (C...
seqfveq 12687 Equality of sequences. (C...
seqfeq 12688 Equality of sequences. (C...
seqshft2 12689 Shifting the index set of ...
seqres 12690 Restricting its characteri...
serf 12691 An infinite series of comp...
serfre 12692 An infinite series of real...
monoord 12693 Ordering relation for a mo...
monoord2 12694 Ordering relation for a mo...
sermono 12695 The partial sums in an inf...
seqsplit 12696 Split a sequence into two ...
seq1p 12697 Removing the first term fr...
seqcaopr3 12698 Lemma for ~ seqcaopr2 . (...
seqcaopr2 12699 The sum of two infinite se...
seqcaopr 12700 The sum of two infinite se...
seqf1olem2a 12701 Lemma for ~ seqf1o . (Con...
seqf1olem1 12702 Lemma for ~ seqf1o . (Con...
seqf1olem2 12703 Lemma for ~ seqf1o . (Con...
seqf1o 12704 Rearrange a sum via an arb...
seradd 12705 The sum of two infinite se...
sersub 12706 The difference of two infi...
seqid3 12707 A sequence that consists e...
seqid 12708 Discard the first few term...
seqid2 12709 The last few terms of a se...
seqhomo 12710 Apply a homomorphism to a ...
seqz 12711 If the operation ` .+ ` ha...
seqfeq4 12712 Equality of series under d...
seqfeq3 12713 Equality of series under d...
seqdistr 12714 The distributive property ...
ser0 12715 The value of the partial s...
ser0f 12716 A zero-valued infinite ser...
serge0 12717 A finite sum of nonnegativ...
serle 12718 Comparison of partial sums...
ser1const 12719 Value of the partial serie...
seqof 12720 Distribute function operat...
seqof2 12721 Distribute function operat...
expval 12724 Value of exponentiation to...
expnnval 12725 Value of exponentiation to...
exp0 12726 Value of a complex number ...
0exp0e1 12727 ` 0 ^ 0 = 1 ` (common case...
exp1 12728 Value of a complex number ...
expp1 12729 Value of a complex number ...
expneg 12730 Value of a complex number ...
expneg2 12731 Value of a complex number ...
expn1 12732 A number to the negative o...
expcllem 12733 Lemma for proving nonnegat...
expcl2lem 12734 Lemma for proving integer ...
nnexpcl 12735 Closure of exponentiation ...
nn0expcl 12736 Closure of exponentiation ...
zexpcl 12737 Closure of exponentiation ...
qexpcl 12738 Closure of exponentiation ...
reexpcl 12739 Closure of exponentiation ...
expcl 12740 Closure law for nonnegativ...
rpexpcl 12741 Closure law for exponentia...
reexpclz 12742 Closure of exponentiation ...
qexpclz 12743 Closure of exponentiation ...
m1expcl2 12744 Closure of exponentiation ...
m1expcl 12745 Closure of exponentiation ...
expclzlem 12746 Closure law for integer ex...
expclz 12747 Closure law for integer ex...
nn0expcli 12748 Closure of exponentiation ...
nn0sqcl 12749 The square of a nonnegativ...
expm1t 12750 Exponentiation in terms of...
1exp 12751 Value of one raised to a n...
expeq0 12752 Positive integer exponenti...
expne0 12753 Positive integer exponenti...
expne0i 12754 Nonnegative integer expone...
expgt0 12755 Nonnegative integer expone...
expnegz 12756 Value of a complex number ...
0exp 12757 Value of zero raised to a ...
expge0 12758 Nonnegative integer expone...
expge1 12759 Nonnegative integer expone...
expgt1 12760 Positive integer exponenti...
mulexp 12761 Positive integer exponenti...
mulexpz 12762 Integer exponentiation of ...
exprec 12763 Nonnegative integer expone...
expadd 12764 Sum of exponents law for n...
expaddzlem 12765 Lemma for ~ expaddz . (Co...
expaddz 12766 Sum of exponents law for i...
expmul 12767 Product of exponents law f...
expmulz 12768 Product of exponents law f...
m1expeven 12769 Exponentiation of negative...
expsub 12770 Exponent subtraction law f...
expp1z 12771 Value of a nonzero complex...
expm1 12772 Value of a complex number ...
expdiv 12773 Nonnegative integer expone...
ltexp2a 12774 Ordering relationship for ...
expcan 12775 Cancellation law for expon...
ltexp2 12776 Ordering law for exponenti...
leexp2 12777 Ordering law for exponenti...
leexp2a 12778 Weak ordering relationship...
ltexp2r 12779 The power of a positive nu...
leexp2r 12780 Weak ordering relationship...
leexp1a 12781 Weak mantissa ordering rel...
exple1 12782 Nonnegative integer expone...
expubnd 12783 An upper bound on ` A ^ N ...
sqval 12784 Value of the square of a c...
sqneg 12785 The square of the negative...
sqsubswap 12786 Swap the order of subtract...
sqcl 12787 Closure of square. (Contr...
sqmul 12788 Distribution of square ove...
sqeq0 12789 A number is zero iff its s...
sqdiv 12790 Distribution of square ove...
sqdivid 12791 The square of a nonzero nu...
sqne0 12792 A number is nonzero iff it...
resqcl 12793 Closure of the square of a...
sqgt0 12794 The square of a nonzero re...
nnsqcl 12795 The naturals are closed un...
zsqcl 12796 Integers are closed under ...
qsqcl 12797 The square of a rational i...
sq11 12798 The square function is one...
lt2sq 12799 The square function on non...
le2sq 12800 The square function on non...
le2sq2 12801 The square of a 'less than...
sqge0 12802 A square of a real is nonn...
zsqcl2 12803 The square of an integer i...
sumsqeq0 12804 Two real numbers are equal...
sqvali 12805 Value of square. Inferenc...
sqcli 12806 Closure of square. (Contr...
sqeq0i 12807 A number is zero iff its s...
sqrecii 12808 Square of reciprocal. (Co...
sqmuli 12809 Distribution of square ove...
sqdivi 12810 Distribution of square ove...
resqcli 12811 Closure of square in reals...
sqgt0i 12812 The square of a nonzero re...
sqge0i 12813 A square of a real is nonn...
lt2sqi 12814 The square function on non...
le2sqi 12815 The square function on non...
sq11i 12816 The square function is one...
sq0 12817 The square of 0 is 0. (Co...
sq0i 12818 If a number is zero, its s...
sq0id 12819 If a number is zero, its s...
sq1 12820 The square of 1 is 1. (Co...
neg1sqe1 12821 ` -u 1 ` squared is 1 (com...
sq2 12822 The square of 2 is 4. (Co...
sq3 12823 The square of 3 is 9. (Co...
sq4e2t8 12824 The square of 4 is 2 times...
cu2 12825 The cube of 2 is 8. (Cont...
irec 12826 The reciprocal of ` _i ` ....
i2 12827 ` _i ` squared. (Contribu...
i3 12828 ` _i ` cubed. (Contribute...
i4 12829 ` _i ` to the fourth power...
nnlesq 12830 A positive integer is less...
iexpcyc 12831 Taking ` _i ` to the ` K `...
expnass 12832 A counterexample showing t...
sqlecan 12833 Cancel one factor of a squ...
subsq 12834 Factor the difference of t...
subsq2 12835 Express the difference of ...
binom2i 12836 The square of a binomial. ...
subsqi 12837 Factor the difference of t...
sqeqori 12838 The squares of two complex...
subsq0i 12839 The two solutions to the d...
sqeqor 12840 The squares of two complex...
binom2 12841 The square of a binomial. ...
binom21 12842 Special case of ~ binom2 w...
binom2sub 12843 Expand the square of a sub...
binom2sub1 12844 Special case of ~ binom2su...
binom2subi 12845 Expand the square of a sub...
mulbinom2 12846 The square of a binomial w...
binom3 12847 The cube of a binomial. (...
sq01 12848 If a complex number equals...
zesq 12849 An integer is even iff its...
nnesq 12850 A positive integer is even...
crreczi 12851 Reciprocal of a complex nu...
bernneq 12852 Bernoulli's inequality, du...
bernneq2 12853 Variation of Bernoulli's i...
bernneq3 12854 A corollary of ~ bernneq ....
expnbnd 12855 Exponentiation with a mant...
expnlbnd 12856 The reciprocal of exponent...
expnlbnd2 12857 The reciprocal of exponent...
expmulnbnd 12858 Exponentiation with a mant...
digit2 12859 Two ways to express the ` ...
digit1 12860 Two ways to express the ` ...
modexp 12861 Exponentiation property of...
discr1 12862 A nonnegative quadratic fo...
discr 12863 If a quadratic polynomial ...
exp0d 12864 Value of a complex number ...
exp1d 12865 Value of a complex number ...
expeq0d 12866 Positive integer exponenti...
sqvald 12867 Value of square. Inferenc...
sqcld 12868 Closure of square. (Contr...
sqeq0d 12869 A number is zero iff its s...
expcld 12870 Closure law for nonnegativ...
expp1d 12871 Value of a complex number ...
expaddd 12872 Sum of exponents law for n...
expmuld 12873 Product of exponents law f...
sqrecd 12874 Square of reciprocal. (Co...
expclzd 12875 Closure law for integer ex...
expne0d 12876 Nonnegative integer expone...
expnegd 12877 Value of a complex number ...
exprecd 12878 Nonnegative integer expone...
expp1zd 12879 Value of a nonzero complex...
expm1d 12880 Value of a complex number ...
expsubd 12881 Exponent subtraction law f...
sqmuld 12882 Distribution of square ove...
sqdivd 12883 Distribution of square ove...
expdivd 12884 Nonnegative integer expone...
mulexpd 12885 Positive integer exponenti...
0expd 12886 Value of zero raised to a ...
reexpcld 12887 Closure of exponentiation ...
expge0d 12888 Nonnegative integer expone...
expge1d 12889 Nonnegative integer expone...
sqoddm1div8 12890 A squared odd number minus...
nnsqcld 12891 The naturals are closed un...
nnexpcld 12892 Closure of exponentiation ...
nn0expcld 12893 Closure of exponentiation ...
rpexpcld 12894 Closure law for exponentia...
ltexp2rd 12895 The power of a positive nu...
reexpclzd 12896 Closure of exponentiation ...
resqcld 12897 Closure of square in reals...
sqge0d 12898 A square of a real is nonn...
sqgt0d 12899 The square of a nonzero re...
ltexp2d 12900 Ordering relationship for ...
leexp2d 12901 Ordering law for exponenti...
expcand 12902 Ordering relationship for ...
leexp2ad 12903 Ordering relationship for ...
leexp2rd 12904 Ordering relationship for ...
lt2sqd 12905 The square function on non...
le2sqd 12906 The square function on non...
sq11d 12907 The square function is one...
mulsubdivbinom2 12908 The square of a binomial w...
muldivbinom2 12909 The square of a binomial w...
sq10 12910 The square of 10 is 100. ...
sq10e99m1 12911 The square of 10 is 99 plu...
3dec 12912 A "decimal constructor" wh...
sq10OLD 12913 Old version of ~ sq10 . O...
sq10e99m1OLD 12914 Old version of ~ sq10e99m1...
3decOLD 12915 Old version of ~ 3dec . O...
nn0le2msqi 12916 The square function on non...
nn0opthlem1 12917 A rather pretty lemma for ...
nn0opthlem2 12918 Lemma for ~ nn0opthi . (C...
nn0opthi 12919 An ordered pair theorem fo...
nn0opth2i 12920 An ordered pair theorem fo...
nn0opth2 12921 An ordered pair theorem fo...
facnn 12924 Value of the factorial fun...
fac0 12925 The factorial of 0. (Cont...
fac1 12926 The factorial of 1. (Cont...
facp1 12927 The factorial of a success...
fac2 12928 The factorial of 2. (Cont...
fac3 12929 The factorial of 3. (Cont...
fac4 12930 The factorial of 4. (Cont...
facnn2 12931 Value of the factorial fun...
faccl 12932 Closure of the factorial f...
faccld 12933 Closure of the factorial f...
facmapnn 12934 The factorial function res...
facne0 12935 The factorial function is ...
facdiv 12936 A positive integer divides...
facndiv 12937 No positive integer (great...
facwordi 12938 Ordering property of facto...
faclbnd 12939 A lower bound for the fact...
faclbnd2 12940 A lower bound for the fact...
faclbnd3 12941 A lower bound for the fact...
faclbnd4lem1 12942 Lemma for ~ faclbnd4 . Pr...
faclbnd4lem2 12943 Lemma for ~ faclbnd4 . Us...
faclbnd4lem3 12944 Lemma for ~ faclbnd4 . Th...
faclbnd4lem4 12945 Lemma for ~ faclbnd4 . Pr...
faclbnd4 12946 Variant of ~ faclbnd5 prov...
faclbnd5 12947 The factorial function gro...
faclbnd6 12948 Geometric lower bound for ...
facubnd 12949 An upper bound for the fac...
facavg 12950 The product of two factori...
bcval 12953 Value of the binomial coef...
bcval2 12954 Value of the binomial coef...
bcval3 12955 Value of the binomial coef...
bcval4 12956 Value of the binomial coef...
bcrpcl 12957 Closure of the binomial co...
bccmpl 12958 "Complementing" its second...
bcn0 12959 ` N ` choose 0 is 1. Rema...
bc0k 12960 The binomial coefficient "...
bcnn 12961 ` N ` choose ` N ` is 1. ...
bcn1 12962 Binomial coefficient: ` N ...
bcnp1n 12963 Binomial coefficient: ` N ...
bcm1k 12964 The proportion of one bino...
bcp1n 12965 The proportion of one bino...
bcp1nk 12966 The proportion of one bino...
bcval5 12967 Write out the top and bott...
bcn2 12968 Binomial coefficient: ` N ...
bcp1m1 12969 Compute the binomial coeff...
bcpasc 12970 Pascal's rule for the bino...
bccl 12971 A binomial coefficient, in...
bccl2 12972 A binomial coefficient, in...
bcn2m1 12973 Compute the binomial coeff...
bcn2p1 12974 Compute the binomial coeff...
permnn 12975 The number of permutations...
bcnm1 12976 The binomial coefficent of...
4bc3eq4 12977 The value of four choose t...
4bc2eq6 12978 The value of four choose t...
hashkf 12981 The finite part of the siz...
hashgval 12982 The value of the ` # ` fun...
hashginv 12983 ` ``' G ` maps the size fu...
hashinf 12984 The value of the ` # ` fun...
hashbnd 12985 If ` A ` has size bounded ...
hashfxnn0 12986 The size function is a fun...
hashf 12987 The size function maps all...
hashfOLD 12988 Obsolete version of ~ hash...
hashxnn0 12989 The value of the hash func...
hashresfn 12990 Restriction of the domain ...
dmhashres 12991 Restriction of the domain ...
hashnn0pnf 12992 The value of the hash func...
hashnnn0genn0 12993 If the size of a set is no...
hashnemnf 12994 The size of a set is never...
hashv01gt1 12995 The size of a set is eithe...
hashfz1 12996 The set ` ( 1 ... N ) ` ha...
hashen 12997 Two finite sets have the s...
hasheni 12998 Equinumerous sets have the...
hasheqf1o 12999 The size of two finite set...
fiinfnf1o 13000 There is no bijection betw...
focdmex 13001 The codomain of an onto fu...
hasheqf1oi 13002 The size of two sets is eq...
hasheqf1oiOLD 13003 Obsolete version of ~ hash...
hashf1rn 13004 The size of a finite set w...
hashf1rnOLD 13005 Obsolete version of ~ hash...
hasheqf1od 13006 The size of two sets is eq...
fz1eqb 13007 Two possibly-empty 1-based...
hashcard 13008 The size function of the c...
hashcl 13009 Closure of the ` # ` funct...
hashxrcl 13010 Extended real closure of t...
hashclb 13011 Reverse closure of the ` #...
hashvnfin 13012 A set of finite size is a ...
hashnfinnn0 13013 The size of an infinite se...
isfinite4 13014 A finite set is equinumero...
hasheq0 13015 Two ways of saying a finit...
hashneq0 13016 Two ways of saying a set i...
hashgt0n0 13017 If the size of a set is gr...
hashnncl 13018 Positive natural closure o...
hash0 13019 The empty set has size zer...
hashsng 13020 The size of a singleton. ...
hashen1 13021 A set with only one elemen...
hashrabrsn 13022 The size of a restricted c...
hashrabsn01 13023 The size of a restricted c...
hashrabsn1 13024 If the size of a restricte...
hashfn 13025 A function is equinumerous...
fseq1hash 13026 The value of the size func...
hashgadd 13027 ` G ` maps ordinal additio...
hashgval2 13028 A short expression for the...
hashdom 13029 Dominance relation for the...
hashdomi 13030 Non-strict order relation ...
hashsdom 13031 Strict dominance relation ...
hashun 13032 The size of the union of d...
hashun2 13033 The size of the union of f...
hashun3 13034 The size of the union of f...
hashinfxadd 13035 The extended real addition...
hashunx 13036 The size of the union of d...
hashge0 13037 The cardinality of a set i...
hashgt0 13038 The cardinality of a nonem...
hashge1 13039 The cardinality of a nonem...
1elfz0hash 13040 1 is an element of the fin...
hashnn0n0nn 13041 If a nonnegative integer i...
hashunsng 13042 The size of the union of a...
hashprg 13043 The size of an unordered p...
hashprgOLD 13044 Obsolete version of ~ hash...
elprchashprn2 13045 If one element of an unord...
hashprb 13046 The size of an unordered p...
hashprdifel 13047 The elements of an unorder...
prhash2ex 13048 There is (at least) one se...
hashle00 13049 If the size of a set is le...
hashgt0elex 13050 If the size of a set is gr...
hashgt0elexb 13051 The size of a set is great...
hashp1i 13052 Size of a finite ordinal. ...
hash1 13053 Size of a finite ordinal. ...
hash2 13054 Size of a finite ordinal. ...
hash3 13055 Size of a finite ordinal. ...
hash4 13056 Size of a finite ordinal. ...
pr0hash2ex 13057 There is (at least) one se...
hashss 13058 The size of a subset is le...
prsshashgt1 13059 The size of a superset of ...
hashin 13060 The size of the intersecti...
hashssdif 13061 The size of the difference...
hashdif 13062 The size of the difference...
hashdifsn 13063 The size of the difference...
hashdifpr 13064 The size of the difference...
hashsn01 13065 The size of a singleton is...
hashsnle1 13066 The size of a singleton is...
hashsnlei 13067 Get an upper bound on a co...
hash1snb 13068 The size of a set is 1 if ...
euhash1 13069 The size of a set is 1 in ...
hashgt12el 13070 In a set with more than on...
hashgt12el2 13071 In a set with more than on...
hashunlei 13072 Get an upper bound on a co...
hashsslei 13073 Get an upper bound on a co...
hashfz 13074 Value of the numeric cardi...
fzsdom2 13075 Condition for finite range...
hashfzo 13076 Cardinality of a half-open...
hashfzo0 13077 Cardinality of a half-open...
hashfzp1 13078 Value of the numeric cardi...
hashfz0 13079 Value of the numeric cardi...
hashxplem 13080 Lemma for ~ hashxp . (Con...
hashxp 13081 The size of the Cartesian ...
hashmap 13082 The size of the set expone...
hashpw 13083 The size of the power set ...
hashfun 13084 A finite set is a function...
hashimarn 13085 The size of the image of a...
hashimarni 13086 If the size of the image o...
fnfz0hash 13087 The size of a function on ...
ffz0hash 13088 The size of a function on ...
fnfz0hashnn0 13089 The size of a function on ...
ffzo0hash 13090 The size of a function on ...
fnfzo0hash 13091 The size of a function on ...
fnfzo0hashnn0 13092 The value of the size func...
hashbclem 13093 Lemma for ~ hashbc : induc...
hashbc 13094 The binomial coefficient c...
hashfacen 13095 The number of bijections b...
hashf1lem1 13096 Lemma for ~ hashf1 . (Con...
hashf1lem2 13097 Lemma for ~ hashf1 . (Con...
hashf1 13098 The permutation number ` |...
hashfac 13099 A factorial counts the num...
leiso 13100 Two ways to write a strict...
leisorel 13101 Version of ~ isorel for st...
fz1isolem 13102 Lemma for ~ fz1iso . (Con...
fz1iso 13103 Any finite ordered set has...
ishashinf 13104 Any set that is not finite...
seqcoll 13105 The function ` F ` contain...
seqcoll2 13106 The function ` F ` contain...
hashprlei 13107 An unordered pair has at m...
hash2pr 13108 A set of size two is an un...
hash2prde 13109 A set of size two is an un...
hash2exprb 13110 A set of size two is an un...
hash2prb 13111 A set of size two is a pro...
prprrab 13112 The set of proper pairs of...
nehash2 13113 The cardinality of a set w...
hash2prd 13114 A set of size two is an un...
hash2pwpr 13115 If the size of a subset of...
pr2pwpr 13116 The set of subsets of a pa...
hashge2el2dif 13117 A set with size at least 2...
hashge2el2difr 13118 A set with at least 2 diff...
hashge2el2difb 13119 A set has size at least 2 ...
hashtplei 13120 An unordered triple has at...
hashtpg 13121 The size of an unordered t...
hashge3el3dif 13122 A set with size at least 3...
elss2prb 13123 An element of the set of s...
hash2sspr 13124 A subset of size two is an...
elss2prOLD 13125 An element of the set of s...
exprelprel 13126 If there is an element of ...
hash3tr 13127 A set of size three is an ...
hash1to3 13128 If the size of a set is be...
fundmge2nop0 13129 A function with a domain c...
fundmge2nop 13130 A function with a domain c...
fun2dmnop0 13131 A function with a domain c...
fun2dmnop 13132 A function with a domain c...
brfi1indlem 13133 Lemma for ~ brfi1ind : Th...
fi1uzind 13134 Properties of an ordered p...
brfi1uzind 13135 Properties of a binary rel...
brfi1ind 13136 Properties of a binary rel...
brfi1indALT 13137 Alternate proof of ~ brfi1...
opfi1uzind 13138 Properties of an ordered p...
opfi1ind 13139 Properties of an ordered p...
fi1uzindOLD 13140 Obsolete version of ~ fi1u...
brfi1uzindOLD 13141 Obsolete version of ~ brfi...
brfi1indOLD 13142 Obsolete version of ~ brfi...
brfi1indALTOLD 13143 Obsolete version of ~ brfi...
opfi1uzindOLD 13144 Obsolete version of ~ opfi...
opfi1indOLD 13145 Obsolete version of ~ opfi...
iswrd 13162 Property of being a word o...
wrdval 13163 Value of the set of words ...
iswrdi 13164 A zero-based sequence is a...
wrdf 13165 A word is a zero-based seq...
iswrdb 13166 A word over an alphabet is...
wrddm 13167 The indices of a word (i.e...
sswrd 13168 The set of words respects ...
snopiswrd 13169 A singleton of an ordered ...
wrdexg 13170 The set of words over a se...
wrdexb 13171 The set of words over a se...
wrdexi 13172 The set of words over a se...
wrdsymbcl 13173 A symbol within a word ove...
wrdfn 13174 A word is a function with ...
wrdv 13175 A word over an alphabet is...
wrdlndm 13176 The length of a word is no...
iswrdsymb 13177 An arbitrary word is a wor...
wrdfin 13178 A word is a finite set. (...
lencl 13179 The length of a word is a ...
lennncl 13180 The length of a nonempty w...
wrdffz 13181 A word is a function from ...
wrdeq 13182 Equality theorem for the s...
wrdeqi 13183 Equality theorem for the s...
iswrddm0 13184 A function with empty doma...
wrd0 13185 The empty set is a word (t...
0wrd0 13186 The empty word is the only...
ffz0iswrd 13187 A sequence with zero-based...
nfwrd 13188 Hypothesis builder for ` W...
csbwrdg 13189 Class substitution for the...
wrdnval 13190 Words of a fixed length ar...
wrdmap 13191 Words as a mapping. (Cont...
hashwrdn 13192 If there is only a finite ...
wrdnfi 13193 If there is only a finite ...
wrdsymb0 13194 A symbol at a position "ou...
wrdlenge1n0 13195 A word with length at leas...
wrdlenge2n0 13196 A word with length at leas...
wrdsymb1 13197 The first symbol of a none...
wrdlen1 13198 A word of length 1 starts ...
fstwrdne 13199 The first symbol of a none...
fstwrdne0 13200 The first symbol of a none...
eqwrd 13201 Two words are equal iff th...
elovmpt2wrd 13202 Implications for the value...
elovmptnn0wrd 13203 Implications for the value...
lsw 13204 Extract the last symbol of...
lsw0 13205 The last symbol of an empt...
lsw0g 13206 The last symbol of an empt...
lsw1 13207 The last symbol of a word ...
lswcl 13208 Closure of the last symbol...
lswlgt0cl 13209 The last symbol of a nonem...
ccatfn 13210 The concatenation operator...
ccatfval 13211 Value of the concatenation...
ccatcl 13212 The concatenation of two w...
ccatlen 13213 The length of a concatenat...
ccatval1 13214 Value of a symbol in the l...
ccatval2 13215 Value of a symbol in the r...
ccatval3 13216 Value of a symbol in the r...
elfzelfzccat 13217 An element of a finite set...
ccatvalfn 13218 The concatenation of two w...
ccatsymb 13219 The symbol at a given posi...
ccatfv0 13220 The first symbol of a conc...
ccatval1lsw 13221 The last symbol of the lef...
ccatlid 13222 Concatenation of a word by...
ccatrid 13223 Concatenation of a word by...
ccatass 13224 Associative law for concat...
ccatrn 13225 The range of a concatenate...
lswccatn0lsw 13226 The last symbol of a word ...
lswccat0lsw 13227 The last symbol of a word ...
ccatalpha 13228 A concatenation of two arb...
ccatrcl1 13229 Reverse closure of a conca...
ids1 13230 Identity function protecti...
s1val 13231 Value of a single-symbol w...
s1rn 13232 The range of a single-symb...
s1eq 13233 Equality theorem for a sin...
s1eqd 13234 Equality theorem for a sin...
s1cl 13235 A singleton word is a word...
s1cld 13236 A singleton word is a word...
s1cli 13237 A singleton word is a word...
s1len 13238 Length of a singleton word...
s1nz 13239 A singleton word is not th...
s1nzOLD 13240 Obsolete proof of ~ s1nz a...
s1dm 13241 The domain of a singleton ...
s1dmALT 13242 Alternate version of ~ s1d...
s1fv 13243 Sole symbol of a singleton...
lsws1 13244 The last symbol of a singl...
eqs1 13245 A word of length 1 is a si...
wrdl1exs1 13246 A word of length 1 is a si...
wrdl1s1 13247 A word of length 1 is a si...
s111 13248 The singleton word functio...
ccatws1cl 13249 The concatenation of a wor...
ccat2s1cl 13250 The concatenation of two s...
ccatws1len 13251 The length of the concaten...
wrdlenccats1lenm1 13252 The length of a word is th...
ccat2s1len 13253 The length of the concaten...
ccatw2s1len 13254 The length of the concaten...
ccats1val1 13255 Value of a symbol in the l...
ccats1val2 13256 Value of the symbol concat...
ccat2s1p1 13257 Extract the first of two c...
ccat2s1p2 13258 Extract the second of two ...
ccatw2s1ass 13259 Associative law for a conc...
ccatws1lenrev 13260 The length of a word conca...
ccatws1n0 13261 The concatenation of a wor...
ccatws1ls 13262 The last symbol of the con...
lswccats1 13263 The last symbol of a word ...
lswccats1fst 13264 The last symbol of a nonem...
ccatw2s1p1 13265 Extract the symbol of the ...
ccatw2s1p2 13266 Extract the second of two ...
ccat2s1fvw 13267 Extract a symbol of a word...
ccat2s1fst 13268 The first symbol of the co...
swrdval 13269 Value of a subword. (Cont...
swrd00 13270 A zero length substring. ...
swrdcl 13271 Closure of the subword ext...
swrdval2 13272 Value of the subword extra...
swrd0val 13273 Value of the subword extra...
swrd0len 13274 Length of a left-anchored ...
swrdlen 13275 Length of an extracted sub...
swrdfv 13276 A symbol in an extracted s...
swrdf 13277 A subword of a word is a f...
swrdvalfn 13278 Value of the subword extra...
swrd0f 13279 A left-anchored subword of...
swrdid 13280 A word is a subword of its...
swrdrn 13281 The range of a subword of ...
swrdn0 13282 A prefixing subword consis...
swrdlend 13283 The value of the subword e...
swrdnd 13284 The value of the subword e...
swrdnd2 13285 Value of the subword extra...
swrd0 13286 A subword of an empty set ...
swrdrlen 13287 Length of a right-anchored...
swrd0len0 13288 Length of a prefix of a wo...
addlenrevswrd 13289 The sum of the lengths of ...
addlenswrd 13290 The sum of the lengths of ...
swrd0fv 13291 A symbol in an left-anchor...
swrd0fv0 13292 The first symbol in a left...
swrdtrcfv 13293 A symbol in a word truncat...
swrdtrcfv0 13294 The first symbol in a word...
swrd0fvlsw 13295 The last symbol in a left-...
swrdeq 13296 Two subwords of words are ...
swrdlen2 13297 Length of an extracted sub...
swrdfv2 13298 A symbol in an extracted s...
swrdsb0eq 13299 Two subwords with the same...
swrdsbslen 13300 Two subwords with the same...
swrdspsleq 13301 Two words have a common su...
swrdtrcfvl 13302 The last symbol in a word ...
swrds1 13303 Extract a single symbol fr...
swrdlsw 13304 Extract the last single sy...
2swrdeqwrdeq 13305 Two words are equal if and...
2swrd1eqwrdeq 13306 Two (nonempty) words are e...
disjxwrd 13307 Sets of words are disjoint...
ccatswrd 13308 Joining two adjacent subwo...
swrdccat1 13309 Recover the left half of a...
swrdccat2 13310 Recover the right half of ...
swrdswrdlem 13311 Lemma for ~ swrdswrd . (C...
swrdswrd 13312 A subword of a subword. (...
swrd0swrd 13313 A prefix of a subword. (C...
swrdswrd0 13314 A subword of a prefix. (C...
swrd0swrd0 13315 A prefix of a prefix. (Co...
swrd0swrdid 13316 A prefix of a prefix with ...
wrdcctswrd 13317 The concatenation of two p...
lencctswrd 13318 The length of two concaten...
lenrevcctswrd 13319 The length of two reversel...
swrdccatwrd 13320 Reconstruct a nonempty wor...
ccats1swrdeq 13321 The last symbol of a word ...
ccatopth 13322 An ~ opth -like theorem fo...
ccatopth2 13323 An ~ opth -like theorem fo...
ccatlcan 13324 Concatenation of words is ...
ccatrcan 13325 Concatenation of words is ...
wrdeqs1cat 13326 Decompose a nonempty word ...
cats1un 13327 Express a word with an ext...
wrdind 13328 Perform induction over the...
wrd2ind 13329 Perform induction over the...
ccats1swrdeqrex 13330 There exists a symbol such...
reuccats1lem 13331 Lemma for ~ reuccats1 . (...
reuccats1 13332 A set of words having the ...
swrdccatfn 13333 The subword of a concatena...
swrdccatin1 13334 The subword of a concatena...
swrdccatin12lem1 13335 Lemma 1 for ~ swrdccatin12...
swrdccatin12lem2a 13336 Lemma 1 for ~ swrdccatin12...
swrdccatin12lem2b 13337 Lemma 2 for ~ swrdccatin12...
swrdccatin2 13338 The subword of a concatena...
swrdccatin12lem2c 13339 Lemma for ~ swrdccatin12le...
swrdccatin12lem2 13340 Lemma 2 for ~ swrdccatin12...
swrdccatin12lem3 13341 Lemma 3 for ~ swrdccatin12...
swrdccatin12 13342 The subword of a concatena...
swrdccat3 13343 The subword of a concatena...
swrdccat 13344 The subword of a concatena...
swrdccat3a 13345 A prefix of a concatenatio...
swrdccat3blem 13346 Lemma for ~ swrdccat3b . ...
swrdccat3b 13347 A suffix of a concatenatio...
swrdccatid 13348 A prefix of a concatenatio...
ccats1swrdeqbi 13349 A word is a prefix of a wo...
swrdccatin1d 13350 The subword of a concatena...
swrdccatin2d 13351 The subword of a concatena...
swrdccatin12d 13352 The subword of a concatena...
splval 13353 Value of the substring rep...
splcl 13354 Closure of the substring r...
splid 13355 Splicing a subword for the...
spllen 13356 The length of a splice. (...
splfv1 13357 Symbols to the left of a s...
splfv2a 13358 Symbols within the replace...
splval2 13359 Value of a splice, assumin...
revval 13360 Value of the word reversin...
revcl 13361 The reverse of a word is a...
revlen 13362 The reverse of a word has ...
revfv 13363 Reverse of a word at a poi...
rev0 13364 The empty word is its own ...
revs1 13365 Singleton words are their ...
revccat 13366 Antiautomorphic property o...
revrev 13367 Reversion is an involution...
reps 13368 Construct a function mappi...
repsundef 13369 A function mapping a half-...
repsconst 13370 Construct a function mappi...
repsf 13371 The constructed function m...
repswsymb 13372 The symbols of a "repeated...
repsw 13373 A function mapping a half-...
repswlen 13374 The length of a "repeated ...
repsw0 13375 The "repeated symbol word"...
repsdf2 13376 Alternative definition of ...
repswsymball 13377 All the symbols of a "repe...
repswsymballbi 13378 A word is a "repeated symb...
repswfsts 13379 The first symbol of a none...
repswlsw 13380 The last symbol of a nonem...
repsw1 13381 The "repeated symbol word"...
repswswrd 13382 A subword of a "repeated s...
repswccat 13383 The concatenation of two "...
repswrevw 13384 The reverse of a "repeated...
cshfn 13387 Perform a cyclical shift f...
cshword 13388 Perform a cyclical shift f...
cshnz 13389 A cyclical shift is the em...
0csh0 13390 Cyclically shifting an emp...
cshw0 13391 A word cyclically shifted ...
cshwmodn 13392 Cyclically shifting a word...
cshwsublen 13393 Cyclically shifting a word...
cshwn 13394 A word cyclically shifted ...
cshwcl 13395 A cyclically shifted word ...
cshwlen 13396 The length of a cyclically...
cshwf 13397 A cyclically shifted word ...
cshwfn 13398 A cyclically shifted word ...
cshwrn 13399 The range of a cyclically ...
cshwidxmod 13400 The symbol at a given inde...
cshwidxmodr 13401 The symbol at a given inde...
cshwidx0mod 13402 The symbol at index 0 of a...
cshwidx0 13403 The symbol at index 0 of a...
cshwidxm1 13404 The symbol at index ((n-N)...
cshwidxm 13405 The symbol at index (n-N) ...
cshwidxn 13406 The symbol at index (n-1) ...
cshf1 13407 Cyclically shifting a word...
cshinj 13408 If a word is injectiv (reg...
repswcshw 13409 A cyclically shifted "repe...
2cshw 13410 Cyclically shifting a word...
2cshwid 13411 Cyclically shifting a word...
lswcshw 13412 The last symbol of a word ...
2cshwcom 13413 Cyclically shifting a word...
cshwleneq 13414 If the results of cyclical...
3cshw 13415 Cyclically shifting a word...
cshweqdif2 13416 If cyclically shifting two...
cshweqdifid 13417 If cyclically shifting a w...
cshweqrep 13418 If cyclically shifting a w...
cshw1 13419 If cyclically shifting a w...
cshw1repsw 13420 If cyclically shifting a w...
cshwsexa 13421 The class of (different!) ...
2cshwcshw 13422 If a word is a cyclically ...
scshwfzeqfzo 13423 For a nonempty word the se...
cshwcshid 13424 A cyclically shifted word ...
cshwcsh2id 13425 A cyclically shifted word ...
cshimadifsn 13426 The image of a cyclically ...
cshimadifsn0 13427 The image of a cyclically ...
wrdco 13428 Mapping a word by a functi...
lenco 13429 Length of a mapped word is...
s1co 13430 Mapping of a singleton wor...
revco 13431 Mapping of words commutes ...
ccatco 13432 Mapping of words commutes ...
cshco 13433 Mapping of words commutes ...
swrdco 13434 Mapping of words commutes ...
lswco 13435 Mapping of (nonempty) word...
repsco 13436 Mapping of words commutes ...
cats1cld 13451 Closure of concatenation w...
cats1co 13452 Closure of concatenation w...
cats1cli 13453 Closure of concatenation w...
cats1fvn 13454 The last symbol of a conca...
cats1fv 13455 A symbol other than the la...
cats1len 13456 The length of concatenatio...
cats1cat 13457 Closure of concatenation w...
cats2cat 13458 Closure of concatenation o...
s2eqd 13459 Equality theorem for a dou...
s3eqd 13460 Equality theorem for a len...
s4eqd 13461 Equality theorem for a len...
s5eqd 13462 Equality theorem for a len...
s6eqd 13463 Equality theorem for a len...
s7eqd 13464 Equality theorem for a len...
s8eqd 13465 Equality theorem for a len...
s2cld 13466 A doubleton word is a word...
s3cld 13467 A length 3 string is a wor...
s4cld 13468 A length 4 string is a wor...
s5cld 13469 A length 5 string is a wor...
s6cld 13470 A length 6 string is a wor...
s7cld 13471 A length 7 string is a wor...
s8cld 13472 A length 7 string is a wor...
s2cl 13473 A doubleton word is a word...
s3cl 13474 A length 3 string is a wor...
s2cli 13475 A doubleton word is a word...
s3cli 13476 A length 3 string is a wor...
s4cli 13477 A length 4 string is a wor...
s5cli 13478 A length 5 string is a wor...
s6cli 13479 A length 6 string is a wor...
s7cli 13480 A length 7 string is a wor...
s8cli 13481 A length 8 string is a wor...
s2fv0 13482 Extract the first symbol f...
s2fv1 13483 Extract the second symbol ...
s2len 13484 The length of a doubleton ...
s2dm 13485 The domain of a doubleton ...
s3fv0 13486 Extract the first symbol f...
s3fv1 13487 Extract the second symbol ...
s3fv2 13488 Extract the third symbol f...
s3len 13489 The length of a length 3 s...
s4fv0 13490 Extract the first symbol f...
s4fv1 13491 Extract the second symbol ...
s4fv2 13492 Extract the third symbol f...
s4fv3 13493 Extract the fourth symbol ...
s4len 13494 The length of a length 4 s...
s5len 13495 The length of a length 5 s...
s6len 13496 The length of a length 6 s...
s7len 13497 The length of a length 7 s...
s8len 13498 The length of a length 8 s...
lsws2 13499 The last symbol of a doubl...
lsws3 13500 The last symbol of a 3 let...
lsws4 13501 The last symbol of a 4 let...
s2prop 13502 A length 2 word is an unor...
s2dmALT 13503 Alternate version of ~ s2d...
s3tpop 13504 A length 3 word is an unor...
s4prop 13505 A length 4 word is a union...
s3fn 13506 A length 3 word is a funct...
funcnvs1 13507 The converse of a singleto...
funcnvs2 13508 The converse of a length 2...
funcnvs3 13509 The converse of a length 3...
funcnvs4 13510 The converse of a length 4...
s2f1o 13511 A length 2 word with mutua...
f1oun2prg 13512 A union of unordered pairs...
s4f1o 13513 A length 4 word with mutua...
s4dom 13514 The domain of a length 4 w...
s2co 13515 Mapping a doubleton word b...
s3co 13516 Mapping a length 3 string ...
s0s1 13517 Concatenation of fixed len...
s1s2 13518 Concatenation of fixed len...
s1s3 13519 Concatenation of fixed len...
s1s4 13520 Concatenation of fixed len...
s1s5 13521 Concatenation of fixed len...
s1s6 13522 Concatenation of fixed len...
s1s7 13523 Concatenation of fixed len...
s2s2 13524 Concatenation of fixed len...
s4s2 13525 Concatenation of fixed len...
s4s3 13526 Concatenation of fixed len...
s4s4 13527 Concatenation of fixed len...
s3s4 13528 Concatenation of fixed len...
s2s5 13529 Concatenation of fixed len...
s5s2 13530 Concatenation of fixed len...
s2eq2s1eq 13531 Two length 2 words are equ...
s2eq2seq 13532 Two length 2 words are equ...
swrds2 13533 Extract two adjacent symbo...
wrdlen2i 13534 Implications of a word of ...
wrd2pr2op 13535 A word of length 2 represe...
wrdlen2 13536 A word of length 2. (Cont...
wrdlen2s2 13537 A word of length 2 as doub...
wrdl2exs2 13538 A word of length 2 is a do...
wrd3tpop 13539 A word of length 3 represe...
wrdlen3s3 13540 A word of length 3 as leng...
repsw2 13541 The "repeated symbol word"...
repsw3 13542 The "repeated symbol word"...
swrd2lsw 13543 Extract the last two symbo...
2swrd2eqwrdeq 13544 Two words of length at lea...
ccatw2s1ccatws2 13545 The concatenation of a wor...
ccat2s1fvwALT 13546 Alternate proof of ~ ccat2...
wwlktovf 13547 Lemma 1 for ~ wrd2f1tovbij...
wwlktovf1 13548 Lemma 2 for ~ wrd2f1tovbij...
wwlktovfo 13549 Lemma 3 for ~ wrd2f1tovbij...
wwlktovf1o 13550 Lemma 4 for ~ wrd2f1tovbij...
wrd2f1tovbij 13551 There is a bijection betwe...
eqwrds3 13552 A word is equal with a len...
wrdl3s3 13553 A word of length 3 is a le...
s3sndisj 13554 The singletons consisting ...
s3iunsndisj 13555 The union of singletons co...
ofccat 13556 Letterwise operations on w...
ofs1 13557 Letterwise operations on a...
ofs2 13558 Letterwise operations on a...
coss12d 13559 Subset deduction for compo...
trrelssd 13560 The composition of subclas...
xpcogend 13561 The most interesting case ...
xpcoidgend 13562 If two classes are not dis...
cotr2g 13563 Two ways of saying that th...
cotr2 13564 Two ways of saying a relat...
cotr3 13565 Two ways of saying a relat...
coemptyd 13566 Deduction about compositio...
xptrrel 13567 The cross product is alway...
0trrel 13568 The empty class is a trans...
cleq1lem 13569 Equality implies bijection...
cleq1 13570 Equality of relations impl...
clsslem 13571 The closure of a subclass ...
trcleq1 13576 Equality of relations impl...
trclsslem 13577 The transitive closure (as...
trcleq2lem 13578 Equality implies bijection...
cvbtrcl 13579 Change of bound variable i...
trcleq12lem 13580 Equality implies bijection...
trclexlem 13581 Existence of relation impl...
trclublem 13582 If a relation exists then ...
trclubi 13583 The Cartesian product of t...
trclubiOLD 13584 Obsolete version of ~ trcl...
trclubgi 13585 The union with the Cartesi...
trclubgiOLD 13586 Obsolete version of ~ trcl...
trclub 13587 The Cartesian product of t...
trclubg 13588 The union with the Cartesi...
trclfv 13589 The transitive closure of ...
brintclab 13590 Two ways to express a bina...
brtrclfv 13591 Two ways of expressing the...
brcnvtrclfv 13592 Two ways of expressing the...
brtrclfvcnv 13593 Two ways of expressing the...
brcnvtrclfvcnv 13594 Two ways of expressing the...
trclfvss 13595 The transitive closure (as...
trclfvub 13596 The transitive closure of ...
trclfvlb 13597 The transitive closure of ...
trclfvcotr 13598 The transitive closure of ...
trclfvlb2 13599 The transitive closure of ...
trclfvlb3 13600 The transitive closure of ...
cotrtrclfv 13601 The transitive closure of ...
trclidm 13602 The transitive closure of ...
trclun 13603 Transitive closure of a un...
trclfvg 13604 The value of the transitiv...
trclfvcotrg 13605 The value of the transitiv...
reltrclfv 13606 The transitive closure of ...
dmtrclfv 13607 The domain of the transiti...
relexp0g 13610 A relation composed zero t...
relexp0 13611 A relation composed zero t...
relexp0d 13612 A relation composed zero t...
relexpsucnnr 13613 A reduction for relation e...
relexp1g 13614 A relation composed once i...
dfid5 13615 Identity relation is equal...
dfid6 13616 Identity relation expresse...
relexpsucr 13617 A reduction for relation e...
relexpsucrd 13618 A reduction for relation e...
relexp1d 13619 A relation composed once i...
relexpsucnnl 13620 A reduction for relation e...
relexpsucl 13621 A reduction for relation e...
relexpsucld 13622 A reduction for relation e...
relexpcnv 13623 Commutation of converse an...
relexpcnvd 13624 Commutation of converse an...
relexp0rel 13625 The exponentiation of a cl...
relexprelg 13626 The exponentiation of a cl...
relexprel 13627 The exponentiation of a re...
relexpreld 13628 The exponentiation of a re...
relexpnndm 13629 The domain of an exponenti...
relexpdmg 13630 The domain of an exponenti...
relexpdm 13631 The domain of an exponenti...
relexpdmd 13632 The domain of an exponenti...
relexpnnrn 13633 The range of an exponentia...
relexprng 13634 The range of an exponentia...
relexprn 13635 The range of an exponentia...
relexprnd 13636 The range of an exponentia...
relexpfld 13637 The field of an exponentia...
relexpfldd 13638 The field of an exponentia...
relexpaddnn 13639 Relation composition becom...
relexpuzrel 13640 The exponentiation of a cl...
relexpaddg 13641 Relation composition becom...
relexpaddd 13642 Relation composition becom...
dfrtrclrec2 13645 If two elements are connec...
rtrclreclem1 13646 The reflexive, transitive ...
rtrclreclem2 13647 The reflexive, transitive ...
rtrclreclem3 13648 The reflexive, transitive ...
rtrclreclem4 13649 The reflexive, transitive ...
dfrtrcl2 13650 The two definitions ` t* `...
relexpindlem 13651 Principle of transitive in...
relexpind 13652 Principle of transitive in...
rtrclind 13653 Principle of transitive in...
shftlem 13656 Two ways to write a shifte...
shftuz 13657 A shift of the upper integ...
shftfval 13658 The value of the sequence ...
shftdm 13659 Domain of a relation shift...
shftfib 13660 Value of a fiber of the re...
shftfn 13661 Functionality and domain o...
shftval 13662 Value of a sequence shifte...
shftval2 13663 Value of a sequence shifte...
shftval3 13664 Value of a sequence shifte...
shftval4 13665 Value of a sequence shifte...
shftval5 13666 Value of a shifted sequenc...
shftf 13667 Functionality of a shifted...
2shfti 13668 Composite shift operations...
shftidt2 13669 Identity law for the shift...
shftidt 13670 Identity law for the shift...
shftcan1 13671 Cancellation law for the s...
shftcan2 13672 Cancellation law for the s...
seqshft 13673 Shifting the index set of ...
sgnval 13676 Value of Signum function. ...
sgn0 13677 Proof that signum of 0 is ...
sgnp 13678 Proof that signum of posit...
sgnrrp 13679 Proof that signum of posit...
sgn1 13680 Proof that the signum of 1...
sgnpnf 13681 Proof that the signum of `...
sgnn 13682 Proof that signum of negat...
sgnmnf 13683 Proof that the signum of `...
cjval 13690 The value of the conjugate...
cjth 13691 The defining property of t...
cjf 13692 Domain and codomain of the...
cjcl 13693 The conjugate of a complex...
reval 13694 The value of the real part...
imval 13695 The value of the imaginary...
imre 13696 The imaginary part of a co...
reim 13697 The real part of a complex...
recl 13698 The real part of a complex...
imcl 13699 The imaginary part of a co...
ref 13700 Domain and codomain of the...
imf 13701 Domain and codomain of the...
crre 13702 The real part of a complex...
crim 13703 The real part of a complex...
replim 13704 Reconstruct a complex numb...
remim 13705 Value of the conjugate of ...
reim0 13706 The imaginary part of a re...
reim0b 13707 A number is real iff its i...
rereb 13708 A number is real iff it eq...
mulre 13709 A product with a nonzero r...
rere 13710 A real number equals its r...
cjreb 13711 A number is real iff it eq...
recj 13712 Real part of a complex con...
reneg 13713 Real part of negative. (C...
readd 13714 Real part distributes over...
resub 13715 Real part distributes over...
remullem 13716 Lemma for ~ remul , ~ immu...
remul 13717 Real part of a product. (...
remul2 13718 Real part of a product. (...
rediv 13719 Real part of a division. ...
imcj 13720 Imaginary part of a comple...
imneg 13721 The imaginary part of a ne...
imadd 13722 Imaginary part distributes...
imsub 13723 Imaginary part distributes...
immul 13724 Imaginary part of a produc...
immul2 13725 Imaginary part of a produc...
imdiv 13726 Imaginary part of a divisi...
cjre 13727 A real number equals its c...
cjcj 13728 The conjugate of the conju...
cjadd 13729 Complex conjugate distribu...
cjmul 13730 Complex conjugate distribu...
ipcnval 13731 Standard inner product on ...
cjmulrcl 13732 A complex number times its...
cjmulval 13733 A complex number times its...
cjmulge0 13734 A complex number times its...
cjneg 13735 Complex conjugate of negat...
addcj 13736 A number plus its conjugat...
cjsub 13737 Complex conjugate distribu...
cjexp 13738 Complex conjugate of posit...
imval2 13739 The imaginary part of a nu...
re0 13740 The real part of zero. (C...
im0 13741 The imaginary part of zero...
re1 13742 The real part of one. (Co...
im1 13743 The imaginary part of one....
rei 13744 The real part of ` _i ` . ...
imi 13745 The imaginary part of ` _i...
cj0 13746 The conjugate of zero. (C...
cji 13747 The complex conjugate of t...
cjreim 13748 The conjugate of a represe...
cjreim2 13749 The conjugate of the repre...
cj11 13750 Complex conjugate is a one...
cjne0 13751 A number is nonzero iff it...
cjdiv 13752 Complex conjugate distribu...
cnrecnv 13753 The inverse to the canonic...
sqeqd 13754 A deduction for showing tw...
recli 13755 The real part of a complex...
imcli 13756 The imaginary part of a co...
cjcli 13757 Closure law for complex co...
replimi 13758 Construct a complex number...
cjcji 13759 The conjugate of the conju...
reim0bi 13760 A number is real iff its i...
rerebi 13761 A real number equals its r...
cjrebi 13762 A number is real iff it eq...
recji 13763 Real part of a complex con...
imcji 13764 Imaginary part of a comple...
cjmulrcli 13765 A complex number times its...
cjmulvali 13766 A complex number times its...
cjmulge0i 13767 A complex number times its...
renegi 13768 Real part of negative. (C...
imnegi 13769 Imaginary part of negative...
cjnegi 13770 Complex conjugate of negat...
addcji 13771 A number plus its conjugat...
readdi 13772 Real part distributes over...
imaddi 13773 Imaginary part distributes...
remuli 13774 Real part of a product. (...
immuli 13775 Imaginary part of a produc...
cjaddi 13776 Complex conjugate distribu...
cjmuli 13777 Complex conjugate distribu...
ipcni 13778 Standard inner product on ...
cjdivi 13779 Complex conjugate distribu...
crrei 13780 The real part of a complex...
crimi 13781 The imaginary part of a co...
recld 13782 The real part of a complex...
imcld 13783 The imaginary part of a co...
cjcld 13784 Closure law for complex co...
replimd 13785 Construct a complex number...
remimd 13786 Value of the conjugate of ...
cjcjd 13787 The conjugate of the conju...
reim0bd 13788 A number is real iff its i...
rerebd 13789 A real number equals its r...
cjrebd 13790 A number is real iff it eq...
cjne0d 13791 A number is nonzero iff it...
recjd 13792 Real part of a complex con...
imcjd 13793 Imaginary part of a comple...
cjmulrcld 13794 A complex number times its...
cjmulvald 13795 A complex number times its...
cjmulge0d 13796 A complex number times its...
renegd 13797 Real part of negative. (C...
imnegd 13798 Imaginary part of negative...
cjnegd 13799 Complex conjugate of negat...
addcjd 13800 A number plus its conjugat...
cjexpd 13801 Complex conjugate of posit...
readdd 13802 Real part distributes over...
imaddd 13803 Imaginary part distributes...
resubd 13804 Real part distributes over...
imsubd 13805 Imaginary part distributes...
remuld 13806 Real part of a product. (...
immuld 13807 Imaginary part of a produc...
cjaddd 13808 Complex conjugate distribu...
cjmuld 13809 Complex conjugate distribu...
ipcnd 13810 Standard inner product on ...
cjdivd 13811 Complex conjugate distribu...
rered 13812 A real number equals its r...
reim0d 13813 The imaginary part of a re...
cjred 13814 A real number equals its c...
remul2d 13815 Real part of a product. (...
immul2d 13816 Imaginary part of a produc...
redivd 13817 Real part of a division. ...
imdivd 13818 Imaginary part of a divisi...
crred 13819 The real part of a complex...
crimd 13820 The imaginary part of a co...
sqrtval 13825 Value of square root funct...
absval 13826 The absolute value (modulu...
rennim 13827 A real number does not lie...
cnpart 13828 The specification of restr...
sqr0lem 13829 Square root of zero. (Con...
sqrt0 13830 Square root of zero. (Con...
sqrlem1 13831 Lemma for ~ 01sqrex . (Co...
sqrlem2 13832 Lemma for ~ 01sqrex . (Co...
sqrlem3 13833 Lemma for ~ 01sqrex . (Co...
sqrlem4 13834 Lemma for ~ 01sqrex . (Co...
sqrlem5 13835 Lemma for ~ 01sqrex . (Co...
sqrlem6 13836 Lemma for ~ 01sqrex . (Co...
sqrlem7 13837 Lemma for ~ 01sqrex . (Co...
01sqrex 13838 Existence of a square root...
resqrex 13839 Existence of a square root...
sqrmo 13840 Uniqueness for the square ...
resqreu 13841 Existence and uniqueness f...
resqrtcl 13842 Closure of the square root...
resqrtthlem 13843 Lemma for ~ resqrtth . (C...
resqrtth 13844 Square root theorem over t...
remsqsqrt 13845 Square of square root. (C...
sqrtge0 13846 The square root function i...
sqrtgt0 13847 The square root function i...
sqrtmul 13848 Square root distributes ov...
sqrtle 13849 Square root is monotonic. ...
sqrtlt 13850 Square root is strictly mo...
sqrt11 13851 The square root function i...
sqrt00 13852 A square root is zero iff ...
rpsqrtcl 13853 The square root of a posit...
sqrtdiv 13854 Square root distributes ov...
sqrtneglem 13855 The square root of a negat...
sqrtneg 13856 The square root of a negat...
sqrtsq2 13857 Relationship between squar...
sqrtsq 13858 Square root of square. (C...
sqrtmsq 13859 Square root of square. (C...
sqrt1 13860 The square root of 1 is 1....
sqrt4 13861 The square root of 4 is 2....
sqrt9 13862 The square root of 9 is 3....
sqrt2gt1lt2 13863 The square root of 2 is bo...
sqrtm1 13864 The imaginary unit is the ...
absneg 13865 Absolute value of negative...
abscl 13866 Real closure of absolute v...
abscj 13867 The absolute value of a nu...
absvalsq 13868 Square of value of absolut...
absvalsq2 13869 Square of value of absolut...
sqabsadd 13870 Square of absolute value o...
sqabssub 13871 Square of absolute value o...
absval2 13872 Value of absolute value fu...
abs0 13873 The absolute value of 0. ...
absi 13874 The absolute value of the ...
absge0 13875 Absolute value is nonnegat...
absrpcl 13876 The absolute value of a no...
abs00 13877 The absolute value of a nu...
abs00ad 13878 A complex number is zero i...
abs00bd 13879 If a complex number is zer...
absreimsq 13880 Square of the absolute val...
absreim 13881 Absolute value of a number...
absmul 13882 Absolute value distributes...
absdiv 13883 Absolute value distributes...
absid 13884 A nonnegative number is it...
abs1 13885 The absolute value of 1. ...
absnid 13886 A negative number is the n...
leabs 13887 A real number is less than...
absor 13888 The absolute value of a re...
absre 13889 Absolute value of a real n...
absresq 13890 Square of the absolute val...
absmod0 13891 ` A ` is divisible by ` B ...
absexp 13892 Absolute value of positive...
absexpz 13893 Absolute value of integer ...
abssq 13894 Square can be moved in and...
sqabs 13895 The squares of two reals a...
absrele 13896 The absolute value of a co...
absimle 13897 The absolute value of a co...
max0add 13898 The sum of the positive an...
absz 13899 A real number is an intege...
nn0abscl 13900 The absolute value of an i...
zabscl 13901 The absolute value of an i...
abslt 13902 Absolute value and 'less t...
absle 13903 Absolute value and 'less t...
abssubne0 13904 If the absolute value of a...
absdiflt 13905 The absolute value of a di...
absdifle 13906 The absolute value of a di...
elicc4abs 13907 Membership in a symmetric ...
lenegsq 13908 Comparison to a nonnegativ...
releabs 13909 The real part of a number ...
recval 13910 Reciprocal expressed with ...
absidm 13911 The absolute value functio...
absgt0 13912 The absolute value of a no...
nnabscl 13913 The absolute value of a no...
abssub 13914 Swapping order of subtract...
abssubge0 13915 Absolute value of a nonneg...
abssuble0 13916 Absolute value of a nonpos...
absmax 13917 The maximum of two numbers...
abstri 13918 Triangle inequality for ab...
abs3dif 13919 Absolute value of differen...
abs2dif 13920 Difference of absolute val...
abs2dif2 13921 Difference of absolute val...
abs2difabs 13922 Absolute value of differen...
abs1m 13923 For any complex number, th...
recan 13924 Cancellation law involving...
absf 13925 Mapping domain and codomai...
abs3lem 13926 Lemma involving absolute v...
abslem2 13927 Lemma involving absolute v...
rddif 13928 The difference between a r...
absrdbnd 13929 Bound on the absolute valu...
fzomaxdiflem 13930 Lemma for ~ fzomaxdif . (...
fzomaxdif 13931 A bound on the separation ...
uzin2 13932 The upper integers are clo...
rexanuz 13933 Combine two different uppe...
rexanre 13934 Combine two different uppe...
rexfiuz 13935 Combine finitely many diff...
rexuz3 13936 Restrict the base of the u...
rexanuz2 13937 Combine two different uppe...
r19.29uz 13938 A version of ~ 19.29 for u...
r19.2uz 13939 A version of ~ r19.2z for ...
rexuzre 13940 Convert an upper real quan...
rexico 13941 Restrict the base of an up...
cau3lem 13942 Lemma for ~ cau3 . (Contr...
cau3 13943 Convert between three-quan...
cau4 13944 Change the base of a Cauch...
caubnd2 13945 A Cauchy sequence of compl...
caubnd 13946 A Cauchy sequence of compl...
sqreulem 13947 Lemma for ~ sqreu : write ...
sqreu 13948 Existence and uniqueness f...
sqrtcl 13949 Closure of the square root...
sqrtthlem 13950 Lemma for ~ sqrtth . (Con...
sqrtf 13951 Mapping domain and codomai...
sqrtth 13952 Square root theorem over t...
sqrtrege0 13953 The square root function m...
eqsqrtor 13954 Solve an equation containi...
eqsqrtd 13955 A deduction for showing th...
eqsqrt2d 13956 A deduction for showing th...
amgm2 13957 Arithmetic-geometric mean ...
sqrtthi 13958 Square root theorem. Theo...
sqrtcli 13959 The square root of a nonne...
sqrtgt0i 13960 The square root of a posit...
sqrtmsqi 13961 Square root of square. (C...
sqrtsqi 13962 Square root of square. (C...
sqsqrti 13963 Square of square root. (C...
sqrtge0i 13964 The square root of a nonne...
absidi 13965 A nonnegative number is it...
absnidi 13966 A negative number is the n...
leabsi 13967 A real number is less than...
absori 13968 The absolute value of a re...
absrei 13969 Absolute value of a real n...
sqrtpclii 13970 The square root of a posit...
sqrtgt0ii 13971 The square root of a posit...
sqrt11i 13972 The square root function i...
sqrtmuli 13973 Square root distributes ov...
sqrtmulii 13974 Square root distributes ov...
sqrtmsq2i 13975 Relationship between squar...
sqrtlei 13976 Square root is monotonic. ...
sqrtlti 13977 Square root is strictly mo...
abslti 13978 Absolute value and 'less t...
abslei 13979 Absolute value and 'less t...
absvalsqi 13980 Square of value of absolut...
absvalsq2i 13981 Square of value of absolut...
abscli 13982 Real closure of absolute v...
absge0i 13983 Absolute value is nonnegat...
absval2i 13984 Value of absolute value fu...
abs00i 13985 The absolute value of a nu...
absgt0i 13986 The absolute value of a no...
absnegi 13987 Absolute value of negative...
abscji 13988 The absolute value of a nu...
releabsi 13989 The real part of a number ...
abssubi 13990 Swapping order of subtract...
absmuli 13991 Absolute value distributes...
sqabsaddi 13992 Square of absolute value o...
sqabssubi 13993 Square of absolute value o...
absdivzi 13994 Absolute value distributes...
abstrii 13995 Triangle inequality for ab...
abs3difi 13996 Absolute value of differen...
abs3lemi 13997 Lemma involving absolute v...
rpsqrtcld 13998 The square root of a posit...
sqrtgt0d 13999 The square root of a posit...
absnidd 14000 A negative number is the n...
leabsd 14001 A real number is less than...
absord 14002 The absolute value of a re...
absred 14003 Absolute value of a real n...
resqrtcld 14004 The square root of a nonne...
sqrtmsqd 14005 Square root of square. (C...
sqrtsqd 14006 Square root of square. (C...
sqrtge0d 14007 The square root of a nonne...
sqrtnegd 14008 The square root of a negat...
absidd 14009 A nonnegative number is it...
sqrtdivd 14010 Square root distributes ov...
sqrtmuld 14011 Square root distributes ov...
sqrtsq2d 14012 Relationship between squar...
sqrtled 14013 Square root is monotonic. ...
sqrtltd 14014 Square root is strictly mo...
sqr11d 14015 The square root function i...
absltd 14016 Absolute value and 'less t...
absled 14017 Absolute value and 'less t...
abssubge0d 14018 Absolute value of a nonneg...
abssuble0d 14019 Absolute value of a nonpos...
absdifltd 14020 The absolute value of a di...
absdifled 14021 The absolute value of a di...
icodiamlt 14022 Two elements in a half-ope...
abscld 14023 Real closure of absolute v...
sqrtcld 14024 Closure of the square root...
sqrtrege0d 14025 The real part of the squar...
sqsqrtd 14026 Square root theorem. Theo...
msqsqrtd 14027 Square root theorem. Theo...
sqr00d 14028 A square root is zero iff ...
absvalsqd 14029 Square of value of absolut...
absvalsq2d 14030 Square of value of absolut...
absge0d 14031 Absolute value is nonnegat...
absval2d 14032 Value of absolute value fu...
abs00d 14033 The absolute value of a nu...
absne0d 14034 The absolute value of a nu...
absrpcld 14035 The absolute value of a no...
absnegd 14036 Absolute value of negative...
abscjd 14037 The absolute value of a nu...
releabsd 14038 The real part of a number ...
absexpd 14039 Absolute value of positive...
abssubd 14040 Swapping order of subtract...
absmuld 14041 Absolute value distributes...
absdivd 14042 Absolute value distributes...
abstrid 14043 Triangle inequality for ab...
abs2difd 14044 Difference of absolute val...
abs2dif2d 14045 Difference of absolute val...
abs2difabsd 14046 Absolute value of differen...
abs3difd 14047 Absolute value of differen...
abs3lemd 14048 Lemma involving absolute v...
limsupgord 14051 Ordering property of the s...
limsupcl 14052 Closure of the superior li...
limsupval 14053 The superior limit of an i...
limsupgf 14054 Closure of the superior li...
limsupgval 14055 Value of the superior limi...
limsupgle 14056 The defining property of t...
limsuple 14057 The defining property of t...
limsuplt 14058 The defining property of t...
limsupval2 14059 The superior limit, relati...
limsupgre 14060 If a sequence of real numb...
limsupbnd1 14061 If a sequence is eventuall...
limsupbnd2 14062 If a sequence is eventuall...
climrel 14071 The limit relation is a re...
rlimrel 14072 The limit relation is a re...
clim 14073 Express the predicate: Th...
rlim 14074 Express the predicate: Th...
rlim2 14075 Rewrite ~ rlim for a mappi...
rlim2lt 14076 Use strictly less-than in ...
rlim3 14077 Restrict the range of the ...
climcl 14078 Closure of the limit of a ...
rlimpm 14079 Closure of a function with...
rlimf 14080 Closure of a function with...
rlimss 14081 Domain closure of a functi...
rlimcl 14082 Closure of the limit of a ...
clim2 14083 Express the predicate: Th...
clim2c 14084 Express the predicate ` F ...
clim0 14085 Express the predicate ` F ...
clim0c 14086 Express the predicate ` F ...
rlim0 14087 Express the predicate ` B ...
rlim0lt 14088 Use strictly less-than in ...
climi 14089 Convergence of a sequence ...
climi2 14090 Convergence of a sequence ...
climi0 14091 Convergence of a sequence ...
rlimi 14092 Convergence at infinity of...
rlimi2 14093 Convergence at infinity of...
ello1 14094 Elementhood in the set of ...
ello12 14095 Elementhood in the set of ...
ello12r 14096 Sufficient condition for e...
lo1f 14097 An eventually upper bounde...
lo1dm 14098 An eventually upper bounde...
lo1bdd 14099 The defining property of a...
ello1mpt 14100 Elementhood in the set of ...
ello1mpt2 14101 Elementhood in the set of ...
ello1d 14102 Sufficient condition for e...
lo1bdd2 14103 If an eventually bounded f...
lo1bddrp 14104 Refine ~ o1bdd2 to give a ...
elo1 14105 Elementhood in the set of ...
elo12 14106 Elementhood in the set of ...
elo12r 14107 Sufficient condition for e...
o1f 14108 An eventually bounded func...
o1dm 14109 An eventually bounded func...
o1bdd 14110 The defining property of a...
lo1o1 14111 A function is eventually b...
lo1o12 14112 A function is eventually b...
elo1mpt 14113 Elementhood in the set of ...
elo1mpt2 14114 Elementhood in the set of ...
elo1d 14115 Sufficient condition for e...
o1lo1 14116 A real function is eventua...
o1lo12 14117 A lower bounded real funct...
o1lo1d 14118 A real eventually bounded ...
icco1 14119 Derive eventual boundednes...
o1bdd2 14120 If an eventually bounded f...
o1bddrp 14121 Refine ~ o1bdd2 to give a ...
climconst 14122 An (eventually) constant s...
rlimconst 14123 A constant sequence conver...
rlimclim1 14124 Forward direction of ~ rli...
rlimclim 14125 A sequence on an upper int...
climrlim2 14126 Produce a real limit from ...
climconst2 14127 A constant sequence conver...
climz 14128 The zero sequence converge...
rlimuni 14129 A real function whose doma...
rlimdm 14130 Two ways to express that a...
climuni 14131 An infinite sequence of co...
fclim 14132 The limit relation is func...
climdm 14133 Two ways to express that a...
climeu 14134 An infinite sequence of co...
climreu 14135 An infinite sequence of co...
climmo 14136 An infinite sequence of co...
rlimres 14137 The restriction of a funct...
lo1res 14138 The restriction of an even...
o1res 14139 The restriction of an even...
rlimres2 14140 The restriction of a funct...
lo1res2 14141 The restriction of a funct...
o1res2 14142 The restriction of a funct...
lo1resb 14143 The restriction of a funct...
rlimresb 14144 The restriction of a funct...
o1resb 14145 The restriction of a funct...
climeq 14146 Two functions that are eve...
lo1eq 14147 Two functions that are eve...
rlimeq 14148 Two functions that are eve...
o1eq 14149 Two functions that are eve...
climmpt 14150 Exhibit a function ` G ` w...
2clim 14151 If two sequences converge ...
climmpt2 14152 Relate an integer limit on...
climshftlem 14153 A shifted function converg...
climres 14154 A function restricted to u...
climshft 14155 A shifted function converg...
serclim0 14156 The zero series converges ...
rlimcld2 14157 If ` D ` is a closed set i...
rlimrege0 14158 The limit of a sequence of...
rlimrecl 14159 The limit of a real sequen...
rlimge0 14160 The limit of a sequence of...
climshft2 14161 A shifted function converg...
climrecl 14162 The limit of a convergent ...
climge0 14163 A nonnegative sequence con...
climabs0 14164 Convergence to zero of the...
o1co 14165 Sufficient condition for t...
o1compt 14166 Sufficient condition for t...
rlimcn1 14167 Image of a limit under a c...
rlimcn1b 14168 Image of a limit under a c...
rlimcn2 14169 Image of a limit under a c...
climcn1 14170 Image of a limit under a c...
climcn2 14171 Image of a limit under a c...
addcn2 14172 Complex number addition is...
subcn2 14173 Complex number subtraction...
mulcn2 14174 Complex number multiplicat...
reccn2 14175 The reciprocal function is...
cn1lem 14176 A sufficient condition for...
abscn2 14177 The absolute value functio...
cjcn2 14178 The complex conjugate func...
recn2 14179 The real part function is ...
imcn2 14180 The imaginary part functio...
climcn1lem 14181 The limit of a continuous ...
climabs 14182 Limit of the absolute valu...
climcj 14183 Limit of the complex conju...
climre 14184 Limit of the real part of ...
climim 14185 Limit of the imaginary par...
rlimmptrcl 14186 Reverse closure for a real...
rlimabs 14187 Limit of the absolute valu...
rlimcj 14188 Limit of the complex conju...
rlimre 14189 Limit of the real part of ...
rlimim 14190 Limit of the imaginary par...
o1of2 14191 Show that a binary operati...
o1add 14192 The sum of two eventually ...
o1mul 14193 The product of two eventua...
o1sub 14194 The difference of two even...
rlimo1 14195 Any function with a finite...
rlimdmo1 14196 A convergent function is e...
o1rlimmul 14197 The product of an eventual...
o1const 14198 A constant function is eve...
lo1const 14199 A constant function is eve...
lo1mptrcl 14200 Reverse closure for an eve...
o1mptrcl 14201 Reverse closure for an eve...
o1add2 14202 The sum of two eventually ...
o1mul2 14203 The product of two eventua...
o1sub2 14204 The product of two eventua...
lo1add 14205 The sum of two eventually ...
lo1mul 14206 The product of an eventual...
lo1mul2 14207 The product of an eventual...
o1dif 14208 If the difference of two f...
lo1sub 14209 The difference of an event...
climadd 14210 Limit of the sum of two co...
climmul 14211 Limit of the product of tw...
climsub 14212 Limit of the difference of...
climaddc1 14213 Limit of a constant ` C ` ...
climaddc2 14214 Limit of a constant ` C ` ...
climmulc2 14215 Limit of a sequence multip...
climsubc1 14216 Limit of a constant ` C ` ...
climsubc2 14217 Limit of a constant ` C ` ...
climle 14218 Comparison of the limits o...
climsqz 14219 Convergence of a sequence ...
climsqz2 14220 Convergence of a sequence ...
rlimadd 14221 Limit of the sum of two co...
rlimsub 14222 Limit of the difference of...
rlimmul 14223 Limit of the product of tw...
rlimdiv 14224 Limit of the quotient of t...
rlimneg 14225 Limit of the negative of a...
rlimle 14226 Comparison of the limits o...
rlimsqzlem 14227 Lemma for ~ rlimsqz and ~ ...
rlimsqz 14228 Convergence of a sequence ...
rlimsqz2 14229 Convergence of a sequence ...
lo1le 14230 Transfer eventual upper bo...
o1le 14231 Transfer eventual boundedn...
rlimno1 14232 A function whose inverse c...
clim2ser 14233 The limit of an infinite s...
clim2ser2 14234 The limit of an infinite s...
iserex 14235 An infinite series converg...
isermulc2 14236 Multiplication of an infin...
climlec2 14237 Comparison of a constant t...
iserle 14238 Comparison of the limits o...
iserge0 14239 The limit of an infinite s...
climub 14240 The limit of a monotonic s...
climserle 14241 The partial sums of a conv...
isershft 14242 Index shift of the limit o...
isercolllem1 14243 Lemma for ~ isercoll . (C...
isercolllem2 14244 Lemma for ~ isercoll . (C...
isercolllem3 14245 Lemma for ~ isercoll . (C...
isercoll 14246 Rearrange an infinite seri...
isercoll2 14247 Generalize ~ isercoll so t...
climsup 14248 A bounded monotonic sequen...
climcau 14249 A converging sequence of c...
climbdd 14250 A converging sequence of c...
caucvgrlem 14251 Lemma for ~ caurcvgr . (C...
caurcvgr 14252 A Cauchy sequence of real ...
caucvgrlem2 14253 Lemma for ~ caucvgr . (Co...
caucvgr 14254 A Cauchy sequence of compl...
caurcvg 14255 A Cauchy sequence of real ...
caurcvg2 14256 A Cauchy sequence of real ...
caucvg 14257 A Cauchy sequence of compl...
caucvgb 14258 A function is convergent i...
serf0 14259 If an infinite series conv...
iseraltlem1 14260 Lemma for ~ iseralt . A d...
iseraltlem2 14261 Lemma for ~ iseralt . The...
iseraltlem3 14262 Lemma for ~ iseralt . Fro...
iseralt 14263 The alternating series tes...
sumex 14266 A sum is a set. (Contribu...
sumeq1 14267 Equality theorem for a sum...
nfsum1 14268 Bound-variable hypothesis ...
nfsum 14269 Bound-variable hypothesis ...
sumeq2w 14270 Equality theorem for sum, ...
sumeq2ii 14271 Equality theorem for sum, ...
sumeq2 14272 Equality theorem for sum. ...
cbvsum 14273 Change bound variable in a...
cbvsumv 14274 Change bound variable in a...
cbvsumi 14275 Change bound variable in a...
sumeq1i 14276 Equality inference for sum...
sumeq2i 14277 Equality inference for sum...
sumeq12i 14278 Equality inference for sum...
sumeq1d 14279 Equality deduction for sum...
sumeq2d 14280 Equality deduction for sum...
sumeq2dv 14281 Equality deduction for sum...
sumeq2sdv 14282 Equality deduction for sum...
2sumeq2dv 14283 Equality deduction for dou...
sumeq12dv 14284 Equality deduction for sum...
sumeq12rdv 14285 Equality deduction for sum...
sum2id 14286 The second class argument ...
sumfc 14287 A lemma to facilitate conv...
fz1f1o 14288 A lemma for working with f...
sumrblem 14289 Lemma for ~ sumrb . (Cont...
fsumcvg 14290 The sequence of partial su...
sumrb 14291 Rebase the starting point ...
summolem3 14292 Lemma for ~ summo . (Cont...
summolem2a 14293 Lemma for ~ summo . (Cont...
summolem2 14294 Lemma for ~ summo . (Cont...
summo 14295 A sum has at most one limi...
zsum 14296 Series sum with index set ...
isum 14297 Series sum with an upper i...
fsum 14298 The value of a sum over a ...
sum0 14299 Any sum over the empty set...
sumz 14300 Any sum of zero over a sum...
fsumf1o 14301 Re-index a finite sum usin...
sumss 14302 Change the index set to a ...
fsumss 14303 Change the index set to a ...
sumss2 14304 Change the index set of a ...
fsumcvg2 14305 The sequence of partial su...
fsumsers 14306 Special case of series sum...
fsumcvg3 14307 A finite sum is convergent...
fsumser 14308 A finite sum expressed in ...
fsumcl2lem 14309 - Lemma for finite sum clo...
fsumcllem 14310 - Lemma for finite sum clo...
fsumcl 14311 Closure of a finite sum of...
fsumrecl 14312 Closure of a finite sum of...
fsumzcl 14313 Closure of a finite sum of...
fsumnn0cl 14314 Closure of a finite sum of...
fsumrpcl 14315 Closure of a finite sum of...
fsumzcl2 14316 A finite sum with integer ...
fsumadd 14317 The sum of two finite sums...
fsumsplit 14318 Split a sum into two parts...
sumsn 14319 A sum of a singleton is th...
fsum1 14320 The finite sum of ` A ( k ...
sumpr 14321 A sum over a pair is the s...
sumtp 14322 A sum over a triple is the...
sumsns 14323 A sum of a singleton is th...
fsumm1 14324 Separate out the last term...
fzosump1 14325 Separate out the last term...
fsum1p 14326 Separate out the first ter...
fsummsnunz 14327 A finite sum with an addit...
fsumsplitsnun 14328 Separate out a term in a f...
fsump1 14329 The addition of the next t...
isumclim 14330 An infinite sum equals the...
isumclim2 14331 A converging series conver...
isumclim3 14332 The sequence of partial fi...
sumnul 14333 The sum of a non-convergen...
isumcl 14334 The sum of a converging in...
isummulc2 14335 An infinite sum multiplied...
isummulc1 14336 An infinite sum multiplied...
isumdivc 14337 An infinite sum divided by...
isumrecl 14338 The sum of a converging in...
isumge0 14339 An infinite sum of nonnega...
isumadd 14340 Addition of infinite sums....
sumsplit 14341 Split a sum into two parts...
fsump1i 14342 Optimized version of ~ fsu...
fsum2dlem 14343 Lemma for ~ fsum2d - induc...
fsum2d 14344 Write a double sum as a su...
fsumxp 14345 Combine two sums into a si...
fsumcnv 14346 Transform a region of summ...
fsumcom2 14347 Interchange order of summa...
fsumcom2OLD 14348 Obsolete proof of ~ fsumco...
fsumcom 14349 Interchange order of summa...
fsum0diaglem 14350 Lemma for ~ fsum0diag . (...
fsum0diag 14351 Two ways to express "the s...
mptfzshft 14352 1-1 onto function in maps-...
fsumrev 14353 Reversal of a finite sum. ...
fsumshft 14354 Index shift of a finite su...
fsumshftm 14355 Negative index shift of a ...
fsumrev2 14356 Reversal of a finite sum. ...
fsum0diag2 14357 Two ways to express "the s...
fsummulc2 14358 A finite sum multiplied by...
fsummulc1 14359 A finite sum multiplied by...
fsumdivc 14360 A finite sum divided by a ...
fsumneg 14361 Negation of a finite sum. ...
fsumsub 14362 Split a finite sum over a ...
fsum2mul 14363 Separate the nested sum of...
fsumconst 14364 The sum of constant terms ...
modfsummodslem1 14365 Lemma 1 for ~ modfsummods ...
modfsummods 14366 Induction step for ~ modfs...
modfsummod 14367 A finite sum modulo a posi...
fsumge0 14368 If all of the terms of a f...
fsumless 14369 A shorter sum of nonnegati...
fsumge1 14370 A sum of nonnegative numbe...
fsum00 14371 A sum of nonnegative numbe...
fsumle 14372 If all of the terms of fin...
fsumlt 14373 If every term in one finit...
fsumabs 14374 Generalized triangle inequ...
telfsumo 14375 Sum of a telescoping serie...
telfsumo2 14376 Sum of a telescoping serie...
telfsum 14377 Sum of a telescoping serie...
telfsum2 14378 Sum of a telescoping serie...
fsumparts 14379 Summation by parts. (Cont...
fsumrelem 14380 Lemma for ~ fsumre , ~ fsu...
fsumre 14381 The real part of a sum. (...
fsumim 14382 The imaginary part of a su...
fsumcj 14383 The complex conjugate of a...
fsumrlim 14384 Limit of a finite sum of c...
fsumo1 14385 The finite sum of eventual...
o1fsum 14386 If ` A ( k ) ` is O(1), th...
seqabs 14387 Generalized triangle inequ...
iserabs 14388 Generalized triangle inequ...
cvgcmp 14389 A comparison test for conv...
cvgcmpub 14390 An upper bound for the lim...
cvgcmpce 14391 A comparison test for conv...
abscvgcvg 14392 An absolutely convergent s...
climfsum 14393 Limit of a finite sum of c...
fsumiun 14394 Sum over a disjoint indexe...
hashiun 14395 The cardinality of a disjo...
hashrabrex 14396 The number of elements in ...
hashuni 14397 The cardinality of a disjo...
qshash 14398 The cardinality of a set w...
ackbijnn 14399 Translate the Ackermann bi...
binomlem 14400 Lemma for ~ binom (binomia...
binom 14401 The binomial theorem: ` ( ...
binom1p 14402 Special case of the binomi...
binom11 14403 Special case of the binomi...
binom1dif 14404 A summation for the differ...
bcxmaslem1 14405 Lemma for ~ bcxmas . (Con...
bcxmas 14406 Parallel summation (Christ...
incexclem 14407 Lemma for ~ incexc . (Con...
incexc 14408 The inclusion/exclusion pr...
incexc2 14409 The inclusion/exclusion pr...
isumshft 14410 Index shift of an infinite...
isumsplit 14411 Split off the first ` N ` ...
isum1p 14412 The infinite sum of a conv...
isumnn0nn 14413 Sum from 0 to infinity in ...
isumrpcl 14414 The infinite sum of positi...
isumle 14415 Comparison of two infinite...
isumless 14416 A finite sum of nonnegativ...
isumsup2 14417 An infinite sum of nonnega...
isumsup 14418 An infinite sum of nonnega...
isumltss 14419 A partial sum of a series ...
climcndslem1 14420 Lemma for ~ climcnds : bou...
climcndslem2 14421 Lemma for ~ climcnds : bou...
climcnds 14422 The Cauchy condensation te...
divrcnv 14423 The sequence of reciprocal...
divcnv 14424 The sequence of reciprocal...
flo1 14425 The floor function satisfi...
divcnvshft 14426 Limit of a ratio function....
supcvg 14427 Extract a sequence ` f ` i...
infcvgaux1i 14428 Auxiliary theorem for appl...
infcvgaux2i 14429 Auxiliary theorem for appl...
harmonic 14430 The harmonic series ` H ` ...
arisum 14431 Arithmetic series sum of t...
arisum2 14432 Arithmetic series sum of t...
trireciplem 14433 Lemma for ~ trirecip . Sh...
trirecip 14434 The sum of the reciprocals...
expcnv 14435 A sequence of powers of a ...
explecnv 14436 A sequence of terms conver...
geoserg 14437 The value of the finite ge...
geoser 14438 The value of the finite ge...
pwm1geoser 14439 The n-th power of a number...
geolim 14440 The partial sums in the in...
geolim2 14441 The partial sums in the ge...
georeclim 14442 The limit of a geometric s...
geo2sum 14443 The value of the finite ge...
geo2sum2 14444 The value of the finite ge...
geo2lim 14445 The value of the infinite ...
geomulcvg 14446 The geometric series conve...
geoisum 14447 The infinite sum of ` 1 + ...
geoisumr 14448 The infinite sum of recipr...
geoisum1 14449 The infinite sum of ` A ^ ...
geoisum1c 14450 The infinite sum of ` A x....
0.999... 14451 The recurring decimal 0.99...
0.999...OLD 14452 Obsolete version of ~ 0.99...
geoihalfsum 14453 Prove that the infinite ge...
cvgrat 14454 Ratio test for convergence...
mertenslem1 14455 Lemma for ~ mertens . (Co...
mertenslem2 14456 Lemma for ~ mertens . (Co...
mertens 14457 Mertens' theorem. If ` A ...
prodf 14458 An infinite product of com...
clim2prod 14459 The limit of an infinite p...
clim2div 14460 The limit of an infinite p...
prodfmul 14461 The product of two infinit...
prodf1 14462 The value of the partial p...
prodf1f 14463 A one-valued infinite prod...
prodfclim1 14464 The constant one product c...
prodfn0 14465 No term of a nonzero infin...
prodfrec 14466 The reciprocal of an infin...
prodfdiv 14467 The quotient of two infini...
ntrivcvg 14468 A non-trivially converging...
ntrivcvgn0 14469 A product that converges t...
ntrivcvgfvn0 14470 Any value of a product seq...
ntrivcvgtail 14471 A tail of a non-trivially ...
ntrivcvgmullem 14472 Lemma for ~ ntrivcvgmul . ...
ntrivcvgmul 14473 The product of two non-tri...
prodex 14476 A product is a set. (Cont...
prodeq1f 14477 Equality theorem for a pro...
prodeq1 14478 Equality theorem for a pro...
nfcprod1 14479 Bound-variable hypothesis ...
nfcprod 14480 Bound-variable hypothesis ...
prodeq2w 14481 Equality theorem for produ...
prodeq2ii 14482 Equality theorem for produ...
prodeq2 14483 Equality theorem for produ...
cbvprod 14484 Change bound variable in a...
cbvprodv 14485 Change bound variable in a...
cbvprodi 14486 Change bound variable in a...
prodeq1i 14487 Equality inference for pro...
prodeq2i 14488 Equality inference for pro...
prodeq12i 14489 Equality inference for pro...
prodeq1d 14490 Equality deduction for pro...
prodeq2d 14491 Equality deduction for pro...
prodeq2dv 14492 Equality deduction for pro...
prodeq2sdv 14493 Equality deduction for pro...
2cprodeq2dv 14494 Equality deduction for dou...
prodeq12dv 14495 Equality deduction for pro...
prodeq12rdv 14496 Equality deduction for pro...
prod2id 14497 The second class argument ...
prodrblem 14498 Lemma for ~ prodrb . (Con...
fprodcvg 14499 The sequence of partial pr...
prodrblem2 14500 Lemma for ~ prodrb . (Con...
prodrb 14501 Rebase the starting point ...
prodmolem3 14502 Lemma for ~ prodmo . (Con...
prodmolem2a 14503 Lemma for ~ prodmo . (Con...
prodmolem2 14504 Lemma for ~ prodmo . (Con...
prodmo 14505 A product has at most one ...
zprod 14506 Series product with index ...
iprod 14507 Series product with an upp...
zprodn0 14508 Nonzero series product wit...
iprodn0 14509 Nonzero series product wit...
fprod 14510 The value of a product ove...
fprodntriv 14511 A non-triviality lemma for...
prod0 14512 A product over the empty s...
prod1 14513 Any product of one over a ...
prodfc 14514 A lemma to facilitate conv...
fprodf1o 14515 Re-index a finite product ...
prodss 14516 Change the index set to a ...
fprodss 14517 Change the index set to a ...
fprodser 14518 A finite product expressed...
fprodcl2lem 14519 Finite product closure lem...
fprodcllem 14520 Finite product closure lem...
fprodcl 14521 Closure of a finite produc...
fprodrecl 14522 Closure of a finite produc...
fprodzcl 14523 Closure of a finite produc...
fprodnncl 14524 Closure of a finite produc...
fprodrpcl 14525 Closure of a finite produc...
fprodnn0cl 14526 Closure of a finite produc...
fprodcllemf 14527 Finite product closure lem...
fprodreclf 14528 Closure of a finite produc...
fprodmul 14529 The product of two finite ...
fproddiv 14530 The quotient of two finite...
prodsn 14531 A product of a singleton i...
fprod1 14532 A finite product of only o...
prodsnf 14533 A product of a singleton i...
climprod1 14534 The limit of a product ove...
fprodsplit 14535 Split a finite product int...
fprodm1 14536 Separate out the last term...
fprod1p 14537 Separate out the first ter...
fprodp1 14538 Multiply in the last term ...
fprodm1s 14539 Separate out the last term...
fprodp1s 14540 Multiply in the last term ...
prodsns 14541 A product of the singleton...
fprodfac 14542 Factorial using product no...
fprodabs 14543 The absolute value of a fi...
fprodeq0 14544 Anything finite product co...
fprodshft 14545 Shift the index of a finit...
fprodrev 14546 Reversal of a finite produ...
fprodconst 14547 The product of constant te...
fprodn0 14548 A finite product of nonzer...
fprod2dlem 14549 Lemma for ~ fprod2d - indu...
fprod2d 14550 Write a double product as ...
fprodxp 14551 Combine two products into ...
fprodcnv 14552 Transform a product region...
fprodcom2 14553 Interchange order of multi...
fprodcom2OLD 14554 Obsolete proof of ~ fprodc...
fprodcom 14555 Interchange product order....
fprod0diag 14556 Two ways to express "the p...
fproddivf 14557 The quotient of two finite...
fprodsplitf 14558 Split a finite product int...
fprodsplitsn 14559 Separate out a term in a f...
fprodsplit1f 14560 Separate out a term in a f...
fprodn0f 14561 A finite product of nonzer...
fprodclf 14562 Closure of a finite produc...
fprodge0 14563 If all the terms of a fini...
fprodeq0g 14564 Any finite product contain...
fprodge1 14565 If all of the terms of a f...
fprodle 14566 If all the terms of two fi...
fprodmodd 14567 If all factors of two fini...
iprodclim 14568 An infinite product equals...
iprodclim2 14569 A converging product conve...
iprodclim3 14570 The sequence of partial fi...
iprodcl 14571 The product of a non-trivi...
iprodrecl 14572 The product of a non-trivi...
iprodmul 14573 Multiplication of infinite...
risefacval 14578 The value of the rising fa...
fallfacval 14579 The value of the falling f...
risefacval2 14580 One-based value of rising ...
fallfacval2 14581 One-based value of falling...
fallfacval3 14582 A product representation o...
risefaccllem 14583 Lemma for rising factorial...
fallfaccllem 14584 Lemma for falling factoria...
risefaccl 14585 Closure law for rising fac...
fallfaccl 14586 Closure law for falling fa...
rerisefaccl 14587 Closure law for rising fac...
refallfaccl 14588 Closure law for falling fa...
nnrisefaccl 14589 Closure law for rising fac...
zrisefaccl 14590 Closure law for rising fac...
zfallfaccl 14591 Closure law for falling fa...
nn0risefaccl 14592 Closure law for rising fac...
rprisefaccl 14593 Closure law for rising fac...
risefallfac 14594 A relationship between ris...
fallrisefac 14595 A relationship between fal...
risefall0lem 14596 Lemma for ~ risefac0 and ~...
risefac0 14597 The value of the rising fa...
fallfac0 14598 The value of the falling f...
risefacp1 14599 The value of the rising fa...
fallfacp1 14600 The value of the falling f...
risefacp1d 14601 The value of the rising fa...
fallfacp1d 14602 The value of the falling f...
risefac1 14603 The value of rising factor...
fallfac1 14604 The value of falling facto...
risefacfac 14605 Relate rising factorial to...
fallfacfwd 14606 The forward difference of ...
0fallfac 14607 The value of the zero fall...
0risefac 14608 The value of the zero risi...
binomfallfaclem1 14609 Lemma for ~ binomfallfac ....
binomfallfaclem2 14610 Lemma for ~ binomfallfac ....
binomfallfac 14611 A version of the binomial ...
binomrisefac 14612 A version of the binomial ...
fallfacval4 14613 Represent the falling fact...
bcfallfac 14614 Binomial coefficient in te...
fallfacfac 14615 Relate falling factorial t...
bpolylem 14618 Lemma for ~ bpolyval . (C...
bpolyval 14619 The value of the Bernoulli...
bpoly0 14620 The value of the Bernoulli...
bpoly1 14621 The value of the Bernoulli...
bpolycl 14622 Closure law for Bernoulli ...
bpolysum 14623 A sum for Bernoulli polyno...
bpolydiflem 14624 Lemma for ~ bpolydif . (C...
bpolydif 14625 Calculate the difference b...
fsumkthpow 14626 A closed-form expression f...
bpoly2 14627 The Bernoulli polynomials ...
bpoly3 14628 The Bernoulli polynomials ...
bpoly4 14629 The Bernoulli polynomials ...
fsumcube 14630 Express the sum of cubes i...
eftcl 14643 Closure of a term in the s...
reeftcl 14644 The terms of the series ex...
eftabs 14645 The absolute value of a te...
eftval 14646 The value of a term in the...
efcllem 14647 Lemma for ~ efcl . The se...
ef0lem 14648 The series defining the ex...
efval 14649 Value of the exponential f...
esum 14650 Value of Euler's constant ...
eff 14651 Domain and codomain of the...
efcl 14652 Closure law for the expone...
efval2 14653 Value of the exponential f...
efcvg 14654 The series that defines th...
efcvgfsum 14655 Exponential function conve...
reefcl 14656 The exponential function i...
reefcld 14657 The exponential function i...
ere 14658 Euler's constant ` _e ` = ...
ege2le3 14659 Lemma for ~ egt2lt3 . (Co...
ef0 14660 Value of the exponential f...
efcj 14661 Exponential function of a ...
efaddlem 14662 Lemma for ~ efadd (exponen...
efadd 14663 Sum of exponents law for e...
fprodefsum 14664 Move the exponential funct...
efcan 14665 Cancellation of law for ex...
efne0 14666 The exponential function n...
efneg 14667 Exponent of a negative num...
eff2 14668 The exponential function m...
efsub 14669 Difference of exponents la...
efexp 14670 Exponential function to an...
efzval 14671 Value of the exponential f...
efgt0 14672 The exponential function o...
rpefcl 14673 The exponential function o...
rpefcld 14674 The exponential function o...
eftlcvg 14675 The tail series of the exp...
eftlcl 14676 Closure of the sum of an i...
reeftlcl 14677 Closure of the sum of an i...
eftlub 14678 An upper bound on the abso...
efsep 14679 Separate out the next term...
effsumlt 14680 The partial sums of the se...
eft0val 14681 The value of the first ter...
ef4p 14682 Separate out the first fou...
efgt1p2 14683 The exponential function o...
efgt1p 14684 The exponential function o...
efgt1 14685 The exponential function o...
eflt 14686 The exponential function o...
efle 14687 The exponential function o...
reef11 14688 The exponential function o...
reeff1 14689 The exponential function m...
eflegeo 14690 The exponential function o...
sinval 14691 Value of the sine function...
cosval 14692 Value of the cosine functi...
sinf 14693 Domain and codomain of the...
cosf 14694 Domain and codomain of the...
sincl 14695 Closure of the sine functi...
coscl 14696 Closure of the cosine func...
tanval 14697 Value of the tangent funct...
tancl 14698 The closure of the tangent...
sincld 14699 Closure of the sine functi...
coscld 14700 Closure of the cosine func...
tancld 14701 Closure of the tangent fun...
tanval2 14702 Express the tangent functi...
tanval3 14703 Express the tangent functi...
resinval 14704 The sine of a real number ...
recosval 14705 The cosine of a real numbe...
efi4p 14706 Separate out the first fou...
resin4p 14707 Separate out the first fou...
recos4p 14708 Separate out the first fou...
resincl 14709 The sine of a real number ...
recoscl 14710 The cosine of a real numbe...
retancl 14711 The closure of the tangent...
resincld 14712 Closure of the sine functi...
recoscld 14713 Closure of the cosine func...
retancld 14714 Closure of the tangent fun...
sinneg 14715 The sine of a negative is ...
cosneg 14716 The cosines of a number an...
tanneg 14717 The tangent of a negative ...
sin0 14718 Value of the sine function...
cos0 14719 Value of the cosine functi...
tan0 14720 The value of the tangent f...
efival 14721 The exponential function i...
efmival 14722 The exponential function i...
sinhval 14723 Value of the hyperbolic si...
coshval 14724 Value of the hyperbolic co...
resinhcl 14725 The hyperbolic sine of a r...
rpcoshcl 14726 The hyperbolic cosine of a...
recoshcl 14727 The hyperbolic cosine of a...
retanhcl 14728 The hyperbolic tangent of ...
tanhlt1 14729 The hyperbolic tangent of ...
tanhbnd 14730 The hyperbolic tangent of ...
efeul 14731 Eulerian representation of...
efieq 14732 The exponentials of two im...
sinadd 14733 Addition formula for sine....
cosadd 14734 Addition formula for cosin...
tanaddlem 14735 A useful intermediate step...
tanadd 14736 Addition formula for tange...
sinsub 14737 Sine of difference. (Cont...
cossub 14738 Cosine of difference. (Co...
addsin 14739 Sum of sines. (Contribute...
subsin 14740 Difference of sines. (Con...
sinmul 14741 Product of sines can be re...
cosmul 14742 Product of cosines can be ...
addcos 14743 Sum of cosines. (Contribu...
subcos 14744 Difference of cosines. (C...
sincossq 14745 Sine squared plus cosine s...
sin2t 14746 Double-angle formula for s...
cos2t 14747 Double-angle formula for c...
cos2tsin 14748 Double-angle formula for c...
sinbnd 14749 The sine of a real number ...
cosbnd 14750 The cosine of a real numbe...
sinbnd2 14751 The sine of a real number ...
cosbnd2 14752 The cosine of a real numbe...
ef01bndlem 14753 Lemma for ~ sin01bnd and ~...
sin01bnd 14754 Bounds on the sine of a po...
cos01bnd 14755 Bounds on the cosine of a ...
cos1bnd 14756 Bounds on the cosine of 1....
cos2bnd 14757 Bounds on the cosine of 2....
sinltx 14758 The sine of a positive rea...
sin01gt0 14759 The sine of a positive rea...
cos01gt0 14760 The cosine of a positive r...
sin02gt0 14761 The sine of a positive rea...
sincos1sgn 14762 The signs of the sine and ...
sincos2sgn 14763 The signs of the sine and ...
sin4lt0 14764 The sine of 4 is negative....
absefi 14765 The absolute value of the ...
absef 14766 The absolute value of the ...
absefib 14767 A number is real iff its i...
efieq1re 14768 A number whose imaginary e...
demoivre 14769 De Moivre's Formula. Proo...
demoivreALT 14770 Alternate proof of ~ demoi...
eirrlem 14771 Lemma for ~ eirr . (Contr...
eirr 14772 ` _e ` is irrational. (Co...
egt2lt3 14773 Euler's constant ` _e ` = ...
epos 14774 Euler's constant ` _e ` is...
epr 14775 Euler's constant ` _e ` is...
ene0 14776 ` _e ` is not 0. (Contrib...
ene1 14777 ` _e ` is not 1. (Contrib...
xpnnen 14778 The Cartesian product of t...
znnenlem 14779 Lemma for ~ znnen . (Cont...
znnen 14780 The set of integers and th...
qnnen 14781 The rational numbers are c...
rpnnen2lem1 14782 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem2 14783 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem3 14784 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem4 14785 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem5 14786 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem6 14787 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem7 14788 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem8 14789 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem9 14790 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem10 14791 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem11 14792 Lemma for ~ rpnnen2 . (Co...
rpnnen2lem12 14793 Lemma for ~ rpnnen2 . (Co...
rpnnen2 14794 The other half of ~ rpnnen...
rpnnen 14795 The cardinality of the con...
rexpen 14796 The real numbers are equin...
cpnnen 14797 The complex numbers are eq...
rucALT 14798 Alternate proof of ~ ruc ....
ruclem1 14799 Lemma for ~ ruc (the reals...
ruclem2 14800 Lemma for ~ ruc . Orderin...
ruclem3 14801 Lemma for ~ ruc . The con...
ruclem4 14802 Lemma for ~ ruc . Initial...
ruclem6 14803 Lemma for ~ ruc . Domain ...
ruclem7 14804 Lemma for ~ ruc . Success...
ruclem8 14805 Lemma for ~ ruc . The int...
ruclem9 14806 Lemma for ~ ruc . The fir...
ruclem10 14807 Lemma for ~ ruc . Every f...
ruclem11 14808 Lemma for ~ ruc . Closure...
ruclem12 14809 Lemma for ~ ruc . The sup...
ruclem13 14810 Lemma for ~ ruc . There i...
ruc 14811 The set of positive intege...
resdomq 14812 The set of rationals is st...
aleph1re 14813 There are at least aleph-o...
aleph1irr 14814 There are at least aleph-o...
cnso 14815 The complex numbers can be...
sqr2irrlem 14816 Lemma for irrationality of...
sqrt2irr 14817 The square root of 2 is ir...
sqrt2re 14818 The square root of 2 exist...
nthruc 14819 The sequence ` NN ` , ` ZZ...
nthruz 14820 The sequence ` NN ` , ` NN...
divides 14823 Define the divides relatio...
dvdsval2 14824 One nonzero integer divide...
dvdsval3 14825 One nonzero integer divide...
dvdszrcl 14826 Reverse closure for the di...
nndivdvds 14827 Strong form of ~ dvdsval2 ...
nndivides 14828 Definition of the divides ...
moddvds 14829 Two ways to say ` A == B `...
dvds0lem 14830 A lemma to assist theorems...
dvds1lem 14831 A lemma to assist theorems...
dvds2lem 14832 A lemma to assist theorems...
iddvds 14833 An integer divides itself....
1dvds 14834 1 divides any integer. Th...
dvds0 14835 Any integer divides 0. Th...
negdvdsb 14836 An integer divides another...
dvdsnegb 14837 An integer divides another...
absdvdsb 14838 An integer divides another...
dvdsabsb 14839 An integer divides another...
0dvds 14840 Only 0 is divisible by 0. ...
dvdsmul1 14841 An integer divides a multi...
dvdsmul2 14842 An integer divides a multi...
iddvdsexp 14843 An integer divides a posit...
muldvds1 14844 If a product divides an in...
muldvds2 14845 If a product divides an in...
dvdscmul 14846 Multiplication by a consta...
dvdsmulc 14847 Multiplication by a consta...
dvdscmulr 14848 Cancellation law for the d...
dvdsmulcr 14849 Cancellation law for the d...
summodnegmod 14850 The sum of two integers mo...
modmulconst 14851 Constant multiplication in...
dvds2ln 14852 If an integer divides each...
dvds2add 14853 If an integer divides each...
dvds2sub 14854 If an integer divides each...
dvds2subd 14855 Natural deduction form of ...
dvdstr 14856 The divides relation is tr...
dvdsmultr1 14857 If an integer divides anot...
dvdsmultr1d 14858 Natural deduction form of ...
dvdsmultr2 14859 If an integer divides anot...
ordvdsmul 14860 If an integer divides eith...
dvdssub2 14861 If an integer divides a di...
dvdsadd 14862 An integer divides another...
dvdsaddr 14863 An integer divides another...
dvdssub 14864 An integer divides another...
dvdssubr 14865 An integer divides another...
dvdsadd2b 14866 Adding a multiple of the b...
dvdsaddre2b 14867 Adding a multiple of the b...
fsumdvds 14868 If every term in a sum is ...
dvdslelem 14869 Lemma for ~ dvdsle . (Con...
dvdsle 14870 The divisors of a positive...
dvdsleabs 14871 The divisors of a nonzero ...
dvdsleabs2 14872 Transfer divisibility to a...
dvdsabseq 14873 If two integers divide eac...
dvdseq 14874 If two nonnegative integer...
divconjdvds 14875 If a nonzero integer ` M `...
dvdsdivcl 14876 The complement of a diviso...
dvdsflip 14877 An involution of the divis...
dvdsssfz1 14878 The set of divisors of a n...
dvds1 14879 The only nonnegative integ...
alzdvds 14880 Only 0 is divisible by all...
dvdsext 14881 Poset extensionality for d...
fzm1ndvds 14882 No number between ` 1 ` an...
fzo0dvdseq 14883 Zero is the only one of th...
fzocongeq 14884 Two different elements of ...
addmodlteqALT 14885 Two nonnegative integers l...
dvdsfac 14886 A positive integer divides...
dvdsexp 14887 A power divides a power wi...
dvdsmod 14888 Any number ` K ` whose mod...
mulmoddvds 14889 If an integer is divisible...
3dvds 14890 A rule for divisibility by...
3dvdsOLD 14891 Obsolete version of ~ 3dvd...
3dvdsdec 14892 A decimal number is divisi...
3dvdsdecOLD 14893 Obsolete proof of ~ 3dvdsd...
3dvds2dec 14894 A decimal number is divisi...
3dvds2decOLD 14895 Old version of ~ 3dvds2dec...
fprodfvdvdsd 14896 A finite product of intege...
fproddvdsd 14897 A finite product of intege...
evenelz 14898 An even number is an integ...
zeo3 14899 An integer is even or odd....
zeo4 14900 An integer is even or odd ...
zeneo 14901 No even integer equals an ...
odd2np1lem 14902 Lemma for ~ odd2np1 . (Co...
odd2np1 14903 An integer is odd iff it i...
even2n 14904 An integer is even iff it ...
oddm1even 14905 An integer is odd iff its ...
oddp1even 14906 An integer is odd iff its ...
oexpneg 14907 The exponential of the neg...
mod2eq0even 14908 An integer is 0 modulo 2 i...
mod2eq1n2dvds 14909 An integer is 1 modulo 2 i...
oddnn02np1 14910 A nonnegative integer is o...
oddge22np1 14911 An integer greater than on...
evennn02n 14912 A nonnegative integer is e...
evennn2n 14913 A positive integer is even...
2tp1odd 14914 A number which is twice an...
mulsucdiv2z 14915 An integer multiplied with...
sqoddm1div8z 14916 A squared odd number minus...
2teven 14917 A number which is twice an...
zeo5 14918 An integer is either even ...
evend2 14919 An integer is even iff its...
oddp1d2 14920 An integer is odd iff its ...
zob 14921 Alternate characterization...
oddm1d2 14922 An integer is odd iff its ...
ltoddhalfle 14923 An integer is less than ha...
halfleoddlt 14924 An integer is greater than...
opoe 14925 The sum of two odds is eve...
omoe 14926 The difference of two odds...
opeo 14927 The sum of an odd and an e...
omeo 14928 The difference of an odd a...
m1expe 14929 Exponentiation of -1 by an...
m1expo 14930 Exponentiation of -1 by an...
m1exp1 14931 Exponentiation of negative...
nn0enne 14932 A positive integer is an e...
nn0ehalf 14933 The half of an even nonneg...
nnehalf 14934 The half of an even positi...
nn0o1gt2 14935 An odd nonnegative integer...
nno 14936 An alternate characterizat...
nn0o 14937 An alternate characterizat...
nn0ob 14938 Alternate characterization...
nn0oddm1d2 14939 A positive integer is odd ...
nnoddm1d2 14940 A positive integer is odd ...
z0even 14941 0 is even. (Contributed b...
n2dvds1 14942 2 does not divide 1 (commo...
n2dvdsm1 14943 2 does not divide -1. Tha...
z2even 14944 2 is even. (Contributed b...
n2dvds3 14945 2 does not divide 3, i.e. ...
z4even 14946 4 is an even number. (Con...
4dvdseven 14947 An integer which is divisi...
sumeven 14948 If every term in a sum is ...
sumodd 14949 If every term in a sum is ...
evensumodd 14950 If every term in a sum wit...
oddsumodd 14951 If every term in a sum wit...
pwp1fsum 14952 The n-th power of a number...
oddpwp1fsum 14953 An odd power of a number i...
divalglem0 14954 Lemma for ~ divalg . (Con...
divalglem1 14955 Lemma for ~ divalg . (Con...
divalglem2 14956 Lemma for ~ divalg . (Con...
divalglem4 14957 Lemma for ~ divalg . (Con...
divalglem5 14958 Lemma for ~ divalg . (Con...
divalglem6 14959 Lemma for ~ divalg . (Con...
divalglem7 14960 Lemma for ~ divalg . (Con...
divalglem8 14961 Lemma for ~ divalg . (Con...
divalglem9 14962 Lemma for ~ divalg . (Con...
divalglem10 14963 Lemma for ~ divalg . (Con...
divalg 14964 The division algorithm (th...
divalgb 14965 Express the division algor...
divalg2 14966 The division algorithm (th...
divalgmod 14967 The result of the ` mod ` ...
divalgmodOLD 14968 Obsolete proof of ~ divalg...
divalgmodcl 14969 The result of the ` mod ` ...
modremain 14970 The result of the modulo o...
ndvdssub 14971 Corollary of the division ...
ndvdsadd 14972 Corollary of the division ...
ndvdsp1 14973 Special case of ~ ndvdsadd...
ndvdsi 14974 A quick test for non-divis...
flodddiv4 14975 The floor of an odd intege...
fldivndvdslt 14976 The floor of an integer di...
flodddiv4lt 14977 The floor of an odd number...
flodddiv4t2lthalf 14978 The floor of an odd number...
bitsfval 14983 Expand the definition of t...
bitsval 14984 Expand the definition of t...
bitsval2 14985 Expand the definition of t...
bitsss 14986 The set of bits of an inte...
bitsf 14987 The ` bits ` function is a...
bits0 14988 Value of the zeroth bit. ...
bits0e 14989 The zeroth bit of an even ...
bits0o 14990 The zeroth bit of an odd n...
bitsp1 14991 The ` M + 1 ` -th bit of `...
bitsp1e 14992 The ` M + 1 ` -th bit of `...
bitsp1o 14993 The ` M + 1 ` -th bit of `...
bitsfzolem 14994 Lemma for ~ bitsfzo . (Co...
bitsfzo 14995 The bits of a number are a...
bitsmod 14996 Truncating the bit sequenc...
bitsfi 14997 Every number is associated...
bitscmp 14998 The bit complement of ` N ...
0bits 14999 The bits of zero. (Contri...
m1bits 15000 The bits of negative one. ...
bitsinv1lem 15001 Lemma for ~ bitsinv1 . (C...
bitsinv1 15002 There is an explicit inver...
bitsinv2 15003 There is an explicit inver...
bitsf1ocnv 15004 The ` bits ` function rest...
bitsf1o 15005 The ` bits ` function rest...
bitsf1 15006 The ` bits ` function is a...
2ebits 15007 The bits of a power of two...
bitsinv 15008 The inverse of the ` bits ...
bitsinvp1 15009 Recursive definition of th...
sadadd2lem2 15010 The core of the proof of ~...
sadfval 15012 Define the addition of two...
sadcf 15013 The carry sequence is a se...
sadc0 15014 The initial element of the...
sadcp1 15015 The carry sequence (which ...
sadval 15016 The full adder sequence is...
sadcaddlem 15017 Lemma for ~ sadcadd . (Co...
sadcadd 15018 Non-recursive definition o...
sadadd2lem 15019 Lemma for ~ sadadd2 . (Co...
sadadd2 15020 Sum of initial segments of...
sadadd3 15021 Sum of initial segments of...
sadcl 15022 The sum of two sequences i...
sadcom 15023 The adder sequence functio...
saddisjlem 15024 Lemma for ~ sadadd . (Con...
saddisj 15025 The sum of disjoint sequen...
sadaddlem 15026 Lemma for ~ sadadd . (Con...
sadadd 15027 For sequences that corresp...
sadid1 15028 The adder sequence functio...
sadid2 15029 The adder sequence functio...
sadasslem 15030 Lemma for ~ sadass . (Con...
sadass 15031 Sequence addition is assoc...
sadeq 15032 Any element of a sequence ...
bitsres 15033 Restrict the bits of a num...
bitsuz 15034 The bits of a number are a...
bitsshft 15035 Shifting a bit sequence to...
smufval 15037 The multiplication of two ...
smupf 15038 The sequence of partial su...
smup0 15039 The initial element of the...
smupp1 15040 The initial element of the...
smuval 15041 Define the addition of two...
smuval2 15042 The partial sum sequence s...
smupvallem 15043 If ` A ` only has elements...
smucl 15044 The product of two sequenc...
smu01lem 15045 Lemma for ~ smu01 and ~ sm...
smu01 15046 Multiplication of a sequen...
smu02 15047 Multiplication of a sequen...
smupval 15048 Rewrite the elements of th...
smup1 15049 Rewrite ~ smupp1 using onl...
smueqlem 15050 Any element of a sequence ...
smueq 15051 Any element of a sequence ...
smumullem 15052 Lemma for ~ smumul . (Con...
smumul 15053 For sequences that corresp...
gcdval 15056 The value of the ` gcd ` o...
gcd0val 15057 The value, by convention, ...
gcdn0val 15058 The value of the ` gcd ` o...
gcdcllem1 15059 Lemma for ~ gcdn0cl , ~ gc...
gcdcllem2 15060 Lemma for ~ gcdn0cl , ~ gc...
gcdcllem3 15061 Lemma for ~ gcdn0cl , ~ gc...
gcdn0cl 15062 Closure of the ` gcd ` ope...
gcddvds 15063 The gcd of two integers di...
dvdslegcd 15064 An integer which divides b...
nndvdslegcd 15065 A positive integer which d...
gcdcl 15066 Closure of the ` gcd ` ope...
gcdnncl 15067 Closure of the ` gcd ` ope...
gcdcld 15068 Closure of the ` gcd ` ope...
gcd2n0cl 15069 Closure of the ` gcd ` ope...
zeqzmulgcd 15070 An integer is the product ...
divgcdz 15071 An integer divided by the ...
gcdf 15072 Domain and codomain of the...
gcdcom 15073 The ` gcd ` operator is co...
divgcdnn 15074 A positive integer divided...
divgcdnnr 15075 A positive integer divided...
gcdeq0 15076 The gcd of two integers is...
gcdn0gt0 15077 The gcd of two integers is...
gcd0id 15078 The gcd of 0 and an intege...
gcdid0 15079 The gcd of an integer and ...
nn0gcdid0 15080 The gcd of a nonnegative i...
gcdneg 15081 Negating one operand of th...
neggcd 15082 Negating one operand of th...
gcdaddmlem 15083 Lemma for ~ gcdaddm . (Co...
gcdaddm 15084 Adding a multiple of one o...
gcdadd 15085 The GCD of two numbers is ...
gcdid 15086 The gcd of a number and it...
gcd1 15087 The gcd of a number with 1...
gcdabs 15088 The gcd of two integers is...
gcdabs1 15089 ` gcd ` of the absolute va...
gcdabs2 15090 ` gcd ` of the absolute va...
modgcd 15091 The gcd remains unchanged ...
1gcd 15092 The GCD of one and an inte...
6gcd4e2 15093 The greatest common diviso...
bezoutlem1 15094 Lemma for ~ bezout . (Con...
bezoutlem2 15095 Lemma for ~ bezout . (Con...
bezoutlem3 15096 Lemma for ~ bezout . (Con...
bezoutlem4 15097 Lemma for ~ bezout . (Con...
bezout 15098 Bézout's identity: ...
dvdsgcd 15099 An integer which divides e...
dvdsgcdb 15100 Biconditional form of ~ dv...
dfgcd2 15101 Alternate definition of th...
gcdass 15102 Associative law for ` gcd ...
mulgcd 15103 Distribute multiplication ...
absmulgcd 15104 Distribute absolute value ...
mulgcdr 15105 Reverse distribution law f...
gcddiv 15106 Division law for GCD. (Con...
gcdmultiple 15107 The GCD of a multiple of a...
gcdmultiplez 15108 Extend ~ gcdmultiple so ` ...
gcdzeq 15109 A positive integer ` A ` i...
gcdeq 15110 ` A ` is equal to its gcd ...
dvdssqim 15111 Unidirectional form of ~ d...
dvdsmulgcd 15112 A divisibility equivalent ...
rpmulgcd 15113 If ` K ` and ` M ` are rel...
rplpwr 15114 If ` A ` and ` B ` are rel...
rppwr 15115 If ` A ` and ` B ` are rel...
sqgcd 15116 Square distributes over GC...
dvdssqlem 15117 Lemma for ~ dvdssq . (Con...
dvdssq 15118 Two numbers are divisible ...
bezoutr 15119 Partial converse to ~ bezo...
bezoutr1 15120 Converse of ~ bezout for t...
nn0seqcvgd 15121 A strictly-decreasing nonn...
seq1st 15122 A sequence whose iteration...
algr0 15123 The value of the algorithm...
algrf 15124 An algorithm is a step fun...
algrp1 15125 The value of the algorithm...
alginv 15126 If ` I ` is an invariant o...
algcvg 15127 One way to prove that an a...
algcvgblem 15128 Lemma for ~ algcvgb . (Co...
algcvgb 15129 Two ways of expressing tha...
algcvga 15130 The countdown function ` C...
algfx 15131 If ` F ` reaches a fixed p...
eucalgval2 15132 The value of the step func...
eucalgval 15133 Euclid's Algorithm ~ eucal...
eucalgf 15134 Domain and codomain of the...
eucalginv 15135 The invariant of the step ...
eucalglt 15136 The second member of the s...
eucalgcvga 15137 Once Euclid's Algorithm ha...
eucalg 15138 Euclid's Algorithm compute...
lcmval 15143 Value of the ` lcm ` opera...
lcmcom 15144 The ` lcm ` operator is co...
lcm0val 15145 The value, by convention, ...
lcmn0val 15146 The value of the ` lcm ` o...
lcmcllem 15147 Lemma for ~ lcmn0cl and ~ ...
lcmn0cl 15148 Closure of the ` lcm ` ope...
dvdslcm 15149 The lcm of two integers is...
lcmledvds 15150 A positive integer which b...
lcmeq0 15151 The lcm of two integers is...
lcmcl 15152 Closure of the ` lcm ` ope...
gcddvdslcm 15153 The greatest common diviso...
lcmneg 15154 Negating one operand of th...
neglcm 15155 Negating one operand of th...
lcmabs 15156 The lcm of two integers is...
lcmgcdlem 15157 Lemma for ~ lcmgcd and ~ l...
lcmgcd 15158 The product of two numbers...
lcmdvds 15159 The lcm of two integers di...
lcmid 15160 The lcm of an integer and ...
lcm1 15161 The lcm of an integer and ...
lcmgcdnn 15162 The product of two positiv...
lcmgcdeq 15163 Two integers' absolute val...
lcmdvdsb 15164 Biconditional form of ~ lc...
lcmass 15165 Associative law for ` lcm ...
3lcm2e6woprm 15166 The least common multiple ...
6lcm4e12 15167 The least common multiple ...
absproddvds 15168 The absolute value of the ...
absprodnn 15169 The absolute value of the ...
fissn0dvds 15170 For each finite subset of ...
fissn0dvdsn0 15171 For each finite subset of ...
lcmfval 15172 Value of the ` _lcm ` func...
lcmf0val 15173 The value, by convention, ...
lcmfn0val 15174 The value of the ` _lcm ` ...
lcmfnnval 15175 The value of the ` _lcm ` ...
lcmfcllem 15176 Lemma for ~ lcmfn0cl and ~...
lcmfn0cl 15177 Closure of the ` _lcm ` fu...
lcmfpr 15178 The value of the ` _lcm ` ...
lcmfcl 15179 Closure of the ` _lcm ` fu...
lcmfnncl 15180 Closure of the ` _lcm ` fu...
lcmfeq0b 15181 The least common multiple ...
dvdslcmf 15182 The least common multiple ...
lcmfledvds 15183 A positive integer which i...
lcmf 15184 Characterization of the le...
lcmf0 15185 The least common multiple ...
lcmfsn 15186 The least common multiple ...
lcmftp 15187 The least common multiple ...
lcmfunsnlem1 15188 Lemma for ~ lcmfdvds and ~...
lcmfunsnlem2lem1 15189 Lemma 1 for ~ lcmfunsnlem2...
lcmfunsnlem2lem2 15190 Lemma 2 for ~ lcmfunsnlem2...
lcmfunsnlem2 15191 Lemma for ~ lcmfunsn and ~...
lcmfunsnlem 15192 Lemma for ~ lcmfdvds and ~...
lcmfdvds 15193 The least common multiple ...
lcmfdvdsb 15194 Biconditional form of ~ lc...
lcmfunsn 15195 The ` _lcm ` function for ...
lcmfun 15196 The ` _lcm ` function for ...
lcmfass 15197 Associative law for the ` ...
lcmf2a3a4e12 15198 The least common multiple ...
lcmflefac 15199 The least common multiple ...
coprmgcdb 15200 Two positive integers are ...
ncoprmgcdne1b 15201 Two positive integers are ...
ncoprmgcdgt1b 15202 Two positive integers are ...
coprmdvds1 15203 If two positive integers a...
coprmdvds 15204 Euclid's Lemma (see ProofW...
coprmdvdsOLD 15205 If an integer divides the ...
coprmdvds2 15206 If an integer is divisible...
mulgcddvds 15207 One half of ~ rpmulgcd2 , ...
rpmulgcd2 15208 If ` M ` is relatively pri...
qredeq 15209 Two equal reduced fraction...
qredeu 15210 Every rational number has ...
rpmul 15211 If ` K ` is relatively pri...
rpdvds 15212 If ` K ` is relatively pri...
coprmprod 15213 The product of the element...
coprmproddvdslem 15214 Lemma for ~ coprmproddvds ...
coprmproddvds 15215 If a positive integer is d...
congr 15216 Definition of congruence b...
divgcdcoprm0 15217 Integers divided by gcd ar...
divgcdcoprmex 15218 Integers divided by gcd ar...
cncongr1 15219 One direction of the bicon...
cncongr2 15220 The other direction of the...
cncongr 15221 Cancellability of Congruen...
cncongrcoprm 15222 Corollary 1 of Cancellabil...
isprm 15225 The predicate "is a prime ...
prmnn 15226 A prime number is a positi...
prmz 15227 A prime number is an integ...
prmssnn 15228 The prime numbers are a su...
prmex 15229 The set of prime numbers e...
1nprm 15230 1 is not a prime number. ...
1idssfct 15231 The positive divisors of a...
isprm2lem 15232 Lemma for ~ isprm2 . (Con...
isprm2 15233 The predicate "is a prime ...
isprm3 15234 The predicate "is a prime ...
isprm4 15235 The predicate "is a prime ...
prmind2 15236 A variation on ~ prmind as...
prmind 15237 Perform induction over the...
dvdsprime 15238 If ` M ` divides a prime, ...
nprm 15239 A product of two integers ...
nprmi 15240 An inference for composite...
dvdsnprmd 15241 If a number is divisible b...
prm2orodd 15242 A prime number is either 2...
2prm 15243 2 is a prime number. (Con...
3prm 15244 3 is a prime number. (Con...
4nprm 15245 4 is not a prime number. ...
prmuz2 15246 A prime number is an integ...
prmgt1 15247 A prime number is an integ...
prmm2nn0 15248 Subtracting 2 from a prime...
oddprmgt2 15249 An odd prime is greater th...
oddprmge3 15250 An odd prime is greater th...
prmn2uzge3OLD 15251 Obsolete version of ~ oddp...
sqnprm 15252 A square is never prime. ...
dvdsprm 15253 An integer greater than or...
exprmfct 15254 Every integer greater than...
prmdvdsfz 15255 Each integer greater than ...
nprmdvds1 15256 No prime number divides 1....
isprm5 15257 One need only check prime ...
isprm7 15258 One need only check prime ...
maxprmfct 15259 The set of prime factors o...
divgcdodd 15260 Either ` A / ( A gcd B ) `...
coprm 15261 A prime number either divi...
prmrp 15262 Unequal prime numbers are ...
euclemma 15263 Euclid's lemma. A prime n...
isprm6 15264 A number is prime iff it s...
prmdvdsexp 15265 A prime divides a positive...
prmdvdsexpb 15266 A prime divides a positive...
prmdvdsexpr 15267 If a prime divides a nonne...
prmexpb 15268 Two positive prime powers ...
prmfac1 15269 The factorial of a number ...
rpexp 15270 If two numbers ` A ` and `...
rpexp1i 15271 Relative primality passes ...
rpexp12i 15272 Relative primality passes ...
prmndvdsfaclt 15273 A prime number does not di...
ncoprmlnprm 15274 If two positive integers a...
cncongrprm 15275 Corollary 2 of Cancellabil...
isevengcd2 15276 The predicate "is an even ...
isoddgcd1 15277 The predicate "is an odd n...
3lcm2e6 15278 The least common multiple ...
qnumval 15283 Value of the canonical num...
qdenval 15284 Value of the canonical den...
qnumdencl 15285 Lemma for ~ qnumcl and ~ q...
qnumcl 15286 The canonical numerator of...
qdencl 15287 The canonical denominator ...
fnum 15288 Canonical numerator define...
fden 15289 Canonical denominator defi...
qnumdenbi 15290 Two numbers are the canoni...
qnumdencoprm 15291 The canonical representati...
qeqnumdivden 15292 Recover a rational number ...
qmuldeneqnum 15293 Multiplying a rational by ...
divnumden 15294 Calculate the reduced form...
divdenle 15295 Reducing a quotient never ...
qnumgt0 15296 A rational is positive iff...
qgt0numnn 15297 A rational is positive iff...
nn0gcdsq 15298 Squaring commutes with GCD...
zgcdsq 15299 ~ nn0gcdsq extended to int...
numdensq 15300 Squaring a rational square...
numsq 15301 Square commutes with canon...
densq 15302 Square commutes with canon...
qden1elz 15303 A rational is an integer i...
zsqrtelqelz 15304 If an integer has a ration...
nonsq 15305 Any integer strictly betwe...
phival 15310 Value of the Euler ` phi `...
phicl2 15311 Bounds and closure for the...
phicl 15312 Closure for the value of t...
phibndlem 15313 Lemma for ~ phibnd . (Con...
phibnd 15314 A slightly tighter bound o...
phicld 15315 Closure for the value of t...
phi1 15316 Value of the Euler ` phi `...
dfphi2 15317 Alternate definition of th...
hashdvds 15318 The number of numbers in a...
phiprmpw 15319 Value of the Euler ` phi `...
phiprm 15320 Value of the Euler ` phi `...
crth 15321 The Chinese Remainder Theo...
phimullem 15322 Lemma for ~ phimul . (Con...
phimul 15323 The Euler ` phi ` function...
eulerthlem1 15324 Lemma for ~ eulerth . (Co...
eulerthlem2 15325 Lemma for ~ eulerth . (Co...
eulerth 15326 Euler's theorem, a general...
fermltl 15327 Fermat's little theorem. ...
prmdiv 15328 Show an explicit expressio...
prmdiveq 15329 The modular inverse of ` A...
prmdivdiv 15330 The (modular) inverse of t...
hashgcdlem 15331 A correspondence between e...
hashgcdeq 15332 Number of initial positive...
phisum 15333 The divisor sum identity o...
odzval 15334 Value of the order functio...
odzcllem 15335 - Lemma for ~ odzcl , show...
odzcl 15336 The order of a group eleme...
odzid 15337 Any element raised to the ...
odzdvds 15338 The only powers of ` A ` t...
odzphi 15339 The order of any group ele...
modprm1div 15340 A prime number divides an ...
m1dvdsndvds 15341 If an integer minus 1 is d...
modprminv 15342 Show an explicit expressio...
modprminveq 15343 The modular inverse of ` A...
vfermltl 15344 Variant of Fermat's little...
vfermltlALT 15345 Alternate proof of ~ vferm...
powm2modprm 15346 If an integer minus 1 is d...
reumodprminv 15347 For any prime number and f...
modprm0 15348 For two positive integers ...
nnnn0modprm0 15349 For a positive integer and...
modprmn0modprm0 15350 For an integer not being 0...
coprimeprodsq 15351 If three numbers are copri...
coprimeprodsq2 15352 If three numbers are copri...
oddprm 15353 A prime not equal to ` 2 `...
nnoddn2prm 15354 A prime not equal to ` 2 `...
oddn2prm 15355 A prime not equal to ` 2 `...
nnoddn2prmb 15356 A number is a prime number...
prm23lt5 15357 A prime less than 5 is eit...
prm23ge5 15358 A prime is either 2 or 3 o...
pythagtriplem1 15359 Lemma for ~ pythagtrip . ...
pythagtriplem2 15360 Lemma for ~ pythagtrip . ...
pythagtriplem3 15361 Lemma for ~ pythagtrip . ...
pythagtriplem4 15362 Lemma for ~ pythagtrip . ...
pythagtriplem10 15363 Lemma for ~ pythagtrip . ...
pythagtriplem6 15364 Lemma for ~ pythagtrip . ...
pythagtriplem7 15365 Lemma for ~ pythagtrip . ...
pythagtriplem8 15366 Lemma for ~ pythagtrip . ...
pythagtriplem9 15367 Lemma for ~ pythagtrip . ...
pythagtriplem11 15368 Lemma for ~ pythagtrip . ...
pythagtriplem12 15369 Lemma for ~ pythagtrip . ...
pythagtriplem13 15370 Lemma for ~ pythagtrip . ...
pythagtriplem14 15371 Lemma for ~ pythagtrip . ...
pythagtriplem15 15372 Lemma for ~ pythagtrip . ...
pythagtriplem16 15373 Lemma for ~ pythagtrip . ...
pythagtriplem17 15374 Lemma for ~ pythagtrip . ...
pythagtriplem18 15375 Lemma for ~ pythagtrip . ...
pythagtriplem19 15376 Lemma for ~ pythagtrip . ...
pythagtrip 15377 Parameterize the Pythagore...
iserodd 15378 Collect the odd terms in a...
pclem 15381 - Lemma for the prime powe...
pcprecl 15382 Closure of the prime power...
pcprendvds 15383 Non-divisibility property ...
pcprendvds2 15384 Non-divisibility property ...
pcpre1 15385 Value of the prime power p...
pcpremul 15386 Multiplicative property of...
pcval 15387 The value of the prime pow...
pceulem 15388 Lemma for ~ pceu . (Contr...
pceu 15389 Uniqueness for the prime p...
pczpre 15390 Connect the prime count pr...
pczcl 15391 Closure of the prime power...
pccl 15392 Closure of the prime power...
pccld 15393 Closure of the prime power...
pcmul 15394 Multiplication property of...
pcdiv 15395 Division property of the p...
pcqmul 15396 Multiplication property of...
pc0 15397 The value of the prime pow...
pc1 15398 Value of the prime count f...
pcqcl 15399 Closure of the general pri...
pcqdiv 15400 Division property of the p...
pcrec 15401 Prime power of a reciproca...
pcexp 15402 Prime power of an exponent...
pcxcl 15403 Extended real closure of t...
pcge0 15404 The prime count of an inte...
pczdvds 15405 Defining property of the p...
pcdvds 15406 Defining property of the p...
pczndvds 15407 Defining property of the p...
pcndvds 15408 Defining property of the p...
pczndvds2 15409 The remainder after dividi...
pcndvds2 15410 The remainder after dividi...
pcdvdsb 15411 ` P ^ A ` divides ` N ` if...
pcelnn 15412 There are a positive numbe...
pceq0 15413 There are zero powers of a...
pcidlem 15414 The prime count of a prime...
pcid 15415 The prime count of a prime...
pcneg 15416 The prime count of a negat...
pcabs 15417 The prime count of an abso...
pcdvdstr 15418 The prime count increases ...
pcgcd1 15419 The prime count of a GCD i...
pcgcd 15420 The prime count of a GCD i...
pc2dvds 15421 A characterization of divi...
pc11 15422 The prime count function, ...
pcz 15423 The prime count function c...
pcprmpw2 15424 Self-referential expressio...
pcprmpw 15425 Self-referential expressio...
dvdsprmpweq 15426 If a positive integer divi...
dvdsprmpweqnn 15427 If an integer greater than...
dvdsprmpweqle 15428 If a positive integer divi...
difsqpwdvds 15429 If the difference of two s...
pcaddlem 15430 Lemma for ~ pcadd . The o...
pcadd 15431 An inequality for the prim...
pcadd2 15432 The inequality of ~ pcadd ...
pcmptcl 15433 Closure for the prime powe...
pcmpt 15434 Construct a function with ...
pcmpt2 15435 Dividing two prime count m...
pcmptdvds 15436 The partial products of th...
pcprod 15437 The product of the primes ...
sumhash 15438 The sum of 1 over a set is...
fldivp1 15439 The difference between the...
pcfaclem 15440 Lemma for ~ pcfac . (Cont...
pcfac 15441 Calculate the prime count ...
pcbc 15442 Calculate the prime count ...
qexpz 15443 If a power of a rational n...
expnprm 15444 A second or higher power o...
oddprmdvds 15445 Every positive integer whi...
prmpwdvds 15446 A relation involving divis...
pockthlem 15447 Lemma for ~ pockthg . (Co...
pockthg 15448 The generalized Pocklingto...
pockthi 15449 Pocklington's theorem, whi...
unbenlem 15450 Lemma for ~ unben . (Cont...
unben 15451 An unbounded set of positi...
infpnlem1 15452 Lemma for ~ infpn . The s...
infpnlem2 15453 Lemma for ~ infpn . For a...
infpn 15454 There exist infinitely man...
infpn2 15455 There exist infinitely man...
prmunb 15456 The primes are unbounded. ...
prminf 15457 There are an infinite numb...
prmreclem1 15458 Lemma for ~ prmrec . Prop...
prmreclem2 15459 Lemma for ~ prmrec . Ther...
prmreclem3 15460 Lemma for ~ prmrec . The ...
prmreclem4 15461 Lemma for ~ prmrec . Show...
prmreclem5 15462 Lemma for ~ prmrec . Here...
prmreclem6 15463 Lemma for ~ prmrec . If t...
prmrec 15464 The sum of the reciprocals...
1arithlem1 15465 Lemma for ~ 1arith . (Con...
1arithlem2 15466 Lemma for ~ 1arith . (Con...
1arithlem3 15467 Lemma for ~ 1arith . (Con...
1arithlem4 15468 Lemma for ~ 1arith . (Con...
1arith 15469 Fundamental theorem of ari...
1arith2 15470 Fundamental theorem of ari...
elgz 15473 Elementhood in the gaussia...
gzcn 15474 A gaussian integer is a co...
zgz 15475 An integer is a gaussian i...
igz 15476 ` _i ` is a gaussian integ...
gznegcl 15477 The gaussian integers are ...
gzcjcl 15478 The gaussian integers are ...
gzaddcl 15479 The gaussian integers are ...
gzmulcl 15480 The gaussian integers are ...
gzreim 15481 Construct a gaussian integ...
gzsubcl 15482 The gaussian integers are ...
gzabssqcl 15483 The squared norm of a gaus...
4sqlem5 15484 Lemma for ~ 4sq . (Contri...
4sqlem6 15485 Lemma for ~ 4sq . (Contri...
4sqlem7 15486 Lemma for ~ 4sq . (Contri...
4sqlem8 15487 Lemma for ~ 4sq . (Contri...
4sqlem9 15488 Lemma for ~ 4sq . (Contri...
4sqlem10 15489 Lemma for ~ 4sq . (Contri...
4sqlem1 15490 Lemma for ~ 4sq . The set...
4sqlem2 15491 Lemma for ~ 4sq . Change ...
4sqlem3 15492 Lemma for ~ 4sq . Suffici...
4sqlem4a 15493 Lemma for ~ 4sqlem4 . (Co...
4sqlem4 15494 Lemma for ~ 4sq . We can ...
mul4sqlem 15495 Lemma for ~ mul4sq : algeb...
mul4sq 15496 Euler's four-square identi...
4sqlem11 15497 Lemma for ~ 4sq . Use the...
4sqlem12 15498 Lemma for ~ 4sq . For any...
4sqlem13 15499 Lemma for ~ 4sq . (Contri...
4sqlem14 15500 Lemma for ~ 4sq . (Contri...
4sqlem15 15501 Lemma for ~ 4sq . (Contri...
4sqlem16 15502 Lemma for ~ 4sq . (Contri...
4sqlem17 15503 Lemma for ~ 4sq . (Contri...
4sqlem18 15504 Lemma for ~ 4sq . Inducti...
4sqlem19 15505 Lemma for ~ 4sq . The pro...
4sq 15506 Lagrange's four-square the...
vdwapfval 15513 Define the arithmetic prog...
vdwapf 15514 The arithmetic progression...
vdwapval 15515 Value of the arithmetic pr...
vdwapun 15516 Remove the first element o...
vdwapid1 15517 The first element of an ar...
vdwap0 15518 Value of a length-1 arithm...
vdwap1 15519 Value of a length-1 arithm...
vdwmc 15520 The predicate " The ` <. R...
vdwmc2 15521 Expand out the definition ...
vdwpc 15522 The predicate " The colori...
vdwlem1 15523 Lemma for ~ vdw . (Contri...
vdwlem2 15524 Lemma for ~ vdw . (Contri...
vdwlem3 15525 Lemma for ~ vdw . (Contri...
vdwlem4 15526 Lemma for ~ vdw . (Contri...
vdwlem5 15527 Lemma for ~ vdw . (Contri...
vdwlem6 15528 Lemma for ~ vdw . (Contri...
vdwlem7 15529 Lemma for ~ vdw . (Contri...
vdwlem8 15530 Lemma for ~ vdw . (Contri...
vdwlem9 15531 Lemma for ~ vdw . (Contri...
vdwlem10 15532 Lemma for ~ vdw . Set up ...
vdwlem11 15533 Lemma for ~ vdw . (Contri...
vdwlem12 15534 Lemma for ~ vdw . ` K = 2 ...
vdwlem13 15535 Lemma for ~ vdw . Main in...
vdw 15536 Van der Waerden's theorem....
vdwnnlem1 15537 Corollary of ~ vdw , and l...
vdwnnlem2 15538 Lemma for ~ vdwnn . The s...
vdwnnlem3 15539 Lemma for ~ vdwnn . (Cont...
vdwnn 15540 Van der Waerden's theorem,...
ramtlecl 15542 The set ` T ` of numbers w...
hashbcval 15544 Value of the "binomial set...
hashbccl 15545 The binomial set is a fini...
hashbcss 15546 Subset relation for the bi...
hashbc0 15547 The set of subsets of size...
hashbc2 15548 The size of the binomial s...
0hashbc 15549 There are no subsets of th...
ramval 15550 The value of the Ramsey nu...
ramcl2lem 15551 Lemma for extended real cl...
ramtcl 15552 The Ramsey number has the ...
ramtcl2 15553 The Ramsey number is an in...
ramtub 15554 The Ramsey number is a low...
ramub 15555 The Ramsey number is a low...
ramub2 15556 It is sufficient to check ...
rami 15557 The defining property of a...
ramcl2 15558 The Ramsey number is eithe...
ramxrcl 15559 The Ramsey number is an ex...
ramubcl 15560 If the Ramsey number is up...
ramlb 15561 Establish a lower bound on...
0ram 15562 The Ramsey number when ` M...
0ram2 15563 The Ramsey number when ` M...
ram0 15564 The Ramsey number when ` R...
0ramcl 15565 Lemma for ~ ramcl : Exist...
ramz2 15566 The Ramsey number when ` F...
ramz 15567 The Ramsey number when ` F...
ramub1lem1 15568 Lemma for ~ ramub1 . (Con...
ramub1lem2 15569 Lemma for ~ ramub1 . (Con...
ramub1 15570 Inductive step for Ramsey'...
ramcl 15571 Ramsey's theorem: the Rams...
ramsey 15572 Ramsey's theorem with the ...
prmoval 15575 Value of the primorial fun...
prmocl 15576 Closure of the primorial f...
prmone0 15577 The primorial function is ...
prmo0 15578 The primorial of 0. (Cont...
prmo1 15579 The primorial of 1. (Cont...
prmop1 15580 The primorial of a success...
prmonn2 15581 Value of the primorial fun...
prmo2 15582 The primorial of 2. (Cont...
prmo3 15583 The primorial of 3. (Cont...
prmdvdsprmo 15584 The primorial of a number ...
prmdvdsprmop 15585 The primorial of a number ...
fvprmselelfz 15586 The value of the prime sel...
fvprmselgcd1 15587 The greatest common diviso...
prmolefac 15588 The primorial of a positiv...
prmodvdslcmf 15589 The primorial of a nonnega...
prmolelcmf 15590 The primorial of a positiv...
prmgaplem1 15591 Lemma for ~ prmgap : The ...
prmgaplem2 15592 Lemma for ~ prmgap : The ...
prmgaplcmlem1 15593 Lemma for ~ prmgaplcm : T...
prmgaplcmlem2 15594 Lemma for ~ prmgaplcm : T...
prmgaplem3 15595 Lemma for ~ prmgap . (Con...
prmgaplem4 15596 Lemma for ~ prmgap . (Con...
prmgaplem5 15597 Lemma for ~ prmgap : for e...
prmgaplem6 15598 Lemma for ~ prmgap : for e...
prmgaplem7 15599 Lemma for ~ prmgap . (Con...
prmgaplem8 15600 Lemma for ~ prmgap . (Con...
prmgap 15601 The prime gap theorem: for...
prmgaplcm 15602 Alternate proof of ~ prmga...
prmgapprmolem 15603 Lemma for ~ prmgapprmo : ...
prmgapprmo 15604 Alternate proof of ~ prmga...
dec2dvds 15605 Divisibility by two is obv...
dec5dvds 15606 Divisibility by five is ob...
dec5dvds2 15607 Divisibility by five is ob...
dec5nprm 15608 Divisibility by five is ob...
dec2nprm 15609 Divisibility by two is obv...
modxai 15610 Add exponents in a power m...
mod2xi 15611 Double exponents in a powe...
modxp1i 15612 Add one to an exponent in ...
mod2xnegi 15613 Version of ~ mod2xi with a...
modsubi 15614 Subtract from within a mod...
gcdi 15615 Calculate a GCD via Euclid...
gcdmodi 15616 Calculate a GCD via Euclid...
decexp2 15617 Calculate a power of two. ...
numexp0 15618 Calculate an integer power...
numexp1 15619 Calculate an integer power...
numexpp1 15620 Calculate an integer power...
numexp2x 15621 Double an integer power. ...
decsplit0b 15622 Split a decimal number int...
decsplit0 15623 Split a decimal number int...
decsplit1 15624 Split a decimal number int...
decsplit 15625 Split a decimal number int...
decsplit0bOLD 15626 Obsolete version of ~ decs...
decsplit0OLD 15627 Obsolete version of ~ decs...
decsplit1OLD 15628 Obsolete version of ~ decs...
decsplitOLD 15629 Obsolete version of ~ decs...
karatsuba 15630 The Karatsuba multiplicati...
karatsubaOLD 15631 Obsolete version of ~ kara...
2exp4 15632 Two to the fourth power is...
2exp6 15633 Two to the sixth power is ...
2exp8 15634 Two to the eighth power is...
2exp16 15635 Two to the sixteenth power...
3exp3 15636 Three to the third power i...
2expltfac 15637 The factorial grows faster...
cshwsidrepsw 15638 If cyclically shifting a w...
cshwsidrepswmod0 15639 If cyclically shifting a w...
cshwshashlem1 15640 If cyclically shifting a w...
cshwshashlem2 15641 If cyclically shifting a w...
cshwshashlem3 15642 If cyclically shifting a w...
cshwsdisj 15643 The singletons resulting b...
cshwsiun 15644 The set of (different!) wo...
cshwsex 15645 The class of (different!) ...
cshws0 15646 The size of the set of (di...
cshwrepswhash1 15647 The size of the set of (di...
cshwshashnsame 15648 If a word (not consisting ...
cshwshash 15649 If a word has a length bei...
prmlem0 15650 Lemma for ~ prmlem1 and ~ ...
prmlem1a 15651 A quick proof skeleton to ...
prmlem1 15652 A quick proof skeleton to ...
5prm 15653 5 is a prime number. (Con...
6nprm 15654 6 is not a prime number. ...
7prm 15655 7 is a prime number. (Con...
8nprm 15656 8 is not a prime number. ...
9nprm 15657 9 is not a prime number. ...
10nprm 15658 10 is not a prime number. ...
10nprmOLD 15659 Obsolete version of ~ 10np...
11prm 15660 11 is a prime number. (Co...
13prm 15661 13 is a prime number. (Co...
17prm 15662 17 is a prime number. (Co...
19prm 15663 19 is a prime number. (Co...
23prm 15664 23 is a prime number. (Co...
prmlem2 15665 Our last proving session g...
37prm 15666 37 is a prime number. (Co...
43prm 15667 43 is a prime number. (Co...
83prm 15668 83 is a prime number. (Co...
139prm 15669 139 is a prime number. (C...
163prm 15670 163 is a prime number. (C...
317prm 15671 317 is a prime number. (C...
631prm 15672 631 is a prime number. (C...
prmo4 15673 The primorial of 4. (Cont...
prmo5 15674 The primorial of 5. (Cont...
prmo6 15675 The primorial of 6. (Cont...
1259lem1 15676 Lemma for ~ 1259prm . Cal...
1259lem2 15677 Lemma for ~ 1259prm . Cal...
1259lem3 15678 Lemma for ~ 1259prm . Cal...
1259lem4 15679 Lemma for ~ 1259prm . Cal...
1259lem5 15680 Lemma for ~ 1259prm . Cal...
1259prm 15681 1259 is a prime number. (...
2503lem1 15682 Lemma for ~ 2503prm . Cal...
2503lem2 15683 Lemma for ~ 2503prm . Cal...
2503lem3 15684 Lemma for ~ 2503prm . Cal...
2503prm 15685 2503 is a prime number. (...
4001lem1 15686 Lemma for ~ 4001prm . Cal...
4001lem2 15687 Lemma for ~ 4001prm . Cal...
4001lem3 15688 Lemma for ~ 4001prm . Cal...
4001lem4 15689 Lemma for ~ 4001prm . Cal...
4001prm 15690 4001 is a prime number. (...
brstruct 15703 The structure relation is ...
isstruct2 15704 The property of being a st...
isstruct 15705 The property of being a st...
structcnvcnv 15706 Two ways to express the re...
structfun 15707 Convert between two kinds ...
structfn 15708 Convert between two kinds ...
slotfn 15709 A slot is a function on se...
strfvnd 15710 Deduction version of ~ str...
wunndx 15711 Closure of the index extra...
strfvn 15712 Value of a structure compo...
strfvss 15713 A structure component extr...
wunstr 15714 Closure of a structure ind...
ndxarg 15715 Get the numeric argument f...
ndxid 15716 A structure component extr...
strndxid 15717 The value of a structure c...
reldmsets 15718 The structure override ope...
setsvalg 15719 Value of the structure rep...
setsval 15720 Value of the structure rep...
setsidvald 15721 Value of the structure rep...
fvsetsid 15722 The value of the structure...
fsets 15723 The structure replacement ...
setsdm 15724 The domain of a structure ...
setsfun 15725 A structure with replaceme...
setsfun0 15726 A structure with replaceme...
setsstruct 15727 An extensible structure wi...
wunsets 15728 Closure of structure repla...
setsres 15729 The structure replacement ...
setsabs 15730 Replacing the same compone...
setscom 15731 Component-setting is commu...
strfvd 15732 Deduction version of ~ str...
strfv2d 15733 Deduction version of ~ str...
strfv2 15734 A variation on ~ strfv to ...
strfv 15735 Extract a structure compon...
strfv3 15736 Variant on ~ strfv for lar...
strssd 15737 Deduction version of ~ str...
strss 15738 Propagate component extrac...
str0 15739 All components of the empt...
base0 15740 The base set of the empty ...
strfvi 15741 Structure slot extractors ...
setsid 15742 Value of the structure rep...
setsnid 15743 Value of the structure rep...
sbcie2s 15744 A special version of class...
sbcie3s 15745 A special version of class...
baseval 15746 Value of the base set extr...
baseid 15747 Utility theorem: index-ind...
elbasfv 15748 Utility theorem: reverse c...
elbasov 15749 Utility theorem: reverse c...
strov2rcl 15750 Partial reverse closure fo...
basendx 15751 Index value of the base se...
basendxnn 15752 The index value of the bas...
reldmress 15753 The structure restriction ...
ressval 15754 Value of structure restric...
ressid2 15755 General behavior of trivia...
ressval2 15756 Value of nontrivial struct...
ressbas 15757 Base set of a structure re...
ressbas2 15758 Base set of a structure re...
ressbasss 15759 The base set of a restrict...
resslem 15760 Other elements of a struct...
ress0 15761 All restrictions of the nu...
ressid 15762 Behavior of trivial restri...
ressinbas 15763 Restriction only cares abo...
ressval3d 15764 Value of structure restric...
ressress 15765 Restriction composition la...
ressabs 15766 Restriction absorption law...
wunress 15767 Closure of structure restr...
dfpleOLD 15789 Obsolete version of ~ df-p...
strlemor0 15795 Structure definition utili...
strlemor1 15796 Add one element to the end...
strlemor2 15797 Add two elements to the en...
strlemor3 15798 Add three elements to the ...
strleun 15799 Combine two structures int...
strle1 15800 Make a structure from a si...
strle2 15801 Make a structure from a pa...
strle3 15802 Make a structure from a tr...
plusgndx 15803 Index value of the ~ df-pl...
plusgid 15804 Utility theorem: index-ind...
1strstr 15805 A constructed one-slot str...
1strbas 15806 The base set of a construc...
1strwunbndx 15807 A constructed one-slot str...
1strwun 15808 A constructed one-slot str...
2strstr 15809 A constructed two-slot str...
2strbas 15810 The base set of a construc...
2strop 15811 The other slot of a constr...
2strstr1 15812 A constructed two-slot str...
2strbas1 15813 The base set of a construc...
2strop1 15814 The other slot of a constr...
grpstr 15815 A constructed group is a s...
grpbase 15816 The base set of a construc...
grpplusg 15817 The operation of a constru...
ressplusg 15818 ` +g ` is unaffected by re...
grpbasex 15819 The base of an explicitly ...
grpplusgx 15820 The operation of an explic...
mulrndx 15821 Index value of the ~ df-mu...
mulrid 15822 Utility theorem: index-ind...
rngstr 15823 A constructed ring is a st...
rngbase 15824 The base set of a construc...
rngplusg 15825 The additive operation of ...
rngmulr 15826 The multiplicative operati...
starvndx 15827 Index value of the ~ df-st...
starvid 15828 Utility theorem: index-ind...
ressmulr 15829 ` .r ` is unaffected by re...
ressstarv 15830 ` *r ` is unaffected by re...
srngfn 15831 A constructed star ring is...
srngbase 15832 The base set of a construc...
srngplusg 15833 The addition operation of ...
srngmulr 15834 The multiplication operati...
srnginvl 15835 The involution function of...
scandx 15836 Index value of the ~ df-sc...
scaid 15837 Utility theorem: index-ind...
vscandx 15838 Index value of the ~ df-vs...
vscaid 15839 Utility theorem: index-ind...
lmodstr 15840 A constructed left module ...
lmodbase 15841 The base set of a construc...
lmodplusg 15842 The additive operation of ...
lmodsca 15843 The set of scalars of a co...
lmodvsca 15844 The scalar product operati...
ipndx 15845 Index value of the ~ df-ip...
ipid 15846 Utility theorem: index-ind...
ipsstr 15847 Lemma to shorten proofs of...
ipsbase 15848 The base set of a construc...
ipsaddg 15849 The additive operation of ...
ipsmulr 15850 The multiplicative operati...
ipssca 15851 The set of scalars of a co...
ipsvsca 15852 The scalar product operati...
ipsip 15853 The multiplicative operati...
resssca 15854 ` Scalar ` is unaffected b...
ressvsca 15855 ` .s ` is unaffected by re...
ressip 15856 The inner product is unaff...
phlstr 15857 A constructed pre-Hilbert ...
phlbase 15858 The base set of a construc...
phlplusg 15859 The additive operation of ...
phlsca 15860 The ring of scalars of a c...
phlvsca 15861 The scalar product operati...
phlip 15862 The inner product (Hermiti...
tsetndx 15863 Index value of the ~ df-ts...
tsetid 15864 Utility theorem: index-ind...
topgrpstr 15865 A constructed topological ...
topgrpbas 15866 The base set of a construc...
topgrpplusg 15867 The additive operation of ...
topgrptset 15868 The topology of a construc...
resstset 15869 ` TopSet ` is unaffected b...
plendx 15870 Index value of the ~ df-pl...
plendxOLD 15871 Obsolete version of ~ df-p...
pleid 15872 Utility theorem: self-refe...
pleidOLD 15873 Obsolete version of ~ otps...
otpsstr 15874 Functionality of a topolog...
otpsbas 15875 The base set of a topologi...
otpstset 15876 The open sets of a topolog...
otpsle 15877 The order of a topological...
otpsstrOLD 15878 Obsolete version of ~ otps...
otpsbasOLD 15879 Obsolete version of ~ otps...
otpstsetOLD 15880 Obsolete version of ~ otps...
otpsleOLD 15881 Obsolete version of ~ otps...
ressle 15882 ` le ` is unaffected by re...
ocndx 15883 Index value of the ~ df-oc...
ocid 15884 Utility theorem: index-ind...
dsndx 15885 Index value of the ~ df-ds...
dsid 15886 Utility theorem: index-ind...
unifndx 15887 Index value of the ~ df-un...
unifid 15888 Utility theorem: index-ind...
odrngstr 15889 Functionality of an ordere...
odrngbas 15890 The base set of an ordered...
odrngplusg 15891 The addition operation of ...
odrngmulr 15892 The multiplication operati...
odrngtset 15893 The open sets of an ordere...
odrngle 15894 The order of an ordered me...
odrngds 15895 The metric of an ordered m...
ressds 15896 ` dist ` is unaffected by ...
homndx 15897 Index value of the ~ df-ho...
homid 15898 Utility theorem: index-ind...
ccondx 15899 Index value of the ~ df-cc...
ccoid 15900 Utility theorem: index-ind...
resshom 15901 ` Hom ` is unaffected by r...
ressco 15902 ` comp ` is unaffected by ...
slotsbhcdif 15903 The slots ` Base ` , ` Hom...
restfn 15908 The subspace topology oper...
topnfn 15909 The topology extractor fun...
restval 15910 The subspace topology indu...
elrest 15911 The predicate "is an open ...
elrestr 15912 Sufficient condition for b...
0rest 15913 Value of the structure res...
restid2 15914 The subspace topology over...
restsspw 15915 The subspace topology is a...
firest 15916 The finite intersections o...
restid 15917 The subspace topology of t...
topnval 15918 Value of the topology extr...
topnid 15919 Value of the topology extr...
topnpropd 15920 The topology extractor fun...
reldmprds 15932 The structure product is a...
prdsbasex 15934 Lemma for structure produc...
imasvalstr 15935 Structure product value is...
prdsvalstr 15936 Structure product value is...
prdsvallem 15937 Lemma for ~ prdsbas and si...
prdsval 15938 Value of the structure pro...
prdssca 15939 Scalar ring of a structure...
prdsbas 15940 Base set of a structure pr...
prdsplusg 15941 Addition in a structure pr...
prdsmulr 15942 Multiplication in a struct...
prdsvsca 15943 Scalar multiplication in a...
prdsip 15944 Inner product in a structu...
prdsle 15945 Structure product weak ord...
prdsless 15946 Closure of the order relat...
prdsds 15947 Structure product distance...
prdsdsfn 15948 Structure product distance...
prdstset 15949 Structure product topology...
prdshom 15950 Structure product hom-sets...
prdsco 15951 Structure product composit...
prdsbas2 15952 The base set of a structur...
prdsbasmpt 15953 A constructed tuple is a p...
prdsbasfn 15954 Points in the structure pr...
prdsbasprj 15955 Each point in a structure ...
prdsplusgval 15956 Value of a componentwise s...
prdsplusgfval 15957 Value of a structure produ...
prdsmulrval 15958 Value of a componentwise r...
prdsmulrfval 15959 Value of a structure produ...
prdsleval 15960 Value of the product order...
prdsdsval 15961 Value of the metric in a s...
prdsvscaval 15962 Scalar multiplication in a...
prdsvscafval 15963 Scalar multiplication of a...
prdsbas3 15964 The base set of an indexed...
prdsbasmpt2 15965 A constructed tuple is a p...
prdsbascl 15966 An element of the base has...
prdsdsval2 15967 Value of the metric in a s...
prdsdsval3 15968 Value of the metric in a s...
pwsval 15969 Value of a structure power...
pwsbas 15970 Base set of a structure po...
pwselbasb 15971 Membership in the base set...
pwselbas 15972 An element of a structure ...
pwsplusgval 15973 Value of addition in a str...
pwsmulrval 15974 Value of multiplication in...
pwsle 15975 Ordering in a structure po...
pwsleval 15976 Ordering in a structure po...
pwsvscafval 15977 Scalar multiplication in a...
pwsvscaval 15978 Scalar multiplication of a...
pwssca 15979 The ring of scalars of a s...
pwsdiagel 15980 Membership of diagonal ele...
pwssnf1o 15981 Triviality of singleton po...
imasval 15994 Value of an image structur...
imasbas 15995 The base set of an image s...
imasds 15996 The distance function of a...
imasdsfn 15997 The distance function is a...
imasdsval 15998 The distance function of a...
imasdsval2 15999 The distance function of a...
imasplusg 16000 The group operation in an ...
imasmulr 16001 The ring multiplication in...
imassca 16002 The scalar field of an ima...
imasvsca 16003 The scalar multiplication ...
imasip 16004 The inner product of an im...
imastset 16005 The topology of an image s...
imasle 16006 The ordering of an image s...
f1ocpbllem 16007 Lemma for ~ f1ocpbl . (Co...
f1ocpbl 16008 An injection is compatible...
f1ovscpbl 16009 An injection is compatible...
f1olecpbl 16010 An injection is compatible...
imasaddfnlem 16011 The image structure operat...
imasaddvallem 16012 The operation of an image ...
imasaddflem 16013 The image set operations a...
imasaddfn 16014 The image structure's grou...
imasaddval 16015 The value of an image stru...
imasaddf 16016 The image structure's grou...
imasmulfn 16017 The image structure's ring...
imasmulval 16018 The value of an image stru...
imasmulf 16019 The image structure's ring...
imasvscafn 16020 The image structure's scal...
imasvscaval 16021 The value of an image stru...
imasvscaf 16022 The image structure's scal...
imasless 16023 The order relation defined...
imasleval 16024 The value of the image str...
qusval 16025 Value of a quotient struct...
quslem 16026 The function in ~ qusval i...
qusin 16027 Restrict the equivalence r...
qusbas 16028 Base set of a quotient str...
quss 16029 The scalar field of a quot...
divsfval 16030 Value of the function in ~...
ercpbllem 16031 Lemma for ~ ercpbl . (Con...
ercpbl 16032 Translate the function com...
erlecpbl 16033 Translate the relation com...
qusaddvallem 16034 Value of an operation defi...
qusaddflem 16035 The operation of a quotien...
qusaddval 16036 The base set of an image s...
qusaddf 16037 The base set of an image s...
qusmulval 16038 The base set of an image s...
qusmulf 16039 The base set of an image s...
xpsc 16040 A short expression for the...
xpscg 16041 A short expression for the...
xpscfn 16042 The pair function is a fun...
xpsc0 16043 The pair function maps ` 0...
xpsc1 16044 The pair function maps ` 1...
xpscfv 16045 The value of the pair func...
xpsfrnel 16046 Elementhood in the target ...
xpsfeq 16047 A function on ` 2o ` is de...
xpsfrnel2 16048 Elementhood in the target ...
xpscf 16049 Equivalent condition for t...
xpsfval 16050 The value of the function ...
xpsff1o 16051 The function appearing in ...
xpsfrn 16052 A short expression for the...
xpsfrn2 16053 A short expression for the...
xpsff1o2 16054 The function appearing in ...
xpsval 16055 Value of the binary struct...
xpslem 16056 The indexed structure prod...
xpsbas 16057 The base set of the binary...
xpsaddlem 16058 Lemma for ~ xpsadd and ~ x...
xpsadd 16059 Value of the addition oper...
xpsmul 16060 Value of the multiplicatio...
xpssca 16061 Value of the scalar field ...
xpsvsca 16062 Value of the scalar multip...
xpsless 16063 Closure of the ordering in...
xpsle 16064 Value of the ordering in a...
ismre 16073 Property of being a Moore ...
fnmre 16074 The Moore collection gener...
mresspw 16075 A Moore collection is a su...
mress 16076 A Moore-closed subset is a...
mre1cl 16077 In any Moore collection th...
mreintcl 16078 A nonempty collection of c...
mreiincl 16079 A nonempty indexed interse...
mrerintcl 16080 The relative intersection ...
mreriincl 16081 The relative intersection ...
mreincl 16082 Two closed sets have a clo...
mreuni 16083 Since the entire base set ...
mreunirn 16084 Two ways to express the no...
ismred 16085 Properties that determine ...
ismred2 16086 Properties that determine ...
mremre 16087 The Moore collections of s...
submre 16088 The subcollection of a clo...
mrcflem 16089 The domain and range of th...
fnmrc 16090 Moore-closure is a well-be...
mrcfval 16091 Value of the function expr...
mrcf 16092 The Moore closure is a fun...
mrcval 16093 Evaluation of the Moore cl...
mrccl 16094 The Moore closure of a set...
mrcsncl 16095 The Moore closure of a sin...
mrcid 16096 The closure of a closed se...
mrcssv 16097 The closure of a set is a ...
mrcidb 16098 A set is closed iff it is ...
mrcss 16099 Closure preserves subset o...
mrcssid 16100 The closure of a set is a ...
mrcidb2 16101 A set is closed iff it con...
mrcidm 16102 The closure operation is i...
mrcsscl 16103 The closure is the minimal...
mrcuni 16104 Idempotence of closure und...
mrcun 16105 Idempotence of closure und...
mrcssvd 16106 The Moore closure of a set...
mrcssd 16107 Moore closure preserves su...
mrcssidd 16108 A set is contained in its ...
mrcidmd 16109 Moore closure is idempoten...
mressmrcd 16110 In a Moore system, if a se...
submrc 16111 In a closure system which ...
mrieqvlemd 16112 In a Moore system, if ` Y ...
mrisval 16113 Value of the set of indepe...
ismri 16114 Criterion for a set to be ...
ismri2 16115 Criterion for a subset of ...
ismri2d 16116 Criterion for a subset of ...
ismri2dd 16117 Definition of independence...
mriss 16118 An independent set of a Mo...
mrissd 16119 An independent set of a Mo...
ismri2dad 16120 Consequence of a set in a ...
mrieqvd 16121 In a Moore system, a set i...
mrieqv2d 16122 In a Moore system, a set i...
mrissmrcd 16123 In a Moore system, if an i...
mrissmrid 16124 In a Moore system, subsets...
mreexd 16125 In a Moore system, the clo...
mreexmrid 16126 In a Moore system whose cl...
mreexexlemd 16127 This lemma is used to gene...
mreexexlem2d 16128 Used in ~ mreexexlem4d to ...
mreexexlem3d 16129 Base case of the induction...
mreexexlem4d 16130 Induction step of the indu...
mreexexd 16131 Exchange-type theorem. In...
mreexexdOLD 16132 Obsolete proof of ~ mreexe...
mreexdomd 16133 In a Moore system whose cl...
mreexfidimd 16134 In a Moore system whose cl...
isacs 16135 A set is an algebraic clos...
acsmre 16136 Algebraic closure systems ...
isacs2 16137 In the definition of an al...
acsfiel 16138 A set is closed in an alge...
acsfiel2 16139 A set is closed in an alge...
acsmred 16140 An algebraic closure syste...
isacs1i 16141 A closure system determine...
mreacs 16142 Algebraicity is a composab...
acsfn 16143 Algebraicity of a conditio...
acsfn0 16144 Algebraicity of a point cl...
acsfn1 16145 Algebraicity of a one-argu...
acsfn1c 16146 Algebraicity of a one-argu...
acsfn2 16147 Algebraicity of a two-argu...
iscat 16156 The predicate "is a catego...
iscatd 16157 Properties that determine ...
catidex 16158 Each object in a category ...
catideu 16159 Each object in a category ...
cidfval 16160 Each object in a category ...
cidval 16161 Each object in a category ...
cidffn 16162 The identity arrow constru...
cidfn 16163 The identity arrow operato...
catidd 16164 Deduce the identity arrow ...
iscatd2 16165 Version of ~ iscatd with a...
catidcl 16166 Each object in a category ...
catlid 16167 Left identity property of ...
catrid 16168 Right identity property of...
catcocl 16169 Closure of a composition a...
catass 16170 Associativity of compositi...
0catg 16171 Any structure with an empt...
0cat 16172 The empty set is a categor...
homffval 16173 Value of the functionalize...
fnhomeqhomf 16174 If the Hom-set operation i...
homfval 16175 Value of the functionalize...
homffn 16176 The functionalized Hom-set...
homfeq 16177 Condition for two categori...
homfeqd 16178 If two structures have the...
homfeqbas 16179 Deduce equality of base se...
homfeqval 16180 Value of the functionalize...
comfffval 16181 Value of the functionalize...
comffval 16182 Value of the functionalize...
comfval 16183 Value of the functionalize...
comfffval2 16184 Value of the functionalize...
comffval2 16185 Value of the functionalize...
comfval2 16186 Value of the functionalize...
comfffn 16187 The functionalized composi...
comffn 16188 The functionalized composi...
comfeq 16189 Condition for two categori...
comfeqd 16190 Condition for two categori...
comfeqval 16191 Equality of two compositio...
catpropd 16192 Two structures with the sa...
cidpropd 16193 Two structures with the sa...
oppcval 16196 Value of the opposite cate...
oppchomfval 16197 Hom-sets of the opposite c...
oppchom 16198 Hom-sets of the opposite c...
oppccofval 16199 Composition in the opposit...
oppcco 16200 Composition in the opposit...
oppcbas 16201 Base set of an opposite ca...
oppccatid 16202 Lemma for ~ oppccat . (Co...
oppchomf 16203 Hom-sets of the opposite c...
oppcid 16204 Identity function of an op...
oppccat 16205 An opposite category is a ...
2oppcbas 16206 The double opposite catego...
2oppchomf 16207 The double opposite catego...
2oppccomf 16208 The double opposite catego...
oppchomfpropd 16209 If two categories have the...
oppccomfpropd 16210 If two categories have the...
monfval 16215 Definition of a monomorphi...
ismon 16216 Definition of a monomorphi...
ismon2 16217 Write out the monomorphism...
monhom 16218 A monomorphism is a morphi...
moni 16219 Property of a monomorphism...
monpropd 16220 If two categories have the...
oppcmon 16221 A monomorphism in the oppo...
oppcepi 16222 An epimorphism in the oppo...
isepi 16223 Definition of an epimorphi...
isepi2 16224 Write out the epimorphism ...
epihom 16225 An epimorphism is a morphi...
epii 16226 Property of an epimorphism...
sectffval 16233 Value of the section opera...
sectfval 16234 Value of the section relat...
sectss 16235 The section relation is a ...
issect 16236 The property " ` F ` is a ...
issect2 16237 Property of being a sectio...
sectcan 16238 If ` G ` is a section of `...
sectco 16239 Composition of two section...
isofval 16240 Function value of the func...
invffval 16241 Value of the inverse relat...
invfval 16242 Value of the inverse relat...
isinv 16243 Value of the inverse relat...
invss 16244 The inverse relation is a ...
invsym 16245 The inverse relation is sy...
invsym2 16246 The inverse relation is sy...
invfun 16247 The inverse relation is a ...
isoval 16248 The isomorphisms are the d...
inviso1 16249 If ` G ` is an inverse to ...
inviso2 16250 If ` G ` is an inverse to ...
invf 16251 The inverse relation is a ...
invf1o 16252 The inverse relation is a ...
invinv 16253 The inverse of the inverse...
invco 16254 The composition of two iso...
dfiso2 16255 Alternate definition of an...
dfiso3 16256 Alternate definition of an...
inveq 16257 If there are two inverses ...
isofn 16258 The function value of the ...
isohom 16259 An isomorphism is a homomo...
isoco 16260 The composition of two iso...
oppcsect 16261 A section in the opposite ...
oppcsect2 16262 A section in the opposite ...
oppcinv 16263 An inverse in the opposite...
oppciso 16264 An isomorphism in the oppo...
sectmon 16265 If ` F ` is a section of `...
monsect 16266 If ` F ` is a monomorphism...
sectepi 16267 If ` F ` is a section of `...
episect 16268 If ` F ` is an epimorphism...
sectid 16269 The identity is a section ...
invid 16270 The inverse of the identit...
idiso 16271 The identity is an isomorp...
idinv 16272 The inverse of the identit...
invisoinvl 16273 The inverse of an isomorph...
invisoinvr 16274 The inverse of an isomorph...
invcoisoid 16275 The inverse of an isomorph...
isocoinvid 16276 The inverse of an isomorph...
rcaninv 16277 Right cancellation of an i...
cicfval 16280 The set of isomorphic obje...
brcic 16281 The relation "is isomorphi...
cic 16282 Objects ` X ` and ` Y ` in...
brcici 16283 Prove that two objects are...
cicref 16284 Isomorphism is reflexive. ...
ciclcl 16285 Isomorphism implies the le...
cicrcl 16286 Isomorphism implies the ri...
cicsym 16287 Isomorphism is symmetric. ...
cictr 16288 Isomorphism is transitive....
cicer 16289 Isomorphism is an equivale...
sscrel 16296 The subcategory subset rel...
brssc 16297 The subcategory subset rel...
sscpwex 16298 An analogue of ~ pwex for ...
subcrcl 16299 Reverse closure for the su...
sscfn1 16300 The subcategory subset rel...
sscfn2 16301 The subcategory subset rel...
ssclem 16302 Lemma for ~ ssc1 and simil...
isssc 16303 Value of the subcategory s...
ssc1 16304 Infer subset relation on o...
ssc2 16305 Infer subset relation on m...
sscres 16306 Any function restricted to...
sscid 16307 The subcategory subset rel...
ssctr 16308 The subcategory subset rel...
ssceq 16309 The subcategory subset rel...
rescval 16310 Value of the category rest...
rescval2 16311 Value of the category rest...
rescbas 16312 Base set of the category r...
reschom 16313 Hom-sets of the category r...
reschomf 16314 Hom-sets of the category r...
rescco 16315 Composition in the categor...
rescabs 16316 Restriction absorption law...
rescabs2 16317 Restriction absorption law...
issubc 16318 Elementhood in the set of ...
issubc2 16319 Elementhood in the set of ...
0ssc 16320 For any category ` C ` , t...
0subcat 16321 For any category ` C ` , t...
catsubcat 16322 For any category ` C ` , `...
subcssc 16323 An element in the set of s...
subcfn 16324 An element in the set of s...
subcss1 16325 The objects of a subcatego...
subcss2 16326 The morphisms of a subcate...
subcidcl 16327 The identity of the origin...
subccocl 16328 A subcategory is closed un...
subccatid 16329 A subcategory is a categor...
subcid 16330 The identity in a subcateg...
subccat 16331 A subcategory is a categor...
issubc3 16332 Alternate definition of a ...
fullsubc 16333 The full subcategory gener...
fullresc 16334 The category formed by str...
resscat 16335 A category restricted to a...
subsubc 16336 A subcategory of a subcate...
relfunc 16345 The set of functors is a r...
funcrcl 16346 Reverse closure for a func...
isfunc 16347 Value of the set of functo...
isfuncd 16348 Deduce that an operation i...
funcf1 16349 The object part of a funct...
funcixp 16350 The morphism part of a fun...
funcf2 16351 The morphism part of a fun...
funcfn2 16352 The morphism part of a fun...
funcid 16353 A functor maps each identi...
funcco 16354 A functor maps composition...
funcsect 16355 The image of a section und...
funcinv 16356 The image of an inverse un...
funciso 16357 The image of an isomorphis...
funcoppc 16358 A functor on categories yi...
idfuval 16359 Value of the identity func...
idfu2nd 16360 Value of the morphism part...
idfu2 16361 Value of the morphism part...
idfu1st 16362 Value of the object part o...
idfu1 16363 Value of the object part o...
idfucl 16364 The identity functor is a ...
cofuval 16365 Value of the composition o...
cofu1st 16366 Value of the object part o...
cofu1 16367 Value of the object part o...
cofu2nd 16368 Value of the morphism part...
cofu2 16369 Value of the morphism part...
cofuval2 16370 Value of the composition o...
cofucl 16371 The composition of two fun...
cofuass 16372 Functor composition is ass...
cofulid 16373 The identity functor is a ...
cofurid 16374 The identity functor is a ...
resfval 16375 Value of the functor restr...
resfval2 16376 Value of the functor restr...
resf1st 16377 Value of the functor restr...
resf2nd 16378 Value of the functor restr...
funcres 16379 A functor restricted to a ...
funcres2b 16380 Condition for a functor to...
funcres2 16381 A functor into a restricte...
wunfunc 16382 A weak universe is closed ...
funcpropd 16383 If two categories have the...
funcres2c 16384 Condition for a functor to...
fullfunc 16389 A full functor is a functo...
fthfunc 16390 A faithful functor is a fu...
relfull 16391 The set of full functors i...
relfth 16392 The set of faithful functo...
isfull 16393 Value of the set of full f...
isfull2 16394 Equivalent condition for a...
fullfo 16395 The morphism map of a full...
fulli 16396 The morphism map of a full...
isfth 16397 Value of the set of faithf...
isfth2 16398 Equivalent condition for a...
isffth2 16399 A fully faithful functor i...
fthf1 16400 The morphism map of a fait...
fthi 16401 The morphism map of a fait...
ffthf1o 16402 The morphism map of a full...
fullpropd 16403 If two categories have the...
fthpropd 16404 If two categories have the...
fulloppc 16405 The opposite functor of a ...
fthoppc 16406 The opposite functor of a ...
ffthoppc 16407 The opposite functor of a ...
fthsect 16408 A faithful functor reflect...
fthinv 16409 A faithful functor reflect...
fthmon 16410 A faithful functor reflect...
fthepi 16411 A faithful functor reflect...
ffthiso 16412 A fully faithful functor r...
fthres2b 16413 Condition for a faithful f...
fthres2c 16414 Condition for a faithful f...
fthres2 16415 A faithful functor into a ...
idffth 16416 The identity functor is a ...
cofull 16417 The composition of two ful...
cofth 16418 The composition of two fai...
coffth 16419 The composition of two ful...
rescfth 16420 The inclusion functor from...
ressffth 16421 The inclusion functor from...
fullres2c 16422 Condition for a full funct...
ffthres2c 16423 Condition for a fully fait...
fnfuc 16428 The ` FuncCat ` operation ...
natfval 16429 Value of the function givi...
isnat 16430 Property of being a natura...
isnat2 16431 Property of being a natura...
natffn 16432 The natural transformation...
natrcl 16433 Reverse closure for a natu...
nat1st2nd 16434 Rewrite the natural transf...
natixp 16435 A natural transformation i...
natcl 16436 A component of a natural t...
natfn 16437 A natural transformation i...
nati 16438 Naturality property of a n...
wunnat 16439 A weak universe is closed ...
catstr 16440 A category structure is a ...
fucval 16441 Value of the functor categ...
fuccofval 16442 Value of the functor categ...
fucbas 16443 The objects of the functor...
fuchom 16444 The morphisms in the funct...
fucco 16445 Value of the composition o...
fuccoval 16446 Value of the functor categ...
fuccocl 16447 The composition of two nat...
fucidcl 16448 The identity natural trans...
fuclid 16449 Left identity of natural t...
fucrid 16450 Right identity of natural ...
fucass 16451 Associativity of natural t...
fuccatid 16452 The functor category is a ...
fuccat 16453 The functor category is a ...
fucid 16454 The identity morphism in t...
fucsect 16455 Two natural transformation...
fucinv 16456 Two natural transformation...
invfuc 16457 If ` V ( x ) ` is an inver...
fuciso 16458 A natural transformation i...
natpropd 16459 If two categories have the...
fucpropd 16460 If two categories have the...
initorcl 16467 Reverse closure for an ini...
termorcl 16468 Reverse closure for a term...
zeroorcl 16469 Reverse closure for a zero...
initoval 16470 The value of the initial o...
termoval 16471 The value of the terminal ...
zerooval 16472 The value of the zero obje...
isinito 16473 The predicate "is an initi...
istermo 16474 The predicate "is a termin...
iszeroo 16475 The predicate "is a zero o...
isinitoi 16476 Implication of a class bei...
istermoi 16477 Implication of a class bei...
initoid 16478 For an initial object, the...
termoid 16479 For a terminal object, the...
initoo 16480 An initial object is an ob...
termoo 16481 A terminal object is an ob...
iszeroi 16482 Implication of a class bei...
2initoinv 16483 Morphisms between two init...
initoeu1 16484 Initial objects are essent...
initoeu1w 16485 Initial objects are essent...
initoeu2lem0 16486 Lemma 0 for ~ initoeu2 . ...
initoeu2lem1 16487 Lemma 1 for ~ initoeu2 . ...
initoeu2lem2 16488 Lemma 2 for ~ initoeu2 . ...
initoeu2 16489 Initial objects are essent...
2termoinv 16490 Morphisms between two term...
termoeu1 16491 Terminal objects are essen...
termoeu1w 16492 Terminal objects are essen...
homarcl 16501 Reverse closure for an arr...
homafval 16502 Value of the disjointified...
homaf 16503 Functionality of the disjo...
homaval 16504 Value of the disjointified...
elhoma 16505 Value of the disjointified...
elhomai 16506 Produce an arrow from a mo...
elhomai2 16507 Produce an arrow from a mo...
homarcl2 16508 Reverse closure for the do...
homarel 16509 An arrow is an ordered pai...
homa1 16510 The first component of an ...
homahom2 16511 The second component of an...
homahom 16512 The second component of an...
homadm 16513 The domain of an arrow wit...
homacd 16514 The codomain of an arrow w...
homadmcd 16515 Decompose an arrow into do...
arwval 16516 The set of arrows is the u...
arwrcl 16517 The first component of an ...
arwhoma 16518 An arrow is contained in t...
homarw 16519 A hom-set is a subset of t...
arwdm 16520 The domain of an arrow is ...
arwcd 16521 The codomain of an arrow i...
dmaf 16522 The domain function is a f...
cdaf 16523 The codomain function is a...
arwhom 16524 The second component of an...
arwdmcd 16525 Decompose an arrow into do...
idafval 16530 Value of the identity arro...
idaval 16531 Value of the identity arro...
ida2 16532 Morphism part of the ident...
idahom 16533 Domain and codomain of the...
idadm 16534 Domain of the identity arr...
idacd 16535 Codomain of the identity a...
idaf 16536 The identity arrow functio...
coafval 16537 The value of the compositi...
eldmcoa 16538 A pair ` <. G , F >. ` is ...
dmcoass 16539 The domain of composition ...
homdmcoa 16540 If ` F : X --> Y ` and ` G...
coaval 16541 Value of composition for c...
coa2 16542 The morphism part of arrow...
coahom 16543 The composition of two com...
coapm 16544 Composition of arrows is a...
arwlid 16545 Left identity of a categor...
arwrid 16546 Right identity of a catego...
arwass 16547 Associativity of compositi...
setcval 16550 Value of the category of s...
setcbas 16551 Set of objects of the cate...
setchomfval 16552 Set of arrows of the categ...
setchom 16553 Set of arrows of the categ...
elsetchom 16554 A morphism of sets is a fu...
setccofval 16555 Composition in the categor...
setcco 16556 Composition in the categor...
setccatid 16557 Lemma for ~ setccat . (Co...
setccat 16558 The category of sets is a ...
setcid 16559 The identity arrow in the ...
setcmon 16560 A monomorphism of sets is ...
setcepi 16561 An epimorphism of sets is ...
setcsect 16562 A section in the category ...
setcinv 16563 An inverse in the category...
setciso 16564 An isomorphism in the cate...
resssetc 16565 The restriction of the cat...
funcsetcres2 16566 A functor into a smaller c...
catcval 16569 Value of the category of c...
catcbas 16570 Set of objects of the cate...
catchomfval 16571 Set of arrows of the categ...
catchom 16572 Set of arrows of the categ...
catccofval 16573 Composition in the categor...
catcco 16574 Composition in the categor...
catccatid 16575 Lemma for ~ catccat . (Co...
catcid 16576 The identity arrow in the ...
catccat 16577 The category of categories...
resscatc 16578 The restriction of the cat...
catcisolem 16579 Lemma for ~ catciso . (Co...
catciso 16580 A functor is an isomorphis...
catcoppccl 16581 The category of categories...
catcfuccl 16582 The category of categories...
fncnvimaeqv 16583 The inverse images of the ...
bascnvimaeqv 16584 The inverse image of the u...
estrcval 16587 Value of the category of e...
estrcbas 16588 Set of objects of the cate...
estrchomfval 16589 Set of morphisms ("arrows"...
estrchom 16590 The morphisms between exte...
elestrchom 16591 A morphism between extensi...
estrccofval 16592 Composition in the categor...
estrcco 16593 Composition in the categor...
estrcbasbas 16594 An element of the base set...
estrccatid 16595 Lemma for ~ estrccat . (C...
estrccat 16596 The category of extensible...
estrcid 16597 The identity arrow in the ...
estrchomfn 16598 The Hom-set operation in t...
estrchomfeqhom 16599 The functionalized Hom-set...
estrreslem1 16600 Lemma 1 for ~ estrres . (...
estrreslem2 16601 Lemma 2 for ~ estrres . (...
estrres 16602 Any restriction of a categ...
funcestrcsetclem1 16603 Lemma 1 for ~ funcestrcset...
funcestrcsetclem2 16604 Lemma 2 for ~ funcestrcset...
funcestrcsetclem3 16605 Lemma 3 for ~ funcestrcset...
funcestrcsetclem4 16606 Lemma 4 for ~ funcestrcset...
funcestrcsetclem5 16607 Lemma 5 for ~ funcestrcset...
funcestrcsetclem6 16608 Lemma 6 for ~ funcestrcset...
funcestrcsetclem7 16609 Lemma 7 for ~ funcestrcset...
funcestrcsetclem8 16610 Lemma 8 for ~ funcestrcset...
funcestrcsetclem9 16611 Lemma 9 for ~ funcestrcset...
funcestrcsetc 16612 The "natural forgetful fun...
fthestrcsetc 16613 The "natural forgetful fun...
fullestrcsetc 16614 The "natural forgetful fun...
equivestrcsetc 16615 The "natural forgetful fun...
setc1strwun 16616 A constructed one-slot str...
funcsetcestrclem1 16617 Lemma 1 for ~ funcsetcestr...
funcsetcestrclem2 16618 Lemma 2 for ~ funcsetcestr...
funcsetcestrclem3 16619 Lemma 3 for ~ funcsetcestr...
embedsetcestrclem 16620 Lemma for ~ embedsetcestrc...
funcsetcestrclem4 16621 Lemma 4 for ~ funcsetcestr...
funcsetcestrclem5 16622 Lemma 5 for ~ funcsetcestr...
funcsetcestrclem6 16623 Lemma 6 for ~ funcsetcestr...
funcsetcestrclem7 16624 Lemma 7 for ~ funcsetcestr...
funcsetcestrclem8 16625 Lemma 8 for ~ funcsetcestr...
funcsetcestrclem9 16626 Lemma 9 for ~ funcsetcestr...
funcsetcestrc 16627 The "embedding functor" fr...
fthsetcestrc 16628 The "embedding functor" fr...
fullsetcestrc 16629 The "embedding functor" fr...
embedsetcestrc 16630 The "embedding functor" fr...
fnxpc 16639 The binary product of cate...
xpcval 16640 Value of the binary produc...
xpcbas 16641 Set of objects of the bina...
xpchomfval 16642 Set of morphisms of the bi...
xpchom 16643 Set of morphisms of the bi...
relxpchom 16644 A hom-set in the binary pr...
xpccofval 16645 Value of composition in th...
xpcco 16646 Value of composition in th...
xpcco1st 16647 Value of composition in th...
xpcco2nd 16648 Value of composition in th...
xpchom2 16649 Value of the set of morphi...
xpcco2 16650 Value of composition in th...
xpccatid 16651 The product of two categor...
xpcid 16652 The identity morphism in t...
xpccat 16653 The product of two categor...
1stfval 16654 Value of the first project...
1stf1 16655 Value of the first project...
1stf2 16656 Value of the first project...
2ndfval 16657 Value of the first project...
2ndf1 16658 Value of the first project...
2ndf2 16659 Value of the first project...
1stfcl 16660 The first projection funct...
2ndfcl 16661 The second projection func...
prfval 16662 Value of the pairing funct...
prf1 16663 Value of the pairing funct...
prf2fval 16664 Value of the pairing funct...
prf2 16665 Value of the pairing funct...
prfcl 16666 The pairing of functors ` ...
prf1st 16667 Cancellation of pairing wi...
prf2nd 16668 Cancellation of pairing wi...
1st2ndprf 16669 Break a functor into a pro...
catcxpccl 16670 The category of categories...
xpcpropd 16671 If two categories have the...
evlfval 16680 Value of the evaluation fu...
evlf2 16681 Value of the evaluation fu...
evlf2val 16682 Value of the evaluation na...
evlf1 16683 Value of the evaluation fu...
evlfcllem 16684 Lemma for ~ evlfcl . (Con...
evlfcl 16685 The evaluation functor is ...
curfval 16686 Value of the curry functor...
curf1fval 16687 Value of the object part o...
curf1 16688 Value of the object part o...
curf11 16689 Value of the double evalua...
curf12 16690 The partially evaluated cu...
curf1cl 16691 The partially evaluated cu...
curf2 16692 Value of the curry functor...
curf2val 16693 Value of a component of th...
curf2cl 16694 The curry functor at a mor...
curfcl 16695 The curry functor of a fun...
curfpropd 16696 If two categories have the...
uncfval 16697 Value of the uncurry funct...
uncfcl 16698 The uncurry operation take...
uncf1 16699 Value of the uncurry funct...
uncf2 16700 Value of the uncurry funct...
curfuncf 16701 Cancellation of curry with...
uncfcurf 16702 Cancellation of uncurry wi...
diagval 16703 Define the diagonal functo...
diagcl 16704 The diagonal functor is a ...
diag1cl 16705 The constant functor of ` ...
diag11 16706 Value of the constant func...
diag12 16707 Value of the constant func...
diag2 16708 Value of the diagonal func...
diag2cl 16709 The diagonal functor at a ...
curf2ndf 16710 As shown in ~ diagval , th...
hofval 16715 Value of the Hom functor, ...
hof1fval 16716 The object part of the Hom...
hof1 16717 The object part of the Hom...
hof2fval 16718 The morphism part of the H...
hof2val 16719 The morphism part of the H...
hof2 16720 The morphism part of the H...
hofcllem 16721 Lemma for ~ hofcl . (Cont...
hofcl 16722 Closure of the Hom functor...
oppchofcl 16723 Closure of the opposite Ho...
yonval 16724 Value of the Yoneda embedd...
yoncl 16725 The Yoneda embedding is a ...
yon1cl 16726 The Yoneda embedding at an...
yon11 16727 Value of the Yoneda embedd...
yon12 16728 Value of the Yoneda embedd...
yon2 16729 Value of the Yoneda embedd...
hofpropd 16730 If two categories have the...
yonpropd 16731 If two categories have the...
oppcyon 16732 Value of the opposite Yone...
oyoncl 16733 The opposite Yoneda embedd...
oyon1cl 16734 The opposite Yoneda embedd...
yonedalem1 16735 Lemma for ~ yoneda . (Con...
yonedalem21 16736 Lemma for ~ yoneda . (Con...
yonedalem3a 16737 Lemma for ~ yoneda . (Con...
yonedalem4a 16738 Lemma for ~ yoneda . (Con...
yonedalem4b 16739 Lemma for ~ yoneda . (Con...
yonedalem4c 16740 Lemma for ~ yoneda . (Con...
yonedalem22 16741 Lemma for ~ yoneda . (Con...
yonedalem3b 16742 Lemma for ~ yoneda . (Con...
yonedalem3 16743 Lemma for ~ yoneda . (Con...
yonedainv 16744 The Yoneda Lemma with expl...
yonffthlem 16745 Lemma for ~ yonffth . (Co...
yoneda 16746 The Yoneda Lemma. There i...
yonffth 16747 The Yoneda Lemma. The Yon...
yoniso 16748 If the codomain is recover...
isprs 16753 Property of being a preord...
prslem 16754 Lemma for ~ prsref and ~ p...
prsref 16755 Less-or-equal is reflexive...
prstr 16756 Less-or-equal is transitiv...
isdrs 16757 Property of being a direct...
drsdir 16758 Direction of a directed se...
drsprs 16759 A directed set is a preset...
drsbn0 16760 The base of a directed set...
drsdirfi 16761 Any _finite_ number of ele...
isdrs2 16762 Directed sets may be defin...
ispos 16770 The predicate "is a poset....
ispos2 16771 A poset is an antisymmetri...
posprs 16772 A poset is a preset. (Con...
posi 16773 Lemma for poset properties...
posref 16774 A poset ordering is reflex...
posasymb 16775 A poset ordering is asymme...
postr 16776 A poset ordering is transi...
0pos 16777 Technical lemma to simplif...
isposd 16778 Properties that determine ...
isposi 16779 Properties that determine ...
isposix 16780 Properties that determine ...
pltfval 16782 Value of the less-than rel...
pltval 16783 Less-than relation. ( ~ d...
pltle 16784 Less-than implies less-tha...
pltne 16785 Less-than relation. ( ~ d...
pltirr 16786 The less-than relation is ...
pleval2i 16787 One direction of ~ pleval2...
pleval2 16788 Less-than-or-equal in term...
pltnle 16789 Less-than implies not inve...
pltval3 16790 Alternate expression for l...
pltnlt 16791 The less-than relation imp...
pltn2lp 16792 The less-than relation has...
plttr 16793 The less-than relation is ...
pltletr 16794 Transitive law for chained...
plelttr 16795 Transitive law for chained...
pospo 16796 Write a poset structure in...
lubfval 16801 Value of the least upper b...
lubdm 16802 Domain of the least upper ...
lubfun 16803 The LUB is a function. (C...
lubeldm 16804 Member of the domain of th...
lubelss 16805 A member of the domain of ...
lubeu 16806 Unique existence proper of...
lubval 16807 Value of the least upper b...
lubcl 16808 The least upper bound func...
lubprop 16809 Properties of greatest low...
luble 16810 The greatest lower bound i...
lublecllem 16811 Lemma for ~ lublecl and ~ ...
lublecl 16812 The set of all elements le...
lubid 16813 The LUB of elements less t...
glbfval 16814 Value of the greatest lowe...
glbdm 16815 Domain of the greatest low...
glbfun 16816 The GLB is a function. (C...
glbeldm 16817 Member of the domain of th...
glbelss 16818 A member of the domain of ...
glbeu 16819 Unique existence proper of...
glbval 16820 Value of the greatest lowe...
glbcl 16821 The least upper bound func...
glbprop 16822 Properties of greatest low...
glble 16823 The greatest lower bound i...
joinfval 16824 Value of join function for...
joinfval2 16825 Value of join function for...
joindm 16826 Domain of join function fo...
joindef 16827 Two ways to say that a joi...
joinval 16828 Join value. Since both si...
joincl 16829 Closure of join of element...
joindmss 16830 Subset property of domain ...
joinval2lem 16831 Lemma for ~ joinval2 and ~...
joinval2 16832 Value of join for a poset ...
joineu 16833 Uniqueness of join of elem...
joinlem 16834 Lemma for join properties....
lejoin1 16835 A join's first argument is...
lejoin2 16836 A join's second argument i...
joinle 16837 A join is less than or equ...
meetfval 16838 Value of meet function for...
meetfval2 16839 Value of meet function for...
meetdm 16840 Domain of meet function fo...
meetdef 16841 Two ways to say that a mee...
meetval 16842 Meet value. Since both si...
meetcl 16843 Closure of meet of element...
meetdmss 16844 Subset property of domain ...
meetval2lem 16845 Lemma for ~ meetval2 and ~...
meetval2 16846 Value of meet for a poset ...
meeteu 16847 Uniqueness of meet of elem...
meetlem 16848 Lemma for meet properties....
lemeet1 16849 A meet's first argument is...
lemeet2 16850 A meet's second argument i...
meetle 16851 A meet is less than or equ...
joincomALT 16852 The join of a poset commut...
joincom 16853 The join of a poset commut...
meetcomALT 16854 The meet of a poset commut...
meetcom 16855 The meet of a poset commut...
istos 16858 The predicate "is a toset....
tosso 16859 Write the totally ordered ...
p0val 16864 Value of poset zero. (Con...
p1val 16865 Value of poset zero. (Con...
p0le 16866 Any element is less than o...
ple1 16867 Any element is less than o...
islat 16870 The predicate "is a lattic...
latcl2 16871 The join and meet of any t...
latlem 16872 Lemma for lattice properti...
latpos 16873 A lattice is a poset. (Co...
latjcl 16874 Closure of join operation ...
latmcl 16875 Closure of meet operation ...
latref 16876 A lattice ordering is refl...
latasymb 16877 A lattice ordering is asym...
latasym 16878 A lattice ordering is asym...
lattr 16879 A lattice ordering is tran...
latasymd 16880 Deduce equality from latti...
lattrd 16881 A lattice ordering is tran...
latjcom 16882 The join of a lattice comm...
latlej1 16883 A join's first argument is...
latlej2 16884 A join's second argument i...
latjle12 16885 A join is less than or equ...
latleeqj1 16886 Less-than-or-equal-to in t...
latleeqj2 16887 Less-than-or-equal-to in t...
latjlej1 16888 Add join to both sides of ...
latjlej2 16889 Add join to both sides of ...
latjlej12 16890 Add join to both sides of ...
latnlej 16891 An idiom to express that a...
latnlej1l 16892 An idiom to express that a...
latnlej1r 16893 An idiom to express that a...
latnlej2 16894 An idiom to express that a...
latnlej2l 16895 An idiom to express that a...
latnlej2r 16896 An idiom to express that a...
latjidm 16897 Lattice join is idempotent...
latmcom 16898 The join of a lattice comm...
latmle1 16899 A meet is less than or equ...
latmle2 16900 A meet is less than or equ...
latlem12 16901 An element is less than or...
latleeqm1 16902 Less-than-or-equal-to in t...
latleeqm2 16903 Less-than-or-equal-to in t...
latmlem1 16904 Add meet to both sides of ...
latmlem2 16905 Add meet to both sides of ...
latmlem12 16906 Add join to both sides of ...
latnlemlt 16907 Negation of less-than-or-e...
latnle 16908 Equivalent expressions for...
latmidm 16909 Lattice join is idempotent...
latabs1 16910 Lattice absorption law. F...
latabs2 16911 Lattice absorption law. F...
latledi 16912 An ortholattice is distrib...
latmlej11 16913 Ordering of a meet and joi...
latmlej12 16914 Ordering of a meet and joi...
latmlej21 16915 Ordering of a meet and joi...
latmlej22 16916 Ordering of a meet and joi...
lubsn 16917 The least upper bound of a...
latjass 16918 Lattice join is associativ...
latj12 16919 Swap 1st and 2nd members o...
latj32 16920 Swap 2nd and 3rd members o...
latj13 16921 Swap 1st and 3rd members o...
latj31 16922 Swap 2nd and 3rd members o...
latjrot 16923 Rotate lattice join of 3 c...
latj4 16924 Rearrangement of lattice j...
latj4rot 16925 Rotate lattice join of 4 c...
latjjdi 16926 Lattice join distributes o...
latjjdir 16927 Lattice join distributes o...
mod1ile 16928 The weak direction of the ...
mod2ile 16929 The weak direction of the ...
isclat 16932 The predicate "is a comple...
clatpos 16933 A complete lattice is a po...
clatlem 16934 Lemma for properties of a ...
clatlubcl 16935 Any subset of the base set...
clatlubcl2 16936 Any subset of the base set...
clatglbcl 16937 Any subset of the base set...
clatglbcl2 16938 Any subset of the base set...
clatl 16939 A complete lattice is a la...
isglbd 16940 Properties that determine ...
lublem 16941 Lemma for the least upper ...
lubub 16942 The LUB of a complete latt...
lubl 16943 The LUB of a complete latt...
lubss 16944 Subset law for least upper...
lubel 16945 An element of a set is les...
lubun 16946 The LUB of a union. (Cont...
clatglb 16947 Properties of greatest low...
clatglble 16948 The greatest lower bound i...
clatleglb 16949 Two ways of expressing "le...
clatglbss 16950 Subset law for greatest lo...
oduval 16953 Value of an order dual str...
oduleval 16954 Value of the less-equal re...
oduleg 16955 Truth of the less-equal re...
odubas 16956 Base set of an order dual ...
pospropd 16957 Posethood is determined on...
odupos 16958 Being a poset is a self-du...
oduposb 16959 Being a poset is a self-du...
meet0 16960 Lemma for ~ odujoin . (Co...
join0 16961 Lemma for ~ odumeet . (Co...
oduglb 16962 Greatest lower bounds in a...
odumeet 16963 Meets in a dual order are ...
odulub 16964 Least upper bounds in a du...
odujoin 16965 Joins in a dual order are ...
odulatb 16966 Being a lattice is self-du...
oduclatb 16967 Being a complete lattice i...
odulat 16968 Being a lattice is self-du...
poslubmo 16969 Least upper bounds in a po...
posglbmo 16970 Greatest lower bounds in a...
poslubd 16971 Properties which determine...
poslubdg 16972 Properties which determine...
posglbd 16973 Properties which determine...
ipostr 16976 The structure of ~ df-ipo ...
ipoval 16977 Value of the inclusion pos...
ipobas 16978 Base set of the inclusion ...
ipolerval 16979 Relation of the inclusion ...
ipotset 16980 Topology of the inclusion ...
ipole 16981 Weak order condition of th...
ipolt 16982 Strict order condition of ...
ipopos 16983 The inclusion poset on a f...
isipodrs 16984 Condition for a family of ...
ipodrscl 16985 Direction by inclusion as ...
ipodrsfi 16986 Finite upper bound propert...
fpwipodrs 16987 The finite subsets of any ...
ipodrsima 16988 The monotone image of a di...
isacs3lem 16989 An algebraic closure syste...
acsdrsel 16990 An algebraic closure syste...
isacs4lem 16991 In a closure system in whi...
isacs5lem 16992 If closure commutes with d...
acsdrscl 16993 In an algebraic closure sy...
acsficl 16994 A closure in an algebraic ...
isacs5 16995 A closure system is algebr...
isacs4 16996 A closure system is algebr...
isacs3 16997 A closure system is algebr...
acsficld 16998 In an algebraic closure sy...
acsficl2d 16999 In an algebraic closure sy...
acsfiindd 17000 In an algebraic closure sy...
acsmapd 17001 In an algebraic closure sy...
acsmap2d 17002 In an algebraic closure sy...
acsinfd 17003 In an algebraic closure sy...
acsdomd 17004 In an algebraic closure sy...
acsinfdimd 17005 In an algebraic closure sy...
acsexdimd 17006 In an algebraic closure sy...
mrelatglb 17007 Greatest lower bounds in a...
mrelatglb0 17008 The empty intersection in ...
mrelatlub 17009 Least upper bounds in a Mo...
mreclatBAD 17010 A Moore space is a complet...
latmass 17011 Lattice meet is associativ...
latdisdlem 17012 Lemma for ~ latdisd . (Co...
latdisd 17013 In a lattice, joins distri...
isdlat 17016 Property of being a distri...
dlatmjdi 17017 In a distributive lattice,...
dlatl 17018 A distributive lattice is ...
odudlatb 17019 The dual of a distributive...
dlatjmdi 17020 In a distributive lattice,...
isps 17025 The predicate "is a poset"...
psrel 17026 A poset is a relation. (C...
psref2 17027 A poset is antisymmetric a...
pstr2 17028 A poset is transitive. (C...
pslem 17029 Lemma for ~ psref and othe...
psdmrn 17030 The domain and range of a ...
psref 17031 A poset is reflexive. (Co...
psrn 17032 The range of a poset equal...
psasym 17033 A poset is antisymmetric. ...
pstr 17034 A poset is transitive. (C...
cnvps 17035 The converse of a poset is...
cnvpsb 17036 The converse of a poset is...
psss 17037 Any subset of a partially ...
psssdm2 17038 Field of a subposet. (Con...
psssdm 17039 Field of a subposet. (Con...
istsr 17040 The predicate is a toset. ...
istsr2 17041 The predicate is a toset. ...
tsrlin 17042 A toset is a linear order....
tsrlemax 17043 Two ways of saying a numbe...
tsrps 17044 A toset is a poset. (Cont...
cnvtsr 17045 The converse of a toset is...
tsrss 17046 Any subset of a totally or...
ledm 17047 domain of ` <_ ` is ` RR* ...
lern 17048 The range of ` <_ ` is ` R...
lefld 17049 The field of the 'less or ...
letsr 17050 The "less than or equal to...
isdir 17055 A condition for a relation...
reldir 17056 A direction is a relation....
dirdm 17057 A direction's domain is eq...
dirref 17058 A direction is reflexive. ...
dirtr 17059 A direction is transitive....
dirge 17060 For any two elements of a ...
tsrdir 17061 A totally ordered set is a...
ismgm 17066 The predicate "is a magma"...
ismgmn0 17067 The predicate "is a magma"...
mgmcl 17068 Closure of the operation o...
isnmgm 17069 A condition for a structur...
plusffval 17070 The group addition operati...
plusfval 17071 The group addition operati...
plusfeq 17072 If the addition operation ...
plusffn 17073 The group addition operati...
mgmplusf 17074 The group addition functio...
issstrmgm 17075 Characterize a substructur...
intopsn 17076 The internal operation for...
mgmb1mgm1 17077 The only magma with a base...
mgm0 17078 Any set with an empty base...
mgm0b 17079 The structure with an empt...
mgm1 17080 The structure with one ele...
opifismgm 17081 A structure with a group a...
mgmidmo 17082 A two-sided identity eleme...
grpidval 17083 The value of the identity ...
grpidpropd 17084 If two structures have the...
fn0g 17085 The group zero extractor i...
0g0 17086 The identity element funct...
ismgmid 17087 The identity element of a ...
mgmidcl 17088 The identity element of a ...
mgmlrid 17089 The identity element of a ...
ismgmid2 17090 Show that a given element ...
grpidd 17091 Deduce the identity elemen...
mgmidsssn0 17092 Property of the set of ide...
gsumvalx 17093 Expand out the substitutio...
gsumval 17094 Expand out the substitutio...
gsumpropd 17095 The group sum depends only...
gsumpropd2lem 17096 Lemma for ~ gsumpropd2 . ...
gsumpropd2 17097 A stronger version of ~ gs...
gsummgmpropd 17098 A stronger version of ~ gs...
gsumress 17099 The group sum in a substru...
gsumval1 17100 Value of the group sum ope...
gsum0 17101 Value of the empty group s...
gsumval2a 17102 Value of the group sum ope...
gsumval2 17103 Value of the group sum ope...
gsumprval 17104 Value of the group sum ope...
gsumpr12val 17105 Value of the group sum ope...
issgrp 17108 The predicate "is a semigr...
issgrpv 17109 The predicate "is a semigr...
issgrpn0 17110 The predicate "is a semigr...
isnsgrp 17111 A condition for a structur...
sgrpmgm 17112 A semigroup is a magma. (...
sgrpass 17113 A semigroup operation is a...
sgrp0 17114 Any set with an empty base...
sgrp0b 17115 The structure with an empt...
sgrp1 17116 The structure with one ele...
ismnddef 17119 The predicate "is a monoid...
ismnd 17120 The predicate "is a monoid...
isnmnd 17121 A condition for a structur...
mndsgrp 17122 A monoid is a semigroup. ...
mndmgm 17123 A monoid is a magma. (Con...
mndcl 17124 Closure of the operation o...
mndass 17125 A monoid operation is asso...
mndid 17126 A monoid has a two-sided i...
mndideu 17127 The two-sided identity ele...
mnd32g 17128 Commutative/associative la...
mnd12g 17129 Commutative/associative la...
mnd4g 17130 Commutative/associative la...
mndidcl 17131 The identity element of a ...
mndplusf 17132 The group addition operati...
mndlrid 17133 A monoid's identity elemen...
mndlid 17134 The identity element of a ...
mndrid 17135 The identity element of a ...
ismndd 17136 Deduce a monoid from its p...
mndpfo 17137 The addition operation of ...
mndfo 17138 The addition operation of ...
mndpropd 17139 If two structures have the...
mndprop 17140 If two structures have the...
issubmnd 17141 Characterize a submonoid b...
ress0g 17142 ` 0g ` is unaffected by re...
submnd0 17143 The zero of a submonoid is...
prdsplusgcl 17144 Structure product pointwis...
prdsidlem 17145 Characterization of identi...
prdsmndd 17146 The product of a family of...
prds0g 17147 Zero in a product of monoi...
pwsmnd 17148 The structure power of a m...
pws0g 17149 Zero in a product of monoi...
imasmnd2 17150 The image structure of a m...
imasmnd 17151 The image structure of a m...
imasmndf1 17152 The image of a monoid unde...
xpsmnd 17153 The binary product of mono...
mnd1 17154 The (smallest) structure r...
mnd1id 17155 The singleton element of a...
ismhm 17160 Property of a monoid homom...
mhmrcl1 17161 Reverse closure of a monoi...
mhmrcl2 17162 Reverse closure of a monoi...
mhmf 17163 A monoid homomorphism is a...
mhmpropd 17164 Monoid homomorphism depend...
mhmlin 17165 A monoid homomorphism comm...
mhm0 17166 A monoid homomorphism pres...
idmhm 17167 The identity homomorphism ...
mhmf1o 17168 A monoid homomorphism is b...
submrcl 17169 Reverse closure for submon...
issubm 17170 Expand definition of a sub...
issubm2 17171 Submonoids are subsets tha...
issubmd 17172 Deduction for proving a su...
submss 17173 Submonoids are subsets of ...
submid 17174 Every monoid is trivially ...
subm0cl 17175 Submonoids contain zero. ...
submcl 17176 Submonoids are closed unde...
submmnd 17177 Submonoids are themselves ...
submbas 17178 The base set of a submonoi...
subm0 17179 Submonoids have the same i...
subsubm 17180 A submonoid of a submonoid...
0mhm 17181 The constant zero linear f...
resmhm 17182 Restriction of a monoid ho...
resmhm2 17183 One direction of ~ resmhm2...
resmhm2b 17184 Restriction of the codomai...
mhmco 17185 The composition of monoid ...
mhmima 17186 The homomorphic image of a...
mhmeql 17187 The equalizer of two monoi...
submacs 17188 Submonoids are an algebrai...
mrcmndind 17189 (( From SO's determinants ...
prdspjmhm 17190 A projection from a produc...
pwspjmhm 17191 A projection from a produc...
pwsdiagmhm 17192 Diagonal monoid homomorphi...
pwsco1mhm 17193 Right composition with a f...
pwsco2mhm 17194 Left composition with a mo...
gsumvallem2 17195 Lemma for properties of th...
gsumsubm 17196 Evaluate a group sum in a ...
gsumz 17197 Value of a group sum over ...
gsumwsubmcl 17198 Closure of the composite i...
gsumws1 17199 A singleton composite reco...
gsumwcl 17200 Closure of the composite o...
gsumccat 17201 Homomorphic property of co...
gsumws2 17202 Valuation of a pair in a m...
gsumccatsn 17203 Homomorphic property of co...
gsumspl 17204 The primary purpose of the...
gsumwmhm 17205 Behavior of homomorphisms ...
gsumwspan 17206 The submonoid generated by...
frmdval 17211 Value of the free monoid c...
frmdbas 17212 The base set of a free mon...
frmdelbas 17213 An element of the base set...
frmdplusg 17214 The monoid operation of a ...
frmdadd 17215 Value of the monoid operat...
vrmdfval 17216 The canonical injection fr...
vrmdval 17217 The value of the generatin...
vrmdf 17218 The mapping from the index...
frmdmnd 17219 A free monoid is a monoid....
frmd0 17220 The identity of the free m...
frmdsssubm 17221 The set of words taking va...
frmdgsum 17222 Any word in a free monoid ...
frmdss2 17223 A subset of generators is ...
frmdup1 17224 Any assignment of the gene...
frmdup2 17225 The evaluation map has the...
frmdup3lem 17226 Lemma for ~ frmdup3 . (Co...
frmdup3 17227 Universal property of the ...
mgm2nsgrplem1 17228 Lemma 1 for ~ mgm2nsgrp : ...
mgm2nsgrplem2 17229 Lemma 2 for ~ mgm2nsgrp . ...
mgm2nsgrplem3 17230 Lemma 3 for ~ mgm2nsgrp . ...
mgm2nsgrplem4 17231 Lemma 4 for ~ mgm2nsgrp : ...
mgm2nsgrp 17232 A small magma (with two el...
sgrp2nmndlem1 17233 Lemma 1 for ~ sgrp2nmnd : ...
sgrp2nmndlem2 17234 Lemma 2 for ~ sgrp2nmnd . ...
sgrp2nmndlem3 17235 Lemma 3 for ~ sgrp2nmnd . ...
sgrp2rid2 17236 A small semigroup (with tw...
sgrp2rid2ex 17237 A small semigroup (with tw...
sgrp2nmndlem4 17238 Lemma 4 for ~ sgrp2nmnd : ...
sgrp2nmndlem5 17239 Lemma 5 for ~ sgrp2nmnd : ...
sgrp2nmnd 17240 A small semigroup (with tw...
mgmnsgrpex 17241 There is a magma which is ...
sgrpnmndex 17242 There is a semigroup which...
sgrpssmgm 17243 The class of all semigroup...
mndsssgrp 17244 The class of all monoids i...
isgrp 17251 The predicate "is a group....
grpmnd 17252 A group is a monoid. (Con...
grpcl 17253 Closure of the operation o...
grpass 17254 A group operation is assoc...
grpinvex 17255 Every member of a group ha...
grpideu 17256 The two-sided identity ele...
grpplusf 17257 The group addition operati...
grpplusfo 17258 The group addition operati...
resgrpplusfrn 17259 The underlying set of a gr...
grppropd 17260 If two structures have the...
grpprop 17261 If two structures have the...
grppropstr 17262 Generalize a specific 2-el...
grpss 17263 Show that a structure exte...
isgrpd2e 17264 Deduce a group from its pr...
isgrpd2 17265 Deduce a group from its pr...
isgrpde 17266 Deduce a group from its pr...
isgrpd 17267 Deduce a group from its pr...
isgrpi 17268 Properties that determine ...
grpsgrp 17269 A group is a semigroup. (...
dfgrp2 17270 Alternate definition of a ...
dfgrp2e 17271 Alternate definition of a ...
isgrpix 17272 Properties that determine ...
grpidcl 17273 The identity element of a ...
grpbn0 17274 The base set of a group is...
grplid 17275 The identity element of a ...
grprid 17276 The identity element of a ...
grpn0 17277 A group is not empty. (Co...
grprcan 17278 Right cancellation law for...
grpinveu 17279 The left inverse element o...
grpid 17280 Two ways of saying that an...
isgrpid2 17281 Properties showing that an...
grpidd2 17282 Deduce the identity elemen...
grpinvfval 17283 The inverse function of a ...
grpinvval 17284 The inverse of a group ele...
grpinvfn 17285 Functionality of the group...
grpinvfvi 17286 The group inverse function...
grpsubfval 17287 Group subtraction (divisio...
grpsubval 17288 Group subtraction (divisio...
grpinvf 17289 The group inversion operat...
grpinvcl 17290 A group element's inverse ...
grplinv 17291 The left inverse of a grou...
grprinv 17292 The right inverse of a gro...
grpinvid1 17293 The inverse of a group ele...
grpinvid2 17294 The inverse of a group ele...
isgrpinv 17295 Properties showing that a ...
grplrinv 17296 In a group, every member h...
grpidinv2 17297 A group's properties using...
grpidinv 17298 A group has a left and rig...
grpinvid 17299 The inverse of the identit...
grplcan 17300 Left cancellation law for ...
grpasscan1 17301 An associative cancellatio...
grpasscan2 17302 An associative cancellatio...
grpidrcan 17303 If right adding an element...
grpidlcan 17304 If left adding an element ...
grpinvinv 17305 Double inverse law for gro...
grpinvcnv 17306 The group inverse is its o...
grpinv11 17307 The group inverse is one-t...
grpinvf1o 17308 The group inverse is a one...
grpinvnz 17309 The inverse of a nonzero g...
grpinvnzcl 17310 The inverse of a nonzero g...
grpsubinv 17311 Subtraction of an inverse....
grplmulf1o 17312 Left multiplication by a g...
grpinvpropd 17313 If two structures have the...
grpidssd 17314 If the base set of a group...
grpinvssd 17315 If the base set of a group...
grpinvadd 17316 The inverse of the group o...
grpsubf 17317 Functionality of group sub...
grpsubcl 17318 Closure of group subtracti...
grpsubrcan 17319 Right cancellation law for...
grpinvsub 17320 Inverse of a group subtrac...
grpinvval2 17321 A ~ df-neg -like equation ...
grpsubid 17322 Subtraction of a group ele...
grpsubid1 17323 Subtraction of the identit...
grpsubeq0 17324 If the difference between ...
grpsubadd0sub 17325 Subtraction expressed as a...
grpsubadd 17326 Relationship between group...
grpsubsub 17327 Double group subtraction. ...
grpaddsubass 17328 Associative-type law for g...
grppncan 17329 Cancellation law for subtr...
grpnpcan 17330 Cancellation law for subtr...
grpsubsub4 17331 Double group subtraction (...
grppnpcan2 17332 Cancellation law for mixed...
grpnpncan 17333 Cancellation law for group...
grpnpncan0 17334 Cancellation law for group...
grpnnncan2 17335 Cancellation law for group...
dfgrp3lem 17336 Lemma for ~ dfgrp3 . (Con...
dfgrp3 17337 Alternate definition of a ...
dfgrp3e 17338 Alternate definition of a ...
grplactfval 17339 The left group action of e...
grplactval 17340 The value of the left grou...
grplactcnv 17341 The left group action of e...
grplactf1o 17342 The left group action of e...
grpsubpropd 17343 Weak property deduction fo...
grpsubpropd2 17344 Strong property deduction ...
grp1 17345 The (smallest) structure r...
grp1inv 17346 The inverse function of th...
prdsinvlem 17347 Characterization of invers...
prdsgrpd 17348 The product of a family of...
prdsinvgd 17349 Negation in a product of g...
pwsgrp 17350 The product of a family of...
pwsinvg 17351 Negation in a group power....
pwssub 17352 Subtraction in a group pow...
imasgrp2 17353 The image structure of a g...
imasgrp 17354 The image structure of a g...
imasgrpf1 17355 The image of a group under...
qusgrp2 17356 Prove that a quotient stru...
xpsgrp 17357 The binary product of grou...
mhmlem 17358 Lemma for ~ mhmmnd and ~ g...
mhmid 17359 A surjective monoid morphi...
mhmmnd 17360 The image of a monoid ` G ...
mhmfmhm 17361 The function fulfilling th...
ghmgrp 17362 The image of a group ` G `...
mulgfval 17365 Group multiple (exponentia...
mulgval 17366 Value of the group multipl...
mulgfn 17367 Functionality of the group...
mulgfvi 17368 The group multiple operati...
mulg0 17369 Group multiple (exponentia...
mulgnn 17370 Group multiple (exponentia...
mulg1 17371 Group multiple (exponentia...
mulgnnp1 17372 Group multiple (exponentia...
mulg2 17373 Group multiple (exponentia...
mulgnegnn 17374 Group multiple (exponentia...
mulgnn0p1 17375 Group multiple (exponentia...
mulgnnsubcl 17376 Closure of the group multi...
mulgnn0subcl 17377 Closure of the group multi...
mulgsubcl 17378 Closure of the group multi...
mulgnncl 17379 Closure of the group multi...
mulgnnclOLD 17380 Obsolete proof of ~ mulgnn...
mulgnn0cl 17381 Closure of the group multi...
mulgcl 17382 Closure of the group multi...
mulgneg 17383 Group multiple (exponentia...
mulgnegneg 17384 The inverse of a negative ...
mulgm1 17385 Group multiple (exponentia...
mulgaddcomlem 17386 Lemma for ~ mulgaddcom . ...
mulgaddcom 17387 The group multiple operato...
mulginvcom 17388 The group multiple operato...
mulginvinv 17389 The group multiple operato...
mulgnn0z 17390 A group multiple of the id...
mulgz 17391 A group multiple of the id...
mulgnndir 17392 Sum of group multiples, fo...
mulgnndirOLD 17393 Obsolete proof of ~ mulgnn...
mulgnn0dir 17394 Sum of group multiples, ge...
mulgdirlem 17395 Lemma for ~ mulgdir . (Co...
mulgdir 17396 Sum of group multiples, ge...
mulgp1 17397 Group multiple (exponentia...
mulgneg2 17398 Group multiple (exponentia...
mulgnnass 17399 Product of group multiples...
mulgnnassOLD 17400 Obsolete proof of ~ mulgnn...
mulgnn0ass 17401 Product of group multiples...
mulgass 17402 Product of group multiples...
mulgassr 17403 Reversed product of group ...
mulgmodid 17404 Casting out multiples of t...
mulgsubdir 17405 Subtraction of a group ele...
mhmmulg 17406 A homomorphism of monoids ...
mulgpropd 17407 Two structures with the sa...
submmulgcl 17408 Closure of the group multi...
submmulg 17409 A group multiple is the sa...
pwsmulg 17410 Value of a group multiple ...
issubg 17417 The subgroup predicate. (...
subgss 17418 A subgroup is a subset. (...
subgid 17419 A group is a subgroup of i...
subggrp 17420 A subgroup is a group. (C...
subgbas 17421 The base of the restricted...
subgrcl 17422 Reverse closure for the su...
subg0 17423 A subgroup of a group must...
subginv 17424 The inverse of an element ...
subg0cl 17425 The group identity is an e...
subginvcl 17426 The inverse of an element ...
subgcl 17427 A subgroup is closed under...
subgsubcl 17428 A subgroup is closed under...
subgsub 17429 The subtraction of element...
subgmulgcl 17430 Closure of the group multi...
subgmulg 17431 A group multiple is the sa...
issubg2 17432 Characterize the subgroups...
issubgrpd2 17433 Prove a subgroup by closur...
issubgrpd 17434 Prove a subgroup by closur...
issubg3 17435 A subgroup is a symmetric ...
issubg4 17436 A subgroup is a nonempty s...
grpissubg 17437 If the base set of a group...
resgrpisgrp 17438 If the base set of a group...
subgsubm 17439 A subgroup is a submonoid....
subsubg 17440 A subgroup of a subgroup i...
subgint 17441 The intersection of a none...
0subg 17442 The zero subgroup of an ar...
cycsubgcl 17443 The set of integer powers ...
cycsubgss 17444 The cyclic subgroup genera...
cycsubg 17445 The cyclic group generated...
isnsg 17446 Property of being a normal...
isnsg2 17447 Weaken the condition of ~ ...
nsgbi 17448 Defining property of a nor...
nsgsubg 17449 A normal subgroup is a sub...
nsgconj 17450 The conjugation of an elem...
isnsg3 17451 A subgroup is normal iff t...
subgacs 17452 Subgroups are an algebraic...
nsgacs 17453 Normal subgroups form an a...
cycsubg2 17454 The subgroup generated by ...
cycsubg2cl 17455 Any multiple of an element...
elnmz 17456 Elementhood in the normali...
nmzbi 17457 Defining property of the n...
nmzsubg 17458 The normalizer N_G(S) of a...
ssnmz 17459 A subgroup is a subset of ...
isnsg4 17460 A subgroup is normal iff i...
nmznsg 17461 Any subgroup is a normal s...
0nsg 17462 The zero subgroup is norma...
nsgid 17463 The whole group is a norma...
releqg 17464 The left coset equivalence...
eqgfval 17465 Value of the subgroup left...
eqgval 17466 Value of the subgroup left...
eqger 17467 The subgroup coset equival...
eqglact 17468 A left coset can be expres...
eqgid 17469 The left coset containing ...
eqgen 17470 Each coset is equipotent t...
eqgcpbl 17471 The subgroup coset equival...
qusgrp 17472 If ` Y ` is a normal subgr...
quseccl 17473 Closure of the quotient ma...
qusadd 17474 Value of the group operati...
qus0 17475 Value of the group identit...
qusinv 17476 Value of the group inverse...
qussub 17477 Value of the group subtrac...
lagsubg2 17478 Lagrange's theorem for fin...
lagsubg 17479 Lagrange theorem for Group...
reldmghm 17482 Lemma for group homomorphi...
isghm 17483 Property of being a homomo...
isghm3 17484 Property of a group homomo...
ghmgrp1 17485 A group homomorphism is on...
ghmgrp2 17486 A group homomorphism is on...
ghmf 17487 A group homomorphism is a ...
ghmlin 17488 A homomorphism of groups i...
ghmid 17489 A homomorphism of groups p...
ghminv 17490 A homomorphism of groups p...
ghmsub 17491 Linearity of subtraction t...
isghmd 17492 Deduction for a group homo...
ghmmhm 17493 A group homomorphism is a ...
ghmmhmb 17494 Group homomorphisms and mo...
ghmmulg 17495 A homomorphism of monoids ...
ghmrn 17496 The range of a homomorphis...
0ghm 17497 The constant zero linear f...
idghm 17498 The identity homomorphism ...
resghm 17499 Restriction of a homomorph...
resghm2 17500 One direction of ~ resghm2...
resghm2b 17501 Restriction of the codomai...
ghmghmrn 17502 A group homomorphism from ...
ghmco 17503 The composition of group h...
ghmima 17504 The image of a subgroup un...
ghmpreima 17505 The inverse image of a sub...
ghmeql 17506 The equalizer of two group...
ghmnsgima 17507 The image of a normal subg...
ghmnsgpreima 17508 The inverse image of a nor...
ghmker 17509 The kernel of a homomorphi...
ghmeqker 17510 Two source points map to t...
pwsdiagghm 17511 Diagonal homomorphism into...
ghmf1 17512 Two ways of saying a group...
ghmf1o 17513 A bijective group homomorp...
conjghm 17514 Conjugation is an automorp...
conjsubg 17515 A conjugated subgroup is a...
conjsubgen 17516 A conjugated subgroup is e...
conjnmz 17517 A subgroup is unchanged un...
conjnmzb 17518 Alternative condition for ...
conjnsg 17519 A normal subgroup is uncha...
qusghm 17520 If ` Y ` is a normal subgr...
ghmpropd 17521 Group homomorphism depends...
gimfn 17526 The group isomorphism func...
isgim 17527 An isomorphism of groups i...
gimf1o 17528 An isomorphism of groups i...
gimghm 17529 An isomorphism of groups i...
isgim2 17530 A group isomorphism is a h...
subggim 17531 Behavior of subgroups unde...
gimcnv 17532 The converse of a bijectiv...
gimco 17533 The composition of group i...
brgic 17534 The relation "is isomorphi...
brgici 17535 Prove isomorphic by an exp...
gicref 17536 Isomorphism is reflexive. ...
giclcl 17537 Isomorphism implies the le...
gicrcl 17538 Isomorphism implies the ri...
gicsym 17539 Isomorphism is symmetric. ...
gictr 17540 Isomorphism is transitive....
gicer 17541 Isomorphism is an equivale...
gicerOLD 17542 Obsolete proof of ~ gicer ...
gicen 17543 Isomorphic groups have equ...
gicsubgen 17544 A less trivial example of ...
isga 17547 The predicate "is a (left)...
gagrp 17548 The left argument of a gro...
gaset 17549 The right argument of a gr...
gagrpid 17550 The identity of the group ...
gaf 17551 The mapping of the group a...
gafo 17552 A group action is onto its...
gaass 17553 An "associative" property ...
ga0 17554 The action of a group on t...
gaid 17555 The trivial action of a gr...
subgga 17556 A subgroup acts on its par...
gass 17557 A subset of a group action...
gasubg 17558 The restriction of a group...
gaid2 17559 A group operation is a lef...
galcan 17560 The action of a particular...
gacan 17561 Group inverses cancel in a...
gapm 17562 The action of a particular...
gaorb 17563 The orbit equivalence rela...
gaorber 17564 The orbit equivalence rela...
gastacl 17565 The stabilizer subgroup in...
gastacos 17566 Write the coset relation f...
orbstafun 17567 Existence and uniqueness f...
orbstaval 17568 Value of the function at a...
orbsta 17569 The Orbit-Stabilizer theor...
orbsta2 17570 Relation between the size ...
cntrval 17575 Substitute definition of t...
cntzfval 17576 First level substitution f...
cntzval 17577 Definition substitution fo...
elcntz 17578 Elementhood in the central...
cntzel 17579 Membership in a centralize...
cntzsnval 17580 Special substitution for t...
elcntzsn 17581 Value of the centralizer o...
sscntz 17582 A centralizer expression f...
cntzrcl 17583 Reverse closure for elemen...
cntzssv 17584 The centralizer is uncondi...
cntzi 17585 Membership in a centralize...
cntri 17586 Defining property of the c...
resscntz 17587 Centralizer in a substruct...
cntz2ss 17588 Centralizers reverse the s...
cntzrec 17589 Reciprocity relationship f...
cntziinsn 17590 Express any centralizer as...
cntzsubm 17591 Centralizers in a monoid a...
cntzsubg 17592 Centralizers in a group ar...
cntzidss 17593 If the elements of ` S ` c...
cntzmhm 17594 Centralizers in a monoid a...
cntzmhm2 17595 Centralizers in a monoid a...
cntrsubgnsg 17596 A central subgroup is norm...
cntrnsg 17597 The center of a group is a...
oppgval 17600 Value of the opposite grou...
oppgplusfval 17601 Value of the addition oper...
oppgplus 17602 Value of the addition oper...
oppglem 17603 Lemma for ~ oppgbas . (Co...
oppgbas 17604 Base set of an opposite gr...
oppgtset 17605 Topology of an opposite gr...
oppgtopn 17606 Topology of an opposite gr...
oppgmnd 17607 The opposite of a monoid i...
oppgmndb 17608 Bidirectional form of ~ op...
oppgid 17609 Zero in a monoid is a symm...
oppggrp 17610 The opposite of a group is...
oppggrpb 17611 Bidirectional form of ~ op...
oppginv 17612 Inverses in a group are a ...
invoppggim 17613 The inverse is an antiauto...
oppggic 17614 Every group is (naturally)...
oppgsubm 17615 Being a submonoid is a sym...
oppgsubg 17616 Being a subgroup is a symm...
oppgcntz 17617 A centralizer in a group i...
oppgcntr 17618 The center of a group is t...
gsumwrev 17619 A sum in an opposite monoi...
symgval 17622 The value of the symmetric...
symgbas 17623 The base set of the symmet...
elsymgbas2 17624 Two ways of saying a funct...
elsymgbas 17625 Two ways of saying a funct...
symgbasf1o 17626 Elements in the symmetric ...
symgbasf 17627 A permutation (element of ...
symghash 17628 The symmetric group on ` n...
symgbasfi 17629 The symmetric group on a f...
symgfv 17630 The function value of a pe...
symgfvne 17631 The function values of a p...
symgplusg 17632 The group operation of a s...
symgov 17633 The value of the group ope...
symgcl 17634 The group operation of the...
symgmov1 17635 For a permutation of a set...
symgmov2 17636 For a permutation of a set...
symgbas0 17637 The base set of the symmet...
symg1hash 17638 The symmetric group on a s...
symg1bas 17639 The symmetric group on a s...
symg2hash 17640 The symmetric group on a (...
symg2bas 17641 The symmetric group on a p...
symgtset 17642 The topology of the symmet...
symggrp 17643 The symmetric group on a s...
symgid 17644 The group identity element...
symginv 17645 The group inverse in the s...
galactghm 17646 The currying of a group ac...
lactghmga 17647 The converse of ~ galactgh...
symgtopn 17648 The topology of the symmet...
symgga 17649 The symmetric group induce...
pgrpsubgsymgbi 17650 Every permutation group is...
pgrpsubgsymg 17651 Every permutation group is...
idresperm 17652 The identity function rest...
idressubgsymg 17653 The singleton containing o...
idrespermg 17654 The structure with the sin...
cayleylem1 17655 Lemma for ~ cayley . (Con...
cayleylem2 17656 Lemma for ~ cayley . (Con...
cayley 17657 Cayley's Theorem (construc...
cayleyth 17658 Cayley's Theorem (existenc...
symgfix2 17659 If a permutation does not ...
symgextf 17660 The extension of a permuta...
symgextfv 17661 The function value of the ...
symgextfve 17662 The function value of the ...
symgextf1lem 17663 Lemma for ~ symgextf1 . (...
symgextf1 17664 The extension of a permuta...
symgextfo 17665 The extension of a permuta...
symgextf1o 17666 The extension of a permuta...
symgextsymg 17667 The extension of a permuta...
symgextres 17668 The restriction of the ext...
gsumccatsymgsn 17669 Homomorphic property of co...
gsmsymgrfixlem1 17670 Lemma 1 for ~ gsmsymgrfix ...
gsmsymgrfix 17671 The composition of permuta...
fvcosymgeq 17672 The values of two composit...
gsmsymgreqlem1 17673 Lemma 1 for ~ gsmsymgreq ....
gsmsymgreqlem2 17674 Lemma 2 for ~ gsmsymgreq ....
gsmsymgreq 17675 Two combination of permuta...
symgfixelq 17676 A permutation of a set fix...
symgfixels 17677 The restriction of a permu...
symgfixelsi 17678 The restriction of a permu...
symgfixf 17679 The mapping of a permutati...
symgfixf1 17680 The mapping of a permutati...
symgfixfolem1 17681 Lemma 1 for ~ symgfixfo . ...
symgfixfo 17682 The mapping of a permutati...
symgfixf1o 17683 The mapping of a permutati...
f1omvdmvd 17686 A permutation of any class...
f1omvdcnv 17687 A permutation and its inve...
mvdco 17688 Composing two permutations...
f1omvdconj 17689 Conjugation of a permutati...
f1otrspeq 17690 A transposition is charact...
f1omvdco2 17691 If exactly one of two perm...
f1omvdco3 17692 If a point is moved by exa...
pmtrfval 17693 The function generating tr...
pmtrval 17694 A generated transposition,...
pmtrfv 17695 General value of mapping a...
pmtrprfv 17696 In a transposition of two ...
pmtrprfv3 17697 In a transposition of two ...
pmtrf 17698 Functionality of a transpo...
pmtrmvd 17699 A transposition moves prec...
pmtrrn 17700 Transposing two points giv...
pmtrfrn 17701 A transposition (as a kind...
pmtrffv 17702 Mapping of a point under a...
pmtrrn2 17703 For any transposition ther...
pmtrfinv 17704 A transposition function i...
pmtrfmvdn0 17705 A transposition moves at l...
pmtrff1o 17706 A transposition function i...
pmtrfcnv 17707 A transposition function i...
pmtrfb 17708 An intrinsic characterizat...
pmtrfconj 17709 Any conjugate of a transpo...
symgsssg 17710 The symmetric group has su...
symgfisg 17711 The symmetric group has a ...
symgtrf 17712 Transpositions are element...
symggen 17713 The span of the transposit...
symggen2 17714 A finite permutation group...
symgtrinv 17715 To invert a permutation re...
pmtr3ncomlem1 17716 Lemma 1 for ~ pmtr3ncom . ...
pmtr3ncomlem2 17717 Lemma 2 for ~ pmtr3ncom . ...
pmtr3ncom 17718 Transpositions over sets w...
pmtrdifellem1 17719 Lemma 1 for ~ pmtrdifel . ...
pmtrdifellem2 17720 Lemma 2 for ~ pmtrdifel . ...
pmtrdifellem3 17721 Lemma 3 for ~ pmtrdifel . ...
pmtrdifellem4 17722 Lemma 4 for ~ pmtrdifel . ...
pmtrdifel 17723 A transposition of element...
pmtrdifwrdellem1 17724 Lemma 1 for ~ pmtrdifwrdel...
pmtrdifwrdellem2 17725 Lemma 2 for ~ pmtrdifwrdel...
pmtrdifwrdellem3 17726 Lemma 3 for ~ pmtrdifwrdel...
pmtrdifwrdel2lem1 17727 Lemma 1 for ~ pmtrdifwrdel...
pmtrdifwrdel 17728 A sequence of transpositio...
pmtrdifwrdel2 17729 A sequence of transpositio...
pmtrprfval 17730 The transpositions on a pa...
pmtrprfvalrn 17731 The range of the transposi...
psgnunilem1 17736 Lemma for ~ psgnuni . Giv...
psgnunilem5 17737 Lemma for ~ psgnuni . It ...
psgnunilem2 17738 Lemma for ~ psgnuni . Ind...
psgnunilem3 17739 Lemma for ~ psgnuni . Any...
psgnunilem4 17740 Lemma for ~ psgnuni . An ...
m1expaddsub 17741 Addition and subtraction o...
psgnuni 17742 If the same permutation ca...
psgnfval 17743 Function definition of the...
psgnfn 17744 Functionality and domain o...
psgndmsubg 17745 The finitary permutations ...
psgneldm 17746 Property of being a finita...
psgneldm2 17747 The finitary permutations ...
psgneldm2i 17748 A sequence of transpositio...
psgneu 17749 A finitary permutation has...
psgnval 17750 Value of the permutation s...
psgnvali 17751 A finitary permutation has...
psgnvalii 17752 Any representation of a pe...
psgnpmtr 17753 All transpositions are odd...
psgn0fv0 17754 The permutation sign funct...
sygbasnfpfi 17755 The class of non-fixed poi...
psgnfvalfi 17756 Function definition of the...
psgnvalfi 17757 Value of the permutation s...
psgnran 17758 The range of the permutati...
gsmtrcl 17759 The group sum of transposi...
psgnfitr 17760 A permutation of a finite ...
psgnfieu 17761 A permutation of a finite ...
pmtrsn 17762 The value of the transposi...
psgnsn 17763 The permutation sign funct...
psgnprfval 17764 The permutation sign funct...
psgnprfval1 17765 The permutation sign of th...
psgnprfval2 17766 The permutation sign of th...
odfval 17775 Value of the order functio...
odval 17776 Second substitution for th...
odlem1 17777 The group element order is...
odcl 17778 The order of a group eleme...
odf 17779 Functionality of the group...
odid 17780 Any element to the power o...
odlem2 17781 Any positive annihilator o...
odmodnn0 17782 Reduce the argument of a g...
mndodconglem 17783 Lemma for ~ mndodcong . (...
mndodcong 17784 If two multipliers are con...
mndodcongi 17785 If two multipliers are con...
oddvdsnn0 17786 The only multiples of ` A ...
odnncl 17787 If a nonzero multiple of a...
odmod 17788 Reduce the argument of a g...
oddvds 17789 The only multiples of ` A ...
oddvdsi 17790 Any group element is annih...
odcong 17791 If two multipliers are con...
odeq 17792 The ~ oddvds property uniq...
odval2 17793 A non-conditional definiti...
odmulgid 17794 A relationship between the...
odmulg2 17795 The order of a multiple di...
odmulg 17796 Relationship between the o...
odmulgeq 17797 A multiple of a point of f...
odbezout 17798 If ` N ` is coprime to the...
od1 17799 The order of the group ide...
odeq1 17800 The group identity is the ...
odinv 17801 The order of the inverse o...
odf1 17802 The multiples of an elemen...
odinf 17803 The multiples of an elemen...
dfod2 17804 An alternative definition ...
odcl2 17805 The order of an element of...
oddvds2 17806 The order of an element of...
submod 17807 The order of an element is...
subgod 17808 The order of an element is...
odsubdvds 17809 The order of an element of...
odf1o1 17810 An element with zero order...
odf1o2 17811 An element with nonzero or...
odhash 17812 An element of zero order g...
odhash2 17813 If an element has nonzero ...
odhash3 17814 An element which generates...
odngen 17815 A cyclic subgroup of size ...
gexval 17816 Value of the exponent of a...
gexlem1 17817 The group element order is...
gexcl 17818 The exponent of a group is...
gexid 17819 Any element to the power o...
gexlem2 17820 Any positive annihilator o...
gexdvdsi 17821 Any group element is annih...
gexdvds 17822 The only ` N ` that annihi...
gexdvds2 17823 An integer divides the gro...
gexod 17824 Any group element is annih...
gexcl3 17825 If the order of every grou...
gexnnod 17826 Every group element has fi...
gexcl2 17827 The exponent of a finite g...
gexdvds3 17828 The exponent of a finite g...
gex1 17829 A group or monoid has expo...
ispgp 17830 A group is a ` P ` -group ...
pgpprm 17831 Reverse closure for the fi...
pgpgrp 17832 Reverse closure for the se...
pgpfi1 17833 A finite group with order ...
pgp0 17834 The identity subgroup is a...
subgpgp 17835 A subgroup of a p-group is...
sylow1lem1 17836 Lemma for ~ sylow1 . The ...
sylow1lem2 17837 Lemma for ~ sylow1 . The ...
sylow1lem3 17838 Lemma for ~ sylow1 . One ...
sylow1lem4 17839 Lemma for ~ sylow1 . The ...
sylow1lem5 17840 Lemma for ~ sylow1 . Usin...
sylow1 17841 Sylow's first theorem. If...
odcau 17842 Cauchy's theorem for the o...
pgpfi 17843 The converse to ~ pgpfi1 ....
pgpfi2 17844 Alternate version of ~ pgp...
pgphash 17845 The order of a p-group. (...
isslw 17846 The property of being a Sy...
slwprm 17847 Reverse closure for the fi...
slwsubg 17848 A Sylow ` P ` -subgroup is...
slwispgp 17849 Defining property of a Syl...
slwpss 17850 A proper superset of a Syl...
slwpgp 17851 A Sylow ` P ` -subgroup is...
pgpssslw 17852 Every ` P ` -subgroup is c...
slwn0 17853 Every finite group contain...
subgslw 17854 A Sylow subgroup that is c...
sylow2alem1 17855 Lemma for ~ sylow2a . An ...
sylow2alem2 17856 Lemma for ~ sylow2a . All...
sylow2a 17857 A named lemma of Sylow's s...
sylow2blem1 17858 Lemma for ~ sylow2b . Eva...
sylow2blem2 17859 Lemma for ~ sylow2b . Lef...
sylow2blem3 17860 Sylow's second theorem. P...
sylow2b 17861 Sylow's second theorem. A...
slwhash 17862 A sylow subgroup has cardi...
fislw 17863 The sylow subgroups of a f...
sylow2 17864 Sylow's second theorem. S...
sylow3lem1 17865 Lemma for ~ sylow3 , first...
sylow3lem2 17866 Lemma for ~ sylow3 , first...
sylow3lem3 17867 Lemma for ~ sylow3 , first...
sylow3lem4 17868 Lemma for ~ sylow3 , first...
sylow3lem5 17869 Lemma for ~ sylow3 , secon...
sylow3lem6 17870 Lemma for ~ sylow3 , secon...
sylow3 17871 Sylow's third theorem. Th...
lsmfval 17876 The subgroup sum function ...
lsmvalx 17877 Subspace sum value (for a ...
lsmelvalx 17878 Subspace sum membership (f...
lsmelvalix 17879 Subspace sum membership (f...
oppglsm 17880 The subspace sum operation...
lsmssv 17881 Subgroup sum is a subset o...
lsmless1x 17882 Subset implies subgroup su...
lsmless2x 17883 Subset implies subgroup su...
lsmub1x 17884 Subgroup sum is an upper b...
lsmub2x 17885 Subgroup sum is an upper b...
lsmval 17886 Subgroup sum value (for a ...
lsmelval 17887 Subgroup sum membership (f...
lsmelvali 17888 Subgroup sum membership (f...
lsmelvalm 17889 Subgroup sum membership an...
lsmelvalmi 17890 Membership of vector subtr...
lsmsubm 17891 The sum of two commuting s...
lsmsubg 17892 The sum of two commuting s...
lsmcom2 17893 Subgroup sum commutes. (C...
lsmub1 17894 Subgroup sum is an upper b...
lsmub2 17895 Subgroup sum is an upper b...
lsmunss 17896 Union of subgroups is a su...
lsmless1 17897 Subset implies subgroup su...
lsmless2 17898 Subset implies subgroup su...
lsmless12 17899 Subset implies subgroup su...
lsmidm 17900 Subgroup sum is idempotent...
lsmlub 17901 The least upper bound prop...
lsmss1 17902 Subgroup sum with a subset...
lsmss1b 17903 Subgroup sum with a subset...
lsmss2 17904 Subgroup sum with a subset...
lsmss2b 17905 Subgroup sum with a subset...
lsmass 17906 Subgroup sum is associativ...
lsm01 17907 Subgroup sum with the zero...
lsm02 17908 Subgroup sum with the zero...
subglsm 17909 The subgroup sum evaluated...
lssnle 17910 Equivalent expressions for...
lsmmod 17911 The modular law holds for ...
lsmmod2 17912 Modular law dual for subgr...
lsmpropd 17913 If two structures have the...
cntzrecd 17914 Commute the "subgroups com...
lsmcntz 17915 The "subgroups commute" pr...
lsmcntzr 17916 The "subgroups commute" pr...
lsmdisj 17917 Disjointness from a subgro...
lsmdisj2 17918 Association of the disjoin...
lsmdisj3 17919 Association of the disjoin...
lsmdisjr 17920 Disjointness from a subgro...
lsmdisj2r 17921 Association of the disjoin...
lsmdisj3r 17922 Association of the disjoin...
lsmdisj2a 17923 Association of the disjoin...
lsmdisj2b 17924 Association of the disjoin...
lsmdisj3a 17925 Association of the disjoin...
lsmdisj3b 17926 Association of the disjoin...
subgdisj1 17927 Vectors belonging to disjo...
subgdisj2 17928 Vectors belonging to disjo...
subgdisjb 17929 Vectors belonging to disjo...
pj1fval 17930 The left projection functi...
pj1val 17931 The left projection functi...
pj1eu 17932 Uniqueness of a left proje...
pj1f 17933 The left projection functi...
pj2f 17934 The right projection funct...
pj1id 17935 Any element of a direct su...
pj1eq 17936 Any element of a direct su...
pj1lid 17937 The left projection functi...
pj1rid 17938 The left projection functi...
pj1ghm 17939 The left projection functi...
pj1ghm2 17940 The left projection functi...
lsmhash 17941 The order of the direct pr...
efgmval 17948 Value of the formal invers...
efgmf 17949 The formal inverse operati...
efgmnvl 17950 The inversion function on ...
efgrcl 17951 Lemma for ~ efgval . (Con...
efglem 17952 Lemma for ~ efgval . (Con...
efgval 17953 Value of the free group co...
efger 17954 Value of the free group co...
efgi 17955 Value of the free group co...
efgi0 17956 Value of the free group co...
efgi1 17957 Value of the free group co...
efgtf 17958 Value of the free group co...
efgtval 17959 Value of the extension fun...
efgval2 17960 Value of the free group co...
efgi2 17961 Value of the free group co...
efgtlen 17962 Value of the free group co...
efginvrel2 17963 The inverse of the reverse...
efginvrel1 17964 The inverse of the reverse...
efgsf 17965 Value of the auxiliary fun...
efgsdm 17966 Elementhood in the domain ...
efgsval 17967 Value of the auxiliary fun...
efgsdmi 17968 Property of the last link ...
efgsval2 17969 Value of the auxiliary fun...
efgsrel 17970 The start and end of any e...
efgs1 17971 A singleton of an irreduci...
efgs1b 17972 Every extension sequence e...
efgsp1 17973 If ` F ` is an extension s...
efgsres 17974 An initial segment of an e...
efgsfo 17975 For any word, there is a s...
efgredlema 17976 The reduced word that form...
efgredlemf 17977 Lemma for ~ efgredleme . ...
efgredlemg 17978 Lemma for ~ efgred . (Con...
efgredleme 17979 Lemma for ~ efgred . (Con...
efgredlemd 17980 The reduced word that form...
efgredlemc 17981 The reduced word that form...
efgredlemb 17982 The reduced word that form...
efgredlem 17983 The reduced word that form...
efgred 17984 The reduced word that form...
efgrelexlema 17985 If two words ` A , B ` are...
efgrelexlemb 17986 If two words ` A , B ` are...
efgrelex 17987 If two words ` A , B ` are...
efgredeu 17988 There is a unique reduced ...
efgred2 17989 Two extension sequences ha...
efgcpbllema 17990 Lemma for ~ efgrelex . De...
efgcpbllemb 17991 Lemma for ~ efgrelex . Sh...
efgcpbl 17992 Two extension sequences ha...
efgcpbl2 17993 Two extension sequences ha...
frgpval 17994 Value of the free group co...
frgpcpbl 17995 Compatibility of the group...
frgp0 17996 The free group is a group....
frgpeccl 17997 Closure of the quotient ma...
frgpgrp 17998 The free group is a group....
frgpadd 17999 Addition in the free group...
frgpinv 18000 The inverse of an element ...
frgpmhm 18001 The "natural map" from wor...
vrgpfval 18002 The canonical injection fr...
vrgpval 18003 The value of the generatin...
vrgpf 18004 The mapping from the index...
vrgpinv 18005 The inverse of a generatin...
frgpuptf 18006 Any assignment of the gene...
frgpuptinv 18007 Any assignment of the gene...
frgpuplem 18008 Any assignment of the gene...
frgpupf 18009 Any assignment of the gene...
frgpupval 18010 Any assignment of the gene...
frgpup1 18011 Any assignment of the gene...
frgpup2 18012 The evaluation map has the...
frgpup3lem 18013 The evaluation map has the...
frgpup3 18014 Universal property of the ...
0frgp 18015 The free group on zero gen...
isabl 18020 The predicate "is an Abeli...
ablgrp 18021 An Abelian group is a grou...
ablcmn 18022 An Abelian group is a comm...
iscmn 18023 The predicate "is a commut...
isabl2 18024 The predicate "is an Abeli...
cmnpropd 18025 If two structures have the...
ablpropd 18026 If two structures have the...
ablprop 18027 If two structures have the...
iscmnd 18028 Properties that determine ...
isabld 18029 Properties that determine ...
isabli 18030 Properties that determine ...
cmnmnd 18031 A commutative monoid is a ...
cmncom 18032 A commutative monoid is co...
ablcom 18033 An Abelian group operation...
cmn32 18034 Commutative/associative la...
cmn4 18035 Commutative/associative la...
cmn12 18036 Commutative/associative la...
abl32 18037 Commutative/associative la...
ablinvadd 18038 The inverse of an Abelian ...
ablsub2inv 18039 Abelian group subtraction ...
ablsubadd 18040 Relationship between Abeli...
ablsub4 18041 Commutative/associative su...
abladdsub4 18042 Abelian group addition/sub...
abladdsub 18043 Associative-type law for g...
ablpncan2 18044 Cancellation law for subtr...
ablpncan3 18045 A cancellation law for com...
ablsubsub 18046 Law for double subtraction...
ablsubsub4 18047 Law for double subtraction...
ablpnpcan 18048 Cancellation law for mixed...
ablnncan 18049 Cancellation law for group...
ablsub32 18050 Swap the second and third ...
ablnnncan 18051 Cancellation law for group...
ablnnncan1 18052 Cancellation law for group...
ablsubsub23 18053 Swap subtrahend and result...
mulgnn0di 18054 Group multiple of a sum, f...
mulgdi 18055 Group multiple of a sum. ...
mulgmhm 18056 The map from ` x ` to ` n ...
mulgghm 18057 The map from ` x ` to ` n ...
mulgsubdi 18058 Group multiple of a differ...
ghmfghm 18059 The function fulfilling th...
ghmcmn 18060 The image of a commutative...
ghmabl 18061 The image of an abelian gr...
invghm 18062 The inversion map is a gro...
eqgabl 18063 Value of the subgroup cose...
subgabl 18064 A subgroup of an abelian g...
subcmn 18065 A submonoid of a commutati...
submcmn 18066 A submonoid of a commutati...
submcmn2 18067 A submonoid is commutative...
cntzcmn 18068 The centralizer of any sub...
cntzcmnss 18069 Any subset in a commutativ...
cntzspan 18070 If the generators commute,...
cntzcmnf 18071 Discharge the centralizer ...
ghmplusg 18072 The pointwise sum of two l...
ablnsg 18073 Every subgroup of an abeli...
odadd1 18074 The order of a product in ...
odadd2 18075 The order of a product in ...
odadd 18076 The order of a product is ...
gex2abl 18077 A group with exponent 2 (o...
gexexlem 18078 Lemma for ~ gexex . (Cont...
gexex 18079 In an abelian group with f...
torsubg 18080 The set of all elements of...
oddvdssubg 18081 The set of all elements wh...
lsmcomx 18082 Subgroup sum commutes (ext...
ablcntzd 18083 All subgroups in an abelia...
lsmcom 18084 Subgroup sum commutes. (C...
lsmsubg2 18085 The sum of two subgroups i...
lsm4 18086 Commutative/associative la...
prdscmnd 18087 The product of a family of...
prdsabld 18088 The product of a family of...
pwscmn 18089 The structure power on a c...
pwsabl 18090 The structure power on an ...
qusabl 18091 If ` Y ` is a subgroup of ...
abl1 18092 The (smallest) structure r...
abln0 18093 Abelian groups (and theref...
cnaddablx 18094 The complex numbers are an...
cnaddabl 18095 The complex numbers are an...
cnaddid 18096 The group identity element...
cnaddinv 18097 Value of the group inverse...
zaddablx 18098 The integers are an Abelia...
frgpnabllem1 18099 Lemma for ~ frgpnabl . (C...
frgpnabllem2 18100 Lemma for ~ frgpnabl . (C...
frgpnabl 18101 The free group on two or m...
iscyg 18104 Definition of a cyclic gro...
iscyggen 18105 The property of being a cy...
iscyggen2 18106 The property of being a cy...
iscyg2 18107 A cyclic group is a group ...
cyggeninv 18108 The inverse of a cyclic ge...
cyggenod 18109 An element is the generato...
cyggenod2 18110 In an infinite cyclic grou...
iscyg3 18111 Definition of a cyclic gro...
iscygd 18112 Definition of a cyclic gro...
iscygodd 18113 Show that a group with an ...
cyggrp 18114 A cyclic group is a group....
cygabl 18115 A cyclic group is abelian....
cygctb 18116 A cyclic group is countabl...
0cyg 18117 The trivial group is cycli...
prmcyg 18118 A group with prime order i...
lt6abl 18119 A group with fewer than ` ...
ghmcyg 18120 The image of a cyclic grou...
cyggex2 18121 The exponent of a cyclic g...
cyggex 18122 The exponent of a finite c...
cyggexb 18123 A finite abelian group is ...
giccyg 18124 Cyclicity is a group prope...
cycsubgcyg 18125 The cyclic subgroup genera...
cycsubgcyg2 18126 The cyclic subgroup genera...
gsumval3a 18127 Value of the group sum ope...
gsumval3eu 18128 The group sum as defined i...
gsumval3lem1 18129 Lemma 1 for ~ gsumval3 . ...
gsumval3lem2 18130 Lemma 2 for ~ gsumval3 . ...
gsumval3 18131 Value of the group sum ope...
gsumcllem 18132 Lemma for ~ gsumcl and rel...
gsumzres 18133 Extend a finite group sum ...
gsumzcl2 18134 Closure of a finite group ...
gsumzcl 18135 Closure of a finite group ...
gsumzf1o 18136 Re-index a finite group su...
gsumres 18137 Extend a finite group sum ...
gsumcl2 18138 Closure of a finite group ...
gsumcl 18139 Closure of a finite group ...
gsumf1o 18140 Re-index a finite group su...
gsumzsubmcl 18141 Closure of a group sum in ...
gsumsubmcl 18142 Closure of a group sum in ...
gsumsubgcl 18143 Closure of a group sum in ...
gsumzaddlem 18144 The sum of two group sums....
gsumzadd 18145 The sum of two group sums....
gsumadd 18146 The sum of two group sums....
gsummptfsadd 18147 The sum of two group sums ...
gsummptfidmadd 18148 The sum of two group sums ...
gsummptfidmadd2 18149 The sum of two group sums ...
gsumzsplit 18150 Split a group sum into two...
gsumsplit 18151 Split a group sum into two...
gsumsplit2 18152 Split a group sum into two...
gsummptfidmsplit 18153 Split a group sum expresse...
gsummptfidmsplitres 18154 Split a group sum expresse...
gsummptfzsplit 18155 Split a group sum expresse...
gsummptfzsplitl 18156 Split a group sum expresse...
gsumconst 18157 Sum of a constant series. ...
gsumconstf 18158 Sum of a constant series. ...
gsummptshft 18159 Index shift of a finite gr...
gsumzmhm 18160 Apply a group homomorphism...
gsummhm 18161 Apply a group homomorphism...
gsummhm2 18162 Apply a group homomorphism...
gsummptmhm 18163 Apply a group homomorphism...
gsummulglem 18164 Lemma for ~ gsummulg and ~...
gsummulg 18165 Nonnegative multiple of a ...
gsummulgz 18166 Integer multiple of a grou...
gsumzoppg 18167 The opposite of a group su...
gsumzinv 18168 Inverse of a group sum. (...
gsuminv 18169 Inverse of a group sum. (...
gsummptfidminv 18170 Inverse of a group sum exp...
gsumsub 18171 The difference of two grou...
gsummptfssub 18172 The difference of two grou...
gsummptfidmsub 18173 The difference of two grou...
gsumsnfd 18174 Group sum of a singleton, ...
gsumsnd 18175 Group sum of a singleton, ...
gsumsnf 18176 Group sum of a singleton, ...
gsumsn 18177 Group sum of a singleton. ...
gsumzunsnd 18178 Append an element to a fin...
gsumunsnfd 18179 Append an element to a fin...
gsumunsnd 18180 Append an element to a fin...
gsumunsnf 18181 Append an element to a fin...
gsumunsn 18182 Append an element to a fin...
gsumdifsnd 18183 Extract a summand from a f...
gsumpt 18184 Sum of a family that is no...
gsummptf1o 18185 Re-index a finite group su...
gsummptun 18186 Group sum of a disjoint un...
gsummpt1n0 18187 If only one summand in a f...
gsummptif1n0 18188 If only one summand in a f...
gsummptcl 18189 Closure of a finite group ...
gsummptfif1o 18190 Re-index a finite group su...
gsummptfzcl 18191 Closure of a finite group ...
gsum2dlem1 18192 Lemma 1 for ~ gsum2d . (C...
gsum2dlem2 18193 Lemma for ~ gsum2d . (Con...
gsum2d 18194 Write a sum over a two-dim...
gsum2d2lem 18195 Lemma for ~ gsum2d2 : show...
gsum2d2 18196 Write a group sum over a t...
gsumcom2 18197 Two-dimensional commutatio...
gsumxp 18198 Write a group sum over a c...
gsumcom 18199 Commute the arguments of a...
prdsgsum 18200 Finite commutative sums in...
pwsgsum 18201 Finite commutative sums in...
fsfnn0gsumfsffz 18202 Replacing a finitely suppo...
nn0gsumfz 18203 Replacing a finitely suppo...
nn0gsumfz0 18204 Replacing a finitely suppo...
gsummptnn0fz 18205 A final group sum over a f...
gsummptnn0fzv 18206 A final group sum over a f...
gsummptnn0fzfv 18207 A final group sum over a f...
telgsumfzslem 18208 Lemma for ~ telgsumfzs (in...
telgsumfzs 18209 Telescoping group sum rang...
telgsumfz 18210 Telescoping group sum rang...
telgsumfz0s 18211 Telescoping finite group s...
telgsumfz0 18212 Telescoping finite group s...
telgsums 18213 Telescoping finitely suppo...
telgsum 18214 Telescoping finitely suppo...
reldmdprd 18219 The domain of the internal...
dmdprd 18220 The domain of definition o...
dmdprdd 18221 Show that a given family i...
dprddomprc 18222 A family of subgroups inde...
dprddomcld 18223 If a family of subgroups i...
dprdval0prc 18224 The internal direct produc...
dprdval 18225 The value of the internal ...
eldprd 18226 A class ` A ` is an intern...
dprdgrp 18227 Reverse closure for the in...
dprdf 18228 The function ` S ` is a fa...
dprdf2 18229 The function ` S ` is a fa...
dprdcntz 18230 The function ` S ` is a fa...
dprddisj 18231 The function ` S ` is a fa...
dprdw 18232 The property of being a fi...
dprdwd 18233 A mapping being a finitely...
dprdff 18234 A finitely supported funct...
dprdfcl 18235 A finitely supported funct...
dprdffsupp 18236 A finitely supported funct...
dprdfcntz 18237 A function on the elements...
dprdssv 18238 The internal direct produc...
dprdfid 18239 A function mapping all but...
eldprdi 18240 The domain of definition o...
dprdfinv 18241 Take the inverse of a grou...
dprdfadd 18242 Take the sum of group sums...
dprdfsub 18243 Take the difference of gro...
dprdfeq0 18244 The zero function is the o...
dprdf11 18245 Two group sums over a dire...
dprdsubg 18246 The internal direct produc...
dprdub 18247 Each factor is a subset of...
dprdlub 18248 The direct product is smal...
dprdspan 18249 The direct product is the ...
dprdres 18250 Restriction of a direct pr...
dprdss 18251 Create a direct product by...
dprdz 18252 A family consisting entire...
dprd0 18253 The empty family is an int...
dprdf1o 18254 Rearrange the index set of...
dprdf1 18255 Rearrange the index set of...
subgdmdprd 18256 A direct product in a subg...
subgdprd 18257 A direct product in a subg...
dprdsn 18258 A singleton family is an i...
dmdprdsplitlem 18259 Lemma for ~ dmdprdsplit . ...
dprdcntz2 18260 The function ` S ` is a fa...
dprddisj2 18261 The function ` S ` is a fa...
dprd2dlem2 18262 The direct product of a co...
dprd2dlem1 18263 The direct product of a co...
dprd2da 18264 The direct product of a co...
dprd2db 18265 The direct product of a co...
dprd2d2 18266 The direct product of a co...
dmdprdsplit2lem 18267 Lemma for ~ dmdprdsplit . ...
dmdprdsplit2 18268 The direct product splits ...
dmdprdsplit 18269 The direct product splits ...
dprdsplit 18270 The direct product is the ...
dmdprdpr 18271 A singleton family is an i...
dprdpr 18272 A singleton family is an i...
dpjlem 18273 Lemma for theorems about d...
dpjcntz 18274 The two subgroups that app...
dpjdisj 18275 The two subgroups that app...
dpjlsm 18276 The two subgroups that app...
dpjfval 18277 Value of the direct produc...
dpjval 18278 Value of the direct produc...
dpjf 18279 The ` X ` -th index projec...
dpjidcl 18280 The key property of projec...
dpjeq 18281 Decompose a group sum into...
dpjid 18282 The key property of projec...
dpjlid 18283 The ` X ` -th index projec...
dpjrid 18284 The ` Y ` -th index projec...
dpjghm 18285 The direct product is the ...
dpjghm2 18286 The direct product is the ...
ablfacrplem 18287 Lemma for ~ ablfacrp2 . (...
ablfacrp 18288 A finite abelian group who...
ablfacrp2 18289 The factors ` K , L ` of ~...
ablfac1lem 18290 Lemma for ~ ablfac1b . Sa...
ablfac1a 18291 The factors of ~ ablfac1b ...
ablfac1b 18292 Any abelian group is the d...
ablfac1c 18293 The factors of ~ ablfac1b ...
ablfac1eulem 18294 Lemma for ~ ablfac1eu . (...
ablfac1eu 18295 The factorization of ~ abl...
pgpfac1lem1 18296 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem2 18297 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem3a 18298 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem3 18299 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem4 18300 Lemma for ~ pgpfac1 . (Co...
pgpfac1lem5 18301 Lemma for ~ pgpfac1 . (Co...
pgpfac1 18302 Factorization of a finite ...
pgpfaclem1 18303 Lemma for ~ pgpfac . (Con...
pgpfaclem2 18304 Lemma for ~ pgpfac . (Con...
pgpfaclem3 18305 Lemma for ~ pgpfac . (Con...
pgpfac 18306 Full factorization of a fi...
ablfaclem1 18307 Lemma for ~ ablfac . (Con...
ablfaclem2 18308 Lemma for ~ ablfac . (Con...
ablfaclem3 18309 Lemma for ~ ablfac . (Con...
ablfac 18310 The Fundamental Theorem of...
ablfac2 18311 Choose generators for each...
fnmgp 18314 The multiplicative group o...
mgpval 18315 Value of the multiplicatio...
mgpplusg 18316 Value of the group operati...
mgplem 18317 Lemma for ~ mgpbas . (Con...
mgpbas 18318 Base set of the multiplica...
mgpsca 18319 The multiplication monoid ...
mgptset 18320 Topology component of the ...
mgptopn 18321 Topology of the multiplica...
mgpds 18322 Distance function of the m...
mgpress 18323 Subgroup commutes with the...
ringidval 18326 The value of the unity ele...
dfur2 18327 The multiplicative identit...
issrg 18330 The predicate "is a semiri...
srgcmn 18331 A semiring is a commutativ...
srgmnd 18332 A semiring is a monoid. (...
srgmgp 18333 A semiring is a monoid und...
srgi 18334 Properties of a semiring. ...
srgcl 18335 Closure of the multiplicat...
srgass 18336 Associative law for the mu...
srgideu 18337 The unit element of a semi...
srgfcl 18338 Functionality of the multi...
srgdi 18339 Distributive law for the m...
srgdir 18340 Distributive law for the m...
srgidcl 18341 The unit element of a semi...
srg0cl 18342 The zero element of a semi...
srgidmlem 18343 Lemma for ~ srglidm and ~ ...
srglidm 18344 The unit element of a semi...
srgridm 18345 The unit element of a semi...
issrgid 18346 Properties showing that an...
srgacl 18347 Closure of the addition op...
srgcom 18348 Commutativity of the addit...
srgrz 18349 The zero of a semiring is ...
srglz 18350 The zero of a semiring is ...
srgisid 18351 In a semiring, the only le...
srg1zr 18352 The only semiring with a b...
srgen1zr 18353 The only semiring with one...
srgmulgass 18354 An associative property be...
srgpcomp 18355 If two elements of a semir...
srgpcompp 18356 If two elements of a semir...
srgpcomppsc 18357 If two elements of a semir...
srglmhm 18358 Left-multiplication in a s...
srgrmhm 18359 Right-multiplication in a ...
srgsummulcr 18360 A finite semiring sum mult...
sgsummulcl 18361 A finite semiring sum mult...
srg1expzeq1 18362 The exponentiation (by a n...
srgbinomlem1 18363 Lemma 1 for ~ srgbinomlem ...
srgbinomlem2 18364 Lemma 2 for ~ srgbinomlem ...
srgbinomlem3 18365 Lemma 3 for ~ srgbinomlem ...
srgbinomlem4 18366 Lemma 4 for ~ srgbinomlem ...
srgbinomlem 18367 Lemma for ~ srgbinom . In...
srgbinom 18368 The binomial theorem for c...
csrgbinom 18369 The binomial theorem for c...
isring 18374 The predicate "is a (unita...
ringgrp 18375 A ring is a group. (Contr...
ringmgp 18376 A ring is a monoid under m...
iscrng 18377 A commutative ring is a ri...
crngmgp 18378 A commutative ring's multi...
ringmnd 18379 A ring is a monoid under a...
ringmgm 18380 A ring is a magma. (Contr...
crngring 18381 A commutative ring is a ri...
mgpf 18382 Restricted functionality o...
ringi 18383 Properties of a unital rin...
ringcl 18384 Closure of the multiplicat...
crngcom 18385 A commutative ring's multi...
iscrng2 18386 A commutative ring is a ri...
ringass 18387 Associative law for the mu...
ringideu 18388 The unit element of a ring...
ringdi 18389 Distributive law for the m...
ringdir 18390 Distributive law for the m...
ringidcl 18391 The unit element of a ring...
ring0cl 18392 The zero element of a ring...
ringidmlem 18393 Lemma for ~ ringlidm and ~...
ringlidm 18394 The unit element of a ring...
ringridm 18395 The unit element of a ring...
isringid 18396 Properties showing that an...
ringid 18397 The multiplication operati...
ringadd2 18398 A ring element plus itself...
rngo2times 18399 A ring element plus itself...
ringidss 18400 A subset of the multiplica...
ringacl 18401 Closure of the addition op...
ringcom 18402 Commutativity of the addit...
ringabl 18403 A ring is an Abelian group...
ringcmn 18404 A ring is a commutative mo...
ringpropd 18405 If two structures have the...
crngpropd 18406 If two structures have the...
ringprop 18407 If two structures have the...
isringd 18408 Properties that determine ...
iscrngd 18409 Properties that determine ...
ringlz 18410 The zero of a unital ring ...
ringrz 18411 The zero of a unital ring ...
ringsrg 18412 Any ring is also a semirin...
ring1eq0 18413 If one and zero are equal,...
ring1ne0 18414 If a ring has at least two...
ringinvnz1ne0 18415 In a unitary ring, a left ...
ringinvnzdiv 18416 In a unitary ring, a left ...
ringnegl 18417 Negation in a ring is the ...
rngnegr 18418 Negation in a ring is the ...
ringmneg1 18419 Negation of a product in a...
ringmneg2 18420 Negation of a product in a...
ringm2neg 18421 Double negation of a produ...
ringsubdi 18422 Ring multiplication distri...
rngsubdir 18423 Ring multiplication distri...
mulgass2 18424 An associative property be...
ring1 18425 The (smallest) structure r...
ringn0 18426 Rings exist. (Contributed...
ringlghm 18427 Left-multiplication in a r...
ringrghm 18428 Right-multiplication in a ...
gsummulc1 18429 A finite ring sum multipli...
gsummulc2 18430 A finite ring sum multipli...
gsummgp0 18431 If one factor in a finite ...
gsumdixp 18432 Distribute a binary produc...
prdsmgp 18433 The multiplicative monoid ...
prdsmulrcl 18434 A structure product of rin...
prdsringd 18435 A product of rings is a ri...
prdscrngd 18436 A product of commutative r...
prds1 18437 Value of the ring unit in ...
pwsring 18438 A structure power of a rin...
pws1 18439 Value of the ring unit in ...
pwscrng 18440 A structure power of a com...
pwsmgp 18441 The multiplicative group o...
imasring 18442 The image structure of a r...
qusring2 18443 The quotient structure of ...
crngbinom 18444 The binomial theorem for c...
opprval 18447 Value of the opposite ring...
opprmulfval 18448 Value of the multiplicatio...
opprmul 18449 Value of the multiplicatio...
crngoppr 18450 In a commutative ring, the...
opprlem 18451 Lemma for ~ opprbas and ~ ...
opprbas 18452 Base set of an opposite ri...
oppradd 18453 Addition operation of an o...
opprring 18454 An opposite ring is a ring...
opprringb 18455 Bidirectional form of ~ op...
oppr0 18456 Additive identity of an op...
oppr1 18457 Multiplicative identity of...
opprneg 18458 The negative function in a...
opprsubg 18459 Being a subgroup is a symm...
mulgass3 18460 An associative property be...
reldvdsr 18467 The divides relation is a ...
dvdsrval 18468 Value of the divides relat...
dvdsr 18469 Value of the divides relat...
dvdsr2 18470 Value of the divides relat...
dvdsrmul 18471 A left-multiple of ` X ` i...
dvdsrcl 18472 Closure of a dividing elem...
dvdsrcl2 18473 Closure of a dividing elem...
dvdsrid 18474 An element in a (unital) r...
dvdsrtr 18475 Divisibility is transitive...
dvdsrmul1 18476 The divisibility relation ...
dvdsrneg 18477 An element divides its neg...
dvdsr01 18478 In a ring, zero is divisib...
dvdsr02 18479 Only zero is divisible by ...
isunit 18480 Property of being a unit o...
1unit 18481 The multiplicative identit...
unitcl 18482 A unit is an element of th...
unitss 18483 The set of units is contai...
opprunit 18484 Being a unit is a symmetri...
crngunit 18485 Property of being a unit i...
dvdsunit 18486 A divisor of a unit is a u...
unitmulcl 18487 The product of units is a ...
unitmulclb 18488 Reversal of ~ unitmulcl in...
unitgrpbas 18489 The base set of the group ...
unitgrp 18490 The group of units is a gr...
unitabl 18491 The group of units of a co...
unitgrpid 18492 The identity of the multip...
unitsubm 18493 The group of units is a su...
invrfval 18496 Multiplicative inverse fun...
unitinvcl 18497 The inverse of a unit exis...
unitinvinv 18498 The inverse of the inverse...
ringinvcl 18499 The inverse of a unit is a...
unitlinv 18500 A unit times its inverse i...
unitrinv 18501 A unit times its inverse i...
1rinv 18502 The inverse of the identit...
0unit 18503 The additive identity is a...
unitnegcl 18504 The negative of a unit is ...
dvrfval 18507 Division operation in a ri...
dvrval 18508 Division operation in a ri...
dvrcl 18509 Closure of division operat...
unitdvcl 18510 The units are closed under...
dvrid 18511 A cancellation law for div...
dvr1 18512 A cancellation law for div...
dvrass 18513 An associative law for div...
dvrcan1 18514 A cancellation law for div...
dvrcan3 18515 A cancellation law for div...
dvreq1 18516 A cancellation law for div...
ringinvdv 18517 Write the inverse function...
rngidpropd 18518 The ring identity depends ...
dvdsrpropd 18519 The divisibility relation ...
unitpropd 18520 The set of units depends o...
invrpropd 18521 The ring inverse function ...
isirred 18522 An irreducible element of ...
isnirred 18523 The property of being a no...
isirred2 18524 Expand out the class diffe...
opprirred 18525 Irreducibility is symmetri...
irredn0 18526 The additive identity is n...
irredcl 18527 An irreducible element is ...
irrednu 18528 An irreducible element is ...
irredn1 18529 The multiplicative identit...
irredrmul 18530 The product of an irreduci...
irredlmul 18531 The product of a unit and ...
irredmul 18532 If product of two elements...
irredneg 18533 The negative of an irreduc...
irrednegb 18534 An element is irreducible ...
dfrhm2 18540 The property of a ring hom...
rhmrcl1 18542 Reverse closure of a ring ...
rhmrcl2 18543 Reverse closure of a ring ...
isrhm 18544 A function is a ring homom...
rhmmhm 18545 A ring homomorphism is a h...
isrim0 18546 An isomorphism of rings is...
rimrcl 18547 Reverse closure for an iso...
rhmghm 18548 A ring homomorphism is an ...
rhmf 18549 A ring homomorphism is a f...
rhmmul 18550 A homomorphism of rings pr...
isrhm2d 18551 Demonstration of ring homo...
isrhmd 18552 Demonstration of ring homo...
rhm1 18553 Ring homomorphisms are req...
idrhm 18554 The identity homomorphism ...
rhmf1o 18555 A ring homomorphism is bij...
isrim 18556 An isomorphism of rings is...
rimf1o 18557 An isomorphism of rings is...
rimrhm 18558 An isomorphism of rings is...
rimgim 18559 An isomorphism of rings is...
rhmco 18560 The composition of ring ho...
pwsco1rhm 18561 Right composition with a f...
pwsco2rhm 18562 Left composition with a ri...
f1rhm0to0 18563 If a ring homomorphism ` F...
f1rhm0to0ALT 18564 Alternate proof for ~ f1rh...
rim0to0 18565 A ring isomorphism maps th...
kerf1hrm 18566 A ring homomorphism ` F ` ...
brric 18567 The relation "is isomorphi...
brric2 18568 The relation "is isomorphi...
ricgic 18569 If two rings are (ring) is...
isdrng 18574 The predicate "is a divisi...
drngunit 18575 Elementhood in the set of ...
drngui 18576 The set of units of a divi...
drngring 18577 A division ring is a ring....
drnggrp 18578 A division ring is a group...
isfld 18579 A field is a commutative d...
isdrng2 18580 A division ring can equiva...
drngprop 18581 If two structures have the...
drngmgp 18582 A division ring contains a...
drngmcl 18583 The product of two nonzero...
drngid 18584 A division ring's unit is ...
drngunz 18585 A division ring's unit is ...
drngid2 18586 Properties showing that an...
drnginvrcl 18587 Closure of the multiplicat...
drnginvrn0 18588 The multiplicative inverse...
drnginvrl 18589 Property of the multiplica...
drnginvrr 18590 Property of the multiplica...
drngmul0or 18591 A product is zero iff one ...
drngmulne0 18592 A product is nonzero iff b...
drngmuleq0 18593 An element is zero iff its...
opprdrng 18594 The opposite of a division...
isdrngd 18595 Properties that determine ...
isdrngrd 18596 Properties that determine ...
drngpropd 18597 If two structures have the...
fldpropd 18598 If two structures have the...
issubrg 18603 The subring predicate. (C...
subrgss 18604 A subring is a subset. (C...
subrgid 18605 Every ring is a subring of...
subrgring 18606 A subring is a ring. (Con...
subrgcrng 18607 A subring of a commutative...
subrgrcl 18608 Reverse closure for a subr...
subrgsubg 18609 A subring is a subgroup. ...
subrg0 18610 A subring always has the s...
subrg1cl 18611 A subring contains the mul...
subrgbas 18612 Base set of a subring stru...
subrg1 18613 A subring always has the s...
subrgacl 18614 A subring is closed under ...
subrgmcl 18615 A subgroup is closed under...
subrgsubm 18616 A subring is a submonoid o...
subrgdvds 18617 If an element divides anot...
subrguss 18618 A unit of a subring is a u...
subrginv 18619 A subring always has the s...
subrgdv 18620 A subring always has the s...
subrgunit 18621 An element of a ring is a ...
subrgugrp 18622 The units of a subring for...
issubrg2 18623 Characterize the subrings ...
opprsubrg 18624 Being a subring is a symme...
subrgint 18625 The intersection of a none...
subrgin 18626 The intersection of two su...
subrgmre 18627 The subrings of a ring are...
issubdrg 18628 Characterize the subfields...
subsubrg 18629 A subring of a subring is ...
subsubrg2 18630 The set of subrings of a s...
issubrg3 18631 A subring is an additive s...
resrhm 18632 Restriction of a ring homo...
rhmeql 18633 The equalizer of two ring ...
rhmima 18634 The homomorphic image of a...
cntzsubr 18635 Centralizers in a ring are...
pwsdiagrhm 18636 Diagonal homomorphism into...
subrgpropd 18637 If two structures have the...
rhmpropd 18638 Ring homomorphism depends ...
abvfval 18641 Value of the set of absolu...
isabv 18642 Elementhood in the set of ...
isabvd 18643 Properties that determine ...
abvrcl 18644 Reverse closure for the ab...
abvfge0 18645 An absolute value is a fun...
abvf 18646 An absolute value is a fun...
abvcl 18647 An absolute value is a fun...
abvge0 18648 The absolute value of a nu...
abveq0 18649 The value of an absolute v...
abvne0 18650 The absolute value of a no...
abvgt0 18651 The absolute value of a no...
abvmul 18652 An absolute value distribu...
abvtri 18653 An absolute value satisfie...
abv0 18654 The absolute value of zero...
abv1z 18655 The absolute value of one ...
abv1 18656 The absolute value of one ...
abvneg 18657 The absolute value of a ne...
abvsubtri 18658 An absolute value satisfie...
abvrec 18659 The absolute value distrib...
abvdiv 18660 The absolute value distrib...
abvdom 18661 Any ring with an absolute ...
abvres 18662 The restriction of an abso...
abvtrivd 18663 The trivial absolute value...
abvtriv 18664 The trivial absolute value...
abvpropd 18665 If two structures have the...
staffval 18670 The functionalization of t...
stafval 18671 The functionalization of t...
staffn 18672 The functionalization is e...
issrng 18673 The predicate "is a star r...
srngrhm 18674 The involution function in...
srngring 18675 A star ring is a ring. (C...
srngcnv 18676 The involution function in...
srngf1o 18677 The involution function in...
srngcl 18678 The involution function in...
srngnvl 18679 The involution function in...
srngadd 18680 The involution function in...
srngmul 18681 The involution function in...
srng1 18682 The conjugate of the ring ...
srng0 18683 The conjugate of the ring ...
issrngd 18684 Properties that determine ...
idsrngd 18685 A commutative ring is a st...
islmod 18690 The predicate "is a left m...
lmodlema 18691 Lemma for properties of a ...
islmodd 18692 Properties that determine ...
lmodgrp 18693 A left module is a group. ...
lmodring 18694 The scalar component of a ...
lmodfgrp 18695 The scalar component of a ...
lmodbn0 18696 The base set of a left mod...
lmodacl 18697 Closure of ring addition f...
lmodmcl 18698 Closure of ring multiplica...
lmodsn0 18699 The set of scalars in a le...
lmodvacl 18700 Closure of vector addition...
lmodass 18701 Left module vector sum is ...
lmodlcan 18702 Left cancellation law for ...
lmodvscl 18703 Closure of scalar product ...
scaffval 18704 The scalar multiplication ...
scafval 18705 The scalar multiplication ...
scafeq 18706 If the scalar multiplicati...
scaffn 18707 The scalar multiplication ...
lmodscaf 18708 The scalar multiplication ...
lmodvsdi 18709 Distributive law for scala...
lmodvsdir 18710 Distributive law for scala...
lmodvsass 18711 Associative law for scalar...
lmod0cl 18712 The ring zero in a left mo...
lmod1cl 18713 The ring unit in a left mo...
lmodvs1 18714 Scalar product with ring u...
lmod0vcl 18715 The zero vector is a vecto...
lmod0vlid 18716 Left identity law for the ...
lmod0vrid 18717 Right identity law for the...
lmod0vid 18718 Identity equivalent to the...
lmod0vs 18719 Zero times a vector is the...
lmodvs0 18720 Anything times the zero ve...
lmodvsmmulgdi 18721 Distributive law for a gro...
lmodfopnelem1 18722 Lemma 1 for ~ lmodfopne . ...
lmodfopnelem2 18723 Lemma 2 for ~ lmodfopne . ...
lmodfopne 18724 The (functionalized) opera...
lcomf 18725 A linear-combination sum i...
lcomfsupp 18726 A linear-combination sum i...
lmodvnegcl 18727 Closure of vector negative...
lmodvnegid 18728 Addition of a vector with ...
lmodvneg1 18729 Minus 1 times a vector is ...
lmodvsneg 18730 Multiplication of a vector...
lmodvsubcl 18731 Closure of vector subtract...
lmodcom 18732 Left module vector sum is ...
lmodabl 18733 A left module is an abelia...
lmodcmn 18734 A left module is a commuta...
lmodnegadd 18735 Distribute negation throug...
lmod4 18736 Commutative/associative la...
lmodvsubadd 18737 Relationship between vecto...
lmodvaddsub4 18738 Vector addition/subtractio...
lmodvpncan 18739 Addition/subtraction cance...
lmodvnpcan 18740 Cancellation law for vecto...
lmodvsubval2 18741 Value of vector subtractio...
lmodsubvs 18742 Subtraction of a scalar pr...
lmodsubdi 18743 Scalar multiplication dist...
lmodsubdir 18744 Scalar multiplication dist...
lmodsubeq0 18745 If the difference between ...
lmodsubid 18746 Subtraction of a vector fr...
lmodvsghm 18747 Scalar multiplication of t...
lmodprop2d 18748 If two structures have the...
lmodpropd 18749 If two structures have the...
gsumvsmul 18750 Pull a scalar multiplicati...
mptscmfsupp0 18751 A mapping to a scalar prod...
mptscmfsuppd 18752 A function mapping to a sc...
lssset 18755 The set of all (not necess...
islss 18756 The predicate "is a subspa...
islssd 18757 Properties that determine ...
lssss 18758 A subspace is a set of vec...
lssel 18759 A subspace member is a vec...
lss1 18760 The set of vectors in a le...
lssuni 18761 The union of all subspaces...
lssn0 18762 A subspace is not empty. ...
00lss 18763 The empty structure has no...
lsscl 18764 Closure property of a subs...
lssvsubcl 18765 Closure of vector subtract...
lssvancl1 18766 Non-closure: if one vector...
lssvancl2 18767 Non-closure: if one vector...
lss0cl 18768 The zero vector belongs to...
lsssn0 18769 The singleton of the zero ...
lss0ss 18770 The zero subspace is inclu...
lssle0 18771 No subspace is smaller tha...
lssne0 18772 A nonzero subspace has a n...
lssneln0 18773 A vector which doesn't bel...
lssssr 18774 Conclude subspace ordering...
lssvacl 18775 Closure of vector addition...
lssvscl 18776 Closure of scalar product ...
lssvnegcl 18777 Closure of negative vector...
lsssubg 18778 All subspaces are subgroup...
lsssssubg 18779 All subspaces are subgroup...
islss3 18780 A linear subspace of a mod...
lsslmod 18781 A submodule is a module. ...
lsslss 18782 The subspaces of a subspac...
islss4 18783 A linear subspace is a sub...
lss1d 18784 One-dimensional subspace (...
lssintcl 18785 The intersection of a none...
lssincl 18786 The intersection of two su...
lssmre 18787 The subspaces of a module ...
lssacs 18788 Submodules are an algebrai...
prdsvscacl 18789 Pointwise scalar multiplic...
prdslmodd 18790 The product of a family of...
pwslmod 18791 The product of a family of...
lspfval 18794 The span function for a le...
lspf 18795 The span operator on a lef...
lspval 18796 The span of a set of vecto...
lspcl 18797 The span of a set of vecto...
lspsncl 18798 The span of a singleton is...
lspprcl 18799 The span of a pair is a su...
lsptpcl 18800 The span of an unordered t...
lspsnsubg 18801 The span of a singleton is...
00lsp 18802 ~ fvco4i lemma for linear ...
lspid 18803 The span of a subspace is ...
lspssv 18804 A span is a set of vectors...
lspss 18805 Span preserves subset orde...
lspssid 18806 A set of vectors is a subs...
lspidm 18807 The span of a set of vecto...
lspun 18808 The span of union is the s...
lspssp 18809 If a set of vectors is a s...
mrclsp 18810 Moore closure generalizes ...
lspsnss 18811 The span of the singleton ...
lspsnel3 18812 A member of the span of th...
lspprss 18813 The span of a pair of vect...
lspsnid 18814 A vector belongs to the sp...
lspsnel6 18815 Relationship between a vec...
lspsnel5 18816 Relationship between a vec...
lspsnel5a 18817 Relationship between a vec...
lspprid1 18818 A member of a pair of vect...
lspprid2 18819 A member of a pair of vect...
lspprvacl 18820 The sum of two vectors bel...
lssats2 18821 A way to express atomistic...
lspsneli 18822 A scalar product with a ve...
lspsn 18823 Span of the singleton of a...
lspsnel 18824 Member of span of the sing...
lspsnvsi 18825 Span of a scalar product o...
lspsnss2 18826 Comparable spans of single...
lspsnneg 18827 Negation does not change t...
lspsnsub 18828 Swapping subtraction order...
lspsn0 18829 Span of the singleton of t...
lsp0 18830 Span of the empty set. (C...
lspuni0 18831 Union of the span of the e...
lspun0 18832 The span of a union with t...
lspsneq0 18833 Span of the singleton is t...
lspsneq0b 18834 Equal singleton spans impl...
lmodindp1 18835 Two independent (non-colin...
lsslsp 18836 Spans in submodules corres...
lss0v 18837 The zero vector in a submo...
lsspropd 18838 If two structures have the...
lsppropd 18839 If two structures have the...
reldmlmhm 18846 Lemma for module homomorph...
lmimfn 18847 Lemma for module isomorphi...
islmhm 18848 Property of being a homomo...
islmhm3 18849 Property of a module homom...
lmhmlem 18850 Non-quantified consequence...
lmhmsca 18851 A homomorphism of left mod...
lmghm 18852 A homomorphism of left mod...
lmhmlmod2 18853 A homomorphism of left mod...
lmhmlmod1 18854 A homomorphism of left mod...
lmhmf 18855 A homomorphism of left mod...
lmhmlin 18856 A homomorphism of left mod...
lmodvsinv 18857 Multiplication of a vector...
lmodvsinv2 18858 Multiplying a negated vect...
islmhm2 18859 A one-equation proof of li...
islmhmd 18860 Deduction for a module hom...
0lmhm 18861 The constant zero linear f...
idlmhm 18862 The identity function on a...
invlmhm 18863 The negative function on a...
lmhmco 18864 The composition of two mod...
lmhmplusg 18865 The pointwise sum of two l...
lmhmvsca 18866 The pointwise scalar produ...
lmhmf1o 18867 A bijective module homomor...
lmhmima 18868 The image of a subspace un...
lmhmpreima 18869 The inverse image of a sub...
lmhmlsp 18870 Homomorphisms preserve spa...
lmhmrnlss 18871 The range of a homomorphis...
lmhmkerlss 18872 The kernel of a homomorphi...
reslmhm 18873 Restriction of a homomorph...
reslmhm2 18874 Expansion of the codomain ...
reslmhm2b 18875 Expansion of the codomain ...
lmhmeql 18876 The equalizer of two modul...
lspextmo 18877 A linear function is compl...
pwsdiaglmhm 18878 Diagonal homomorphism into...
pwssplit0 18879 Splitting for structure po...
pwssplit1 18880 Splitting for structure po...
pwssplit2 18881 Splitting for structure po...
pwssplit3 18882 Splitting for structure po...
islmim 18883 An isomorphism of left mod...
lmimf1o 18884 An isomorphism of left mod...
lmimlmhm 18885 An isomorphism of modules ...
lmimgim 18886 An isomorphism of modules ...
islmim2 18887 An isomorphism of left mod...
lmimcnv 18888 The converse of a bijectiv...
brlmic 18889 The relation "is isomorphi...
brlmici 18890 Prove isomorphic by an exp...
lmiclcl 18891 Isomorphism implies the le...
lmicrcl 18892 Isomorphism implies the ri...
lmicsym 18893 Module isomorphism is symm...
lmhmpropd 18894 Module homomorphism depend...
islbs 18897 The predicate " ` B ` is a...
lbsss 18898 A basis is a set of vector...
lbsel 18899 An element of a basis is a...
lbssp 18900 The span of a basis is the...
lbsind 18901 A basis is linearly indepe...
lbsind2 18902 A basis is linearly indepe...
lbspss 18903 No proper subset of a basi...
lsmcl 18904 The sum of two subspaces i...
lsmspsn 18905 Member of subspace sum of ...
lsmelval2 18906 Subspace sum membership in...
lsmsp 18907 Subspace sum in terms of s...
lsmsp2 18908 Subspace sum of spans of s...
lsmssspx 18909 Subspace sum (in its exten...
lsmpr 18910 The span of a pair of vect...
lsppreli 18911 A vector expressed as a su...
lsmelpr 18912 Two ways to say that a vec...
lsppr0 18913 The span of a vector paire...
lsppr 18914 Span of a pair of vectors....
lspprel 18915 Member of the span of a pa...
lspprabs 18916 Absorption of vector sum i...
lspvadd 18917 The span of a vector sum i...
lspsntri 18918 Triangle-type inequality f...
lspsntrim 18919 Triangle-type inequality f...
lbspropd 18920 If two structures have the...
pj1lmhm 18921 The left projection functi...
pj1lmhm2 18922 The left projection functi...
islvec 18925 The predicate "is a left v...
lvecdrng 18926 The set of scalars of a le...
lveclmod 18927 A left vector space is a l...
lsslvec 18928 A vector subspace is a vec...
lvecvs0or 18929 If a scalar product is zer...
lvecvsn0 18930 A scalar product is nonzer...
lssvs0or 18931 If a scalar product belong...
lvecvscan 18932 Cancellation law for scala...
lvecvscan2 18933 Cancellation law for scala...
lvecinv 18934 Invert coefficient of scal...
lspsnvs 18935 A nonzero scalar product d...
lspsneleq 18936 Membership relation that i...
lspsncmp 18937 Comparable spans of nonzer...
lspsnne1 18938 Two ways to express that v...
lspsnne2 18939 Two ways to express that v...
lspsnnecom 18940 Swap two vectors with diff...
lspabs2 18941 Absorption law for span of...
lspabs3 18942 Absorption law for span of...
lspsneq 18943 Equal spans of singletons ...
lspsneu 18944 Nonzero vectors with equal...
lspsnel4 18945 A member of the span of th...
lspdisj 18946 The span of a vector not i...
lspdisjb 18947 A nonzero vector is not in...
lspdisj2 18948 Unequal spans are disjoint...
lspfixed 18949 Show membership in the spa...
lspexch 18950 Exchange property for span...
lspexchn1 18951 Exchange property for span...
lspexchn2 18952 Exchange property for span...
lspindpi 18953 Partial independence prope...
lspindp1 18954 Alternate way to say 3 vec...
lspindp2l 18955 Alternate way to say 3 vec...
lspindp2 18956 Alternate way to say 3 vec...
lspindp3 18957 Independence of 2 vectors ...
lspindp4 18958 (Partial) independence of ...
lvecindp 18959 Compute the ` X ` coeffici...
lvecindp2 18960 Sums of independent vector...
lspsnsubn0 18961 Unequal singleton spans im...
lsmcv 18962 Subspace sum has the cover...
lspsolvlem 18963 Lemma for ~ lspsolv . (Co...
lspsolv 18964 If ` X ` is in the span of...
lssacsex 18965 In a vector space, subspac...
lspsnat 18966 There is no subspace stric...
lspsncv0 18967 The span of a singleton co...
lsppratlem1 18968 Lemma for ~ lspprat . Let...
lsppratlem2 18969 Lemma for ~ lspprat . Sho...
lsppratlem3 18970 Lemma for ~ lspprat . In ...
lsppratlem4 18971 Lemma for ~ lspprat . In ...
lsppratlem5 18972 Lemma for ~ lspprat . Com...
lsppratlem6 18973 Lemma for ~ lspprat . Neg...
lspprat 18974 A proper subspace of the s...
islbs2 18975 An equivalent formulation ...
islbs3 18976 An equivalent formulation ...
lbsacsbs 18977 Being a basis in a vector ...
lvecdim 18978 The dimension theorem for ...
lbsextlem1 18979 Lemma for ~ lbsext . The ...
lbsextlem2 18980 Lemma for ~ lbsext . Sinc...
lbsextlem3 18981 Lemma for ~ lbsext . A ch...
lbsextlem4 18982 Lemma for ~ lbsext . ~ lbs...
lbsextg 18983 For any linearly independe...
lbsext 18984 For any linearly independe...
lbsexg 18985 Every vector space has a b...
lbsex 18986 Every vector space has a b...
lvecprop2d 18987 If two structures have the...
lvecpropd 18988 If two structures have the...
sraval 18997 Lemma for ~ srabase throug...
sralem 18998 Lemma for ~ srabase and si...
srabase 18999 Base set of a subring alge...
sraaddg 19000 Additive operation of a su...
sramulr 19001 Multiplicative operation o...
srasca 19002 The set of scalars of a su...
sravsca 19003 The scalar product operati...
sraip 19004 The inner product operatio...
sratset 19005 Topology component of a su...
sratopn 19006 Topology component of a su...
srads 19007 Distance function of a sub...
sralmod 19008 The subring algebra is a l...
sralmod0 19009 The subring module inherit...
issubrngd2 19010 Prove a subring by closure...
rlmfn 19011 ` ringLMod ` is a function...
rlmval 19012 Value of the ring module. ...
lidlval 19013 Value of the set of ring i...
rspval 19014 Value of the ring span fun...
rlmval2 19015 Value of the ring module e...
rlmbas 19016 Base set of the ring modul...
rlmplusg 19017 Vector addition in the rin...
rlm0 19018 Zero vector in the ring mo...
rlmsub 19019 Subtraction in the ring mo...
rlmmulr 19020 Ring multiplication in the...
rlmsca 19021 Scalars in the ring module...
rlmsca2 19022 Scalars in the ring module...
rlmvsca 19023 Scalar multiplication in t...
rlmtopn 19024 Topology component of the ...
rlmds 19025 Metric component of the ri...
rlmlmod 19026 The ring module is a modul...
rlmlvec 19027 The ring module over a div...
rlmvneg 19028 Vector negation in the rin...
rlmscaf 19029 Functionalized scalar mult...
ixpsnbasval 19030 The value of an infinite C...
lidlss 19031 An ideal is a subset of th...
islidl 19032 Predicate of being a (left...
lidl0cl 19033 An ideal contains 0. (Con...
lidlacl 19034 An ideal is closed under a...
lidlnegcl 19035 An ideal contains negative...
lidlsubg 19036 An ideal is a subgroup of ...
lidlsubcl 19037 An ideal is closed under s...
lidlmcl 19038 An ideal is closed under l...
lidl1el 19039 An ideal contains 1 iff it...
lidl0 19040 Every ring contains a zero...
lidl1 19041 Every ring contains a unit...
lidlacs 19042 The ideal system is an alg...
rspcl 19043 The span of a set of ring ...
rspssid 19044 The span of a set of ring ...
rsp1 19045 The span of the identity e...
rsp0 19046 The span of the zero eleme...
rspssp 19047 The ideal span of a set of...
mrcrsp 19048 Moore closure generalizes ...
lidlnz 19049 A nonzero ideal contains a...
drngnidl 19050 A division ring has only t...
lidlrsppropd 19051 The left ideals and ring s...
2idlval 19054 Definition of a two-sided ...
2idlcpbl 19055 The coset equivalence rela...
qus1 19056 The multiplicative identit...
qusring 19057 If ` S ` is a two-sided id...
qusrhm 19058 If ` S ` is a two-sided id...
crngridl 19059 In a commutative ring, the...
crng2idl 19060 In a commutative ring, a t...
quscrng 19061 The quotient of a commutat...
lpival 19066 Value of the set of princi...
islpidl 19067 Property of being a princi...
lpi0 19068 The zero ideal is always p...
lpi1 19069 The unit ideal is always p...
islpir 19070 Principal ideal rings are ...
lpiss 19071 Principal ideals are a sub...
islpir2 19072 Principal ideal rings are ...
lpirring 19073 Principal ideal rings are ...
drnglpir 19074 Division rings are princip...
rspsn 19075 Membership in principal id...
lidldvgen 19076 An element generates an id...
lpigen 19077 An ideal is principal iff ...
isnzr 19080 Property of a nonzero ring...
nzrnz 19081 One and zero are different...
nzrring 19082 A nonzero ring is a ring. ...
drngnzr 19083 All division rings are non...
isnzr2 19084 Equivalent characterizatio...
isnzr2hash 19085 Equivalent characterizatio...
opprnzr 19086 The opposite of a nonzero ...
ringelnzr 19087 A ring is nonzero if it ha...
nzrunit 19088 A unit is nonzero in any n...
subrgnzr 19089 A subring of a nonzero rin...
0ringnnzr 19090 A ring is a zero ring iff ...
0ring 19091 If a ring has only one ele...
0ring01eq 19092 In a ring with only one el...
01eq0ring 19093 If the zero and the identi...
0ring01eqbi 19094 In a unital ring the zero ...
rng1nnzr 19095 The (smallest) structure r...
ring1zr 19096 The only (unital) ring wit...
rngen1zr 19097 The only (unital) ring wit...
ringen1zr 19098 The only unital ring with ...
rng1nfld 19099 The zero ring is not a fie...
rrgval 19108 Value of the set or left-r...
isrrg 19109 Membership in the set of l...
rrgeq0i 19110 Property of a left-regular...
rrgeq0 19111 Left-multiplication by a l...
rrgsupp 19112 Left multiplication by a l...
rrgss 19113 Left-regular elements are ...
unitrrg 19114 Units are regular elements...
isdomn 19115 Expand definition of a dom...
domnnzr 19116 A domain is a nonzero ring...
domnring 19117 A domain is a ring. (Cont...
domneq0 19118 In a domain, a product is ...
domnmuln0 19119 In a domain, a product of ...
isdomn2 19120 A ring is a domain iff all...
domnrrg 19121 In a domain, any nonzero e...
opprdomn 19122 The opposite of a domain i...
abvn0b 19123 Another characterization o...
drngdomn 19124 A division ring is a domai...
isidom 19125 An integral domain is a co...
fldidom 19126 A field is an integral dom...
fidomndrnglem 19127 Lemma for ~ fidomndrng . ...
fidomndrng 19128 A finite domain is a divis...
fiidomfld 19129 A finite integral domain i...
isassa 19136 The properties of an assoc...
assalem 19137 The properties of an assoc...
assaass 19138 Left-associative property ...
assaassr 19139 Right-associative property...
assalmod 19140 An associative algebra is ...
assaring 19141 An associative algebra is ...
assasca 19142 An associative algebra's s...
assa2ass 19143 Left- and right-associativ...
isassad 19144 Sufficient condition for b...
issubassa 19145 The subalgebras of an asso...
sraassa 19146 The subring algebra over a...
rlmassa 19147 The ring module over a com...
assapropd 19148 If two structures have the...
aspval 19149 Value of the algebraic clo...
asplss 19150 The algebraic span of a se...
aspid 19151 The algebraic span of a su...
aspsubrg 19152 The algebraic span of a se...
aspss 19153 Span preserves subset orde...
aspssid 19154 A set of vectors is a subs...
asclfval 19155 Function value of the alge...
asclval 19156 Value of a mapped algebra ...
asclfn 19157 Unconditional functionalit...
asclf 19158 The algebra scalars functi...
asclghm 19159 The algebra scalars functi...
asclmul1 19160 Left multiplication by a l...
asclmul2 19161 Right multiplication by a ...
asclinvg 19162 The group inverse (negatio...
asclrhm 19163 The scalar injection is a ...
rnascl 19164 The set of injected scalar...
ressascl 19165 The injection of scalars i...
issubassa2 19166 A subring of a unital alge...
asclpropd 19167 If two structures have the...
aspval2 19168 The algebraic closure is t...
assamulgscmlem1 19169 Lemma 1 for ~ assamulgscm ...
assamulgscmlem2 19170 Lemma for ~ assamulgscm (i...
assamulgscm 19171 Exponentiation of a scalar...
reldmpsr 19182 The multivariate power ser...
psrval 19183 Value of the multivariate ...
psrvalstr 19184 The multivariate power ser...
psrbag 19185 Elementhood in the set of ...
psrbagf 19186 A finite bag is a function...
snifpsrbag 19187 A bag containing one eleme...
fczpsrbag 19188 The constant function equa...
psrbaglesupp 19189 The support of a dominated...
psrbaglecl 19190 The set of finite bags is ...
psrbagaddcl 19191 The sum of two finite bags...
psrbagcon 19192 The analogue of the statem...
psrbaglefi 19193 There are finitely many ba...
psrbagconcl 19194 The complement of a bag is...
psrbagconf1o 19195 Bag complementation is a b...
gsumbagdiaglem 19196 Lemma for ~ gsumbagdiag . ...
gsumbagdiag 19197 Two-dimensional commutatio...
psrass1lem 19198 A group sum commutation us...
psrbas 19199 The base set of the multiv...
psrelbas 19200 An element of the set of p...
psrelbasfun 19201 An element of the set of p...
psrplusg 19202 The addition operation of ...
psradd 19203 The addition operation of ...
psraddcl 19204 Closure of the power serie...
psrmulr 19205 The multiplication operati...
psrmulfval 19206 The multiplication operati...
psrmulval 19207 The multiplication operati...
psrmulcllem 19208 Closure of the power serie...
psrmulcl 19209 Closure of the power serie...
psrsca 19210 The scalar field of the mu...
psrvscafval 19211 The scalar multiplication ...
psrvsca 19212 The scalar multiplication ...
psrvscaval 19213 The scalar multiplication ...
psrvscacl 19214 Closure of the power serie...
psr0cl 19215 The zero element of the ri...
psr0lid 19216 The zero element of the ri...
psrnegcl 19217 The negative function in t...
psrlinv 19218 The negative function in t...
psrgrp 19219 The ring of power series i...
psr0 19220 The zero element of the ri...
psrneg 19221 The negative function of t...
psrlmod 19222 The ring of power series i...
psr1cl 19223 The identity element of th...
psrlidm 19224 The identity element of th...
psrridm 19225 The identity element of th...
psrass1 19226 Associative identity for t...
psrdi 19227 Distributive law for the r...
psrdir 19228 Distributive law for the r...
psrass23l 19229 Associative identity for t...
psrcom 19230 Commutative law for the ri...
psrass23 19231 Associative identities for...
psrring 19232 The ring of power series i...
psr1 19233 The identity element of th...
psrcrng 19234 The ring of power series i...
psrassa 19235 The ring of power series i...
resspsrbas 19236 A restricted power series ...
resspsradd 19237 A restricted power series ...
resspsrmul 19238 A restricted power series ...
resspsrvsca 19239 A restricted power series ...
subrgpsr 19240 A subring of the base ring...
mvrfval 19241 Value of the generating el...
mvrval 19242 Value of the generating el...
mvrval2 19243 Value of the generating el...
mvrid 19244 The ` X i ` -th coefficien...
mvrf 19245 The power series variable ...
mvrf1 19246 The power series variable ...
mvrcl2 19247 A power series variable is...
reldmmpl 19248 The multivariate polynomia...
mplval 19249 Value of the set of multiv...
mplbas 19250 Base set of the set of mul...
mplelbas 19251 Property of being a polyno...
mplval2 19252 Self-referential expressio...
mplbasss 19253 The set of polynomials is ...
mplelf 19254 A polynomial is defined as...
mplsubglem 19255 If ` A ` is an ideal of se...
mpllsslem 19256 If ` A ` is an ideal of su...
mplsubglem2 19257 Lemma for ~ mplsubg and ~ ...
mplsubg 19258 The set of polynomials is ...
mpllss 19259 The set of polynomials is ...
mplsubrglem 19260 Lemma for ~ mplsubrg . (C...
mplsubrg 19261 The set of polynomials is ...
mpl0 19262 The zero polynomial. (Con...
mpladd 19263 The addition operation on ...
mplmul 19264 The multiplication operati...
mpl1 19265 The identity element of th...
mplsca 19266 The scalar field of a mult...
mplvsca2 19267 The scalar multiplication ...
mplvsca 19268 The scalar multiplication ...
mplvscaval 19269 The scalar multiplication ...
mvrcl 19270 A power series variable is...
mplgrp 19271 The polynomial ring is a g...
mpllmod 19272 The polynomial ring is a l...
mplring 19273 The polynomial ring is a r...
mplcrng 19274 The polynomial ring is a c...
mplassa 19275 The polynomial ring is an ...
ressmplbas2 19276 The base set of a restrict...
ressmplbas 19277 A restricted polynomial al...
ressmpladd 19278 A restricted polynomial al...
ressmplmul 19279 A restricted polynomial al...
ressmplvsca 19280 A restricted power series ...
subrgmpl 19281 A subring of the base ring...
subrgmvr 19282 The variables in a subring...
subrgmvrf 19283 The variables in a polynom...
mplmon 19284 A monomial is a polynomial...
mplmonmul 19285 The product of two monomia...
mplcoe1 19286 Decompose a polynomial int...
mplcoe3 19287 Decompose a monomial in on...
mplcoe5lem 19288 Lemma for ~ mplcoe4 . (Co...
mplcoe5 19289 Decompose a monomial into ...
mplcoe2 19290 Decompose a monomial into ...
mplbas2 19291 An alternative expression ...
ltbval 19292 Value of the well-order on...
ltbwe 19293 The finite bag order is a ...
reldmopsr 19294 Lemma for ordered power se...
opsrval 19295 The value of the "ordered ...
opsrle 19296 An alternative expression ...
opsrval2 19297 Self-referential expressio...
opsrbaslem 19298 Get a component of the ord...
opsrbaslemOLD 19299 Obsolete version of ~ opsr...
opsrbas 19300 The base set of the ordere...
opsrplusg 19301 The addition operation of ...
opsrmulr 19302 The multiplication operati...
opsrvsca 19303 The scalar product operati...
opsrsca 19304 The scalar ring of the ord...
opsrtoslem1 19305 Lemma for ~ opsrtos . (Co...
opsrtoslem2 19306 Lemma for ~ opsrtos . (Co...
opsrtos 19307 The ordered power series s...
opsrso 19308 The ordered power series s...
opsrcrng 19309 The ring of ordered power ...
opsrassa 19310 The ring of ordered power ...
mplrcl 19311 Reverse closure for the po...
mplelsfi 19312 A polynomial treated as a ...
mvrf2 19313 The power series/polynomia...
mplmon2 19314 Express a scaled monomial....
psrbag0 19315 The empty bag is a bag. (...
psrbagsn 19316 A singleton bag is a bag. ...
mplascl 19317 Value of the scalar inject...
mplasclf 19318 The scalar injection is a ...
subrgascl 19319 The scalar injection funct...
subrgasclcl 19320 The scalars in a polynomia...
mplmon2cl 19321 A scaled monomial is a pol...
mplmon2mul 19322 Product of scaled monomial...
mplind 19323 Prove a property of polyno...
mplcoe4 19324 Decompose a polynomial int...
evlslem4 19329 The support of a tensor pr...
psrbagfsupp 19330 Finite bags have finite no...
psrbagev1 19331 A bag of multipliers provi...
psrbagev2 19332 Closure of a sum using a b...
evlslem2 19333 A linear function on the p...
evlslem6 19334 Lemma for ~ evlseu . Fini...
evlslem3 19335 Lemma for ~ evlseu . Poly...
evlslem1 19336 Lemma for ~ evlseu , give ...
evlseu 19337 For a given interpretation...
reldmevls 19338 Well-behaved binary operat...
mpfrcl 19339 Reverse closure for the se...
evlsval 19340 Value of the polynomial ev...
evlsval2 19341 Characterizing properties ...
evlsrhm 19342 Polynomial evaluation is a...
evlssca 19343 Polynomial evaluation maps...
evlsvar 19344 Polynomial evaluation maps...
evlval 19345 Value of the simple/same r...
evlrhm 19346 The simple evaluation map ...
evlsscasrng 19347 The evaluation of a scalar...
evlsca 19348 Simple polynomial evaluati...
evlsvarsrng 19349 The evaluation of the vari...
evlvar 19350 Simple polynomial evaluati...
mpfconst 19351 Constants are multivariate...
mpfproj 19352 Projections are multivaria...
mpfsubrg 19353 Polynomial functions are a...
mpff 19354 Polynomial functions are f...
mpfaddcl 19355 The sum of multivariate po...
mpfmulcl 19356 The product of multivariat...
mpfind 19357 Prove a property of polyno...
psr1baslem 19376 The set of finite bags on ...
psr1val 19377 Value of the ring of univa...
psr1crng 19378 The ring of univariate pow...
psr1assa 19379 The ring of univariate pow...
psr1tos 19380 The ordered power series s...
psr1bas2 19381 The base set of the ring o...
psr1bas 19382 The base set of the ring o...
vr1val 19383 The value of the generator...
vr1cl2 19384 The variable ` X ` is a me...
ply1val 19385 The value of the set of un...
ply1bas 19386 The value of the base set ...
ply1lss 19387 Univariate polynomials for...
ply1subrg 19388 Univariate polynomials for...
ply1crng 19389 The ring of univariate pol...
ply1assa 19390 The ring of univariate pol...
psr1bascl 19391 A univariate power series ...
psr1basf 19392 Univariate power series ba...
ply1basf 19393 Univariate polynomial base...
ply1bascl 19394 A univariate polynomial is...
ply1bascl2 19395 A univariate polynomial is...
coe1fval 19396 Value of the univariate po...
coe1fv 19397 Value of an evaluated coef...
fvcoe1 19398 Value of a multivariate co...
coe1fval3 19399 Univariate power series co...
coe1f2 19400 Functionality of univariat...
coe1fval2 19401 Univariate polynomial coef...
coe1f 19402 Functionality of univariat...
coe1fvalcl 19403 A coefficient of a univari...
coe1sfi 19404 Finite support of univaria...
coe1fsupp 19405 The coefficient vector of ...
mptcoe1fsupp 19406 A mapping involving coeffi...
coe1ae0 19407 The coefficient vector of ...
vr1cl 19408 The generator of a univari...
opsr0 19409 Zero in the ordered power ...
opsr1 19410 One in the ordered power s...
mplplusg 19411 Value of addition in a pol...
mplmulr 19412 Value of multiplication in...
psr1plusg 19413 Value of addition in a uni...
psr1vsca 19414 Value of scalar multiplica...
psr1mulr 19415 Value of multiplication in...
ply1plusg 19416 Value of addition in a uni...
ply1vsca 19417 Value of scalar multiplica...
ply1mulr 19418 Value of multiplication in...
ressply1bas2 19419 The base set of a restrict...
ressply1bas 19420 A restricted polynomial al...
ressply1add 19421 A restricted polynomial al...
ressply1mul 19422 A restricted polynomial al...
ressply1vsca 19423 A restricted power series ...
subrgply1 19424 A subring of the base ring...
gsumply1subr 19425 Evaluate a group sum in a ...
psrbaspropd 19426 Property deduction for pow...
psrplusgpropd 19427 Property deduction for pow...
mplbaspropd 19428 Property deduction for pol...
psropprmul 19429 Reversing multiplication i...
ply1opprmul 19430 Reversing multiplication i...
00ply1bas 19431 Lemma for ~ ply1basfvi and...
ply1basfvi 19432 Protection compatibility o...
ply1plusgfvi 19433 Protection compatibility o...
ply1baspropd 19434 Property deduction for uni...
ply1plusgpropd 19435 Property deduction for uni...
opsrring 19436 Ordered power series form ...
opsrlmod 19437 Ordered power series form ...
psr1ring 19438 Univariate power series fo...
ply1ring 19439 Univariate polynomials for...
psr1lmod 19440 Univariate power series fo...
psr1sca 19441 Scalars of a univariate po...
psr1sca2 19442 Scalars of a univariate po...
ply1lmod 19443 Univariate polynomials for...
ply1sca 19444 Scalars of a univariate po...
ply1sca2 19445 Scalars of a univariate po...
ply1mpl0 19446 The univariate polynomial ...
ply10s0 19447 Zero times a univariate po...
ply1mpl1 19448 The univariate polynomial ...
ply1ascl 19449 The univariate polynomial ...
subrg1ascl 19450 The scalar injection funct...
subrg1asclcl 19451 The scalars in a polynomia...
subrgvr1 19452 The variables in a subring...
subrgvr1cl 19453 The variables in a polynom...
coe1z 19454 The coefficient vector of ...
coe1add 19455 The coefficient vector of ...
coe1addfv 19456 A particular coefficient o...
coe1subfv 19457 A particular coefficient o...
coe1mul2lem1 19458 An equivalence for ~ coe1m...
coe1mul2lem2 19459 An equivalence for ~ coe1m...
coe1mul2 19460 The coefficient vector of ...
coe1mul 19461 The coefficient vector of ...
ply1moncl 19462 Closure of the expression ...
ply1tmcl 19463 Closure of the expression ...
coe1tm 19464 Coefficient vector of a po...
coe1tmfv1 19465 Nonzero coefficient of a p...
coe1tmfv2 19466 Zero coefficient of a poly...
coe1tmmul2 19467 Coefficient vector of a po...
coe1tmmul 19468 Coefficient vector of a po...
coe1tmmul2fv 19469 Function value of a right-...
coe1pwmul 19470 Coefficient vector of a po...
coe1pwmulfv 19471 Function value of a right-...
ply1scltm 19472 A scalar is a term with ze...
coe1sclmul 19473 Coefficient vector of a po...
coe1sclmulfv 19474 A single coefficient of a ...
coe1sclmul2 19475 Coefficient vector of a po...
ply1sclf 19476 A scalar polynomial is a p...
ply1sclcl 19477 The value of the algebra s...
coe1scl 19478 Coefficient vector of a sc...
ply1sclid 19479 Recover the base scalar fr...
ply1sclf1 19480 The polynomial scalar func...
ply1scl0 19481 The zero scalar is zero. ...
ply1scln0 19482 Nonzero scalars create non...
ply1scl1 19483 The one scalar is the unit...
ply1idvr1 19484 The identity of a polynomi...
cply1mul 19485 The product of two constan...
ply1coefsupp 19486 The decomposition of a uni...
ply1coe 19487 Decompose a univariate pol...
eqcoe1ply1eq 19488 Two polynomials over the s...
ply1coe1eq 19489 Two polynomials over the s...
cply1coe0 19490 All but the first coeffici...
cply1coe0bi 19491 A polynomial is constant (...
coe1fzgsumdlem 19492 Lemma for ~ coe1fzgsumd (i...
coe1fzgsumd 19493 Value of an evaluated coef...
gsumsmonply1 19494 A finite group sum of scal...
gsummoncoe1 19495 A coefficient of the polyn...
gsumply1eq 19496 Two univariate polynomials...
lply1binom 19497 The binomial theorem for l...
lply1binomsc 19498 The binomial theorem for l...
reldmevls1 19503 Well-behaved binary operat...
ply1frcl 19504 Reverse closure for the se...
evls1fval 19505 Value of the univariate po...
evls1val 19506 Value of the univariate po...
evls1rhmlem 19507 Lemma for ~ evl1rhm and ~ ...
evls1rhm 19508 Polynomial evaluation is a...
evls1sca 19509 Univariate polynomial eval...
evls1gsumadd 19510 Univariate polynomial eval...
evls1gsummul 19511 Univariate polynomial eval...
evls1varpw 19512 Univariate polynomial eval...
evl1fval 19513 Value of the simple/same r...
evl1val 19514 Value of the simple/same r...
evl1fval1lem 19515 Lemma for ~ evl1fval1 . (...
evl1fval1 19516 Value of the simple/same r...
evl1rhm 19517 Polynomial evaluation is a...
fveval1fvcl 19518 The function value of the ...
evl1sca 19519 Polynomial evaluation maps...
evl1scad 19520 Polynomial evaluation buil...
evl1var 19521 Polynomial evaluation maps...
evl1vard 19522 Polynomial evaluation buil...
evls1var 19523 Univariate polynomial eval...
evls1scasrng 19524 The evaluation of a scalar...
evls1varsrng 19525 The evaluation of the vari...
evl1addd 19526 Polynomial evaluation buil...
evl1subd 19527 Polynomial evaluation buil...
evl1muld 19528 Polynomial evaluation buil...
evl1vsd 19529 Polynomial evaluation buil...
evl1expd 19530 Polynomial evaluation buil...
pf1const 19531 Constants are polynomial f...
pf1id 19532 The identity is a polynomi...
pf1subrg 19533 Polynomial functions are a...
pf1rcl 19534 Reverse closure for the se...
pf1f 19535 Polynomial functions are f...
mpfpf1 19536 Convert a multivariate pol...
pf1mpf 19537 Convert a univariate polyn...
pf1addcl 19538 The sum of multivariate po...
pf1mulcl 19539 The product of multivariat...
pf1ind 19540 Prove a property of polyno...
evl1gsumdlem 19541 Lemma for ~ evl1gsumd (ind...
evl1gsumd 19542 Polynomial evaluation buil...
evl1gsumadd 19543 Univariate polynomial eval...
evl1gsumaddval 19544 Value of a univariate poly...
evl1gsummul 19545 Univariate polynomial eval...
evl1varpw 19546 Univariate polynomial eval...
evl1varpwval 19547 Value of a univariate poly...
evl1scvarpw 19548 Univariate polynomial eval...
evl1scvarpwval 19549 Value of a univariate poly...
evl1gsummon 19550 Value of a univariate poly...
cnfldstr 19569 The field of complex numbe...
cnfldex 19570 The field of complex numbe...
cnfldbas 19571 The base set of the field ...
cnfldadd 19572 The addition operation of ...
cnfldmul 19573 The multiplication operati...
cnfldcj 19574 The conjugation operation ...
cnfldtset 19575 The topology component of ...
cnfldle 19576 The ordering of the field ...
cnfldds 19577 The metric of the field of...
cnfldunif 19578 The uniform structure comp...
xrsstr 19579 The extended real structur...
xrsex 19580 The extended real structur...
xrsbas 19581 The base set of the extend...
xrsadd 19582 The addition operation of ...
xrsmul 19583 The multiplication operati...
xrstset 19584 The topology component of ...
xrsle 19585 The ordering of the extend...
cncrng 19586 The complex numbers form a...
cnring 19587 The complex numbers form a...
xrsmcmn 19588 The multiplicative group o...
cnfld0 19589 The zero element of the fi...
cnfld1 19590 The unit element of the fi...
cnfldneg 19591 The additive inverse in th...
cnfldplusf 19592 The functionalized additio...
cnfldsub 19593 The subtraction operator i...
cndrng 19594 The complex numbers form a...
cnflddiv 19595 The division operation in ...
cnfldinv 19596 The multiplicative inverse...
cnfldmulg 19597 The group multiple functio...
cnfldexp 19598 The exponentiation operato...
cnsrng 19599 The complex numbers form a...
xrsmgm 19600 The (additive group of the...
xrsnsgrp 19601 The (additive group of the...
xrsmgmdifsgrp 19602 The (additive group of the...
xrs1mnd 19603 The extended real numbers,...
xrs10 19604 The zero of the extended r...
xrs1cmn 19605 The extended real numbers ...
xrge0subm 19606 The nonnegative extended r...
xrge0cmn 19607 The nonnegative extended r...
xrsds 19608 The metric of the extended...
xrsdsval 19609 The metric of the extended...
xrsdsreval 19610 The metric of the extended...
xrsdsreclblem 19611 Lemma for ~ xrsdsreclb . ...
xrsdsreclb 19612 The metric of the extended...
cnsubmlem 19613 Lemma for ~ nn0subm and fr...
cnsubglem 19614 Lemma for ~ resubdrg and f...
cnsubrglem 19615 Lemma for ~ resubdrg and f...
cnsubdrglem 19616 Lemma for ~ resubdrg and f...
qsubdrg 19617 The rational numbers form ...
zsubrg 19618 The integers form a subrin...
gzsubrg 19619 The gaussian integers form...
nn0subm 19620 The nonnegative integers f...
rege0subm 19621 The nonnegative reals form...
absabv 19622 The regular absolute value...
zsssubrg 19623 The integers are a subset ...
qsssubdrg 19624 The rational numbers are a...
cnsubrg 19625 There are no subrings of t...
cnmgpabl 19626 The unit group of the comp...
cnmgpid 19627 The group identity element...
cnmsubglem 19628 Lemma for ~ rpmsubg and fr...
rpmsubg 19629 The positive reals form a ...
gzrngunitlem 19630 Lemma for ~ gzrngunit . (...
gzrngunit 19631 The units on ` ZZ [ _i ] `...
gsumfsum 19632 Relate a group sum on ` CC...
regsumfsum 19633 Relate a group sum on ` ( ...
expmhm 19634 Exponentiation is a monoid...
nn0srg 19635 The nonnegative integers f...
rge0srg 19636 The nonnegative real numbe...
zringcrng 19639 The ring of integers is a ...
zringring 19640 The ring of integers is a ...
zringabl 19641 The ring of integers is an...
zringgrp 19642 The ring of integers is an...
zringbas 19643 The integers are the base ...
zringplusg 19644 The addition operation of ...
zringmulg 19645 The multiplication (group ...
zringmulr 19646 The multiplication operati...
zring0 19647 The neutral element of the...
zring1 19648 The multiplicative neutral...
zringnzr 19649 The ring of integers is a ...
dvdsrzring 19650 Ring divisibility in the r...
zringlpirlem1 19651 Lemma for ~ zringlpir . A...
zringlpirlem2 19652 Lemma for ~ zringlpir . A...
zringlpirlem3 19653 Lemma for ~ zringlpir . A...
zringinvg 19654 The additive inverse of an...
zringunit 19655 The units of ` ZZ ` are th...
zringlpir 19656 The integers are a princip...
zringndrg 19657 The integers are not a div...
zringcyg 19658 The integers are a cyclic ...
zringmpg 19659 The multiplication group o...
prmirredlem 19660 A positive integer is irre...
dfprm2 19661 The positive irreducible e...
prmirred 19662 The irreducible elements o...
expghm 19663 Exponentiation is a group ...
mulgghm2 19664 The powers of a group elem...
mulgrhm 19665 The powers of the element ...
mulgrhm2 19666 The powers of the element ...
zrhval 19675 Define the unique homomorp...
zrhval2 19676 Alternate value of the ` Z...
zrhmulg 19677 Value of the ` ZRHom ` hom...
zrhrhmb 19678 The ` ZRHom ` homomorphism...
zrhrhm 19679 The ` ZRHom ` homomorphism...
zrh1 19680 Interpretation of 1 in a r...
zrh0 19681 Interpretation of 0 in a r...
zrhpropd 19682 The ` ZZ ` ring homomorphi...
zlmval 19683 Augment an abelian group w...
zlmlem 19684 Lemma for ~ zlmbas and ~ z...
zlmbas 19685 Base set of a ` ZZ ` -modu...
zlmplusg 19686 Group operation of a ` ZZ ...
zlmmulr 19687 Ring operation of a ` ZZ `...
zlmsca 19688 Scalar ring of a ` ZZ ` -m...
zlmvsca 19689 Scalar multiplication oper...
zlmlmod 19690 The ` ZZ ` -module operati...
zlmassa 19691 The ` ZZ ` -module operati...
chrval 19692 Definition substitution of...
chrcl 19693 Closure of the characteris...
chrid 19694 The canonical ` ZZ ` ring ...
chrdvds 19695 The ` ZZ ` ring homomorphi...
chrcong 19696 If two integers are congru...
chrnzr 19697 Nonzero rings are precisel...
chrrhm 19698 The characteristic restric...
domnchr 19699 The characteristic of a do...
znlidl 19700 The set ` n ZZ ` is an ide...
zncrng2 19701 The value of the ` Z/nZ ` ...
znval 19702 The value of the ` Z/nZ ` ...
znle 19703 The value of the ` Z/nZ ` ...
znval2 19704 Self-referential expressio...
znbaslem 19705 Lemma for ~ znbas . (Cont...
znbaslemOLD 19706 Obsolete version of ~ znba...
znbas2 19707 The base set of ` Z/nZ ` i...
znadd 19708 The additive structure of ...
znmul 19709 The multiplicative structu...
znzrh 19710 The ` ZZ ` ring homomorphi...
znbas 19711 The base set of ` Z/nZ ` s...
zncrng 19712 ` Z/nZ ` is a commutative ...
znzrh2 19713 The ` ZZ ` ring homomorphi...
znzrhval 19714 The ` ZZ ` ring homomorphi...
znzrhfo 19715 The ` ZZ ` ring homomorphi...
zncyg 19716 The group ` ZZ / n ZZ ` is...
zndvds 19717 Express equality of equiva...
zndvds0 19718 Special case of ~ zndvds w...
znf1o 19719 The function ` F ` enumera...
zzngim 19720 The ` ZZ ` ring homomorphi...
znle2 19721 The ordering of the ` Z/nZ...
znleval 19722 The ordering of the ` Z/nZ...
znleval2 19723 The ordering of the ` Z/nZ...
zntoslem 19724 Lemma for ~ zntos . (Cont...
zntos 19725 The ` Z/nZ ` structure is ...
znhash 19726 The ` Z/nZ ` structure has...
znfi 19727 The ` Z/nZ ` structure is ...
znfld 19728 The ` Z/nZ ` structure is ...
znidomb 19729 The ` Z/nZ ` structure is ...
znchr 19730 Cyclic rings are defined b...
znunit 19731 The units of ` Z/nZ ` are ...
znunithash 19732 The size of the unit group...
znrrg 19733 The regular elements of ` ...
cygznlem1 19734 Lemma for ~ cygzn . (Cont...
cygznlem2a 19735 Lemma for ~ cygzn . (Cont...
cygznlem2 19736 Lemma for ~ cygzn . (Cont...
cygznlem3 19737 A cyclic group with ` n ` ...
cygzn 19738 A cyclic group with ` n ` ...
cygth 19739 The "fundamental theorem o...
cyggic 19740 Cyclic groups are isomorph...
frgpcyg 19741 A free group is cyclic iff...
cnmsgnsubg 19742 The signs form a multiplic...
cnmsgnbas 19743 The base set of the sign s...
cnmsgngrp 19744 The group of signs under m...
psgnghm 19745 The sign is a homomorphism...
psgnghm2 19746 The sign is a homomorphism...
psgninv 19747 The sign of a permutation ...
psgnco 19748 Multiplicativity of the pe...
zrhpsgnmhm 19749 Embedding of permutation s...
zrhpsgninv 19750 The embedded sign of a per...
evpmss 19751 Even permutations are perm...
psgnevpmb 19752 A class is an even permuta...
psgnodpm 19753 A permutation which is odd...
psgnevpm 19754 A permutation which is eve...
psgnodpmr 19755 If a permutation has sign ...
zrhpsgnevpm 19756 The sign of an even permut...
zrhpsgnodpm 19757 The sign of an odd permuta...
zrhcofipsgn 19758 Composition of a ` ZRHom `...
zrhpsgnelbas 19759 Embedding of permutation s...
zrhcopsgnelbas 19760 Embedding of permutation s...
evpmodpmf1o 19761 The function for performin...
pmtrodpm 19762 A transposition is an odd ...
psgnfix1 19763 A permutation of a finite ...
psgnfix2 19764 A permutation of a finite ...
psgndiflemB 19765 Lemma 1 for ~ psgndif . (...
psgndiflemA 19766 Lemma 2 for ~ psgndif . (...
psgndif 19767 Embedding of permutation s...
zrhcopsgndif 19768 Embedding of permutation s...
rebase 19771 The base of the field of r...
remulg 19772 The multiplication (group ...
resubdrg 19773 The real numbers form a di...
resubgval 19774 Subtraction in the field o...
replusg 19775 The addition operation of ...
remulr 19776 The multiplication operati...
re0g 19777 The neutral element of the...
re1r 19778 The multiplicative neutral...
rele2 19779 The ordering relation of t...
relt 19780 The ordering relation of t...
reds 19781 The distance of the field ...
redvr 19782 The division operation of ...
retos 19783 The real numbers are a tot...
refld 19784 The real numbers form a fi...
refldcj 19785 The conjugation operation ...
recrng 19786 The real numbers form a st...
regsumsupp 19787 The group sum over the rea...
isphl 19792 The predicate "is a genera...
phllvec 19793 A pre-Hilbert space is a l...
phllmod 19794 A pre-Hilbert space is a l...
phlsrng 19795 The scalar ring of a pre-H...
phllmhm 19796 The inner product of a pre...
ipcl 19797 Closure of the inner produ...
ipcj 19798 Conjugate of an inner prod...
iporthcom 19799 Orthogonality (meaning inn...
ip0l 19800 Inner product with a zero ...
ip0r 19801 Inner product with a zero ...
ipeq0 19802 The inner product of a vec...
ipdir 19803 Distributive law for inner...
ipdi 19804 Distributive law for inner...
ip2di 19805 Distributive law for inner...
ipsubdir 19806 Distributive law for inner...
ipsubdi 19807 Distributive law for inner...
ip2subdi 19808 Distributive law for inner...
ipass 19809 Associative law for inner ...
ipassr 19810 "Associative" law for seco...
ipassr2 19811 "Associative" law for inne...
ipffval 19812 The inner product operatio...
ipfval 19813 The inner product operatio...
ipfeq 19814 If the inner product opera...
ipffn 19815 The inner product operatio...
phlipf 19816 The inner product operatio...
ip2eq 19817 Two vectors are equal iff ...
isphld 19818 Properties that determine ...
phlpropd 19819 If two structures have the...
ssipeq 19820 The inner product on a sub...
phssipval 19821 The inner product on a sub...
phssip 19822 The inner product (as a fu...
ocvfval 19829 The orthocomplement operat...
ocvval 19830 Value of the orthocompleme...
elocv 19831 Elementhood in the orthoco...
ocvi 19832 Property of a member of th...
ocvss 19833 The orthocomplement of a s...
ocvocv 19834 A set is contained in its ...
ocvlss 19835 The orthocomplement of a s...
ocv2ss 19836 Orthocomplements reverse s...
ocvin 19837 An orthocomplement has tri...
ocvsscon 19838 Two ways to say that ` S `...
ocvlsp 19839 The orthocomplement of a l...
ocv0 19840 The orthocomplement of the...
ocvz 19841 The orthocomplement of the...
ocv1 19842 The orthocomplement of the...
unocv 19843 The orthocomplement of a u...
iunocv 19844 The orthocomplement of an ...
cssval 19845 The set of closed subspace...
iscss 19846 The predicate "is a closed...
cssi 19847 Property of a closed subsp...
cssss 19848 A closed subspace is a sub...
iscss2 19849 It is sufficient to prove ...
ocvcss 19850 The orthocomplement of any...
cssincl 19851 The zero subspace is a clo...
css0 19852 The zero subspace is a clo...
css1 19853 The whole space is a close...
csslss 19854 A closed subspace of a pre...
lsmcss 19855 A subset of a pre-Hilbert ...
cssmre 19856 The closed subspaces of a ...
mrccss 19857 The Moore closure correspo...
thlval 19858 Value of the Hilbert latti...
thlbas 19859 Base set of the Hilbert la...
thlle 19860 Ordering on the Hilbert la...
thlleval 19861 Ordering on the Hilbert la...
thloc 19862 Orthocomplement on the Hil...
pjfval 19869 The value of the projectio...
pjdm 19870 A subspace is in the domai...
pjpm 19871 The projection map is a pa...
pjfval2 19872 Value of the projection ma...
pjval 19873 Value of the projection ma...
pjdm2 19874 A subspace is in the domai...
pjff 19875 A projection is a linear o...
pjf 19876 A projection is a function...
pjf2 19877 A projection is a function...
pjfo 19878 A projection is a surjecti...
pjcss 19879 A projection subspace is a...
ocvpj 19880 The orthocomplement of a p...
ishil 19881 The predicate "is a Hilber...
ishil2 19882 The predicate "is a Hilber...
isobs 19883 The predicate "is an ortho...
obsip 19884 The inner product of two e...
obsipid 19885 A basis element has unit l...
obsrcl 19886 Reverse closure for an ort...
obsss 19887 An orthonormal basis is a ...
obsne0 19888 A basis element is nonzero...
obsocv 19889 An orthonormal basis has t...
obs2ocv 19890 The double orthocomplement...
obselocv 19891 A basis element is in the ...
obs2ss 19892 A basis has no proper subs...
obslbs 19893 An orthogonal basis is a l...
reldmdsmm 19896 The direct sum is a well-b...
dsmmval 19897 Value of the module direct...
dsmmbase 19898 Base set of the module dir...
dsmmval2 19899 Self-referential definitio...
dsmmbas2 19900 Base set of the direct sum...
dsmmfi 19901 For finite products, the d...
dsmmelbas 19902 Membership in the finitely...
dsmm0cl 19903 The all-zero vector is con...
dsmmacl 19904 The finite hull is closed ...
prdsinvgd2 19905 Negation of a single coord...
dsmmsubg 19906 The finite hull of a produ...
dsmmlss 19907 The finite hull of a produ...
dsmmlmod 19908 The direct sum of a family...
frlmval 19911 Value of the free module. ...
frlmlmod 19912 The free module is a modul...
frlmpws 19913 The free module as a restr...
frlmlss 19914 The base set of the free m...
frlmpwsfi 19915 The finite free module is ...
frlmsca 19916 The ring of scalars of a f...
frlm0 19917 Zero in a free module (rin...
frlmbas 19918 Base set of the free modul...
frlmelbas 19919 Membership in the base set...
frlmrcl 19920 If a free module is inhabi...
frlmbasfsupp 19921 Elements of the free modul...
frlmbasmap 19922 Elements of the free modul...
frlmbasf 19923 Elements of the free modul...
frlmfibas 19924 The base set of the finite...
elfrlmbasn0 19925 If the dimension of a free...
frlmplusgval 19926 Addition in a free module....
frlmsubgval 19927 Subtraction in a free modu...
frlmvscafval 19928 Scalar multiplication in a...
frlmvscaval 19929 Scalar multiplication in a...
frlmgsum 19930 Finite commutative sums in...
frlmsplit2 19931 Restriction is homomorphic...
frlmsslss 19932 A subset of a free module ...
frlmsslss2 19933 A subset of a free module ...
frlmbas3 19934 An element of the base set...
mpt2frlmd 19935 Elements of the free modul...
frlmip 19936 The inner product of a fre...
frlmipval 19937 The inner product of a fre...
frlmphllem 19938 Lemma for ~ frlmphl . (Co...
frlmphl 19939 Conditions for a free modu...
uvcfval 19942 Value of the unit-vector g...
uvcval 19943 Value of a single unit vec...
uvcvval 19944 Value of a unit vector coo...
uvcvvcl 19945 A coodinate of a unit vect...
uvcvvcl2 19946 A unit vector coordinate i...
uvcvv1 19947 The unit vector is one at ...
uvcvv0 19948 The unit vector is zero at...
uvcff 19949 Domain and range of the un...
uvcf1 19950 In a nonzero ring, each un...
uvcresum 19951 Any element of a free modu...
frlmssuvc1 19952 A scalar multiple of a uni...
frlmssuvc2 19953 A nonzero scalar multiple ...
frlmsslsp 19954 A subset of a free module ...
frlmlbs 19955 The unit vectors comprise ...
frlmup1 19956 Any assignment of unit vec...
frlmup2 19957 The evaluation map has the...
frlmup3 19958 The range of such an evalu...
frlmup4 19959 Universal property of the ...
ellspd 19960 The elements of the span o...
elfilspd 19961 Simplified version of ~ el...
rellindf 19966 The independent-family pre...
islinds 19967 Property of an independent...
linds1 19968 An independent set of vect...
linds2 19969 An independent set of vect...
islindf 19970 Property of an independent...
islinds2 19971 Expanded property of an in...
islindf2 19972 Property of an independent...
lindff 19973 Functional property of a l...
lindfind 19974 A linearly independent fam...
lindsind 19975 A linearly independent set...
lindfind2 19976 In a linearly independent ...
lindsind2 19977 In a linearly independent ...
lindff1 19978 A linearly independent fam...
lindfrn 19979 The range of an independen...
f1lindf 19980 Rearranging and deleting e...
lindfres 19981 Any restriction of an inde...
lindsss 19982 Any subset of an independe...
f1linds 19983 A family constructed from ...
islindf3 19984 In a nonzero ring, indepen...
lindfmm 19985 Linear independence of a f...
lindsmm 19986 Linear independence of a s...
lindsmm2 19987 The monomorphic image of a...
lsslindf 19988 Linear independence is unc...
lsslinds 19989 Linear independence is unc...
islbs4 19990 A basis is an independent ...
lbslinds 19991 A basis is independent. (...
islinds3 19992 A subset is linearly indep...
islinds4 19993 A set is independent in a ...
lmimlbs 19994 The isomorphic image of a ...
lmiclbs 19995 Having a basis is an isomo...
islindf4 19996 A family is independent if...
islindf5 19997 A family is independent if...
indlcim 19998 An independent, spanning f...
lbslcic 19999 A module with a basis is i...
lmisfree 20000 A module has a basis iff i...
lvecisfrlm 20001 Every vector space is isom...
lmimco 20002 The composition of two iso...
lmictra 20003 Module isomorphism is tran...
uvcf1o 20004 In a nonzero ring, the map...
uvcendim 20005 In a nonzero ring, the num...
frlmisfrlm 20006 A free module is isomorphi...
frlmiscvec 20007 Every free module is isomo...
mamufval 20010 Functional value of the ma...
mamuval 20011 Multiplication of two matr...
mamufv 20012 A cell in the multiplicati...
mamudm 20013 The domain of the matrix m...
mamufacex 20014 Every solution of the equa...
mamures 20015 Rows in a matrix product a...
mndvcl 20016 Tuple-wise additive closur...
mndvass 20017 Tuple-wise associativity i...
mndvlid 20018 Tuple-wise left identity i...
mndvrid 20019 Tuple-wise right identity ...
grpvlinv 20020 Tuple-wise left inverse in...
grpvrinv 20021 Tuple-wise right inverse i...
mhmvlin 20022 Tuple extension of monoid ...
ringvcl 20023 Tuple-wise multiplication ...
gsumcom3 20024 A commutative law for fini...
gsumcom3fi 20025 A commutative law for fini...
mamucl 20026 Operation closure of matri...
mamuass 20027 Matrix multiplication is a...
mamudi 20028 Matrix multiplication dist...
mamudir 20029 Matrix multiplication dist...
mamuvs1 20030 Matrix multiplication dist...
mamuvs2 20031 Matrix multiplication dist...
matbas0pc 20034 There is no matrix with a ...
matbas0 20035 There is no matrix for a n...
matval 20036 Value of the matrix algebr...
matrcl 20037 Reverse closure for the ma...
matbas 20038 The matrix ring has the sa...
matplusg 20039 The matrix ring has the sa...
matsca 20040 The matrix ring has the sa...
matvsca 20041 The matrix ring has the sa...
mat0 20042 The matrix ring has the sa...
matinvg 20043 The matrix ring has the sa...
mat0op 20044 Value of a zero matrix as ...
matsca2 20045 The scalars of the matrix ...
matbas2 20046 The base set of the matrix...
matbas2i 20047 A matrix is a function. (...
matbas2d 20048 The base set of the matrix...
eqmat 20049 Two square matrices of the...
matecl 20050 Each entry (according to W...
matecld 20051 Each entry (according to W...
matplusg2 20052 Addition in the matrix rin...
matvsca2 20053 Scalar multiplication in t...
matlmod 20054 The matrix ring is a linea...
matgrp 20055 The matrix ring is a group...
matvscl 20056 Closure of the scalar mult...
matsubg 20057 The matrix ring has the sa...
matplusgcell 20058 Addition in the matrix rin...
matsubgcell 20059 Subtraction in the matrix ...
matinvgcell 20060 Additive inversion in the ...
matvscacell 20061 Scalar multiplication in t...
matgsum 20062 Finite commutative sums in...
matmulr 20063 Multiplication in the matr...
mamumat1cl 20064 The identity matrix (as op...
mat1comp 20065 The components of the iden...
mamulid 20066 The identity matrix (as op...
mamurid 20067 The identity matrix (as op...
matring 20068 Existence of the matrix ri...
matassa 20069 Existence of the matrix al...
matmulcell 20070 Multiplication in the matr...
mpt2matmul 20071 Multiplication of two N x ...
mat1 20072 Value of an identity matri...
mat1ov 20073 Entries of an identity mat...
mat1bas 20074 The identity matrix is a m...
matsc 20075 The identity matrix multip...
ofco2 20076 Distribution law for the f...
oftpos 20077 The transposition of the v...
mattposcl 20078 The transpose of a square ...
mattpostpos 20079 The transpose of the trans...
mattposvs 20080 The transposition of a mat...
mattpos1 20081 The transposition of the i...
tposmap 20082 The transposition of an I ...
mamutpos 20083 Behavior of transposes in ...
mattposm 20084 Multiplying two transposed...
matgsumcl 20085 Closure of a group sum ove...
madetsumid 20086 The identity summand in th...
matepmcl 20087 Each entry of a matrix wit...
matepm2cl 20088 Each entry of a matrix wit...
madetsmelbas 20089 A summand of the determina...
madetsmelbas2 20090 A summand of the determina...
mat0dimbas0 20091 The empty set is the one a...
mat0dim0 20092 The zero of the algebra of...
mat0dimid 20093 The identity of the algebr...
mat0dimscm 20094 The scalar multiplication ...
mat0dimcrng 20095 The algebra of matrices wi...
mat1dimelbas 20096 A matrix with dimension 1 ...
mat1dimbas 20097 A matrix with dimension 1 ...
mat1dim0 20098 The zero of the algebra of...
mat1dimid 20099 The identity of the algebr...
mat1dimscm 20100 The scalar multiplication ...
mat1dimmul 20101 The ring multiplication in...
mat1dimcrng 20102 The algebra of matrices wi...
mat1f1o 20103 There is a 1-1 function fr...
mat1rhmval 20104 The value of the ring homo...
mat1rhmelval 20105 The value of the ring homo...
mat1rhmcl 20106 The value of the ring homo...
mat1f 20107 There is a function from a...
mat1ghm 20108 There is a group homomorph...
mat1mhm 20109 There is a monoid homomorp...
mat1rhm 20110 There is a ring homomorphi...
mat1rngiso 20111 There is a ring isomorphis...
mat1ric 20112 A ring is isomorphic to th...
dmatval 20117 The set of ` N ` x ` N ` d...
dmatel 20118 A ` N ` x ` N ` diagonal m...
dmatmat 20119 An ` N ` x ` N ` diagonal ...
dmatid 20120 The identity matrix is a d...
dmatelnd 20121 An extradiagonal entry of ...
dmatmul 20122 The product of two diagona...
dmatsubcl 20123 The difference of two diag...
dmatsgrp 20124 The set of diagonal matric...
dmatmulcl 20125 The product of two diagona...
dmatsrng 20126 The set of diagonal matric...
dmatcrng 20127 The subring of diagonal ma...
dmatscmcl 20128 The multiplication of a di...
scmatval 20129 The set of ` N ` x ` N ` s...
scmatel 20130 An ` N ` x ` N ` scalar ma...
scmatscmid 20131 A scalar matrix can be exp...
scmatscmide 20132 An entry of a scalar matri...
scmatscmiddistr 20133 Distributive law for scala...
scmatmat 20134 An ` N ` x ` N ` scalar ma...
scmate 20135 An entry of an ` N ` x ` N...
scmatmats 20136 The set of an ` N ` x ` N ...
scmateALT 20137 Alternate proof of ~ scmat...
scmatscm 20138 The multiplication of a ma...
scmatid 20139 The identity matrix is a s...
scmatdmat 20140 A scalar matrix is a diago...
scmataddcl 20141 The sum of two scalar matr...
scmatsubcl 20142 The difference of two scal...
scmatmulcl 20143 The product of two scalar ...
scmatsgrp 20144 The set of scalar matrices...
scmatsrng 20145 The set of scalar matrices...
scmatcrng 20146 The subring of scalar matr...
scmatsgrp1 20147 The set of scalar matrices...
scmatsrng1 20148 The set of scalar matrices...
smatvscl 20149 Closure of the scalar mult...
scmatlss 20150 The set of scalar matrices...
scmatstrbas 20151 The set of scalar matrices...
scmatrhmval 20152 The value of the ring homo...
scmatrhmcl 20153 The value of the ring homo...
scmatf 20154 There is a function from a...
scmatfo 20155 There is a function from a...
scmatf1 20156 There is a 1-1 function fr...
scmatf1o 20157 There is a bijection betwe...
scmatghm 20158 There is a group homomorph...
scmatmhm 20159 There is a monoid homomorp...
scmatrhm 20160 There is a ring homomorphi...
scmatrngiso 20161 There is a ring isomorphis...
scmatric 20162 A ring is isomorphic to ev...
mat0scmat 20163 The empty matrix over a ri...
mat1scmat 20164 A 1-dimensional matrix ove...
mvmulfval 20167 Functional value of the ma...
mvmulval 20168 Multiplication of a vector...
mvmulfv 20169 A cell/element in the vect...
mavmulval 20170 Multiplication of a vector...
mavmulfv 20171 A cell/element in the vect...
mavmulcl 20172 Multiplication of an NxN m...
1mavmul 20173 Multiplication of the iden...
mavmulass 20174 Associativity of the multi...
mavmuldm 20175 The domain of the matrix v...
mavmulsolcl 20176 Every solution of the equa...
mavmul0 20177 Multiplication of a 0-dime...
mavmul0g 20178 The result of the 0-dimens...
mvmumamul1 20179 The multiplication of an M...
mavmumamul1 20180 The multiplication of an N...
marrepfval 20185 First substitution for the...
marrepval0 20186 Second substitution for th...
marrepval 20187 Third substitution for the...
marrepeval 20188 An entry of a matrix with ...
marrepcl 20189 Closure of the row replace...
marepvfval 20190 First substitution for the...
marepvval0 20191 Second substitution for th...
marepvval 20192 Third substitution for the...
marepveval 20193 An entry of a matrix with ...
marepvcl 20194 Closure of the column repl...
ma1repvcl 20195 Closure of the column repl...
ma1repveval 20196 An entry of an identity ma...
mulmarep1el 20197 Element by element multipl...
mulmarep1gsum1 20198 The sum of element by elem...
mulmarep1gsum2 20199 The sum of element by elem...
1marepvmarrepid 20200 Replacing the ith row by 0...
submabas 20203 Any subset of the index se...
submafval 20204 First substitution for a s...
submaval0 20205 Second substitution for a ...
submaval 20206 Third substitution for a s...
submaeval 20207 An entry of a submatrix of...
1marepvsma1 20208 The submatrix of the ident...
mdetfval 20211 First substitution for the...
mdetleib 20212 Full substitution of our d...
mdetleib2 20213 Leibniz' formula can also ...
nfimdetndef 20214 The determinant is not def...
mdetfval1 20215 First substitution of an a...
mdetleib1 20216 Full substitution of an al...
mdet0pr 20217 The determinant for 0-dime...
mdet0f1o 20218 The determinant for 0-dime...
mdet0fv0 20219 The determinant of a 0-dim...
mdetf 20220 Functionality of the deter...
mdetcl 20221 The determinant evaluates ...
m1detdiag 20222 The determinant of a 1-dim...
mdetdiaglem 20223 Lemma for ~ mdetdiag . Pr...
mdetdiag 20224 The determinant of a diago...
mdetdiagid 20225 The determinant of a diago...
mdet1 20226 The determinant of the ide...
mdetrlin 20227 The determinant function i...
mdetrsca 20228 The determinant function i...
mdetrsca2 20229 The determinant function i...
mdetr0 20230 The determinant of a matri...
mdet0 20231 The determinant of the zer...
mdetrlin2 20232 The determinant function i...
mdetralt 20233 The determinant function i...
mdetralt2 20234 The determinant function i...
mdetero 20235 The determinant function i...
mdettpos 20236 Determinant is invariant u...
mdetunilem1 20237 Lemma for ~ mdetuni . (Co...
mdetunilem2 20238 Lemma for ~ mdetuni . (Co...
mdetunilem3 20239 Lemma for ~ mdetuni . (Co...
mdetunilem4 20240 Lemma for ~ mdetuni . (Co...
mdetunilem5 20241 Lemma for ~ mdetuni . (Co...
mdetunilem6 20242 Lemma for ~ mdetuni . (Co...
mdetunilem7 20243 Lemma for ~ mdetuni . (Co...
mdetunilem8 20244 Lemma for ~ mdetuni . (Co...
mdetunilem9 20245 Lemma for ~ mdetuni . (Co...
mdetuni0 20246 Lemma for ~ mdetuni . (Co...
mdetuni 20247 According to the definitio...
mdetmul 20248 Multiplicativity of the de...
m2detleiblem1 20249 Lemma 1 for ~ m2detleib . ...
m2detleiblem5 20250 Lemma 5 for ~ m2detleib . ...
m2detleiblem6 20251 Lemma 6 for ~ m2detleib . ...
m2detleiblem7 20252 Lemma 7 for ~ m2detleib . ...
m2detleiblem2 20253 Lemma 2 for ~ m2detleib . ...
m2detleiblem3 20254 Lemma 3 for ~ m2detleib . ...
m2detleiblem4 20255 Lemma 4 for ~ m2detleib . ...
m2detleib 20256 Leibniz' Formula for 2x2-m...
mndifsplit 20261 Lemma for ~ maducoeval2 . ...
madufval 20262 First substitution for the...
maduval 20263 Second substitution for th...
maducoeval 20264 An entry of the adjunct (c...
maducoeval2 20265 An entry of the adjunct (c...
maduf 20266 Creating the adjunct of ma...
madutpos 20267 The adjuct of a transposed...
madugsum 20268 The determinant of a matri...
madurid 20269 Multiplying a matrix with ...
madulid 20270 Multiplying the adjunct of...
minmar1fval 20271 First substitution for the...
minmar1val0 20272 Second substitution for th...
minmar1val 20273 Third substitution for the...
minmar1eval 20274 An entry of a matrix for a...
minmar1marrep 20275 The minor matrix is a spec...
minmar1cl 20276 Closure of the row replace...
maducoevalmin1 20277 The coefficients of an adj...
symgmatr01lem 20278 Lemma for ~ symgmatr01 . ...
symgmatr01 20279 Applying a permutation tha...
gsummatr01lem1 20280 Lemma A for ~ gsummatr01 ....
gsummatr01lem2 20281 Lemma B for ~ gsummatr01 ....
gsummatr01lem3 20282 Lemma 1 for ~ gsummatr01 ....
gsummatr01lem4 20283 Lemma 2 for ~ gsummatr01 ....
gsummatr01 20284 Lemma 1 for ~ smadiadetlem...
marep01ma 20285 Replacing a row of a squar...
smadiadetlem0 20286 Lemma 0 for ~ smadiadet : ...
smadiadetlem1 20287 Lemma 1 for ~ smadiadet : ...
smadiadetlem1a 20288 Lemma 1a for ~ smadiadet :...
smadiadetlem2 20289 Lemma 2 for ~ smadiadet : ...
smadiadetlem3lem0 20290 Lemma 0 for ~ smadiadetlem...
smadiadetlem3lem1 20291 Lemma 1 for ~ smadiadetlem...
smadiadetlem3lem2 20292 Lemma 2 for ~ smadiadetlem...
smadiadetlem3 20293 Lemma 3 for ~ smadiadet . ...
smadiadetlem4 20294 Lemma 4 for ~ smadiadet . ...
smadiadet 20295 The determinant of a subma...
smadiadetglem1 20296 Lemma 1 for ~ smadiadetg ....
smadiadetglem2 20297 Lemma 2 for ~ smadiadetg ....
smadiadetg 20298 The determinant of a squar...
smadiadetg0 20299 Lemma for ~ smadiadetr : v...
smadiadetr 20300 The determinant of a squar...
invrvald 20301 If a matrix multiplied wit...
matinv 20302 The inverse of a matrix is...
matunit 20303 A matrix is a unit in the ...
slesolvec 20304 Every solution of a system...
slesolinv 20305 The solution of a system o...
slesolinvbi 20306 The solution of a system o...
slesolex 20307 Every system of linear equ...
cramerimplem1 20308 Lemma 1 for ~ cramerimp : ...
cramerimplem2 20309 Lemma 2 for ~ cramerimp : ...
cramerimplem3 20310 Lemma 3 for ~ cramerimp : ...
cramerimp 20311 One direction of Cramer's ...
cramerlem1 20312 Lemma 1 for ~ cramer . (C...
cramerlem2 20313 Lemma 2 for ~ cramer . (C...
cramerlem3 20314 Lemma 3 for ~ cramer . (C...
cramer0 20315 Special case of Cramer's r...
cramer 20316 Cramer's rule. According ...
pmatring 20317 The set of polynomial matr...
pmatlmod 20318 The set of polynomial matr...
pmat0op 20319 The zero polynomial matrix...
pmat1op 20320 The identity polynomial ma...
pmat1ovd 20321 Entries of the identity po...
pmat0opsc 20322 The zero polynomial matrix...
pmat1opsc 20323 The identity polynomial ma...
pmat1ovscd 20324 Entries of the identity po...
pmatcoe1fsupp 20325 For a polynomial matrix th...
1pmatscmul 20326 The scalar product of the ...
cpmat 20333 Value of the constructor o...
cpmatpmat 20334 A constant polynomial matr...
cpmatel 20335 Property of a constant pol...
cpmatelimp 20336 Implication of a set being...
cpmatel2 20337 Another property of a cons...
cpmatelimp2 20338 Another implication of a s...
1elcpmat 20339 The identity of the ring o...
cpmatacl 20340 The set of all constant po...
cpmatinvcl 20341 The set of all constant po...
cpmatmcllem 20342 Lemma for ~ cpmatmcl . (C...
cpmatmcl 20343 The set of all constant po...
cpmatsubgpmat 20344 The set of all constant po...
cpmatsrgpmat 20345 The set of all constant po...
0elcpmat 20346 The zero of the ring of al...
mat2pmatfval 20347 Value of the matrix transf...
mat2pmatval 20348 The result of a matrix tra...
mat2pmatvalel 20349 A (matrix) element of the ...
mat2pmatbas 20350 The result of a matrix tra...
mat2pmatbas0 20351 The result of a matrix tra...
mat2pmatf 20352 The matrix transformation ...
mat2pmatf1 20353 The matrix transformation ...
mat2pmatghm 20354 The transformation of matr...
mat2pmatmul 20355 The transformation of matr...
mat2pmat1 20356 The transformation of the ...
mat2pmatmhm 20357 The transformation of matr...
mat2pmatrhm 20358 The transformation of matr...
mat2pmatlin 20359 The transformation of matr...
0mat2pmat 20360 The transformed zero matri...
idmatidpmat 20361 The transformed identity m...
d0mat2pmat 20362 The transformed empty set ...
d1mat2pmat 20363 The transformation of a ma...
mat2pmatscmxcl 20364 A transformed matrix multi...
m2cpm 20365 The result of a matrix tra...
m2cpmf 20366 The matrix transformation ...
m2cpmf1 20367 The matrix transformation ...
m2cpmghm 20368 The transformation of matr...
m2cpmmhm 20369 The transformation of matr...
m2cpmrhm 20370 The transformation of matr...
m2pmfzmap 20371 The transformed values of ...
m2pmfzgsumcl 20372 Closure of the sum of scal...
cpm2mfval 20373 Value of the inverse matri...
cpm2mval 20374 The result of an inverse m...
cpm2mvalel 20375 A (matrix) element of the ...
cpm2mf 20376 The inverse matrix transfo...
m2cpminvid 20377 The inverse transformation...
m2cpminvid2lem 20378 Lemma for ~ m2cpminvid2 . ...
m2cpminvid2 20379 The transformation applied...
m2cpmfo 20380 The matrix transformation ...
m2cpmf1o 20381 The matrix transformation ...
m2cpmrngiso 20382 The transformation of matr...
matcpmric 20383 The ring of matrices over ...
m2cpminv 20384 The inverse matrix transfo...
m2cpminv0 20385 The inverse matrix transfo...
decpmatval0 20388 The matrix consisting of t...
decpmatval 20389 The matrix consisting of t...
decpmate 20390 An entry of the matrix con...
decpmatcl 20391 Closure of the decompositi...
decpmataa0 20392 The matrix consisting of t...
decpmatfsupp 20393 The mapping to the matrice...
decpmatid 20394 The matrix consisting of t...
decpmatmullem 20395 Lemma for ~ decpmatmul . ...
decpmatmul 20396 The matrix consisting of t...
decpmatmulsumfsupp 20397 Lemma 0 for ~ pm2mpmhm . ...
pmatcollpw1lem1 20398 Lemma 1 for ~ pmatcollpw1 ...
pmatcollpw1lem2 20399 Lemma 2 for ~ pmatcollpw1 ...
pmatcollpw1 20400 Write a polynomial matrix ...
pmatcollpw2lem 20401 Lemma for ~ pmatcollpw2 . ...
pmatcollpw2 20402 Write a polynomial matrix ...
monmatcollpw 20403 The matrix consisting of t...
pmatcollpwlem 20404 Lemma for ~ pmatcollpw . ...
pmatcollpw 20405 Write a polynomial matrix ...
pmatcollpwfi 20406 Write a polynomial matrix ...
pmatcollpw3lem 20407 Lemma for ~ pmatcollpw3 an...
pmatcollpw3 20408 Write a polynomial matrix ...
pmatcollpw3fi 20409 Write a polynomial matrix ...
pmatcollpw3fi1lem1 20410 Lemma 1 for ~ pmatcollpw3f...
pmatcollpw3fi1lem2 20411 Lemma 2 for ~ pmatcollpw3f...
pmatcollpw3fi1 20412 Write a polynomial matrix ...
pmatcollpwscmatlem1 20413 Lemma 1 for ~ pmatcollpwsc...
pmatcollpwscmatlem2 20414 Lemma 2 for ~ pmatcollpwsc...
pmatcollpwscmat 20415 Write a scalar matrix over...
pm2mpf1lem 20418 Lemma for ~ pm2mpf1 . (Co...
pm2mpval 20419 Value of the transformatio...
pm2mpfval 20420 A polynomial matrix transf...
pm2mpcl 20421 The transformation of poly...
pm2mpf 20422 The transformation of poly...
pm2mpf1 20423 The transformation of poly...
pm2mpcoe1 20424 A coefficient of the polyn...
idpm2idmp 20425 The transformation of the ...
mptcoe1matfsupp 20426 The mapping extracting the...
mply1topmatcllem 20427 Lemma for ~ mply1topmatcl ...
mply1topmatval 20428 A polynomial over matrices...
mply1topmatcl 20429 A polynomial over matrices...
mp2pm2mplem1 20430 Lemma 1 for ~ mp2pm2mp . ...
mp2pm2mplem2 20431 Lemma 2 for ~ mp2pm2mp . ...
mp2pm2mplem3 20432 Lemma 3 for ~ mp2pm2mp . ...
mp2pm2mplem4 20433 Lemma 4 for ~ mp2pm2mp . ...
mp2pm2mplem5 20434 Lemma 5 for ~ mp2pm2mp . ...
mp2pm2mp 20435 A polynomial over matrices...
pm2mpghmlem2 20436 Lemma 2 for ~ pm2mpghm . ...
pm2mpghmlem1 20437 Lemma 1 for pm2mpghm . (C...
pm2mpfo 20438 The transformation of poly...
pm2mpf1o 20439 The transformation of poly...
pm2mpghm 20440 The transformation of poly...
pm2mpgrpiso 20441 The transformation of poly...
pm2mpmhmlem1 20442 Lemma 1 for ~ pm2mpmhm . ...
pm2mpmhmlem2 20443 Lemma 2 for ~ pm2mpmhm . ...
pm2mpmhm 20444 The transformation of poly...
pm2mprhm 20445 The transformation of poly...
pm2mprngiso 20446 The transformation of poly...
pmmpric 20447 The ring of polynomial mat...
monmat2matmon 20448 The transformation of a po...
pm2mp 20449 The transformation of a su...
chmatcl 20452 Closure of the characteris...
chmatval 20453 The entries of the charact...
chpmatfval 20454 Value of the characteristi...
chpmatval 20455 The characteristic polynom...
chpmatply1 20456 The characteristic polynom...
chpmatval2 20457 The characteristic polynom...
chpmat0d 20458 The characteristic polynom...
chpmat1dlem 20459 Lemma for ~ chpmat1d . (C...
chpmat1d 20460 The characteristic polynom...
chpdmatlem0 20461 Lemma 0 for ~ chpdmat . (...
chpdmatlem1 20462 Lemma 1 for ~ chpdmat . (...
chpdmatlem2 20463 Lemma 2 for ~ chpdmat . (...
chpdmatlem3 20464 Lemma 3 for ~ chpdmat . (...
chpdmat 20465 The characteristic polynom...
chpscmat 20466 The characteristic polynom...
chpscmat0 20467 The characteristic polynom...
chpscmatgsumbin 20468 The characteristic polynom...
chpscmatgsummon 20469 The characteristic polynom...
chp0mat 20470 The characteristic polynom...
chpidmat 20471 The characteristic polynom...
chmaidscmat 20472 The characteristic polynom...
fvmptnn04if 20473 The function values of a m...
fvmptnn04ifa 20474 The function value of a ma...
fvmptnn04ifb 20475 The function value of a ma...
fvmptnn04ifc 20476 The function value of a ma...
fvmptnn04ifd 20477 The function value of a ma...
chfacfisf 20478 The "characteristic factor...
chfacfisfcpmat 20479 The "characteristic factor...
chfacffsupp 20480 The "characteristic factor...
chfacfscmulcl 20481 Closure of a scaled value ...
chfacfscmul0 20482 A scaled value of the "cha...
chfacfscmulfsupp 20483 A mapping of scaled values...
chfacfscmulgsum 20484 Breaking up a sum of value...
chfacfpmmulcl 20485 Closure of the value of th...
chfacfpmmul0 20486 The value of the "characte...
chfacfpmmulfsupp 20487 A mapping of values of the...
chfacfpmmulgsum 20488 Breaking up a sum of value...
chfacfpmmulgsum2 20489 Breaking up a sum of value...
cayhamlem1 20490 Lemma 1 for ~ cayleyhamilt...
cpmadurid 20491 The right-hand fundamental...
cpmidgsum 20492 Representation of the iden...
cpmidgsumm2pm 20493 Representation of the iden...
cpmidpmatlem1 20494 Lemma 1 for ~ cpmidpmat . ...
cpmidpmatlem2 20495 Lemma 2 for ~ cpmidpmat . ...
cpmidpmatlem3 20496 Lemma 3 for ~ cpmidpmat . ...
cpmidpmat 20497 Representation of the iden...
cpmadugsumlemB 20498 Lemma B for ~ cpmadugsum ....
cpmadugsumlemC 20499 Lemma C for ~ cpmadugsum ....
cpmadugsumlemF 20500 Lemma F for ~ cpmadugsum ....
cpmadugsumfi 20501 The product of the charact...
cpmadugsum 20502 The product of the charact...
cpmidgsum2 20503 Representation of the iden...
cpmidg2sum 20504 Equality of two sums repre...
cpmadumatpolylem1 20505 Lemma 1 for ~ cpmadumatpol...
cpmadumatpolylem2 20506 Lemma 2 for ~ cpmadumatpol...
cpmadumatpoly 20507 The product of the charact...
cayhamlem2 20508 Lemma for ~ cayhamlem3 . ...
chcoeffeqlem 20509 Lemma for ~ chcoeffeq . (...
chcoeffeq 20510 The coefficients of the ch...
cayhamlem3 20511 Lemma for ~ cayhamlem4 . ...
cayhamlem4 20512 Lemma for ~ cayleyhamilton...
cayleyhamilton0 20513 The Cayley-Hamilton theore...
cayleyhamilton 20514 The Cayley-Hamilton theore...
cayleyhamiltonALT 20515 Alternate proof of ~ cayle...
cayleyhamilton1 20516 The Cayley-Hamilton theore...
istopg 20525 Express the predicate " ` ...
istop2g 20526 Express the predicate " ` ...
uniopn 20527 The union of a subset of a...
iunopn 20528 The indexed union of a sub...
inopn 20529 The intersection of two op...
fitop 20530 A topology is closed under...
fiinopn 20531 The intersection of a none...
iinopn 20532 The intersection of a none...
unopn 20533 The union of two open sets...
0opn 20534 The empty set is an open s...
0ntop 20535 The empty set is not a top...
topopn 20536 The underlying set of a to...
eltopss 20537 A member of a topology is ...
riinopn 20538 A finite indexed relative ...
rintopn 20539 A finite relative intersec...
istopon 20540 Property of being a topolo...
topontop 20541 A topology on a given base...
toponuni 20542 The base set of a topology...
toponmax 20543 The base set of a topology...
toponss 20544 A member of a topology is ...
toponcom 20545 If ` K ` is a topology on ...
topontopi 20546 A topology on a given base...
toponunii 20547 The base set of a topology...
toptopon 20548 Alternative definition of ...
topgele 20549 The topologies over the sa...
topsn 20550 The only topology on a sin...
istps 20551 Express the predicate "is ...
istps2 20552 Express the predicate "is ...
tpsuni 20553 The base set of a topologi...
tpstop 20554 The topology extractor on ...
tpspropd 20555 A topological space depend...
tpsprop2d 20556 A topological space depend...
topontopn 20557 Express the predicate "is ...
tsettps 20558 If the topology component ...
istpsi 20559 Properties that determine ...
eltpsg 20560 Properties that determine ...
eltpsi 20561 Properties that determine ...
isbasisg 20562 Express the predicate " ` ...
isbasis2g 20563 Express the predicate " ` ...
isbasis3g 20564 Express the predicate " ` ...
basis1 20565 Property of a basis. (Con...
basis2 20566 Property of a basis. (Con...
fiinbas 20567 If a set is closed under f...
basdif0 20568 A basis is not affected by...
baspartn 20569 A disjoint system of sets ...
tgval 20570 The topology generated by ...
tgval2 20571 Definition of a topology g...
eltg 20572 Membership in a topology g...
eltg2 20573 Membership in a topology g...
eltg2b 20574 Membership in a topology g...
eltg4i 20575 An open set in a topology ...
eltg3i 20576 The union of a set of basi...
eltg3 20577 Membership in a topology g...
tgval3 20578 Alternate expression for t...
tg1 20579 Property of a member of a ...
tg2 20580 Property of a member of a ...
bastg 20581 A member of a basis is a s...
unitg 20582 The topology generated by ...
tgss 20583 Subset relation for genera...
tgcl 20584 Show that a basis generate...
tgclb 20585 The property ~ tgcl can be...
tgtopon 20586 A basis generates a topolo...
topbas 20587 A topology is its own basi...
tgtop 20588 A topology is its own basi...
eltop 20589 Membership in a topology, ...
eltop2 20590 Membership in a topology. ...
eltop3 20591 Membership in a topology. ...
fibas 20592 A collection of finite int...
tgdom 20593 A space has no more open s...
tgiun 20594 The indexed union of a set...
tgidm 20595 The topology generator fun...
bastop 20596 Two ways to express that a...
tgtop11 20597 The topology generation fu...
0top 20598 The singleton of the empty...
en1top 20599 ` { (/) } ` is the only to...
en2top 20600 If a topology has two elem...
tgss3 20601 A criterion for determinin...
tgss2 20602 A criterion for determinin...
basgen 20603 Given a topology ` J ` , s...
basgen2 20604 Given a topology ` J ` , s...
2basgen 20605 Conditions that determine ...
tgfiss 20606 If a subbase is included i...
tgdif0 20607 A generated topology is no...
bastop1 20608 A subset of a topology is ...
bastop2 20609 A version of ~ bastop1 tha...
distop 20610 The discrete topology on a...
distopon 20611 The discrete topology on a...
sn0topon 20612 The singleton of the empty...
sn0top 20613 The singleton of the empty...
indislem 20614 A lemma to eliminate some ...
indistopon 20615 The indiscrete topology on...
indistop 20616 The indiscrete topology on...
indisuni 20617 The base set of the indisc...
fctop 20618 The finite complement topo...
fctop2 20619 The finite complement topo...
cctop 20620 The countable complement t...
ppttop 20621 The particular point topol...
pptbas 20622 The particular point topol...
epttop 20623 The excluded point topolog...
indistpsx 20624 The indiscrete topology on...
indistps 20625 The indiscrete topology on...
indistps2 20626 The indiscrete topology on...
indistpsALT 20627 The indiscrete topology on...
indistps2ALT 20628 The indiscrete topology on...
distps 20629 The discrete topology on a...
fncld 20636 The closed-set generator i...
cldval 20637 The set of closed sets of ...
ntrfval 20638 The interior function on t...
clsfval 20639 The closure function on th...
cldrcl 20640 Reverse closure of the clo...
iscld 20641 The predicate " ` S ` is a...
iscld2 20642 A subset of the underlying...
cldss 20643 A closed set is a subset o...
cldss2 20644 The set of closed sets is ...
cldopn 20645 The complement of a closed...
isopn2 20646 A subset of the underlying...
opncld 20647 The complement of an open ...
difopn 20648 The difference of a closed...
topcld 20649 The underlying set of a to...
ntrval 20650 The interior of a subset o...
clsval 20651 The closure of a subset of...
0cld 20652 The empty set is closed. ...
iincld 20653 The indexed intersection o...
intcld 20654 The intersection of a set ...
uncld 20655 The union of two closed se...
cldcls 20656 A closed subset equals its...
incld 20657 The intersection of two cl...
riincld 20658 An indexed relative inters...
iuncld 20659 A finite indexed union of ...
unicld 20660 A finite union of closed s...
clscld 20661 The closure of a subset of...
clsf 20662 The closure function is a ...
ntropn 20663 The interior of a subset o...
clsval2 20664 Express closure in terms o...
ntrval2 20665 Interior expressed in term...
ntrdif 20666 An interior of a complemen...
clsdif 20667 A closure of a complement ...
clsss 20668 Subset relationship for cl...
ntrss 20669 Subset relationship for in...
sscls 20670 A subset of a topology's u...
ntrss2 20671 A subset includes its inte...
ssntr 20672 An open subset of a set is...
clsss3 20673 The closure of a subset of...
ntrss3 20674 The interior of a subset o...
ntrin 20675 A pairwise intersection of...
cmclsopn 20676 The complement of a closur...
cmntrcld 20677 The complement of an inter...
iscld3 20678 A subset is closed iff it ...
iscld4 20679 A subset is closed iff it ...
isopn3 20680 A subset is open iff it eq...
clsidm 20681 The closure operation is i...
ntridm 20682 The interior operation is ...
clstop 20683 The closure of a topology'...
ntrtop 20684 The interior of a topology...
0ntr 20685 A subset with an empty int...
clsss2 20686 If a subset is included in...
elcls 20687 Membership in a closure. ...
elcls2 20688 Membership in a closure. ...
clsndisj 20689 Any open set containing a ...
ntrcls0 20690 A subset whose closure has...
ntreq0 20691 Two ways to say that a sub...
cldmre 20692 The closed sets of a topol...
mrccls 20693 Moore closure generalizes ...
cls0 20694 The closure of the empty s...
ntr0 20695 The interior of the empty ...
isopn3i 20696 An open subset equals its ...
elcls3 20697 Membership in a closure in...
opncldf1 20698 A bijection useful for con...
opncldf2 20699 The values of the open-clo...
opncldf3 20700 The values of the converse...
isclo 20701 A set ` A ` is clopen iff ...
isclo2 20702 A set ` A ` is clopen iff ...
discld 20703 The open sets of a discret...
sn0cld 20704 The closed sets of the top...
indiscld 20705 The closed sets of an indi...
mretopd 20706 A Moore collection which i...
toponmre 20707 The topologies over a give...
cldmreon 20708 The closed sets of a topol...
iscldtop 20709 A family is the closed set...
mreclatdemoBAD 20710 The closed subspaces of a ...
neifval 20713 The neighborhood function ...
neif 20714 The neighborhood function ...
neiss2 20715 A set with a neighborhood ...
neival 20716 The set of neighborhoods o...
isnei 20717 The predicate " ` N ` is a...
neiint 20718 An intuitive definition of...
isneip 20719 The predicate " ` N ` is a...
neii1 20720 A neighborhood is included...
neisspw 20721 The neighborhoods of any s...
neii2 20722 Property of a neighborhood...
neiss 20723 Any neighborhood of a set ...
ssnei 20724 A set is included in its n...
elnei 20725 A point belongs to any of ...
0nnei 20726 The empty set is not a nei...
neips 20727 A neighborhood of a set is...
opnneissb 20728 An open set is a neighborh...
opnssneib 20729 Any superset of an open se...
ssnei2 20730 Any subset of ` X ` contai...
neindisj 20731 Any neighborhood of an ele...
opnneiss 20732 An open set is a neighborh...
opnneip 20733 An open set is a neighborh...
opnnei 20734 A set is open iff it is a ...
tpnei 20735 The underlying set of a to...
neiuni 20736 The union of the neighborh...
neindisj2 20737 A point ` P ` belongs to t...
topssnei 20738 A finer topology has more ...
innei 20739 The intersection of two ne...
opnneiid 20740 Only an open set is a neig...
neissex 20741 For any neighborhood ` N `...
0nei 20742 The empty set is a neighbo...
neipeltop 20743 Lemma for ~ neiptopreu . ...
neiptopuni 20744 Lemma for ~ neiptopreu . ...
neiptoptop 20745 Lemma for ~ neiptopreu . ...
neiptopnei 20746 Lemma for ~ neiptopreu . ...
neiptopreu 20747 If, to each element ` P ` ...
lpfval 20752 The limit point function o...
lpval 20753 The set of limit points of...
islp 20754 The predicate " ` P ` is a...
lpsscls 20755 The limit points of a subs...
lpss 20756 The limit points of a subs...
lpdifsn 20757 ` P ` is a limit point of ...
lpss3 20758 Subset relationship for li...
islp2 20759 The predicate " ` P ` is a...
islp3 20760 The predicate " ` P ` is a...
maxlp 20761 A point is a limit point o...
clslp 20762 The closure of a subset of...
islpi 20763 A point belonging to a set...
cldlp 20764 A subset of a topological ...
isperf 20765 Definition of a perfect sp...
isperf2 20766 Definition of a perfect sp...
isperf3 20767 A perfect space is a topol...
perflp 20768 The limit points of a perf...
perfi 20769 Property of a perfect spac...
perftop 20770 A perfect space is a topol...
restrcl 20771 Reverse closure for the su...
restbas 20772 A subspace topology basis ...
tgrest 20773 A subspace can be generate...
resttop 20774 A subspace topology is a t...
resttopon 20775 A subspace topology is a t...
restuni 20776 The underlying set of a su...
stoig 20777 The topological space buil...
restco 20778 Composition of subspaces. ...
restabs 20779 Equivalence of being a sub...
restin 20780 When the subspace region i...
restuni2 20781 The underlying set of a su...
resttopon2 20782 The underlying set of a su...
rest0 20783 The subspace topology indu...
restsn 20784 The only subspace topology...
restsn2 20785 The subspace topology indu...
restcld 20786 A closed set of a subspace...
restcldi 20787 A closed set is closed in ...
restcldr 20788 A set which is closed in t...
restopnb 20789 If ` B ` is an open subset...
ssrest 20790 If ` K ` is a finer topolo...
restopn2 20791 The if ` A ` is open, then...
restdis 20792 A subspace of a discrete t...
restfpw 20793 The restriction of the set...
neitr 20794 The neighborhood of a trac...
restcls 20795 A closure in a subspace to...
restntr 20796 An interior in a subspace ...
restlp 20797 The limit points of a subs...
restperf 20798 Perfection of a subspace. ...
perfopn 20799 An open subset of a perfec...
resstopn 20800 The topology of a restrict...
resstps 20801 A restricted topological s...
ordtbaslem 20802 Lemma for ~ ordtbas . In ...
ordtval 20803 Value of the order topolog...
ordtuni 20804 Value of the order topolog...
ordtbas2 20805 Lemma for ~ ordtbas . (Co...
ordtbas 20806 In a total order, the fini...
ordttopon 20807 Value of the order topolog...
ordtopn1 20808 An upward ray ` ( P , +oo ...
ordtopn2 20809 A downward ray ` ( -oo , P...
ordtopn3 20810 An open interval ` ( A , B...
ordtcld1 20811 A downward ray ` ( -oo , P...
ordtcld2 20812 An upward ray ` [ P , +oo ...
ordtcld3 20813 A closed interval ` [ A , ...
ordttop 20814 The order topology is a to...
ordtcnv 20815 The order dual generates t...
ordtrest 20816 The subspace topology of a...
ordtrest2lem 20817 Lemma for ~ ordtrest2 . (...
ordtrest2 20818 An interval-closed set ` A...
letopon 20819 The topology of the extend...
letop 20820 The topology of the extend...
letopuni 20821 The topology of the extend...
xrstopn 20822 The topology component of ...
xrstps 20823 The extended real number s...
leordtvallem1 20824 Lemma for ~ leordtval . (...
leordtvallem2 20825 Lemma for ~ leordtval . (...
leordtval2 20826 The topology of the extend...
leordtval 20827 The topology of the extend...
iccordt 20828 A closed interval is close...
iocpnfordt 20829 An unbounded above open in...
icomnfordt 20830 An unbounded above open in...
iooordt 20831 An open interval is open i...
reordt 20832 The real numbers are an op...
lecldbas 20833 The set of closed interval...
pnfnei 20834 A neighborhood of ` +oo ` ...
mnfnei 20835 A neighborhood of ` -oo ` ...
ordtrestixx 20836 The restriction of the les...
ordtresticc 20837 The restriction of the les...
lmrel 20844 The topological space conv...
lmrcl 20845 Reverse closure for the co...
lmfval 20846 The relation "sequence ` f...
cnfval 20847 The set of all continuous ...
cnpfval 20848 The function mapping the p...
iscn 20849 The predicate " ` F ` is a...
cnpval 20850 The set of all functions f...
iscnp 20851 The predicate " ` F ` is a...
iscn2 20852 The predicate " ` F ` is a...
iscnp2 20853 The predicate " ` F ` is a...
cntop1 20854 Reverse closure for a cont...
cntop2 20855 Reverse closure for a cont...
cnptop1 20856 Reverse closure for a func...
cnptop2 20857 Reverse closure for a func...
iscnp3 20858 The predicate " ` F ` is a...
cnprcl 20859 Reverse closure for a func...
cnf 20860 A continuous function is a...
cnpf 20861 A continuous function at p...
cnpcl 20862 The value of a continuous ...
cnf2 20863 A continuous function is a...
cnpf2 20864 A continuous function at p...
cnprcl2 20865 Reverse closure for a func...
tgcn 20866 The continuity predicate w...
tgcnp 20867 The "continuous at a point...
subbascn 20868 The continuity predicate w...
ssidcn 20869 The identity function is a...
cnpimaex 20870 Property of a function con...
idcn 20871 A restricted identity func...
lmbr 20872 Express the binary relatio...
lmbr2 20873 Express the binary relatio...
lmbrf 20874 Express the binary relatio...
lmconst 20875 A constant sequence conver...
lmcvg 20876 Convergence property of a ...
iscnp4 20877 The predicate " ` F ` is a...
cnpnei 20878 A condition for continuity...
cnima 20879 An open subset of the codo...
cnco 20880 The composition of two con...
cnpco 20881 The composition of two con...
cnclima 20882 A closed subset of the cod...
iscncl 20883 A definition of a continuo...
cncls2i 20884 Property of the preimage o...
cnntri 20885 Property of the preimage o...
cnclsi 20886 Property of the image of a...
cncls2 20887 Continuity in terms of clo...
cncls 20888 Continuity in terms of clo...
cnntr 20889 Continuity in terms of int...
cnss1 20890 If the topology ` K ` is f...
cnss2 20891 If the topology ` K ` is f...
cncnpi 20892 A continuous function is c...
cnsscnp 20893 The set of continuous func...
cncnp 20894 A continuous function is c...
cncnp2 20895 A continuous function is c...
cnnei 20896 Continuity in terms of nei...
cnconst2 20897 A constant function is con...
cnconst 20898 A constant function is con...
cnrest 20899 Continuity of a restrictio...
cnrest2 20900 Equivalence of continuity ...
cnrest2r 20901 Equivalence of continuity ...
cnpresti 20902 One direction of ~ cnprest...
cnprest 20903 Equivalence of continuity ...
cnprest2 20904 Equivalence of point-conti...
cndis 20905 Every function is continuo...
cnindis 20906 Every function is continuo...
cnpdis 20907 If ` A ` is an isolated po...
paste 20908 Pasting lemma. If ` A ` a...
lmfpm 20909 If ` F ` converges, then `...
lmfss 20910 Inclusion of a function ha...
lmcl 20911 Closure of a limit. (Cont...
lmss 20912 Limit on a subspace. (Con...
sslm 20913 A finer topology has fewer...
lmres 20914 A function converges iff i...
lmff 20915 If ` F ` converges, there ...
lmcls 20916 Any convergent sequence of...
lmcld 20917 Any convergent sequence of...
lmcnp 20918 The image of a convergent ...
lmcn 20919 The image of a convergent ...
ist0 20934 The predicate "is a T_0 sp...
ist1 20935 The predicate ` J ` is T_1...
ishaus 20936 Express the predicate " ` ...
iscnrm 20937 The property of being comp...
t0sep 20938 Any two topologically indi...
t0dist 20939 Any two distinct points in...
t1sncld 20940 In a T_1 space, one-point ...
t1ficld 20941 In a T_1 space, finite set...
hausnei 20942 Neighborhood property of a...
t0top 20943 A T_0 space is a topologic...
t1top 20944 A T_1 space is a topologic...
haustop 20945 A Hausdorff space is a top...
isreg 20946 The predicate "is a regula...
regtop 20947 A regular space is a topol...
regsep 20948 In a regular space, every ...
isnrm 20949 The predicate "is a normal...
nrmtop 20950 A normal space is a topolo...
cnrmtop 20951 A completely normal space ...
iscnrm2 20952 The property of being comp...
ispnrm 20953 The property of being perf...
pnrmnrm 20954 A perfectly normal space i...
pnrmtop 20955 A perfectly normal space i...
pnrmcld 20956 A closed set in a perfectl...
pnrmopn 20957 An open set in a perfectly...
ist0-2 20958 The predicate "is a T_0 sp...
ist0-3 20959 The predicate "is a T_0 sp...
cnt0 20960 The preimage of a T_0 topo...
ist1-2 20961 An alternate characterizat...
t1t0 20962 A T_1 space is a T_0 space...
ist1-3 20963 A space is T_1 iff every p...
cnt1 20964 The preimage of a T_1 topo...
ishaus2 20965 Express the predicate " ` ...
haust1 20966 A Hausdorff space is a T_1...
hausnei2 20967 The Hausdorff condition st...
cnhaus 20968 The preimage of a Hausdorf...
nrmsep3 20969 In a normal space, given a...
nrmsep2 20970 In a normal space, any two...
nrmsep 20971 In a normal space, disjoin...
isnrm2 20972 An alternate characterizat...
isnrm3 20973 A topological space is nor...
cnrmi 20974 A subspace of a completely...
cnrmnrm 20975 A completely normal space ...
restcnrm 20976 A subspace of a completely...
resthauslem 20977 Lemma for ~ resthaus and s...
lpcls 20978 The limit points of the cl...
perfcls 20979 A subset of a perfect spac...
restt0 20980 A subspace of a T_0 topolo...
restt1 20981 A subspace of a T_1 topolo...
resthaus 20982 A subspace of a Hausdorff ...
t1sep2 20983 Any two points in a T_1 sp...
t1sep 20984 Any two distinct points in...
sncld 20985 A singleton is closed in a...
sshauslem 20986 Lemma for ~ sshaus and sim...
sst0 20987 A topology finer than a T_...
sst1 20988 A topology finer than a T_...
sshaus 20989 A topology finer than a Ha...
regsep2 20990 In a regular space, a clos...
isreg2 20991 A topological space is reg...
dnsconst 20992 If a continuous mapping to...
ordtt1 20993 The order topology is T_1 ...
lmmo 20994 A sequence in a Hausdorff ...
lmfun 20995 The convergence relation i...
dishaus 20996 A discrete topology is Hau...
ordthauslem 20997 Lemma for ~ ordthaus . (C...
ordthaus 20998 The order topology of a to...
iscmp 21001 The predicate "is a compac...
cmpcov 21002 An open cover of a compact...
cmpcov2 21003 Rewrite ~ cmpcov for the c...
cmpcovf 21004 Combine ~ cmpcov with ~ ac...
cncmp 21005 Compactness is respected b...
fincmp 21006 A finite topology is compa...
0cmp 21007 The singleton of the empty...
cmptop 21008 A compact topology is a to...
rncmp 21009 The image of a compact set...
imacmp 21010 The image of a compact set...
discmp 21011 A discrete topology is com...
cmpsublem 21012 Lemma for ~ cmpsub . (Con...
cmpsub 21013 Two equivalent ways of des...
tgcmp 21014 A topology generated by a ...
cmpcld 21015 A closed subset of a compa...
uncmp 21016 The union of two compact s...
fiuncmp 21017 A finite union of compact ...
sscmp 21018 A subset of a compact topo...
hauscmplem 21019 Lemma for ~ hauscmp . (Co...
hauscmp 21020 A compact subspace of a T2...
cmpfi 21021 If a topology is compact a...
cmpfii 21022 In a compact topology, a s...
bwth 21023 The glorious Bolzano-Weier...
iscon 21026 The predicate ` J ` is a c...
iscon2 21027 The predicate ` J ` is a c...
conclo 21028 The only nonempty clopen s...
conndisj 21029 If a topology is connected...
contop 21030 A connected topology is a ...
indiscon 21031 The indiscrete topology (o...
dfcon2 21032 An alternate definition of...
consuba 21033 Connectedness for a subspa...
connsub 21034 Two equivalent ways of say...
cnconn 21035 Connectedness is respected...
nconsubb 21036 Disconnectedness for a sub...
consubclo 21037 If a clopen set meets a co...
conima 21038 The image of a connected s...
concn 21039 A continuous function from...
iunconlem 21040 Lemma for ~ iuncon . (Con...
iuncon 21041 The indexed union of conne...
uncon 21042 The union of two connected...
clscon 21043 The closure of a connected...
concompid 21044 The connected component co...
concompcon 21045 The connected component co...
concompss 21046 The connected component co...
concompcld 21047 The connected component co...
concompclo 21048 The connected component co...
t1conperf 21049 A connected T_1 space is p...
is1stc 21054 The predicate "is a first-...
is1stc2 21055 An equivalent way of sayin...
1stctop 21056 A first-countable topology...
1stcclb 21057 A property of points in a ...
1stcfb 21058 For any point ` A ` in a f...
is2ndc 21059 The property of being seco...
2ndctop 21060 A second-countable topolog...
2ndci 21061 A countable basis generate...
2ndcsb 21062 Having a countable subbase...
2ndcredom 21063 A second-countable space h...
2ndc1stc 21064 A second-countable space i...
1stcrestlem 21065 Lemma for ~ 1stcrest . (C...
1stcrest 21066 A subspace of a first-coun...
2ndcrest 21067 A subspace of a second-cou...
2ndcctbss 21068 If a topology is second-co...
2ndcdisj 21069 Any disjoint family of ope...
2ndcdisj2 21070 Any disjoint collection of...
2ndcomap 21071 A surjective continuous op...
2ndcsep 21072 A second-countable topolog...
dis2ndc 21073 A discrete space is second...
1stcelcls 21074 A point belongs to the clo...
1stccnp 21075 A mapping is continuous at...
1stccn 21076 A mapping ` X --> Y ` , wh...
islly 21081 The property of being a lo...
isnlly 21082 The property of being an n...
llyeq 21083 Equality theorem for the `...
nllyeq 21084 Equality theorem for the `...
llytop 21085 A locally ` A ` space is a...
nllytop 21086 A locally ` A ` space is a...
llyi 21087 The property of a locally ...
nllyi 21088 The property of an n-local...
nlly2i 21089 Eliminate the neighborhood...
llynlly 21090 A locally ` A ` space is n...
llyssnlly 21091 A locally ` A ` space is n...
llyss 21092 The "locally" predicate re...
nllyss 21093 The "n-locally" predicate ...
subislly 21094 The property of a subspace...
restnlly 21095 If the property ` A ` pass...
restlly 21096 If the property ` A ` pass...
islly2 21097 An alternative expression ...
llyrest 21098 An open subspace of a loca...
nllyrest 21099 An open subspace of an n-l...
loclly 21100 If ` A ` is a local proper...
llyidm 21101 Idempotence of the "locall...
nllyidm 21102 Idempotence of the "n-loca...
toplly 21103 A topology is locally a to...
topnlly 21104 A topology is n-locally a ...
hauslly 21105 A Hausdorff space is local...
hausnlly 21106 A Hausdorff space is n-loc...
hausllycmp 21107 A compact Hausdorff space ...
cldllycmp 21108 A closed subspace of a loc...
lly1stc 21109 First-countability is a lo...
dislly 21110 The discrete space ` ~P X ...
disllycmp 21111 A discrete space is locall...
dis1stc 21112 A discrete space is first-...
hausmapdom 21113 If ` X ` is a first-counta...
hauspwdom 21114 Simplify the cardinal ` A ...
refrel 21121 Refinement is a relation. ...
isref 21122 The property of being a re...
refbas 21123 A refinement covers the sa...
refssex 21124 Every set in a refinement ...
ssref 21125 A subcover is a refinement...
refref 21126 Reflexivity of refinement....
reftr 21127 Refinement is transitive. ...
refun0 21128 Adding the empty set prese...
isptfin 21129 The statement "is a point-...
islocfin 21130 The statement "is a locall...
finptfin 21131 A finite cover is a point-...
ptfinfin 21132 A point covered by a point...
finlocfin 21133 A finite cover of a topolo...
locfintop 21134 A locally finite cover cov...
locfinbas 21135 A locally finite cover mus...
locfinnei 21136 A point covered by a local...
lfinpfin 21137 A locally finite cover is ...
lfinun 21138 Adding a finite set preser...
locfincmp 21139 For a compact space, the l...
unisngl 21140 Taking the union of the se...
dissnref 21141 The set of singletons is a...
dissnlocfin 21142 The set of singletons is l...
locfindis 21143 The locally finite covers ...
locfincf 21144 A locally finite cover in ...
comppfsc 21145 A space where every open c...
kgenval 21148 Value of the compact gener...
elkgen 21149 Value of the compact gener...
kgeni 21150 Property of the open sets ...
kgentopon 21151 The compact generator gene...
kgenuni 21152 The base set of the compac...
kgenftop 21153 The compact generator gene...
kgenf 21154 The compact generator is a...
kgentop 21155 A compactly generated spac...
kgenss 21156 The compact generator gene...
kgenhaus 21157 The compact generator gene...
kgencmp 21158 The compact generator topo...
kgencmp2 21159 The compact generator topo...
kgenidm 21160 The compact generator is i...
iskgen2 21161 A space is compactly gener...
iskgen3 21162 Derive the usual definitio...
llycmpkgen2 21163 A locally compact space is...
cmpkgen 21164 A compact space is compact...
llycmpkgen 21165 A locally compact space is...
1stckgenlem 21166 The one-point compactifica...
1stckgen 21167 A first-countable space is...
kgen2ss 21168 The compact generator pres...
kgencn 21169 A function from a compactl...
kgencn2 21170 A function ` F : J --> K `...
kgencn3 21171 The set of continuous func...
kgen2cn 21172 A continuous function is a...
txval 21177 Value of the binary topolo...
txuni2 21178 The underlying set of the ...
txbasex 21179 The basis for the product ...
txbas 21180 The set of Cartesian produ...
eltx 21181 A set in a product is open...
txtop 21182 The product of two topolog...
ptval 21183 The value of the product t...
ptpjpre1 21184 The preimage of a projecti...
elpt 21185 Elementhood in the bases o...
elptr 21186 A basic open set in the pr...
elptr2 21187 A basic open set in the pr...
ptbasid 21188 The base set of the produc...
ptuni2 21189 The base set for the produ...
ptbasin 21190 The basis for a product to...
ptbasin2 21191 The basis for a product to...
ptbas 21192 The basis for a product to...
ptpjpre2 21193 The basis for a product to...
ptbasfi 21194 The basis for the product ...
pttop 21195 The product topology is a ...
ptopn 21196 A basic open set in the pr...
ptopn2 21197 A sub-basic open set in th...
xkotf 21198 Functionality of function ...
xkobval 21199 Alternative expression for...
xkoval 21200 Value of the compact-open ...
xkotop 21201 The compact-open topology ...
xkoopn 21202 A basic open set of the co...
txtopi 21203 The product of two topolog...
txtopon 21204 The underlying set of the ...
txuni 21205 The underlying set of the ...
txunii 21206 The underlying set of the ...
ptuni 21207 The base set for the produ...
ptunimpt 21208 Base set of a product topo...
pttopon 21209 The base set for the produ...
pttoponconst 21210 The base set for a product...
ptuniconst 21211 The base set for a product...
xkouni 21212 The base set of the compac...
xkotopon 21213 The base set of the compac...
ptval2 21214 The value of the product t...
txopn 21215 The product of two open se...
txcld 21216 The product of two closed ...
txcls 21217 Closure of a rectangle in ...
txss12 21218 Subset property of the top...
txbasval 21219 It is sufficient to consid...
neitx 21220 The Cartesian product of t...
txcnpi 21221 Continuity of a two-argume...
tx1cn 21222 Continuity of the first pr...
tx2cn 21223 Continuity of the second p...
ptpjcn 21224 Continuity of a projection...
ptpjopn 21225 The projection map is an o...
ptcld 21226 A closed box in the produc...
ptcldmpt 21227 A closed box in the produc...
ptclsg 21228 The closure of a box in th...
ptcls 21229 The closure of a box in th...
dfac14lem 21230 Lemma for ~ dfac14 . By e...
dfac14 21231 Theorem ~ ptcls is an equi...
xkoccn 21232 The "constant function" fu...
txcnp 21233 If two functions are conti...
ptcnplem 21234 Lemma for ~ ptcnp . (Cont...
ptcnp 21235 If every projection of a f...
upxp 21236 Universal property of the ...
txcnmpt 21237 A map into the product of ...
uptx 21238 Universal property of the ...
txcn 21239 A map into the product of ...
ptcn 21240 If every projection of a f...
prdstopn 21241 Topology of a structure pr...
prdstps 21242 A structure product of top...
pwstps 21243 A structure product of top...
txrest 21244 The subspace of a topologi...
txdis 21245 The topological product of...
txindislem 21246 Lemma for ~ txindis . (Co...
txindis 21247 The topological product of...
txdis1cn 21248 A function is jointly cont...
txlly 21249 If the property ` A ` is p...
txnlly 21250 If the property ` A ` is p...
pthaus 21251 The product of a collectio...
ptrescn 21252 Restriction is a continuou...
txtube 21253 The "tube lemma". If ` X ...
txcmplem1 21254 Lemma for ~ txcmp . (Cont...
txcmplem2 21255 Lemma for ~ txcmp . (Cont...
txcmp 21256 The topological product of...
txcmpb 21257 The topological product of...
hausdiag 21258 A topology is Hausdorff if...
hauseqlcld 21259 In a Hausdorff topology, t...
txhaus 21260 The topological product of...
txlm 21261 Two sequences converge iff...
lmcn2 21262 The image of a convergent ...
tx1stc 21263 The topological product of...
tx2ndc 21264 The topological product of...
txkgen 21265 The topological product of...
xkohaus 21266 If the codomain space is H...
xkoptsub 21267 The compact-open topology ...
xkopt 21268 The compact-open topology ...
xkopjcn 21269 Continuity of a projection...
xkoco1cn 21270 If ` F ` is a continuous f...
xkoco2cn 21271 If ` F ` is a continuous f...
xkococnlem 21272 Continuity of the composit...
xkococn 21273 Continuity of the composit...
cnmptid 21274 The identity function is c...
cnmptc 21275 A constant function is con...
cnmpt11 21276 The composition of continu...
cnmpt11f 21277 The composition of continu...
cnmpt1t 21278 The composition of continu...
cnmpt12f 21279 The composition of continu...
cnmpt12 21280 The composition of continu...
cnmpt1st 21281 The projection onto the fi...
cnmpt2nd 21282 The projection onto the se...
cnmpt2c 21283 A constant function is con...
cnmpt21 21284 The composition of continu...
cnmpt21f 21285 The composition of continu...
cnmpt2t 21286 The composition of continu...
cnmpt22 21287 The composition of continu...
cnmpt22f 21288 The composition of continu...
cnmpt1res 21289 The restriction of a conti...
cnmpt2res 21290 The restriction of a conti...
cnmptcom 21291 The argument converse of a...
cnmptkc 21292 The curried first projecti...
cnmptkp 21293 The evaluation of the inne...
cnmptk1 21294 The composition of a curri...
cnmpt1k 21295 The composition of a one-a...
cnmptkk 21296 The composition of two cur...
xkofvcn 21297 Joint continuity of the fu...
cnmptk1p 21298 The evaluation of a currie...
cnmptk2 21299 The uncurrying of a currie...
xkoinjcn 21300 Continuity of "injection",...
cnmpt2k 21301 The currying of a two-argu...
txcon 21302 The topological product of...
imasnopn 21303 If a relation graph is ope...
imasncld 21304 If a relation graph is clo...
imasncls 21305 If a relation graph is clo...
qtopval 21308 Value of the quotient topo...
qtopval2 21309 Value of the quotient topo...
elqtop 21310 Value of the quotient topo...
qtopres 21311 The quotient topology is u...
qtoptop2 21312 The quotient topology is a...
qtoptop 21313 The quotient topology is a...
elqtop2 21314 Value of the quotient topo...
qtopuni 21315 The base set of the quotie...
elqtop3 21316 Value of the quotient topo...
qtoptopon 21317 The base set of the quotie...
qtopid 21318 A quotient map is a contin...
idqtop 21319 The quotient topology indu...
qtopcmplem 21320 Lemma for ~ qtopcmp and ~ ...
qtopcmp 21321 A quotient of a compact sp...
qtopcon 21322 A quotient of a connected ...
qtopkgen 21323 A quotient of a compactly ...
basqtop 21324 An injection maps bases to...
tgqtop 21325 An injection maps generate...
qtopcld 21326 The property of being a cl...
qtopcn 21327 Universal property of a qu...
qtopss 21328 A surjective continuous fu...
qtopeu 21329 Universal property of the ...
qtoprest 21330 If ` A ` is a saturated op...
qtopomap 21331 If ` F ` is a surjective c...
qtopcmap 21332 If ` F ` is a surjective c...
imastopn 21333 The topology of an image s...
imastps 21334 The image of a topological...
qustps 21335 A quotient structure is a ...
kqfval 21336 Value of the function appe...
kqfeq 21337 Two points in the Kolmogor...
kqffn 21338 The topological indistingu...
kqval 21339 Value of the quotient topo...
kqtopon 21340 The Kolmogorov quotient is...
kqid 21341 The topological indistingu...
ist0-4 21342 The topological indistingu...
kqfvima 21343 When the image set is open...
kqsat 21344 Any open set is saturated ...
kqdisj 21345 A version of ~ imain for t...
kqcldsat 21346 Any closed set is saturate...
kqopn 21347 The topological indistingu...
kqcld 21348 The topological indistingu...
kqt0lem 21349 Lemma for ~ kqt0 . (Contr...
isr0 21350 The property " ` J ` is an...
r0cld 21351 The analogue of the T_1 ax...
regr1lem 21352 Lemma for ~ regr1 . (Cont...
regr1lem2 21353 A Kolmogorov quotient of a...
kqreglem1 21354 A Kolmogorov quotient of a...
kqreglem2 21355 If the Kolmogorov quotient...
kqnrmlem1 21356 A Kolmogorov quotient of a...
kqnrmlem2 21357 If the Kolmogorov quotient...
kqtop 21358 The Kolmogorov quotient is...
kqt0 21359 The Kolmogorov quotient is...
kqf 21360 The Kolmogorov quotient is...
r0sep 21361 The separation property of...
nrmr0reg 21362 A normal R_0 space is also...
regr1 21363 A regular space is R_1, wh...
kqreg 21364 The Kolmogorov quotient of...
kqnrm 21365 The Kolmogorov quotient of...
hmeofn 21370 The set of homeomorphisms ...
hmeofval 21371 The set of all the homeomo...
ishmeo 21372 The predicate F is a homeo...
hmeocn 21373 A homeomorphism is continu...
hmeocnvcn 21374 The converse of a homeomor...
hmeocnv 21375 The converse of a homeomor...
hmeof1o2 21376 A homeomorphism is a 1-1-o...
hmeof1o 21377 A homeomorphism is a 1-1-o...
hmeoima 21378 The image of an open set b...
hmeoopn 21379 Homeomorphisms preserve op...
hmeocld 21380 Homeomorphisms preserve cl...
hmeocls 21381 Homeomorphisms preserve cl...
hmeontr 21382 Homeomorphisms preserve in...
hmeoimaf1o 21383 The function mapping open ...
hmeores 21384 The restriction of a homeo...
hmeoco 21385 The composite of two homeo...
idhmeo 21386 The identity function is a...
hmeocnvb 21387 The converse of a homeomor...
hmeoqtop 21388 A homeomorphism is a quoti...
hmph 21389 Express the predicate ` J ...
hmphi 21390 If there is a homeomorphis...
hmphtop 21391 Reverse closure for the ho...
hmphtop1 21392 The relation "being homeom...
hmphtop2 21393 The relation "being homeom...
hmphref 21394 "Is homeomorphic to" is re...
hmphsym 21395 "Is homeomorphic to" is sy...
hmphtr 21396 "Is homeomorphic to" is tr...
hmpher 21397 "Is homeomorphic to" is an...
hmphen 21398 Homeomorphisms preserve th...
hmphsymb 21399 "Is homeomorphic to" is sy...
haushmphlem 21400 Lemma for ~ haushmph and s...
cmphmph 21401 Compactness is a topologic...
conhmph 21402 Connectedness is a topolog...
t0hmph 21403 T_0 is a topological prope...
t1hmph 21404 T_1 is a topological prope...
haushmph 21405 Hausdorff-ness is a topolo...
reghmph 21406 Regularity is a topologica...
nrmhmph 21407 Normality is a topological...
hmph0 21408 A topology homeomorphic to...
hmphdis 21409 Homeomorphisms preserve to...
hmphindis 21410 Homeomorphisms preserve to...
indishmph 21411 Equinumerous sets equipped...
hmphen2 21412 Homeomorphisms preserve th...
cmphaushmeo 21413 A continuous bijection fro...
ordthmeolem 21414 Lemma for ~ ordthmeo . (C...
ordthmeo 21415 An order isomorphism is a ...
txhmeo 21416 Lift a pair of homeomorphi...
txswaphmeolem 21417 Show inverse for the "swap...
txswaphmeo 21418 There is a homeomorphism f...
pt1hmeo 21419 The canonical homeomorphis...
ptuncnv 21420 Exhibit the converse funct...
ptunhmeo 21421 Define a homeomorphism fro...
xpstopnlem1 21422 The function ` F ` used in...
xpstps 21423 A binary product of topolo...
xpstopnlem2 21424 Lemma for ~ xpstopn . (Co...
xpstopn 21425 The topology on a binary p...
ptcmpfi 21426 A topological product of f...
xkocnv 21427 The inverse of the "curryi...
xkohmeo 21428 The Exponential Law for to...
qtopf1 21429 If a quotient map is injec...
qtophmeo 21430 If two functions on a base...
t0kq 21431 A topological space is T_0...
kqhmph 21432 A topological space is T_0...
ist1-5lem 21433 Lemma for ~ ist1-5 and sim...
t1r0 21434 A T_1 space is R_0. That ...
ist1-5 21435 A topological space is T_1...
ishaus3 21436 A topological space is Hau...
nrmreg 21437 A normal T_1 space is regu...
reghaus 21438 A regular T_0 space is Hau...
nrmhaus 21439 A T_1 normal space is Haus...
elmptrab 21440 Membership in a one-parame...
elmptrab2OLD 21441 Obsolete version of ~ elmp...
elmptrab2 21442 Membership in a one-parame...
isfbas 21443 The predicate " ` F ` is a...
fbasne0 21444 There are no empty filter ...
0nelfb 21445 No filter base contains th...
fbsspw 21446 A filter base on a set is ...
fbelss 21447 An element of the filter b...
fbdmn0 21448 The domain of a filter bas...
isfbas2 21449 The predicate " ` F ` is a...
fbasssin 21450 A filter base contains sub...
fbssfi 21451 A filter base contains sub...
fbssint 21452 A filter base contains sub...
fbncp 21453 A filter base does not con...
fbun 21454 A necessary and sufficient...
fbfinnfr 21455 No filter base containing ...
opnfbas 21456 The collection of open sup...
trfbas2 21457 Conditions for the trace o...
trfbas 21458 Conditions for the trace o...
isfil 21461 The predicate "is a filter...
filfbas 21462 A filter is a filter base....
0nelfil 21463 The empty set doesn't belo...
fileln0 21464 An element of a filter is ...
filsspw 21465 A filter is a subset of th...
filelss 21466 An element of a filter is ...
filss 21467 A filter is closed under t...
filin 21468 A filter is closed under t...
filtop 21469 The underlying set belongs...
isfil2 21470 Derive the standard axioms...
isfildlem 21471 Lemma for ~ isfild . (Con...
isfild 21472 Sufficient condition for a...
filfi 21473 A filter is closed under t...
filinn0 21474 The intersection of two el...
filintn0 21475 A filter has the finite in...
filn0 21476 The empty set is not a fil...
infil 21477 The intersection of two fi...
snfil 21478 A singleton is a filter. ...
fbasweak 21479 A filter base on any set i...
snfbas 21480 Condition for a singleton ...
fsubbas 21481 A condition for a set to g...
fbasfip 21482 A filter base has the fini...
fbunfip 21483 A helpful lemma for showin...
fgval 21484 The filter generating clas...
elfg 21485 A condition for elements o...
ssfg 21486 A filter base is a subset ...
fgss 21487 A bigger base generates a ...
fgss2 21488 A condition for a filter t...
fgfil 21489 A filter generates itself....
elfilss 21490 An element belongs to a fi...
filfinnfr 21491 No filter containing a fin...
fgcl 21492 A generated filter is a fi...
fgabs 21493 Absorption law for filter ...
neifil 21494 The neighborhoods of a non...
filunibas 21495 Recover the base set from ...
filunirn 21496 Two ways to express a filt...
filcon 21497 A filter gives rise to a c...
fbasrn 21498 Given a filter on a domain...
filuni 21499 The union of a nonempty se...
trfil1 21500 Conditions for the trace o...
trfil2 21501 Conditions for the trace o...
trfil3 21502 Conditions for the trace o...
trfilss 21503 If ` A ` is a member of th...
fgtr 21504 If ` A ` is a member of th...
trfg 21505 The trace operation and th...
trnei 21506 The trace, over a set ` A ...
cfinfil 21507 Relative complements of th...
csdfil 21508 The set of all elements wh...
supfil 21509 The supersets of a nonempt...
zfbas 21510 The set of upper sets of i...
uzrest 21511 The restriction of the set...
uzfbas 21512 The set of upper sets of i...
isufil 21517 The property of being an u...
ufilfil 21518 An ultrafilter is a filter...
ufilss 21519 For any subset of the base...
ufilb 21520 The complement is in an ul...
ufilmax 21521 Any filter finer than an u...
isufil2 21522 The maximal property of an...
ufprim 21523 An ultrafilter is a prime ...
trufil 21524 Conditions for the trace o...
filssufilg 21525 A filter is contained in s...
filssufil 21526 A filter is contained in s...
isufl 21527 Define the (strong) ultraf...
ufli 21528 Property of a set that sat...
numufl 21529 Consequence of ~ filssufil...
fiufl 21530 A finite set satisfies the...
acufl 21531 The axiom of choice implie...
ssufl 21532 If ` Y ` is a subset of ` ...
ufileu 21533 If the ultrafilter contain...
filufint 21534 A filter is equal to the i...
uffix 21535 Lemma for ~ fixufil and ~ ...
fixufil 21536 The condition describing a...
uffixfr 21537 An ultrafilter is either f...
uffix2 21538 A classification of fixed ...
uffixsn 21539 The singleton of the gener...
ufildom1 21540 An ultrafilter is generate...
uffinfix 21541 An ultrafilter containing ...
cfinufil 21542 An ultrafilter is free iff...
ufinffr 21543 An infinite subset is cont...
ufilen 21544 Any infinite set has an ul...
ufildr 21545 An ultrafilter gives rise ...
fin1aufil 21546 There are no definable fre...
fmval 21557 Introduce a function that ...
fmfil 21558 A mapping filter is a filt...
fmf 21559 Pushing-forward via a func...
fmss 21560 A finer filter produces a ...
elfm 21561 An element of a mapping fi...
elfm2 21562 An element of a mapping fi...
fmfg 21563 The image filter of a filt...
elfm3 21564 An alternate formulation o...
imaelfm 21565 An image of a filter eleme...
rnelfmlem 21566 Lemma for ~ rnelfm . (Con...
rnelfm 21567 A condition for a filter t...
fmfnfmlem1 21568 Lemma for ~ fmfnfm . (Con...
fmfnfmlem2 21569 Lemma for ~ fmfnfm . (Con...
fmfnfmlem3 21570 Lemma for ~ fmfnfm . (Con...
fmfnfmlem4 21571 Lemma for ~ fmfnfm . (Con...
fmfnfm 21572 A filter finer than an ima...
fmufil 21573 An image filter of an ultr...
fmid 21574 The filter map applied to ...
fmco 21575 Composition of image filte...
ufldom 21576 The ultrafilter lemma prop...
flimval 21577 The set of limit points of...
elflim2 21578 The predicate "is a limit ...
flimtop 21579 Reverse closure for the li...
flimneiss 21580 A filter contains the neig...
flimnei 21581 A filter contains all of t...
flimelbas 21582 A limit point of a filter ...
flimfil 21583 Reverse closure for the li...
flimtopon 21584 Reverse closure for the li...
elflim 21585 The predicate "is a limit ...
flimss2 21586 A limit point of a filter ...
flimss1 21587 A limit point of a filter ...
neiflim 21588 A point is a limit point o...
flimopn 21589 The condition for being a ...
fbflim 21590 A condition for a filter t...
fbflim2 21591 A condition for a filter b...
flimclsi 21592 The convergent points of a...
hausflimlem 21593 If ` A ` and ` B ` are bot...
hausflimi 21594 One direction of ~ hausfli...
hausflim 21595 A condition for a topology...
flimcf 21596 Fineness is properly chara...
flimrest 21597 The set of limit points in...
flimclslem 21598 Lemma for ~ flimcls . (Co...
flimcls 21599 Closure in terms of filter...
flimsncls 21600 If ` A ` is a limit point ...
hauspwpwf1 21601 Lemma for ~ hauspwpwdom . ...
hauspwpwdom 21602 If ` X ` is a Hausdorff sp...
flffval 21603 Given a topology and a fil...
flfval 21604 Given a function from a fi...
flfnei 21605 The property of being a li...
flfneii 21606 A neighborhood of a limit ...
isflf 21607 The property of being a li...
flfelbas 21608 A limit point of a functio...
flffbas 21609 Limit points of a function...
flftg 21610 Limit points of a function...
hausflf 21611 If a function has its valu...
hausflf2 21612 If a convergent function h...
cnpflfi 21613 Forward direction of ~ cnp...
cnpflf2 21614 ` F ` is continuous at poi...
cnpflf 21615 Continuity of a function a...
cnflf 21616 A function is continuous i...
cnflf2 21617 A function is continuous i...
flfcnp 21618 A continuous function pres...
lmflf 21619 The topological limit rela...
txflf 21620 Two sequences converge in ...
flfcnp2 21621 The image of a convergent ...
fclsval 21622 The set of all cluster poi...
isfcls 21623 A cluster point of a filte...
fclsfil 21624 Reverse closure for the cl...
fclstop 21625 Reverse closure for the cl...
fclstopon 21626 Reverse closure for the cl...
isfcls2 21627 A cluster point of a filte...
fclsopn 21628 Write the cluster point co...
fclsopni 21629 An open neighborhood of a ...
fclselbas 21630 A cluster point is in the ...
fclsneii 21631 A neighborhood of a cluste...
fclssscls 21632 The set of cluster points ...
fclsnei 21633 Cluster points in terms of...
supnfcls 21634 The filter of supersets of...
fclsbas 21635 Cluster points in terms of...
fclsss1 21636 A finer topology has fewer...
fclsss2 21637 A finer filter has fewer c...
fclsrest 21638 The set of cluster points ...
fclscf 21639 Characterization of finene...
flimfcls 21640 A limit point is a cluster...
fclsfnflim 21641 A filter clusters at a poi...
flimfnfcls 21642 A filter converges to a po...
fclscmpi 21643 Forward direction of ~ fcl...
fclscmp 21644 A space is compact iff eve...
uffclsflim 21645 The cluster points of an u...
ufilcmp 21646 A space is compact iff eve...
fcfval 21647 The set of cluster points ...
isfcf 21648 The property of being a cl...
fcfnei 21649 The property of being a cl...
fcfelbas 21650 A cluster point of a funct...
fcfneii 21651 A neighborhood of a cluste...
flfssfcf 21652 A limit point of a functio...
uffcfflf 21653 If the domain filter is an...
cnpfcfi 21654 Lemma for ~ cnpfcf . If a...
cnpfcf 21655 A function ` F ` is contin...
cnfcf 21656 Continuity of a function i...
flfcntr 21657 A continuous function's va...
alexsublem 21658 Lemma for ~ alexsub . (Co...
alexsub 21659 The Alexander Subbase Theo...
alexsubb 21660 Biconditional form of the ...
alexsubALTlem1 21661 Lemma for ~ alexsubALT . ...
alexsubALTlem2 21662 Lemma for ~ alexsubALT . ...
alexsubALTlem3 21663 Lemma for ~ alexsubALT . ...
alexsubALTlem4 21664 Lemma for ~ alexsubALT . ...
alexsubALT 21665 The Alexander Subbase Theo...
ptcmplem1 21666 Lemma for ~ ptcmp . (Cont...
ptcmplem2 21667 Lemma for ~ ptcmp . (Cont...
ptcmplem3 21668 Lemma for ~ ptcmp . (Cont...
ptcmplem4 21669 Lemma for ~ ptcmp . (Cont...
ptcmplem5 21670 Lemma for ~ ptcmp . (Cont...
ptcmpg 21671 Tychonoff's theorem: The ...
ptcmp 21672 Tychonoff's theorem: The ...
cnextval 21675 The function applying cont...
cnextfval 21676 The continuous extension o...
cnextrel 21677 In the general case, a con...
cnextfun 21678 If the target space is Hau...
cnextfvval 21679 The value of the continuou...
cnextf 21680 Extension by continuity. ...
cnextcn 21681 Extension by continuity. ...
cnextfres1 21682 ` F ` and its extension by...
cnextfres 21683 ` F ` and its extension by...
istmd 21688 The predicate "is a topolo...
tmdmnd 21689 A topological monoid is a ...
tmdtps 21690 A topological monoid is a ...
istgp 21691 The predicate "is a topolo...
tgpgrp 21692 A topological group is a g...
tgptmd 21693 A topological group is a t...
tgptps 21694 A topological group is a t...
tmdtopon 21695 The topology of a topologi...
tgptopon 21696 The topology of a topologi...
tmdcn 21697 In a topological monoid, t...
tgpcn 21698 In a topological group, th...
tgpinv 21699 In a topological group, th...
grpinvhmeo 21700 The inverse function in a ...
cnmpt1plusg 21701 Continuity of the group su...
cnmpt2plusg 21702 Continuity of the group su...
tmdcn2 21703 Write out the definition o...
tgpsubcn 21704 In a topological group, th...
istgp2 21705 A group with a topology is...
tmdmulg 21706 In a topological monoid, t...
tgpmulg 21707 In a topological group, th...
tgpmulg2 21708 In a topological monoid, t...
tmdgsum 21709 In a topological monoid, t...
tmdgsum2 21710 For any neighborhood ` U `...
oppgtmd 21711 The opposite of a topologi...
oppgtgp 21712 The opposite of a topologi...
distgp 21713 Any group equipped with th...
indistgp 21714 Any group equipped with th...
symgtgp 21715 The symmetric group is a t...
tmdlactcn 21716 The left group action of e...
tgplacthmeo 21717 The left group action of e...
submtmd 21718 A submonoid of a topologic...
subgtgp 21719 A subgroup of a topologica...
subgntr 21720 A subgroup of a topologica...
opnsubg 21721 An open subgroup of a topo...
clssubg 21722 The closure of a subgroup ...
clsnsg 21723 The closure of a normal su...
cldsubg 21724 A subgroup of finite index...
tgpconcompeqg 21725 The connected component co...
tgpconcomp 21726 The identity component, th...
tgpconcompss 21727 The identity component is ...
ghmcnp 21728 A group homomorphism on to...
snclseqg 21729 The coset of the closure o...
tgphaus 21730 A topological group is Hau...
tgpt1 21731 Hausdorff and T1 are equiv...
tgpt0 21732 Hausdorff and T0 are equiv...
qustgpopn 21733 A quotient map in a topolo...
qustgplem 21734 Lemma for ~ qustgp . (Con...
qustgp 21735 The quotient of a topologi...
qustgphaus 21736 The quotient of a topologi...
prdstmdd 21737 The product of a family of...
prdstgpd 21738 The product of a family of...
tsmsfbas 21741 The collection of all sets...
tsmslem1 21742 The finite partial sums of...
tsmsval2 21743 Definition of the topologi...
tsmsval 21744 Definition of the topologi...
tsmspropd 21745 The group sum depends only...
eltsms 21746 The property of being a su...
tsmsi 21747 The property of being a su...
tsmscl 21748 A sum in a topological gro...
haustsms 21749 In a Hausdorff topological...
haustsms2 21750 In a Hausdorff topological...
tsmscls 21751 One half of ~ tgptsmscls ,...
tsmsgsum 21752 The convergent points of a...
tsmsid 21753 If a sum is finite, the us...
haustsmsid 21754 In a Hausdorff topological...
tsms0 21755 The sum of zero is zero. ...
tsmssubm 21756 Evaluate an infinite group...
tsmsres 21757 Extend an infinite group s...
tsmsf1o 21758 Re-index an infinite group...
tsmsmhm 21759 Apply a continuous group h...
tsmsadd 21760 The sum of two infinite gr...
tsmsinv 21761 Inverse of an infinite gro...
tsmssub 21762 The difference of two infi...
tgptsmscls 21763 A sum in a topological gro...
tgptsmscld 21764 The set of limit points to...
tsmssplit 21765 Split a topological group ...
tsmsxplem1 21766 Lemma for ~ tsmsxp . (Con...
tsmsxplem2 21767 Lemma for ~ tsmsxp . (Con...
tsmsxp 21768 Write a sum over a two-dim...
istrg 21777 Express the predicate " ` ...
trgtmd 21778 The multiplicative monoid ...
istdrg 21779 Express the predicate " ` ...
tdrgunit 21780 The unit group of a topolo...
trgtgp 21781 A topological ring is a to...
trgtmd2 21782 A topological ring is a to...
trgtps 21783 A topological ring is a to...
trgring 21784 A topological ring is a ri...
trggrp 21785 A topological ring is a gr...
tdrgtrg 21786 A topological division rin...
tdrgdrng 21787 A topological division rin...
tdrgring 21788 A topological division rin...
tdrgtmd 21789 A topological division rin...
tdrgtps 21790 A topological division rin...
istdrg2 21791 A topological-ring divisio...
mulrcn 21792 The functionalization of t...
invrcn2 21793 The multiplicative inverse...
invrcn 21794 The multiplicative inverse...
cnmpt1mulr 21795 Continuity of ring multipl...
cnmpt2mulr 21796 Continuity of ring multipl...
dvrcn 21797 The division function is c...
istlm 21798 The predicate " ` W ` is a...
vscacn 21799 The scalar multiplication ...
tlmtmd 21800 A topological module is a ...
tlmtps 21801 A topological module is a ...
tlmlmod 21802 A topological module is a ...
tlmtrg 21803 The scalar ring of a topol...
tlmscatps 21804 The scalar ring of a topol...
istvc 21805 A topological vector space...
tvctdrg 21806 The scalar field of a topo...
cnmpt1vsca 21807 Continuity of scalar multi...
cnmpt2vsca 21808 Continuity of scalar multi...
tlmtgp 21809 A topological vector space...
tvctlm 21810 A topological vector space...
tvclmod 21811 A topological vector space...
tvclvec 21812 A topological vector space...
ustfn 21815 The defined uniform struct...
ustval 21816 The class of all uniform s...
isust 21817 The predicate " ` U ` is a...
ustssxp 21818 Entourages are subsets of ...
ustssel 21819 A uniform structure is upw...
ustbasel 21820 The full set is always an ...
ustincl 21821 A uniform structure is clo...
ustdiag 21822 The diagonal set is includ...
ustinvel 21823 If ` V ` is an entourage, ...
ustexhalf 21824 For each entourage ` V ` t...
ustrel 21825 The elements of uniform st...
ustfilxp 21826 A uniform structure on a n...
ustne0 21827 A uniform structure cannot...
ustssco 21828 In an uniform structure, a...
ustexsym 21829 In an uniform structure, f...
ustex2sym 21830 In an uniform structure, f...
ustex3sym 21831 In an uniform structure, f...
ustref 21832 Any element of the base se...
ust0 21833 The unique uniform structu...
ustn0 21834 The empty set is not an un...
ustund 21835 If two intersecting sets `...
ustelimasn 21836 Any point ` A ` is near en...
ustneism 21837 For a point ` A ` in ` X `...
elrnust 21838 First direction for ~ ustb...
ustbas2 21839 Second direction for ~ ust...
ustuni 21840 The set union of a uniform...
ustbas 21841 Recover the base of an uni...
ustimasn 21842 Lemma for ~ ustuqtop . (C...
trust 21843 The trace of a uniform str...
utopval 21846 The topology induced by a ...
elutop 21847 Open sets in the topology ...
utoptop 21848 The topology induced by a ...
utopbas 21849 The base of the topology i...
utoptopon 21850 Topology induced by a unif...
restutop 21851 Restriction of a topology ...
restutopopn 21852 The restriction of the top...
ustuqtoplem 21853 Lemma for ~ ustuqtop . (C...
ustuqtop0 21854 Lemma for ~ ustuqtop . (C...
ustuqtop1 21855 Lemma for ~ ustuqtop , sim...
ustuqtop2 21856 Lemma for ~ ustuqtop . (C...
ustuqtop3 21857 Lemma for ~ ustuqtop , sim...
ustuqtop4 21858 Lemma for ~ ustuqtop . (C...
ustuqtop5 21859 Lemma for ~ ustuqtop . (C...
ustuqtop 21860 For a given uniform struct...
utopsnneiplem 21861 The neighborhoods of a poi...
utopsnneip 21862 The neighborhoods of a poi...
utopsnnei 21863 Images of singletons by en...
utop2nei 21864 For any symmetrical entour...
utop3cls 21865 Relation between a topolog...
utopreg 21866 All Hausdorff uniform spac...
ussval 21873 The uniform structure on u...
ussid 21874 In case the base of the ` ...
isusp 21875 The predicate ` W ` is a u...
ressunif 21876 ` UnifSet ` is unaffected ...
ressuss 21877 Value of the uniform struc...
ressust 21878 The uniform structure of a...
ressusp 21879 The restriction of a unifo...
tusval 21880 The value of the uniform s...
tuslem 21881 Lemma for ~ tusbas , ~ tus...
tusbas 21882 The base set of a construc...
tusunif 21883 The uniform structure of a...
tususs 21884 The uniform structure of a...
tustopn 21885 The topology induced by a ...
tususp 21886 A constructed uniform spac...
tustps 21887 A constructed uniform spac...
uspreg 21888 If a uniform space is Haus...
ucnval 21891 The set of all uniformly c...
isucn 21892 The predicate " ` F ` is a...
isucn2 21893 The predicate " ` F ` is a...
ucnimalem 21894 Reformulate the ` G ` func...
ucnima 21895 An equivalent statement of...
ucnprima 21896 The preimage by a uniforml...
iducn 21897 The identity is uniformly ...
cstucnd 21898 A constant function is uni...
ucncn 21899 Uniform continuity implies...
iscfilu 21902 The predicate " ` F ` is a...
cfilufbas 21903 A Cauchy filter base is a ...
cfiluexsm 21904 For a Cauchy filter base a...
fmucndlem 21905 Lemma for ~ fmucnd . (Con...
fmucnd 21906 The image of a Cauchy filt...
cfilufg 21907 The filter generated by a ...
trcfilu 21908 Condition for the trace of...
cfiluweak 21909 A Cauchy filter base is al...
neipcfilu 21910 In an uniform space, a nei...
iscusp 21913 The predicate " ` W ` is a...
cuspusp 21914 A complete uniform space i...
cuspcvg 21915 In a complete uniform spac...
iscusp2 21916 The predicate " ` W ` is a...
cnextucn 21917 Extension by continuity. ...
ucnextcn 21918 Extension by continuity. ...
ispsmet 21919 Express the predicate " ` ...
psmetdmdm 21920 Recover the base set from ...
psmetf 21921 The distance function of a...
psmetcl 21922 Closure of the distance fu...
psmet0 21923 The distance function of a...
psmettri2 21924 Triangle inequality for th...
psmetsym 21925 The distance function of a...
psmettri 21926 Triangle inequality for th...
psmetge0 21927 The distance function of a...
psmetxrge0 21928 The distance function of a...
psmetres2 21929 Restriction of a pseudomet...
psmetlecl 21930 Real closure of an extende...
distspace 21931 A structure ` G ` with a d...
ismet 21938 Express the predicate " ` ...
isxmet 21939 Express the predicate " ` ...
ismeti 21940 Properties that determine ...
isxmetd 21941 Properties that determine ...
isxmet2d 21942 It is safe to only require...
metflem 21943 Lemma for ~ metf and other...
xmetf 21944 Mapping of the distance fu...
metf 21945 Mapping of the distance fu...
xmetcl 21946 Closure of the distance fu...
metcl 21947 Closure of the distance fu...
ismet2 21948 An extended metric is a me...
metxmet 21949 A metric is an extended me...
xmetdmdm 21950 Recover the base set from ...
metdmdm 21951 Recover the base set from ...
xmetunirn 21952 Two ways to express an ext...
xmeteq0 21953 The value of an extended m...
meteq0 21954 The value of a metric is z...
xmettri2 21955 Triangle inequality for th...
mettri2 21956 Triangle inequality for th...
xmet0 21957 The distance function of a...
met0 21958 The distance function of a...
xmetge0 21959 The distance function of a...
metge0 21960 The distance function of a...
xmetlecl 21961 Real closure of an extende...
xmetsym 21962 The distance function of a...
xmetpsmet 21963 An extended metric is a ps...
xmettpos 21964 The distance function of a...
metsym 21965 The distance function of a...
xmettri 21966 Triangle inequality for th...
mettri 21967 Triangle inequality for th...
xmettri3 21968 Triangle inequality for th...
mettri3 21969 Triangle inequality for th...
xmetrtri 21970 One half of the reverse tr...
xmetrtri2 21971 The reverse triangle inequ...
metrtri 21972 Reverse triangle inequalit...
xmetgt0 21973 The distance function of a...
metgt0 21974 The distance function of a...
metn0 21975 A metric space is nonempty...
xmetres2 21976 Restriction of an extended...
metreslem 21977 Lemma for ~ metres . (Con...
metres2 21978 Lemma for ~ metres . (Con...
xmetres 21979 A restriction of an extend...
metres 21980 A restriction of a metric ...
0met 21981 The empty metric. (Contri...
prdsdsf 21982 The product metric is a fu...
prdsxmetlem 21983 The product metric is an e...
prdsxmet 21984 The product metric is an e...
prdsmet 21985 The product metric is a me...
ressprdsds 21986 Restriction of a product m...
resspwsds 21987 Restriction of a product m...
imasdsf1olem 21988 Lemma for ~ imasdsf1o . (...
imasdsf1o 21989 The distance function is t...
imasf1oxmet 21990 The image of an extended m...
imasf1omet 21991 The image of a metric is a...
xpsdsfn 21992 Closure of the metric in a...
xpsdsfn2 21993 Closure of the metric in a...
xpsxmetlem 21994 Lemma for ~ xpsxmet . (Co...
xpsxmet 21995 A product metric of extend...
xpsdsval 21996 Value of the metric in a b...
xpsmet 21997 The direct product of two ...
blfvalps 21998 The value of the ball func...
blfval 21999 The value of the ball func...
blvalps 22000 The ball around a point ` ...
blval 22001 The ball around a point ` ...
elblps 22002 Membership in a ball. (Co...
elbl 22003 Membership in a ball. (Co...
elbl2ps 22004 Membership in a ball. (Co...
elbl2 22005 Membership in a ball. (Co...
elbl3ps 22006 Membership in a ball, with...
elbl3 22007 Membership in a ball, with...
blcomps 22008 Commute the arguments to t...
blcom 22009 Commute the arguments to t...
xblpnfps 22010 The infinity ball in an ex...
xblpnf 22011 The infinity ball in an ex...
blpnf 22012 The infinity ball in a sta...
bldisj 22013 Two balls are disjoint if ...
blgt0 22014 A nonempty ball implies th...
bl2in 22015 Two balls are disjoint if ...
xblss2ps 22016 One ball is contained in a...
xblss2 22017 One ball is contained in a...
blss2ps 22018 One ball is contained in a...
blss2 22019 One ball is contained in a...
blhalf 22020 A ball of radius ` R / 2 `...
blfps 22021 Mapping of a ball. (Contr...
blf 22022 Mapping of a ball. (Contr...
blrnps 22023 Membership in the range of...
blrn 22024 Membership in the range of...
xblcntrps 22025 A ball contains its center...
xblcntr 22026 A ball contains its center...
blcntrps 22027 A ball contains its center...
blcntr 22028 A ball contains its center...
xbln0 22029 A ball is nonempty iff the...
bln0 22030 A ball is not empty. (Con...
blelrnps 22031 A ball belongs to the set ...
blelrn 22032 A ball belongs to the set ...
blssm 22033 A ball is a subset of the ...
unirnblps 22034 The union of the set of ba...
unirnbl 22035 The union of the set of ba...
blin 22036 The intersection of two ba...
ssblps 22037 The size of a ball increas...
ssbl 22038 The size of a ball increas...
blssps 22039 Any point ` P ` in a ball ...
blss 22040 Any point ` P ` in a ball ...
blssexps 22041 Two ways to express the ex...
blssex 22042 Two ways to express the ex...
ssblex 22043 A nested ball exists whose...
blin2 22044 Given any two balls and a ...
blbas 22045 The balls of a metric spac...
blres 22046 A ball in a restricted met...
xmeterval 22047 Value of the "finitely sep...
xmeter 22048 The "finitely separated" r...
xmetec 22049 The equivalence classes un...
blssec 22050 A ball centered at ` P ` i...
blpnfctr 22051 The infinity ball in an ex...
xmetresbl 22052 An extended metric restric...
mopnval 22053 An open set is a subset of...
mopntopon 22054 The set of open sets of a ...
mopntop 22055 The set of open sets of a ...
mopnuni 22056 The union of all open sets...
elmopn 22057 The defining property of a...
mopnfss 22058 The family of open sets of...
mopnm 22059 The base set of a metric s...
elmopn2 22060 A defining property of an ...
mopnss 22061 An open set of a metric sp...
isxms 22062 Express the predicate " ` ...
isxms2 22063 Express the predicate " ` ...
isms 22064 Express the predicate " ` ...
isms2 22065 Express the predicate " ` ...
xmstopn 22066 The topology component of ...
mstopn 22067 The topology component of ...
xmstps 22068 A metric space is a topolo...
msxms 22069 A metric space is a topolo...
mstps 22070 A metric space is a topolo...
xmsxmet 22071 The distance function, sui...
msmet 22072 The distance function, sui...
msf 22073 Mapping of the distance fu...
xmsxmet2 22074 The distance function, sui...
msmet2 22075 The distance function, sui...
mscl 22076 Closure of the distance fu...
xmscl 22077 Closure of the distance fu...
xmsge0 22078 The distance function in a...
xmseq0 22079 The distance function in a...
xmssym 22080 The distance function in a...
xmstri2 22081 Triangle inequality for th...
mstri2 22082 Triangle inequality for th...
xmstri 22083 Triangle inequality for th...
mstri 22084 Triangle inequality for th...
xmstri3 22085 Triangle inequality for th...
mstri3 22086 Triangle inequality for th...
msrtri 22087 Reverse triangle inequalit...
xmspropd 22088 Property deduction for an ...
mspropd 22089 Property deduction for a m...
setsmsbas 22090 The base set of a construc...
setsmsds 22091 The distance function of a...
setsmstset 22092 The topology of a construc...
setsmstopn 22093 The topology of a construc...
setsxms 22094 The constructed metric spa...
setsms 22095 The constructed metric spa...
tmsval 22096 For any metric there is an...
tmslem 22097 Lemma for ~ tmsbas , ~ tms...
tmsbas 22098 The base set of a construc...
tmsds 22099 The metric of a constructe...
tmstopn 22100 The topology of a construc...
tmsxms 22101 The constructed metric spa...
tmsms 22102 The constructed metric spa...
imasf1obl 22103 The image of a metric spac...
imasf1oxms 22104 The image of a metric spac...
imasf1oms 22105 The image of a metric spac...
prdsbl 22106 A ball in the product metr...
mopni 22107 An open set of a metric sp...
mopni2 22108 An open set of a metric sp...
mopni3 22109 An open set of a metric sp...
blssopn 22110 The balls of a metric spac...
unimopn 22111 The union of a collection ...
mopnin 22112 The intersection of two op...
mopn0 22113 The empty set is an open s...
rnblopn 22114 A ball of a metric space i...
blopn 22115 A ball of a metric space i...
neibl 22116 The neighborhoods around a...
blnei 22117 A ball around a point is a...
lpbl 22118 Every ball around a limit ...
blsscls2 22119 A smaller closed ball is c...
blcld 22120 A "closed ball" in a metri...
blcls 22121 The closure of an open bal...
blsscls 22122 If two concentric balls ha...
metss 22123 Two ways of saying that me...
metequiv 22124 Two ways of saying that tw...
metequiv2 22125 If there is a sequence of ...
metss2lem 22126 Lemma for ~ metss2 . (Con...
metss2 22127 If the metric ` D ` is "st...
comet 22128 The composition of an exte...
stdbdmetval 22129 Value of the standard boun...
stdbdxmet 22130 The standard bounded metri...
stdbdmet 22131 The standard bounded metri...
stdbdbl 22132 The standard bounded metri...
stdbdmopn 22133 The standard bounded metri...
mopnex 22134 The topology generated by ...
methaus 22135 The topology generated by ...
met1stc 22136 The topology generated by ...
met2ndci 22137 A separable metric space (...
met2ndc 22138 A metric space is second-c...
metrest 22139 Two alternate formulations...
ressxms 22140 The restriction of a metri...
ressms 22141 The restriction of a metri...
prdsmslem1 22142 Lemma for ~ prdsms . The ...
prdsxmslem1 22143 Lemma for ~ prdsms . The ...
prdsxmslem2 22144 Lemma for ~ prdsxms . The...
prdsxms 22145 The indexed product struct...
prdsms 22146 The indexed product struct...
pwsxms 22147 The product of a finite fa...
pwsms 22148 The product of a finite fa...
xpsxms 22149 A binary product of metric...
xpsms 22150 A binary product of metric...
tmsxps 22151 Express the product of two...
tmsxpsmopn 22152 Express the product of two...
tmsxpsval 22153 Value of the product of tw...
tmsxpsval2 22154 Value of the product of tw...
metcnp3 22155 Two ways to express that `...
metcnp 22156 Two ways to say a mapping ...
metcnp2 22157 Two ways to say a mapping ...
metcn 22158 Two ways to say a mapping ...
metcnpi 22159 Epsilon-delta property of ...
metcnpi2 22160 Epsilon-delta property of ...
metcnpi3 22161 Epsilon-delta property of ...
txmetcnp 22162 Continuity of a binary ope...
txmetcn 22163 Continuity of a binary ope...
metuval 22164 Value of the uniform struc...
metustel 22165 Define a filter base ` F `...
metustss 22166 Range of the elements of t...
metustrel 22167 Elements of the filter bas...
metustto 22168 Any two elements of the fi...
metustid 22169 The identity diagonal is i...
metustsym 22170 Elements of the filter bas...
metustexhalf 22171 For any element ` A ` of t...
metustfbas 22172 The filter base generated ...
metust 22173 The uniform structure gene...
cfilucfil 22174 Given a metric ` D ` and a...
metuust 22175 The uniform structure gene...
cfilucfil2 22176 Given a metric ` D ` and a...
blval2 22177 The ball around a point ` ...
elbl4 22178 Membership in a ball, alte...
metuel 22179 Elementhood in the uniform...
metuel2 22180 Elementhood in the uniform...
metustbl 22181 The "section" image of an ...
psmetutop 22182 The topology induced by a ...
xmetutop 22183 The topology induced by a ...
xmsusp 22184 If the uniform set of a me...
restmetu 22185 The uniform structure gene...
metucn 22186 Uniform continuity in metr...
dscmet 22187 The discrete metric on any...
dscopn 22188 The discrete metric genera...
nrmmetd 22189 Show that a group norm gen...
abvmet 22190 An absolute value ` F ` ge...
nmfval 22203 The value of the norm func...
nmval 22204 The value of the norm func...
nmfval2 22205 The value of the norm func...
nmval2 22206 The value of the norm func...
nmf2 22207 The norm is a function fro...
nmpropd 22208 Weak property deduction fo...
nmpropd2 22209 Strong property deduction ...
isngp 22210 The property of being a no...
isngp2 22211 The property of being a no...
isngp3 22212 The property of being a no...
ngpgrp 22213 A normed group is a group....
ngpms 22214 A normed group is a metric...
ngpxms 22215 A normed group is a metric...
ngptps 22216 A normed group is a topolo...
ngpmet 22217 The (induced) metric of a ...
ngpds 22218 Value of the distance func...
ngpdsr 22219 Value of the distance func...
ngpds2 22220 Write the distance between...
ngpds2r 22221 Write the distance between...
ngpds3 22222 Write the distance between...
ngpds3r 22223 Write the distance between...
ngprcan 22224 Cancel right addition insi...
ngplcan 22225 Cancel left addition insid...
isngp4 22226 Express the property of be...
ngpinvds 22227 Two elements are the same ...
ngpsubcan 22228 Cancel right subtraction i...
nmf 22229 The norm on a normed group...
nmcl 22230 The norm of a normed group...
nmge0 22231 The norm of a normed group...
nmeq0 22232 The identity is the only e...
nmne0 22233 The norm of a nonzero elem...
nmrpcl 22234 The norm of a nonzero elem...
nminv 22235 The norm of a negated elem...
nmmtri 22236 The triangle inequality fo...
nmsub 22237 The norm of the difference...
nmrtri 22238 Reverse triangle inequalit...
nm2dif 22239 Inequality for the differe...
nmtri 22240 The triangle inequality fo...
nmtri2 22241 Triangle inequality for th...
ngpi 22242 The properties of a normed...
nm0 22243 Norm of the identity eleme...
nmgt0 22244 The norm of a nonzero elem...
sgrim 22245 The induced metric on a su...
sgrimval 22246 The induced metric on a su...
subgnm 22247 The norm in a subgroup. (...
subgnm2 22248 A substructure assigns the...
subgngp 22249 A normed group restricted ...
ngptgp 22250 A normed abelian group is ...
ngppropd 22251 Property deduction for a n...
reldmtng 22252 The function ` toNrmGrp ` ...
tngval 22253 Value of the function whic...
tnglem 22254 Lemma for ~ tngbas and sim...
tngbas 22255 The base set of a structur...
tngplusg 22256 The group addition of a st...
tng0 22257 The group identity of a st...
tngmulr 22258 The ring multiplication of...
tngsca 22259 The scalar ring of a struc...
tngvsca 22260 The scalar multiplication ...
tngip 22261 The inner product operatio...
tngds 22262 The metric function of a s...
tngtset 22263 The topology generated by ...
tngtopn 22264 The topology generated by ...
tngnm 22265 The topology generated by ...
tngngp2 22266 A norm turns a group into ...
tngngpd 22267 Derive the axioms for a no...
tngngp 22268 Derive the axioms for a no...
tnggrpr 22269 If a structure equipped wi...
tngngp3 22270 Alternate definition of a ...
nrmtngdist 22271 The augmentation of a norm...
nrmtngnrm 22272 The augmentation of a norm...
tngngpim 22273 The induced metric of a no...
isnrg 22274 A normed ring is a ring wi...
nrgabv 22275 The norm of a normed ring ...
nrgngp 22276 A normed ring is a normed ...
nrgring 22277 A normed ring is a ring. ...
nmmul 22278 The norm of a product in a...
nrgdsdi 22279 Distribute a distance calc...
nrgdsdir 22280 Distribute a distance calc...
nm1 22281 The norm of one in a nonze...
unitnmn0 22282 The norm of a unit is nonz...
nminvr 22283 The norm of an inverse in ...
nmdvr 22284 The norm of a division in ...
nrgdomn 22285 A nonzero normed ring is a...
nrgtgp 22286 A normed ring is a topolog...
subrgnrg 22287 A normed ring restricted t...
tngnrg 22288 Given any absolute value o...
isnlm 22289 A normed (left) module is ...
nmvs 22290 Defining property of a nor...
nlmngp 22291 A normed module is a norme...
nlmlmod 22292 A normed module is a left ...
nlmnrg 22293 The scalar component of a ...
nlmngp2 22294 The scalar component of a ...
nlmdsdi 22295 Distribute a distance calc...
nlmdsdir 22296 Distribute a distance calc...
nlmmul0or 22297 If a scalar product is zer...
sranlm 22298 The subring algebra over a...
nlmvscnlem2 22299 Lemma for ~ nlmvscn . Com...
nlmvscnlem1 22300 Lemma for ~ nlmvscn . (Co...
nlmvscn 22301 The scalar multiplication ...
rlmnlm 22302 The ring module over a nor...
rlmnm 22303 The norm function in the r...
nrgtrg 22304 A normed ring is a topolog...
nrginvrcnlem 22305 Lemma for ~ nrginvrcn . C...
nrginvrcn 22306 The ring inverse function ...
nrgtdrg 22307 A normed division ring is ...
nlmtlm 22308 A normed module is a topol...
isnvc 22309 A normed vector space is j...
nvcnlm 22310 A normed vector space is a...
nvclvec 22311 A normed vector space is a...
nvclmod 22312 A normed vector space is a...
isnvc2 22313 A normed vector space is j...
nvctvc 22314 A normed vector space is a...
lssnlm 22315 A subspace of a normed mod...
lssnvc 22316 A subspace of a normed vec...
rlmnvc 22317 The ring module over a nor...
ngpocelbl 22318 Membership of an off-cente...
nmoffn 22325 The function producing ope...
reldmnghm 22326 Lemma for normed group hom...
reldmnmhm 22327 Lemma for module homomorph...
nmofval 22328 Value of the operator norm...
nmoval 22329 Value of the operator norm...
nmogelb 22330 Property of the operator n...
nmolb 22331 Any upper bound on the val...
nmolb2d 22332 Any upper bound on the val...
nmof 22333 The operator norm is a fun...
nmocl 22334 The operator norm of an op...
nmoge0 22335 The operator norm of an op...
nghmfval 22336 A normed group homomorphis...
isnghm 22337 A normed group homomorphis...
isnghm2 22338 A normed group homomorphis...
isnghm3 22339 A normed group homomorphis...
bddnghm 22340 A bounded group homomorphi...
nghmcl 22341 A normed group homomorphis...
nmoi 22342 The operator norm achieves...
nmoix 22343 The operator norm is a bou...
nmoi2 22344 The operator norm is a bou...
nmoleub 22345 The operator norm, defined...
nghmrcl1 22346 Reverse closure for a norm...
nghmrcl2 22347 Reverse closure for a norm...
nghmghm 22348 A normed group homomorphis...
nmo0 22349 The operator norm of the z...
nmoeq0 22350 The operator norm is zero ...
nmoco 22351 An upper bound on the oper...
nghmco 22352 The composition of normed ...
nmotri 22353 Triangle inequality for th...
nghmplusg 22354 The sum of two bounded lin...
0nghm 22355 The zero operator is a nor...
nmoid 22356 The operator norm of the i...
idnghm 22357 The identity operator is a...
nmods 22358 Upper bound for the distan...
nghmcn 22359 A normed group homomorphis...
isnmhm 22360 A normed module homomorphi...
nmhmrcl1 22361 Reverse closure for a norm...
nmhmrcl2 22362 Reverse closure for a norm...
nmhmlmhm 22363 A normed module homomorphi...
nmhmnghm 22364 A normed module homomorphi...
nmhmghm 22365 A normed module homomorphi...
isnmhm2 22366 A normed module homomorphi...
nmhmcl 22367 A normed module homomorphi...
idnmhm 22368 The identity operator is a...
0nmhm 22369 The zero operator is a bou...
nmhmco 22370 The composition of bounded...
nmhmplusg 22371 The sum of two bounded lin...
qtopbaslem 22372 The set of open intervals ...
qtopbas 22373 The set of open intervals ...
retopbas 22374 A basis for the standard t...
retop 22375 The standard topology on t...
uniretop 22376 The underlying set of the ...
retopon 22377 The standard topology on t...
retps 22378 The standard topological s...
iooretop 22379 Open intervals are open se...
icccld 22380 Closed intervals are close...
icopnfcld 22381 Right-unbounded closed int...
iocmnfcld 22382 Left-unbounded closed inte...
qdensere 22383 ` QQ ` is dense in the sta...
cnmetdval 22384 Value of the distance func...
cnmet 22385 The absolute value metric ...
cnxmet 22386 The absolute value metric ...
cnbl0 22387 Two ways to write the open...
cnblcld 22388 Two ways to write the clos...
cnfldms 22389 The complex number field i...
cnfldxms 22390 The complex number field i...
cnfldtps 22391 The complex number field i...
cnfldnm 22392 The norm of the field of c...
cnngp 22393 The complex numbers form a...
cnnrg 22394 The complex numbers form a...
cnfldtopn 22395 The topology of the comple...
cnfldtopon 22396 The topology of the comple...
cnfldtop 22397 The topology of the comple...
cnfldhaus 22398 The topology of the comple...
zringnrg 22399 The ring of integers is a ...
remetdval 22400 Value of the distance func...
remet 22401 The absolute value metric ...
rexmet 22402 The absolute value metric ...
bl2ioo 22403 A ball in terms of an open...
ioo2bl 22404 An open interval of reals ...
ioo2blex 22405 An open interval of reals ...
blssioo 22406 The balls of the standard ...
tgioo 22407 The topology generated by ...
qdensere2 22408 ` QQ ` is dense in ` RR ` ...
blcvx 22409 An open ball in the comple...
rehaus 22410 The standard topology on t...
tgqioo 22411 The topology generated by ...
re2ndc 22412 The standard topology on t...
resubmet 22413 The subspace topology indu...
tgioo2 22414 The standard topology on t...
rerest 22415 The subspace topology indu...
tgioo3 22416 The standard topology on t...
xrtgioo 22417 The topology on the extend...
xrrest 22418 The subspace topology indu...
xrrest2 22419 The subspace topology indu...
xrsxmet 22420 The metric on the extended...
xrsdsre 22421 The metric on the extended...
xrsblre 22422 Any ball of the metric of ...
xrsmopn 22423 The metric on the extended...
zcld 22424 The integers are a closed ...
recld2 22425 The real numbers are a clo...
zcld2 22426 The integers are a closed ...
zdis 22427 The integers are a discret...
sszcld 22428 Every subset of the intege...
reperflem 22429 A subset of the real numbe...
reperf 22430 The real numbers are a per...
cnperf 22431 The complex numbers are a ...
iccntr 22432 The interior of a closed i...
icccmplem1 22433 Lemma for ~ icccmp . (Con...
icccmplem2 22434 Lemma for ~ icccmp . (Con...
icccmplem3 22435 Lemma for ~ icccmp . (Con...
icccmp 22436 A closed interval in ` RR ...
reconnlem1 22437 Lemma for ~ reconn . Conn...
reconnlem2 22438 Lemma for ~ reconn . (Con...
reconn 22439 A subset of the reals is c...
retopcon 22440 Corollary of ~ reconn . T...
iccconn 22441 A closed interval is conne...
opnreen 22442 Every nonempty open set is...
rectbntr0 22443 A countable subset of the ...
xrge0gsumle 22444 A finite sum in the nonneg...
xrge0tsms 22445 Any finite or infinite sum...
xrge0tsms2 22446 Any finite or infinite sum...
metdcnlem 22447 The metric function of a m...
xmetdcn2 22448 The metric function of an ...
xmetdcn 22449 The metric function of an ...
metdcn2 22450 The metric function of a m...
metdcn 22451 The metric function of a m...
msdcn 22452 The metric function of a m...
cnmpt1ds 22453 Continuity of the metric f...
cnmpt2ds 22454 Continuity of the metric f...
nmcn 22455 The norm of a normed group...
ngnmcncn 22456 The norm of a normed group...
abscn 22457 The absolute value functio...
metdsval 22458 Value of the "distance to ...
metdsf 22459 The distance from a point ...
metdsge 22460 The distance from the poin...
metds0 22461 If a point is in a set, it...
metdstri 22462 A generalization of the tr...
metdsle 22463 The distance from a point ...
metdsre 22464 The distance from a point ...
metdseq0 22465 The distance from a point ...
metdscnlem 22466 Lemma for ~ metdscn . (Co...
metdscn 22467 The function ` F ` which g...
metdscn2 22468 The function ` F ` which g...
metnrmlem1a 22469 Lemma for ~ metnrm . (Con...
metnrmlem1 22470 Lemma for ~ metnrm . (Con...
metnrmlem2 22471 Lemma for ~ metnrm . (Con...
metnrmlem3 22472 Lemma for ~ metnrm . (Con...
metnrm 22473 A metric space is normal. ...
metreg 22474 A metric space is regular....
addcnlem 22475 Lemma for ~ addcn , ~ subc...
addcn 22476 Complex number addition is...
subcn 22477 Complex number subtraction...
mulcn 22478 Complex number multiplicat...
divcn 22479 Complex number division is...
cnfldtgp 22480 The complex numbers form a...
fsumcn 22481 A finite sum of functions ...
fsum2cn 22482 Version of ~ fsumcn for tw...
expcn 22483 The power function on comp...
divccn 22484 Division by a nonzero cons...
sqcn 22485 The square function on com...
iitopon 22490 The unit interval is a top...
iitop 22491 The unit interval is a top...
iiuni 22492 The base set of the unit i...
dfii2 22493 Alternate definition of th...
dfii3 22494 Alternate definition of th...
dfii4 22495 Alternate definition of th...
dfii5 22496 The unit interval expresse...
iicmp 22497 The unit interval is compa...
iicon 22498 The unit interval is conne...
cncfval 22499 The value of the continuou...
elcncf 22500 Membership in the set of c...
elcncf2 22501 Version of ~ elcncf with a...
cncfrss 22502 Reverse closure of the con...
cncfrss2 22503 Reverse closure of the con...
cncff 22504 A continuous complex funct...
cncfi 22505 Defining property of a con...
elcncf1di 22506 Membership in the set of c...
elcncf1ii 22507 Membership in the set of c...
rescncf 22508 A continuous complex funct...
cncffvrn 22509 Change the codomain of a c...
cncfss 22510 The set of continuous func...
climcncf 22511 Image of a limit under a c...
abscncf 22512 Absolute value is continuo...
recncf 22513 Real part is continuous. ...
imcncf 22514 Imaginary part is continuo...
cjcncf 22515 Complex conjugate is conti...
mulc1cncf 22516 Multiplication by a consta...
divccncf 22517 Division by a constant is ...
cncfco 22518 The composition of two con...
cncfmet 22519 Relate complex function co...
cncfcn 22520 Relate complex function co...
cncfcn1 22521 Relate complex function co...
cncfmptc 22522 A constant function is a c...
cncfmptid 22523 The identity function is a...
cncfmpt1f 22524 Composition of continuous ...
cncfmpt2f 22525 Composition of continuous ...
cncfmpt2ss 22526 Composition of continuous ...
addccncf 22527 Adding a constant is a con...
cdivcncf 22528 Division with a constant n...
negcncf 22529 The negative function is c...
negfcncf 22530 The negative of a continuo...
abscncfALT 22531 Absolute value is continuo...
cncfcnvcn 22532 Rewrite ~ cmphaushmeo for ...
expcncf 22533 The power function on comp...
cnmptre 22534 Lemma for ~ iirevcn and re...
cnmpt2pc 22535 Piecewise definition of a ...
iirev 22536 Reverse the unit interval....
iirevcn 22537 The reversion function is ...
iihalf1 22538 Map the first half of ` II...
iihalf1cn 22539 The first half function is...
iihalf2 22540 Map the second half of ` I...
iihalf2cn 22541 The second half function i...
elii1 22542 Divide the unit interval i...
elii2 22543 Divide the unit interval i...
iimulcl 22544 The unit interval is close...
iimulcn 22545 Multiplication is a contin...
icoopnst 22546 A half-open interval start...
iocopnst 22547 A half-open interval endin...
icchmeo 22548 The natural bijection from...
icopnfcnv 22549 Define a bijection from ` ...
icopnfhmeo 22550 The defined bijection from...
iccpnfcnv 22551 Define a bijection from ` ...
iccpnfhmeo 22552 The defined bijection from...
xrhmeo 22553 The bijection from ` [ -u ...
xrhmph 22554 The extended reals are hom...
xrcmp 22555 The topology of the extend...
xrcon 22556 The topology of the extend...
icccvx 22557 A linear combination of tw...
oprpiece1res1 22558 Restriction to the first p...
oprpiece1res2 22559 Restriction to the second ...
cnrehmeo 22560 The canonical bijection fr...
cnheiborlem 22561 Lemma for ~ cnheibor . (C...
cnheibor 22562 Heine-Borel theorem for co...
cnllycmp 22563 The topology on the comple...
rellycmp 22564 The topology on the reals ...
bndth 22565 The Boundedness Theorem. ...
evth 22566 The Extreme Value Theorem....
evth2 22567 The Extreme Value Theorem,...
lebnumlem1 22568 Lemma for ~ lebnum . The ...
lebnumlem2 22569 Lemma for ~ lebnum . As a...
lebnumlem3 22570 Lemma for ~ lebnum . By t...
lebnum 22571 The Lebesgue number lemma,...
xlebnum 22572 Generalize ~ lebnum to ext...
lebnumii 22573 Specialize the Lebesgue nu...
ishtpy 22579 Membership in the class of...
htpycn 22580 A homotopy is a continuous...
htpyi 22581 A homotopy evaluated at it...
ishtpyd 22582 Deduction for membership i...
htpycom 22583 Given a homotopy from ` F ...
htpyid 22584 A homotopy from a function...
htpyco1 22585 Compose a homotopy with a ...
htpyco2 22586 Compose a homotopy with a ...
htpycc 22587 Concatenate two homotopies...
isphtpy 22588 Membership in the class of...
phtpyhtpy 22589 A path homotopy is a homot...
phtpycn 22590 A path homotopy is a conti...
phtpyi 22591 Membership in the class of...
phtpy01 22592 Two path-homotopic paths h...
isphtpyd 22593 Deduction for membership i...
isphtpy2d 22594 Deduction for membership i...
phtpycom 22595 Given a homotopy from ` F ...
phtpyid 22596 A homotopy from a path to ...
phtpyco2 22597 Compose a path homotopy wi...
phtpycc 22598 Concatenate two path homot...
phtpcrel 22600 The path homotopy relation...
isphtpc 22601 The relation "is path homo...
phtpcer 22602 Path homotopy is an equiva...
phtpcerOLD 22603 Obsolete proof of ~ phtpce...
phtpc01 22604 Path homotopic paths have ...
reparphti 22605 Lemma for ~ reparpht . (C...
reparpht 22606 Reparametrization lemma. ...
phtpcco2 22607 Compose a path homotopy wi...
pcofval 22618 The value of the path conc...
pcoval 22619 The concatenation of two p...
pcovalg 22620 Evaluate the concatenation...
pcoval1 22621 Evaluate the concatenation...
pco0 22622 The starting point of a pa...
pco1 22623 The ending point of a path...
pcoval2 22624 Evaluate the concatenation...
pcocn 22625 The concatenation of two p...
copco 22626 The composition of a conca...
pcohtpylem 22627 Lemma for ~ pcohtpy . (Co...
pcohtpy 22628 Homotopy invariance of pat...
pcoptcl 22629 A constant function is a p...
pcopt 22630 Concatenation with a point...
pcopt2 22631 Concatenation with a point...
pcoass 22632 Order of concatenation doe...
pcorevcl 22633 Closure for a reversed pat...
pcorevlem 22634 Lemma for ~ pcorev . Prov...
pcorev 22635 Concatenation with the rev...
pcorev2 22636 Concatenation with the rev...
pcophtb 22637 The path homotopy equivale...
om1val 22638 The definition of the loop...
om1bas 22639 The base set of the loop s...
om1elbas 22640 Elementhood in the base se...
om1addcl 22641 Closure of the group opera...
om1plusg 22642 The group operation (which...
om1tset 22643 The topology of the loop s...
om1opn 22644 The topology of the loop s...
pi1val 22645 The definition of the fund...
pi1bas 22646 The base set of the fundam...
pi1blem 22647 Lemma for ~ pi1buni . (Co...
pi1buni 22648 Another way to write the l...
pi1bas2 22649 The base set of the fundam...
pi1eluni 22650 Elementhood in the base se...
pi1bas3 22651 The base set of the fundam...
pi1cpbl 22652 The group operation, loop ...
elpi1 22653 The elements of the fundam...
elpi1i 22654 The elements of the fundam...
pi1addf 22655 The group operation of ` p...
pi1addval 22656 The concatenation of two p...
pi1grplem 22657 Lemma for ~ pi1grp . (Con...
pi1grp 22658 The fundamental group is a...
pi1id 22659 The identity element of th...
pi1inv 22660 An inverse in the fundamen...
pi1xfrf 22661 Functionality of the loop ...
pi1xfrval 22662 The value of the loop tran...
pi1xfr 22663 Given a path ` F ` and its...
pi1xfrcnvlem 22664 Given a path ` F ` between...
pi1xfrcnv 22665 Given a path ` F ` between...
pi1xfrgim 22666 The mapping ` G ` between ...
pi1cof 22667 Functionality of the loop ...
pi1coval 22668 The value of the loop tran...
pi1coghm 22669 The mapping ` G ` between ...
isclm 22672 A complex module is a left...
clmsca 22673 A complex module is a left...
clmsubrg 22674 A complex module is a left...
clmlmod 22675 A complex module is a left...
clmgrp 22676 A complex module is an add...
clmabl 22677 A complex module is an abe...
clmring 22678 The scalar ring of a compl...
clmfgrp 22679 The scalar ring of a compl...
clm0 22680 The zero of the scalar rin...
clm1 22681 The identity of the scalar...
clmadd 22682 The addition of the scalar...
clmmul 22683 The multiplication of the ...
clmcj 22684 The conjugation of the sca...
isclmi 22685 Reverse direction of ~ isc...
clmzss 22686 The scalar ring of a compl...
clmsscn 22687 The scalar ring of a compl...
clmsub 22688 Subtraction in the scalar ...
clmneg 22689 Negation in the scalar rin...
clmneg1 22690 Minus one is in the scalar...
clmabs 22691 Norm in the scalar ring of...
clmacl 22692 Closure of ring addition f...
clmmcl 22693 Closure of ring multiplica...
clmsubcl 22694 Closure of ring subtractio...
lmhmclm 22695 The domain of a linear ope...
clmvscl 22696 Closure of scalar product ...
clmvsass 22697 Associative law for scalar...
clmvscom 22698 Commutative law for the sc...
clmvsdir 22699 Distributive law for scala...
clmvsdi 22700 Distributive law for scala...
clmvs1 22701 Scalar product with ring u...
clmvs2 22702 A vector plus itself is tw...
clm0vs 22703 Zero times a vector is the...
clmopfne 22704 The (functionalized) opera...
isclmp 22705 The predicate "is a comple...
isclmi0 22706 Properties that determine ...
clmvneg1 22707 Minus 1 times a vector is ...
clmvsneg 22708 Multiplication of a vector...
clmmulg 22709 The group multiple functio...
clmsubdir 22710 Scalar multiplication dist...
clmpm1dir 22711 Subtractive distributive l...
clmnegneg 22712 Double negative of a vecto...
clmnegsubdi2 22713 Distribution of negative o...
clmsub4 22714 Rearrangement of 4 terms i...
clmvsrinv 22715 A vector minus itself. (C...
clmvslinv 22716 Minus a vector plus itself...
clmvsubval 22717 Value of vector subtractio...
clmvsubval2 22718 Value of vector subtractio...
clmvz 22719 Two ways to express the ne...
zlmclm 22720 The ` ZZ ` -module operati...
clmzlmvsca 22721 The scalar product of a co...
nmoleub2lem 22722 Lemma for ~ nmoleub2a and ...
nmoleub2lem3 22723 Lemma for ~ nmoleub2a and ...
nmoleub2lem2 22724 Lemma for ~ nmoleub2a and ...
nmoleub2a 22725 The operator norm is the s...
nmoleub2b 22726 The operator norm is the s...
nmoleub3 22727 The operator norm is the s...
nmhmcn 22728 A linear operator over a n...
cmodscexp 22729 The powers of ` _i ` belon...
cmodscmulexp 22730 The scalar product of a ve...
cvslvec 22733 A complex vector space is ...
cvsclm 22734 A complex vector space is ...
iscvs 22735 A complex vector space is ...
iscvsp 22736 The predicate "is a comple...
iscvsi 22737 Properties that determine ...
cvsi 22738 The properties of a comple...
cvsunit 22739 Unit group of the scalar r...
cvsdiv 22740 Division of the scalar rin...
cvsdivcl 22741 The scalar field of a comp...
cvsmuleqdivd 22742 An equality involving rati...
cvsdiveqd 22743 An equality involving rati...
cnlmodlem1 22744 Lemma 1 for ~ cnlmod . (C...
cnlmodlem2 22745 Lemma 2 for ~ cnlmod . (C...
cnlmodlem3 22746 Lemma 3 for ~ cnlmod . (C...
cnlmod4 22747 Lemma 4 for ~ cnlmod . (C...
cnlmod 22748 The set of complex numbers...
cnstrcvs 22749 The set of complex numbers...
cnrbas 22750 The set of complex numbers...
cnrlmod 22751 The set of complex numbers...
cnrlvec 22752 The set of complex numbers...
cncvs 22753 The set of complex numbers...
recvs 22754 The set of real numbers (a...
qcvs 22755 The set of rational number...
zclmncvs 22756 The set of integers (as a ...
isncvsngp 22757 The predicate "is a normed...
isncvsngpd 22758 Properties that determine ...
ncvsi 22759 The properties of a normed...
ncvsprp 22760 Proportionality property o...
ncvsge0 22761 The norm of a scalar produ...
ncvsm1 22762 The norm of the negative o...
ncvsdif 22763 The norm of the difference...
ncvspi 22764 The norm of a vector plus ...
ncvs1 22765 From any nonzero vector, c...
cnrnvc 22766 The set of complex numbers...
cnncvs 22767 The set of complex numbers...
cnnm 22768 The norm operation of the ...
ncvspds 22769 Value of the distance func...
cnindmet 22770 The metric induced on the ...
cnncvsaddassdemo 22771 Derive the associative law...
cnncvsmulassdemo 22772 Derive the associative law...
cnncvsabsnegdemo 22773 Derive the absolute value ...
iscph 22778 A complex pre-Hilbert spac...
cphphl 22779 A complex pre-Hilbert spac...
cphnlm 22780 A complex pre-Hilbert spac...
cphngp 22781 A complex pre-Hilbert spac...
cphlmod 22782 A complex pre-Hilbert spac...
cphlvec 22783 A complex pre-Hilbert spac...
cphnvc 22784 A complex pre-Hilbert spac...
cphsubrglem 22785 Lemma for ~ cphsubrg . (C...
cphreccllem 22786 Lemma for ~ cphreccl . (C...
cphsca 22787 A complex pre-Hilbert spac...
cphsubrg 22788 The scalar field of a comp...
cphreccl 22789 The scalar field of a comp...
cphdivcl 22790 The scalar field of a comp...
cphcjcl 22791 The scalar field of a comp...
cphsqrtcl 22792 The scalar field of a comp...
cphabscl 22793 The scalar field of a comp...
cphsqrtcl2 22794 The scalar field of a comp...
cphsqrtcl3 22795 If the scalar field contai...
cphqss 22796 The scalar field of a comp...
cphclm 22797 A complex pre-Hilbert spac...
cphnmvs 22798 Norm of a scalar product. ...
cphipcl 22799 An inner product is a memb...
cphnmfval 22800 The value of the norm in a...
cphnm 22801 The square of the norm is ...
nmsq 22802 The square of the norm is ...
cphnmf 22803 The norm of a vector is a ...
cphnmcl 22804 The norm of a vector is a ...
reipcl 22805 An inner product of an ele...
ipge0 22806 The inner product in a com...
cphipcj 22807 Conjugate of an inner prod...
cphipipcj 22808 An inner product times its...
cphorthcom 22809 Orthogonality (meaning inn...
cphip0l 22810 Inner product with a zero ...
cphip0r 22811 Inner product with a zero ...
cphipeq0 22812 The inner product of a vec...
cphdir 22813 Distributive law for inner...
cphdi 22814 Distributive law for inner...
cph2di 22815 Distributive law for inner...
cphsubdir 22816 Distributive law for inner...
cphsubdi 22817 Distributive law for inner...
cph2subdi 22818 Distributive law for inner...
cphass 22819 Associative law for inner ...
cphassr 22820 "Associative" law for seco...
cph2ass 22821 Move scalar multiplication...
cphassi 22822 Associative law for the fi...
cphassir 22823 "Associative" law for the ...
tchex 22824 Lemma for ~ tchbas and sim...
tchval 22825 Define a function to augme...
tchbas 22826 The base set of a pre-Hilb...
tchplusg 22827 The addition operation of ...
tchsub 22828 The subtraction operation ...
tchmulr 22829 The ring operation of a pr...
tchsca 22830 The scalar field of a pre-...
tchvsca 22831 The scalar multiplication ...
tchip 22832 The inner product of a pre...
tchtopn 22833 The topology of a pre-Hilb...
tchphl 22834 Augmentation of a pre-Hilb...
tchnmfval 22835 The norm of a pre-Hilbert ...
tchnmval 22836 The norm of a pre-Hilbert ...
cphtchnm 22837 The norm of a norm-augment...
tchds 22838 The distance of a pre-Hilb...
tchclm 22839 Lemma for ~ tchcph . (Con...
tchcphlem3 22840 Lemma for ~ tchcph : real ...
ipcau2 22841 The Cauchy-Schwarz inequal...
tchcphlem1 22842 Lemma for ~ tchcph : the t...
tchcphlem2 22843 Lemma for ~ tchcph : homog...
tchcph 22844 The standard definition of...
ipcau 22845 The Cauchy-Schwarz inequal...
nmparlem 22846 Lemma for ~ nmpar . (Cont...
nmpar 22847 A complex pre-Hilbert spac...
cphipval2 22848 Value of the inner product...
4cphipval2 22849 Four times the inner produ...
cphipval 22850 Value of the inner product...
ipcnlem2 22851 The inner product operatio...
ipcnlem1 22852 The inner product operatio...
ipcn 22853 The inner product operatio...
cnmpt1ip 22854 Continuity of inner produc...
cnmpt2ip 22855 Continuity of inner produc...
csscld 22856 A "closed subspace" in a c...
clsocv 22857 The orthogonal complement ...
lmmbr 22864 Express the binary relatio...
lmmbr2 22865 Express the binary relatio...
lmmbr3 22866 Express the binary relatio...
lmmcvg 22867 Convergence property of a ...
lmmbrf 22868 Express the binary relatio...
lmnn 22869 A condition that implies c...
cfilfval 22870 The set of Cauchy filters ...
iscfil 22871 The property of being a Ca...
iscfil2 22872 The property of being a Ca...
cfilfil 22873 A Cauchy filter is a filte...
cfili 22874 Property of a Cauchy filte...
cfil3i 22875 A Cauchy filter contains b...
cfilss 22876 A filter finer than a Cauc...
fgcfil 22877 The Cauchy filter conditio...
fmcfil 22878 The Cauchy filter conditio...
iscfil3 22879 A filter is Cauchy iff it ...
cfilfcls 22880 Similar to ultrafilters ( ...
caufval 22881 The set of Cauchy sequence...
iscau 22882 Express the property " ` F...
iscau2 22883 Express the property " ` F...
iscau3 22884 Express the Cauchy sequenc...
iscau4 22885 Express the property " ` F...
iscauf 22886 Express the property " ` F...
caun0 22887 A metric with a Cauchy seq...
caufpm 22888 Inclusion of a Cauchy sequ...
caucfil 22889 A Cauchy sequence predicat...
iscmet 22890 The property " ` D ` is a ...
cmetcvg 22891 The convergence of a Cauch...
cmetmet 22892 A complete metric space is...
cmetmeti 22893 A complete metric space is...
cmetcaulem 22894 Lemma for ~ cmetcau . (Co...
cmetcau 22895 The convergence of a Cauch...
iscmet3lem3 22896 Lemma for ~ iscmet3 . (Co...
iscmet3lem1 22897 Lemma for ~ iscmet3 . (Co...
iscmet3lem2 22898 Lemma for ~ iscmet3 . (Co...
iscmet3 22899 The property " ` D ` is a ...
iscmet2 22900 A metric ` D ` is complete...
cfilresi 22901 A Cauchy filter on a metri...
cfilres 22902 Cauchy filter on a metric ...
caussi 22903 Cauchy sequence on a metri...
causs 22904 Cauchy sequence on a metri...
equivcfil 22905 If the metric ` D ` is "st...
equivcau 22906 If the metric ` D ` is "st...
lmle 22907 If the distance from each ...
nglmle 22908 If the norm of each member...
lmclim 22909 Relate a limit on the metr...
lmclimf 22910 Relate a limit on the metr...
metelcls 22911 A point belongs to the clo...
metcld 22912 A subset of a metric space...
metcld2 22913 A subset of a metric space...
caubl 22914 Sufficient condition to en...
caublcls 22915 The convergent point of a ...
metcnp4 22916 Two ways to say a mapping ...
metcn4 22917 Two ways to say a mapping ...
iscmet3i 22918 Properties that determine ...
lmcau 22919 Every convergent sequence ...
flimcfil 22920 Every convergent filter in...
cmetss 22921 A subspace of a complete m...
equivcmet 22922 If two metrics are strongl...
relcmpcmet 22923 If ` D ` is a metric space...
cmpcmet 22924 A compact metric space is ...
cfilucfil3 22925 Given a metric ` D ` and a...
cfilucfil4 22926 Given a metric ` D ` and a...
cncmet 22927 The set of complex numbers...
recmet 22928 The real numbers are a com...
bcthlem1 22929 Lemma for ~ bcth . Substi...
bcthlem2 22930 Lemma for ~ bcth . The ba...
bcthlem3 22931 Lemma for ~ bcth . The li...
bcthlem4 22932 Lemma for ~ bcth . Given ...
bcthlem5 22933 Lemma for ~ bcth . The pr...
bcth 22934 Baire's Category Theorem. ...
bcth2 22935 Baire's Category Theorem, ...
bcth3 22936 Baire's Category Theorem, ...
isbn 22943 A Banach space is a normed...
bnsca 22944 The scalar field of a comp...
bnnvc 22945 A Banach space is a normed...
bnnlm 22946 A Banach space is a normed...
bnngp 22947 A Banach space is a normed...
bnlmod 22948 A Banach space is a left m...
bncms 22949 A Banach space is a comple...
iscms 22950 A complete metric space is...
cmscmet 22951 The induced metric on a co...
bncmet 22952 The induced metric on Bana...
cmsms 22953 A complete metric space is...
cmspropd 22954 Property deduction for a c...
cmsss 22955 The restriction of a compl...
lssbn 22956 A subspace of a Banach spa...
cmetcusp1 22957 If the uniform set of a co...
cmetcusp 22958 The uniform space generate...
cncms 22959 The field of complex numbe...
cnflduss 22960 The uniform structure of t...
cnfldcusp 22961 The field of complex numbe...
resscdrg 22962 The real numbers are a sub...
cncdrg 22963 The only complete subfield...
srabn 22964 The subring algebra over a...
rlmbn 22965 The ring module over a com...
ishl 22966 The predicate "is a comple...
hlbn 22967 Every complex Hilbert spac...
hlcph 22968 Every complex Hilbert spac...
hlphl 22969 Every complex Hilbert spac...
hlcms 22970 Every complex Hilbert spac...
hlprlem 22971 Lemma for ~ hlpr . (Contr...
hlress 22972 The scalar field of a comp...
hlpr 22973 The scalar field of a comp...
ishl2 22974 A Hilbert space is a compl...
retopn 22975 The topology of the real n...
recms 22976 The real numbers form a co...
reust 22977 The Uniform structure of t...
recusp 22978 The real numbers form a co...
rrxval 22983 Value of the generalized E...
rrxbase 22984 The base of the generalize...
rrxprds 22985 Expand the definition of t...
rrxip 22986 The inner product of the g...
rrxnm 22987 The norm of the generalize...
rrxcph 22988 Generalized Euclidean real...
rrxds 22989 The distance over generali...
csbren 22990 Cauchy-Schwarz-Bunjakovsky...
trirn 22991 Triangle inequality in R^n...
rrxf 22992 Euclidean vectors as funct...
rrxfsupp 22993 Euclidean vectors are of f...
rrxsuppss 22994 Support of Euclidean vecto...
rrxmvallem 22995 Support of the function us...
rrxmval 22996 The value of the Euclidean...
rrxmfval 22997 The value of the Euclidean...
rrxmetlem 22998 Lemma for ~ rrxmet . (Con...
rrxmet 22999 Euclidean space is a metri...
rrxdstprj1 23000 The distance between two p...
ehlval 23001 Value of the Euclidean spa...
ehlbase 23002 The base of the Euclidean ...
minveclem1 23003 Lemma for ~ minvec . The ...
minveclem4c 23004 Lemma for ~ minvec . The ...
minveclem2 23005 Lemma for ~ minvec . Any ...
minveclem3a 23006 Lemma for ~ minvec . ` D `...
minveclem3b 23007 Lemma for ~ minvec . The ...
minveclem3 23008 Lemma for ~ minvec . The ...
minveclem4a 23009 Lemma for ~ minvec . ` F `...
minveclem4b 23010 Lemma for ~ minvec . The ...
minveclem4 23011 Lemma for ~ minvec . The ...
minveclem5 23012 Lemma for ~ minvec . Disc...
minveclem6 23013 Lemma for ~ minvec . Any ...
minveclem7 23014 Lemma for ~ minvec . Sinc...
minvec 23015 Minimizing vector theorem,...
pjthlem1 23016 Lemma for ~ pjth . (Contr...
pjthlem2 23017 Lemma for ~ pjth . (Contr...
pjth 23018 Projection Theorem: Any H...
pjth2 23019 Projection Theorem with ab...
cldcss 23020 Corollary of the Projectio...
cldcss2 23021 Corollary of the Projectio...
hlhil 23022 Corollary of the Projectio...
mulcncf 23023 The multiplication of two ...
pmltpclem1 23024 Lemma for ~ pmltpc . (Con...
pmltpclem2 23025 Lemma for ~ pmltpc . (Con...
pmltpc 23026 Any function on the reals ...
ivthlem1 23027 Lemma for ~ ivth . The se...
ivthlem2 23028 Lemma for ~ ivth . Show t...
ivthlem3 23029 Lemma for ~ ivth , the int...
ivth 23030 The intermediate value the...
ivth2 23031 The intermediate value the...
ivthle 23032 The intermediate value the...
ivthle2 23033 The intermediate value the...
ivthicc 23034 The interval between any t...
evthicc 23035 Specialization of the Extr...
evthicc2 23036 Combine ~ ivthicc with ~ e...
cniccbdd 23037 A continuous function on a...
ovolfcl 23042 Closure for the interval e...
ovolfioo 23043 Unpack the interval coveri...
ovolficc 23044 Unpack the interval coveri...
ovolficcss 23045 Any (closed) interval cove...
ovolfsval 23046 The value of the interval ...
ovolfsf 23047 Closure for the interval l...
ovolsf 23048 Closure for the partial su...
ovolval 23049 The value of the outer mea...
elovolm 23050 Elementhood in the set ` M...
elovolmr 23051 Sufficient condition for e...
ovolmge0 23052 The set ` M ` is composed ...
ovolcl 23053 The volume of a set is an ...
ovollb 23054 The outer volume is a lowe...
ovolgelb 23055 The outer volume is the gr...
ovolge0 23056 The volume of a set is alw...
ovolf 23057 The domain and range of th...
ovollecl 23058 If an outer volume is boun...
ovolsslem 23059 Lemma for ~ ovolss . (Con...
ovolss 23060 The volume of a set is mon...
ovolsscl 23061 If a set is contained in a...
ovolssnul 23062 A subset of a nullset is n...
ovollb2lem 23063 Lemma for ~ ovollb2 . (Co...
ovollb2 23064 It is often more convenien...
ovolctb 23065 The volume of a denumerabl...
ovolq 23066 The rational numbers have ...
ovolctb2 23067 The volume of a countable ...
ovol0 23068 The empty set has 0 outer ...
ovolfi 23069 A finite set has 0 outer L...
ovolsn 23070 A singleton has 0 outer Le...
ovolunlem1a 23071 Lemma for ~ ovolun . (Con...
ovolunlem1 23072 Lemma for ~ ovolun . (Con...
ovolunlem2 23073 Lemma for ~ ovolun . (Con...
ovolun 23074 The Lebesgue outer measure...
ovolunnul 23075 Adding a nullset does not ...
ovolfiniun 23076 The Lebesgue outer measure...
ovoliunlem1 23077 Lemma for ~ ovoliun . (Co...
ovoliunlem2 23078 Lemma for ~ ovoliun . (Co...
ovoliunlem3 23079 Lemma for ~ ovoliun . (Co...
ovoliun 23080 The Lebesgue outer measure...
ovoliun2 23081 The Lebesgue outer measure...
ovoliunnul 23082 A countable union of nulls...
shft2rab 23083 If ` B ` is a shift of ` A...
ovolshftlem1 23084 Lemma for ~ ovolshft . (C...
ovolshftlem2 23085 Lemma for ~ ovolshft . (C...
ovolshft 23086 The Lebesgue outer measure...
sca2rab 23087 If ` B ` is a scale of ` A...
ovolscalem1 23088 Lemma for ~ ovolsca . (Co...
ovolscalem2 23089 Lemma for ~ ovolshft . (C...
ovolsca 23090 The Lebesgue outer measure...
ovolicc1 23091 The measure of a closed in...
ovolicc2lem1 23092 Lemma for ~ ovolicc2 . (C...
ovolicc2lem2 23093 Lemma for ~ ovolicc2 . (C...
ovolicc2lem3 23094 Lemma for ~ ovolicc2 . (C...
ovolicc2lem4 23095 Lemma for ~ ovolicc2 . (C...
ovolicc2lem5 23096 Lemma for ~ ovolicc2 . (C...
ovolicc2 23097 The measure of a closed in...
ovolicc 23098 The measure of a closed in...
ovolicopnf 23099 The measure of a right-unb...
ovolre 23100 The measure of the real nu...
ismbl 23101 The predicate " ` A ` is L...
ismbl2 23102 From ~ ovolun , it suffice...
volres 23103 A self-referencing abbrevi...
volf 23104 The domain and range of th...
mblvol 23105 The volume of a measurable...
mblss 23106 A measurable set is a subs...
mblsplit 23107 The defining property of m...
volss 23108 The Lebesgue measure is mo...
cmmbl 23109 The complement of a measur...
nulmbl 23110 A nullset is measurable. ...
nulmbl2 23111 A set of outer measure zer...
unmbl 23112 A union of measurable sets...
shftmbl 23113 A shift of a measurable se...
0mbl 23114 The empty set is measurabl...
rembl 23115 The set of all real number...
unidmvol 23116 The union of the Lebesgue ...
inmbl 23117 An intersection of measura...
difmbl 23118 A difference of measurable...
finiunmbl 23119 A finite union of measurab...
volun 23120 The Lebesgue measure funct...
volinun 23121 Addition of non-disjoint s...
volfiniun 23122 The volume of a disjoint f...
iundisj 23123 Rewrite a countable union ...
iundisj2 23124 A disjoint union is disjoi...
voliunlem1 23125 Lemma for ~ voliun . (Con...
voliunlem2 23126 Lemma for ~ voliun . (Con...
voliunlem3 23127 Lemma for ~ voliun . (Con...
iunmbl 23128 The measurable sets are cl...
voliun 23129 The Lebesgue measure funct...
volsuplem 23130 Lemma for ~ volsup . (Con...
volsup 23131 The volume of the limit of...
iunmbl2 23132 The measurable sets are cl...
ioombl1lem1 23133 Lemma for ~ ioombl1 . (Co...
ioombl1lem2 23134 Lemma for ~ ioombl1 . (Co...
ioombl1lem3 23135 Lemma for ~ ioombl1 . (Co...
ioombl1lem4 23136 Lemma for ~ ioombl1 . (Co...
ioombl1 23137 An open right-unbounded in...
icombl1 23138 A closed unbounded-above i...
icombl 23139 A closed-below, open-above...
ioombl 23140 An open real interval is m...
iccmbl 23141 A closed real interval is ...
iccvolcl 23142 A closed real interval has...
ovolioo 23143 The measure of an open int...
ioovolcl 23144 An open real interval has ...
ovolfs2 23145 Alternative expression for...
ioorcl2 23146 An open interval with fini...
ioorf 23147 Define a function from ope...
ioorval 23148 Define a function from ope...
ioorinv2 23149 The function ` F ` is an "...
ioorinv 23150 The function ` F ` is an "...
ioorcl 23151 The function ` F ` does no...
uniiccdif 23152 A union of closed interval...
uniioovol 23153 A disjoint union of open i...
uniiccvol 23154 An almost-disjoint union o...
uniioombllem1 23155 Lemma for ~ uniioombl . (...
uniioombllem2a 23156 Lemma for ~ uniioombl . (...
uniioombllem2 23157 Lemma for ~ uniioombl . (...
uniioombllem3a 23158 Lemma for ~ uniioombl . (...
uniioombllem3 23159 Lemma for ~ uniioombl . (...
uniioombllem4 23160 Lemma for ~ uniioombl . (...
uniioombllem5 23161 Lemma for ~ uniioombl . (...
uniioombllem6 23162 Lemma for ~ uniioombl . (...
uniioombl 23163 A disjoint union of open i...
uniiccmbl 23164 An almost-disjoint union o...
dyadf 23165 The function ` F ` returns...
dyadval 23166 Value of the dyadic ration...
dyadovol 23167 Volume of a dyadic rationa...
dyadss 23168 Two closed dyadic rational...
dyaddisjlem 23169 Lemma for ~ dyaddisj . (C...
dyaddisj 23170 Two closed dyadic rational...
dyadmaxlem 23171 Lemma for ~ dyadmax . (Co...
dyadmax 23172 Any nonempty set of dyadic...
dyadmbllem 23173 Lemma for ~ dyadmbl . (Co...
dyadmbl 23174 Any union of dyadic ration...
opnmbllem 23175 Lemma for ~ opnmbl . (Con...
opnmbl 23176 All open sets are measurab...
opnmblALT 23177 All open sets are measurab...
subopnmbl 23178 Sets which are open in a m...
volsup2 23179 The volume of ` A ` is the...
volcn 23180 The function formed by res...
volivth 23181 The Intermediate Value The...
vitalilem1 23182 Lemma for ~ vitali . (Con...
vitalilem1OLD 23183 Obsolete proof of ~ vitali...
vitalilem2 23184 Lemma for ~ vitali . (Con...
vitalilem3 23185 Lemma for ~ vitali . (Con...
vitalilem4 23186 Lemma for ~ vitali . (Con...
vitalilem5 23187 Lemma for ~ vitali . (Con...
vitali 23188 If the reals can be well-o...
ismbf1 23199 The predicate " ` F ` is a...
mbff 23200 A measurable function is a...
mbfdm 23201 The domain of a measurable...
mbfconstlem 23202 Lemma for ~ mbfconst . (C...
ismbf 23203 The predicate " ` F ` is a...
ismbfcn 23204 A complex function is meas...
mbfima 23205 Definitional property of a...
mbfimaicc 23206 The preimage of any closed...
mbfimasn 23207 The preimage of a point un...
mbfconst 23208 A constant function is mea...
mbfid 23209 The identity function is m...
mbfmptcl 23210 Lemma for the ` MblFn ` pr...
mbfdm2 23211 The domain of a measurable...
ismbfcn2 23212 A complex function is meas...
ismbfd 23213 Deduction to prove measura...
ismbf2d 23214 Deduction to prove measura...
mbfeqalem 23215 Lemma for ~ mbfeqa . (Con...
mbfeqa 23216 If two functions are equal...
mbfres 23217 The restriction of a measu...
mbfres2 23218 Measurability of a piecewi...
mbfss 23219 Change the domain of a mea...
mbfmulc2lem 23220 Multiplication by a consta...
mbfmulc2re 23221 Multiplication by a consta...
mbfmax 23222 The maximum of two functio...
mbfneg 23223 The negative of a measurab...
mbfpos 23224 The positive part of a mea...
mbfposr 23225 Converse to ~ mbfpos . (C...
mbfposb 23226 A function is measurable i...
ismbf3d 23227 Simplified form of ~ ismbf...
mbfimaopnlem 23228 Lemma for ~ mbfimaopn . (...
mbfimaopn 23229 The preimage of any open s...
mbfimaopn2 23230 The preimage of any set op...
cncombf 23231 The composition of a conti...
cnmbf 23232 A continuous function is m...
mbfaddlem 23233 The sum of two measurable ...
mbfadd 23234 The sum of two measurable ...
mbfsub 23235 The difference of two meas...
mbfmulc2 23236 A complex constant times a...
mbfsup 23237 The supremum of a sequence...
mbfinf 23238 The infimum of a sequence ...
mbflimsup 23239 The limit supremum of a se...
mbflimlem 23240 The pointwise limit of a s...
mbflim 23241 The pointwise limit of a s...
0pval 23244 The zero function evaluate...
0plef 23245 Two ways to say that the f...
0pledm 23246 Adjust the domain of the l...
isi1f 23247 The predicate " ` F ` is a...
i1fmbf 23248 Simple functions are measu...
i1ff 23249 A simple function is a fun...
i1frn 23250 A simple function has fini...
i1fima 23251 Any preimage of a simple f...
i1fima2 23252 Any preimage of a simple f...
i1fima2sn 23253 Preimage of a singleton. ...
i1fd 23254 A simplified set of assump...
i1f0rn 23255 Any simple function takes ...
itg1val 23256 The value of the integral ...
itg1val2 23257 The value of the integral ...
itg1cl 23258 Closure of the integral on...
itg1ge0 23259 Closure of the integral on...
i1f0 23260 The zero function is simpl...
itg10 23261 The zero function has zero...
i1f1lem 23262 Lemma for ~ i1f1 and ~ itg...
i1f1 23263 Base case simple functions...
itg11 23264 The integral of an indicat...
itg1addlem1 23265 Decompose a preimage, whic...
i1faddlem 23266 Decompose the preimage of ...
i1fmullem 23267 Decompose the preimage of ...
i1fadd 23268 The sum of two simple func...
i1fmul 23269 The pointwise product of t...
itg1addlem2 23270 Lemma for ~ itg1add . The...
itg1addlem3 23271 Lemma for ~ itg1add . (Co...
itg1addlem4 23272 Lemma for itg1add . (Cont...
itg1addlem5 23273 Lemma for itg1add . (Cont...
itg1add 23274 The integral of a sum of s...
i1fmulclem 23275 Decompose the preimage of ...
i1fmulc 23276 A nonnegative constant tim...
itg1mulc 23277 The integral of a constant...
i1fres 23278 The "restriction" of a sim...
i1fpos 23279 The positive part of a sim...
i1fposd 23280 Deduction form of ~ i1fpos...
i1fsub 23281 The difference of two simp...
itg1sub 23282 The integral of a differen...
itg10a 23283 The integral of a simple f...
itg1ge0a 23284 The integral of an almost ...
itg1lea 23285 Approximate version of ~ i...
itg1le 23286 If one simple function dom...
itg1climres 23287 Restricting the simple fun...
mbfi1fseqlem1 23288 Lemma for ~ mbfi1fseq . (...
mbfi1fseqlem2 23289 Lemma for ~ mbfi1fseq . (...
mbfi1fseqlem3 23290 Lemma for ~ mbfi1fseq . (...
mbfi1fseqlem4 23291 Lemma for ~ mbfi1fseq . T...
mbfi1fseqlem5 23292 Lemma for ~ mbfi1fseq . V...
mbfi1fseqlem6 23293 Lemma for ~ mbfi1fseq . V...
mbfi1fseq 23294 A characterization of meas...
mbfi1flimlem 23295 Lemma for ~ mbfi1flim . (...
mbfi1flim 23296 Any real measurable functi...
mbfmullem2 23297 Lemma for ~ mbfmul . (Con...
mbfmullem 23298 Lemma for ~ mbfmul . (Con...
mbfmul 23299 The product of two measura...
itg2lcl 23300 The set of lower sums is a...
itg2val 23301 Value of the integral on n...
itg2l 23302 Elementhood in the set ` L...
itg2lr 23303 Sufficient condition for e...
xrge0f 23304 A real function is a nonne...
itg2cl 23305 The integral of a nonnegat...
itg2ub 23306 The integral of a nonnegat...
itg2leub 23307 Any upper bound on the int...
itg2ge0 23308 The integral of a nonnegat...
itg2itg1 23309 The integral of a nonnegat...
itg20 23310 The integral of the zero f...
itg2lecl 23311 If an ` S.2 ` integral is ...
itg2le 23312 If one function dominates ...
itg2const 23313 Integral of a constant fun...
itg2const2 23314 When the base set of a con...
itg2seq 23315 Definitional property of t...
itg2uba 23316 Approximate version of ~ i...
itg2lea 23317 Approximate version of ~ i...
itg2eqa 23318 Approximate equality of in...
itg2mulclem 23319 Lemma for ~ itg2mulc . (C...
itg2mulc 23320 The integral of a nonnegat...
itg2splitlem 23321 Lemma for ~ itg2split . (...
itg2split 23322 The ` S.2 ` integral split...
itg2monolem1 23323 Lemma for ~ itg2mono . We...
itg2monolem2 23324 Lemma for ~ itg2mono . (C...
itg2monolem3 23325 Lemma for ~ itg2mono . (C...
itg2mono 23326 The Monotone Convergence T...
itg2i1fseqle 23327 Subject to the conditions ...
itg2i1fseq 23328 Subject to the conditions ...
itg2i1fseq2 23329 In an extension to the res...
itg2i1fseq3 23330 Special case of ~ itg2i1fs...
itg2addlem 23331 Lemma for ~ itg2add . (Co...
itg2add 23332 The ` S.2 ` integral is li...
itg2gt0 23333 If the function ` F ` is s...
itg2cnlem1 23334 Lemma for ~ itgcn . (Cont...
itg2cnlem2 23335 Lemma for ~ itgcn . (Cont...
itg2cn 23336 A sort of absolute continu...
ibllem 23337 Conditioned equality theor...
isibl 23338 The predicate " ` F ` is i...
isibl2 23339 The predicate " ` F ` is i...
iblmbf 23340 An integrable function is ...
iblitg 23341 If a function is integrabl...
dfitg 23342 Evaluate the class substit...
itgex 23343 An integral is a set. (Co...
itgeq1f 23344 Equality theorem for an in...
itgeq1 23345 Equality theorem for an in...
nfitg1 23346 Bound-variable hypothesis ...
nfitg 23347 Bound-variable hypothesis ...
cbvitg 23348 Change bound variable in a...
cbvitgv 23349 Change bound variable in a...
itgeq2 23350 Equality theorem for an in...
itgresr 23351 The domain of an integral ...
itg0 23352 The integral of anything o...
itgz 23353 The integral of zero on an...
itgeq2dv 23354 Equality theorem for an in...
itgmpt 23355 Change bound variable in a...
itgcl 23356 The integral of an integra...
itgvallem 23357 Substitution lemma. (Cont...
itgvallem3 23358 Lemma for ~ itgposval and ...
ibl0 23359 The zero function is integ...
iblcnlem1 23360 Lemma for ~ iblcnlem . (C...
iblcnlem 23361 Expand out the forall in ~...
itgcnlem 23362 Expand out the sum in ~ df...
iblrelem 23363 Integrability of a real fu...
iblposlem 23364 Lemma for ~ iblpos . (Con...
iblpos 23365 Integrability of a nonnega...
iblre 23366 Integrability of a real fu...
itgrevallem1 23367 Lemma for ~ itgposval and ...
itgposval 23368 The integral of a nonnegat...
itgreval 23369 Decompose the integral of ...
itgrecl 23370 Real closure of an integra...
iblcn 23371 Integrability of a complex...
itgcnval 23372 Decompose the integral of ...
itgre 23373 Real part of an integral. ...
itgim 23374 Imaginary part of an integ...
iblneg 23375 The negative of an integra...
itgneg 23376 Negation of an integral. ...
iblss 23377 A subset of an integrable ...
iblss2 23378 Change the domain of an in...
itgitg2 23379 Transfer an integral using...
i1fibl 23380 A simple function is integ...
itgitg1 23381 Transfer an integral using...
itgle 23382 Monotonicity of an integra...
itgge0 23383 The integral of a positive...
itgss 23384 Expand the set of an integ...
itgss2 23385 Expand the set of an integ...
itgeqa 23386 Approximate equality of in...
itgss3 23387 Expand the set of an integ...
itgioo 23388 Equality of integrals on o...
itgless 23389 Expand the integral of a n...
iblconst 23390 A constant function is int...
itgconst 23391 Integral of a constant fun...
ibladdlem 23392 Lemma for ~ ibladd . (Con...
ibladd 23393 Add two integrals over the...
iblsub 23394 Subtract two integrals ove...
itgaddlem1 23395 Lemma for ~ itgadd . (Con...
itgaddlem2 23396 Lemma for ~ itgadd . (Con...
itgadd 23397 Add two integrals over the...
itgsub 23398 Subtract two integrals ove...
itgfsum 23399 Take a finite sum of integ...
iblabslem 23400 Lemma for ~ iblabs . (Con...
iblabs 23401 The absolute value of an i...
iblabsr 23402 A measurable function is i...
iblmulc2 23403 Multiply an integral by a ...
itgmulc2lem1 23404 Lemma for ~ itgmulc2 : pos...
itgmulc2lem2 23405 Lemma for ~ itgmulc2 : rea...
itgmulc2 23406 Multiply an integral by a ...
itgabs 23407 The triangle inequality fo...
itgsplit 23408 The ` S. ` integral splits...
itgspliticc 23409 The ` S. ` integral splits...
itgsplitioo 23410 The ` S. ` integral splits...
bddmulibl 23411 A bounded function times a...
bddibl 23412 A bounded function is inte...
cniccibl 23413 A continuous function on a...
itggt0 23414 The integral of a strictly...
itgcn 23415 Transfer ~ itg2cn to the f...
ditgeq1 23418 Equality theorem for the d...
ditgeq2 23419 Equality theorem for the d...
ditgeq3 23420 Equality theorem for the d...
ditgeq3dv 23421 Equality theorem for the d...
ditgex 23422 A directed integral is a s...
ditg0 23423 Value of the directed inte...
cbvditg 23424 Change bound variable in a...
cbvditgv 23425 Change bound variable in a...
ditgpos 23426 Value of the directed inte...
ditgneg 23427 Value of the directed inte...
ditgcl 23428 Closure of a directed inte...
ditgswap 23429 Reverse a directed integra...
ditgsplitlem 23430 Lemma for ~ ditgsplit . (...
ditgsplit 23431 This theorem is the raison...
reldv 23440 The derivative function is...
limcvallem 23441 Lemma for ~ ellimc . (Con...
limcfval 23442 Value and set bounds on th...
ellimc 23443 Value of the limit predica...
limcrcl 23444 Reverse closure for the li...
limccl 23445 Closure of the limit opera...
limcdif 23446 It suffices to consider fu...
ellimc2 23447 Write the definition of a ...
limcnlp 23448 If ` B ` is not a limit po...
ellimc3 23449 Write the epsilon-delta de...
limcflflem 23450 Lemma for ~ limcflf . (Co...
limcflf 23451 The limit operator can be ...
limcmo 23452 If ` B ` is a limit point ...
limcmpt 23453 Express the limit operator...
limcmpt2 23454 Express the limit operator...
limcresi 23455 Any limit of ` F ` is also...
limcres 23456 If ` B ` is an interior po...
cnplimc 23457 A function is continuous a...
cnlimc 23458 ` F ` is a continuous func...
cnlimci 23459 If ` F ` is a continuous f...
cnmptlimc 23460 If ` F ` is a continuous f...
limccnp 23461 If the limit of ` F ` at `...
limccnp2 23462 The image of a convergent ...
limcco 23463 Composition of two limits....
limciun 23464 A point is a limit of ` F ...
limcun 23465 A point is a limit of ` F ...
dvlem 23466 Closure for a difference q...
dvfval 23467 Value and set bounds on th...
eldv 23468 The differentiable predica...
dvcl 23469 The derivative function ta...
dvbssntr 23470 The set of differentiable ...
dvbss 23471 The set of differentiable ...
dvbsss 23472 The set of differentiable ...
perfdvf 23473 The derivative is a functi...
recnprss 23474 Both ` RR ` and ` CC ` are...
recnperf 23475 Both ` RR ` and ` CC ` are...
dvfg 23476 Explicitly write out the f...
dvf 23477 The derivative is a functi...
dvfcn 23478 The derivative is a functi...
dvreslem 23479 Lemma for ~ dvres . (Cont...
dvres2lem 23480 Lemma for ~ dvres2 . (Con...
dvres 23481 Restriction of a derivativ...
dvres2 23482 Restriction of the base se...
dvres3 23483 Restriction of a complex d...
dvres3a 23484 Restriction of a complex d...
dvidlem 23485 Lemma for ~ dvid and ~ dvc...
dvconst 23486 Derivative of a constant f...
dvid 23487 Derivative of the identity...
dvcnp 23488 The difference quotient is...
dvcnp2 23489 A function is continuous a...
dvcn 23490 A differentiable function ...
dvnfval 23491 Value of the iterated deri...
dvnff 23492 The iterated derivative is...
dvn0 23493 Zero times iterated deriva...
dvnp1 23494 Successor iterated derivat...
dvn1 23495 One times iterated derivat...
dvnf 23496 The N-times derivative is ...
dvnbss 23497 The set of N-times differe...
dvnadd 23498 The ` N ` -th derivative o...
dvn2bss 23499 An N-times differentiable ...
dvnres 23500 Multiple derivative versio...
cpnfval 23501 Condition for n-times cont...
fncpn 23502 The ` C^n ` object is a fu...
elcpn 23503 Condition for n-times cont...
cpnord 23504 ` C^n ` conditions are ord...
cpncn 23505 A ` C^n ` function is cont...
cpnres 23506 The restriction of a ` C^n...
dvaddbr 23507 The sum rule for derivativ...
dvmulbr 23508 The product rule for deriv...
dvadd 23509 The sum rule for derivativ...
dvmul 23510 The product rule for deriv...
dvaddf 23511 The sum rule for everywher...
dvmulf 23512 The product rule for every...
dvcmul 23513 The product rule when one ...
dvcmulf 23514 The product rule when one ...
dvcobr 23515 The chain rule for derivat...
dvco 23516 The chain rule for derivat...
dvcof 23517 The chain rule for everywh...
dvcjbr 23518 The derivative of the conj...
dvcj 23519 The derivative of the conj...
dvfre 23520 The derivative of a real f...
dvnfre 23521 The ` N ` -th derivative o...
dvexp 23522 Derivative of a power func...
dvexp2 23523 Derivative of an exponenti...
dvrec 23524 Derivative of the reciproc...
dvmptres3 23525 Function-builder for deriv...
dvmptid 23526 Function-builder for deriv...
dvmptc 23527 Function-builder for deriv...
dvmptcl 23528 Closure lemma for ~ dvmptc...
dvmptadd 23529 Function-builder for deriv...
dvmptmul 23530 Function-builder for deriv...
dvmptres2 23531 Function-builder for deriv...
dvmptres 23532 Function-builder for deriv...
dvmptcmul 23533 Function-builder for deriv...
dvmptdivc 23534 Function-builder for deriv...
dvmptneg 23535 Function-builder for deriv...
dvmptsub 23536 Function-builder for deriv...
dvmptcj 23537 Function-builder for deriv...
dvmptre 23538 Function-builder for deriv...
dvmptim 23539 Function-builder for deriv...
dvmptntr 23540 Function-builder for deriv...
dvmptco 23541 Function-builder for deriv...
dvmptfsum 23542 Function-builder for deriv...
dvcnvlem 23543 Lemma for ~ dvcnvre . (Co...
dvcnv 23544 A weak version of ~ dvcnvr...
dvexp3 23545 Derivative of an exponenti...
dveflem 23546 Derivative of the exponent...
dvef 23547 Derivative of the exponent...
dvsincos 23548 Derivative of the sine and...
dvsin 23549 Derivative of the sine fun...
dvcos 23550 Derivative of the cosine f...
dvferm1lem 23551 Lemma for ~ dvferm . (Con...
dvferm1 23552 One-sided version of ~ dvf...
dvferm2lem 23553 Lemma for ~ dvferm . (Con...
dvferm2 23554 One-sided version of ~ dvf...
dvferm 23555 Fermat's theorem on statio...
rollelem 23556 Lemma for ~ rolle . (Cont...
rolle 23557 Rolle's theorem. If ` F `...
cmvth 23558 Cauchy's Mean Value Theore...
mvth 23559 The Mean Value Theorem. I...
dvlip 23560 A function with derivative...
dvlipcn 23561 A complex function with de...
dvlip2 23562 Combine the results of ~ d...
c1liplem1 23563 Lemma for ~ c1lip1 . (Con...
c1lip1 23564 C1 functions are Lipschitz...
c1lip2 23565 C1 functions are Lipschitz...
c1lip3 23566 C1 functions are Lipschitz...
dveq0 23567 If a continuous function h...
dv11cn 23568 Two functions defined on a...
dvgt0lem1 23569 Lemma for ~ dvgt0 and ~ dv...
dvgt0lem2 23570 Lemma for ~ dvgt0 and ~ dv...
dvgt0 23571 A function on a closed int...
dvlt0 23572 A function on a closed int...
dvge0 23573 A function on a closed int...
dvle 23574 If ` A ( x ) , C ( x ) ` a...
dvivthlem1 23575 Lemma for ~ dvivth . (Con...
dvivthlem2 23576 Lemma for ~ dvivth . (Con...
dvivth 23577 Darboux' theorem, or the i...
dvne0 23578 A function on a closed int...
dvne0f1 23579 A function on a closed int...
lhop1lem 23580 Lemma for ~ lhop1 . (Cont...
lhop1 23581 L'Hôpital's Rule for...
lhop2 23582 L'Hôpital's Rule for...
lhop 23583 L'Hôpital's Rule. I...
dvcnvrelem1 23584 Lemma for ~ dvcnvre . (Co...
dvcnvrelem2 23585 Lemma for ~ dvcnvre . (Co...
dvcnvre 23586 The derivative rule for in...
dvcvx 23587 A real function with stric...
dvfsumle 23588 Compare a finite sum to an...
dvfsumge 23589 Compare a finite sum to an...
dvfsumabs 23590 Compare a finite sum to an...
dvmptrecl 23591 Real closure of a derivati...
dvfsumrlimf 23592 Lemma for ~ dvfsumrlim . ...
dvfsumlem1 23593 Lemma for ~ dvfsumrlim . ...
dvfsumlem2 23594 Lemma for ~ dvfsumrlim . ...
dvfsumlem3 23595 Lemma for ~ dvfsumrlim . ...
dvfsumlem4 23596 Lemma for ~ dvfsumrlim . ...
dvfsumrlimge0 23597 Lemma for ~ dvfsumrlim . ...
dvfsumrlim 23598 Compare a finite sum to an...
dvfsumrlim2 23599 Compare a finite sum to an...
dvfsumrlim3 23600 Conjoin the statements of ...
dvfsum2 23601 The reverse of ~ dvfsumrli...
ftc1lem1 23602 Lemma for ~ ftc1a and ~ ft...
ftc1lem2 23603 Lemma for ~ ftc1 . (Contr...
ftc1a 23604 The Fundamental Theorem of...
ftc1lem3 23605 Lemma for ~ ftc1 . (Contr...
ftc1lem4 23606 Lemma for ~ ftc1 . (Contr...
ftc1lem5 23607 Lemma for ~ ftc1 . (Contr...
ftc1lem6 23608 Lemma for ~ ftc1 . (Contr...
ftc1 23609 The Fundamental Theorem of...
ftc1cn 23610 Strengthen the assumptions...
ftc2 23611 The Fundamental Theorem of...
ftc2ditglem 23612 Lemma for ~ ftc2ditg . (C...
ftc2ditg 23613 Directed integral analogue...
itgparts 23614 Integration by parts. If ...
itgsubstlem 23615 Lemma for ~ itgsubst . (C...
itgsubst 23616 Integration by ` u ` -subs...
reldmmdeg 23621 Multivariate degree is a b...
tdeglem1 23622 Functionality of the total...
tdeglem3 23623 Additivity of the total de...
tdeglem4 23624 There is only one multi-in...
tdeglem2 23625 Simplification of total de...
mdegfval 23626 Value of the multivariate ...
mdegval 23627 Value of the multivariate ...
mdegleb 23628 Property of being of limit...
mdeglt 23629 If there is an upper limit...
mdegldg 23630 A nonzero polynomial has s...
mdegxrcl 23631 Closure of polynomial degr...
mdegxrf 23632 Functionality of polynomia...
mdegcl 23633 Sharp closure for multivar...
mdeg0 23634 Degree of the zero polynom...
mdegnn0cl 23635 Degree of a nonzero polyno...
degltlem1 23636 Theorem on arithmetic of e...
degltp1le 23637 Theorem on arithmetic of e...
mdegaddle 23638 The degree of a sum is at ...
mdegvscale 23639 The degree of a scalar mul...
mdegvsca 23640 The degree of a scalar mul...
mdegle0 23641 A polynomial has nonpositi...
mdegmullem 23642 Lemma for ~ mdegmulle2 . ...
mdegmulle2 23643 The multivariate degree of...
deg1fval 23644 Relate univariate polynomi...
deg1xrf 23645 Functionality of univariat...
deg1xrcl 23646 Closure of univariate poly...
deg1cl 23647 Sharp closure of univariat...
mdegpropd 23648 Property deduction for pol...
deg1fvi 23649 Univariate polynomial degr...
deg1propd 23650 Property deduction for pol...
deg1z 23651 Degree of the zero univari...
deg1nn0cl 23652 Degree of a nonzero univar...
deg1n0ima 23653 Degree image of a set of p...
deg1nn0clb 23654 A polynomial is nonzero if...
deg1lt0 23655 A polynomial is zero iff i...
deg1ldg 23656 A nonzero univariate polyn...
deg1ldgn 23657 An index at which a polyno...
deg1ldgdomn 23658 A nonzero univariate polyn...
deg1leb 23659 Property of being of limit...
deg1val 23660 Value of the univariate de...
deg1lt 23661 If the degree of a univari...
deg1ge 23662 Conversely, a nonzero coef...
coe1mul3 23663 The coefficient vector of ...
coe1mul4 23664 Value of the "leading" coe...
deg1addle 23665 The degree of a sum is at ...
deg1addle2 23666 If both factors have degre...
deg1add 23667 Exact degree of a sum of t...
deg1vscale 23668 The degree of a scalar tim...
deg1vsca 23669 The degree of a scalar tim...
deg1invg 23670 The degree of the negated ...
deg1suble 23671 The degree of a difference...
deg1sub 23672 Exact degree of a differen...
deg1mulle2 23673 Produce a bound on the pro...
deg1sublt 23674 Subtraction of two polynom...
deg1le0 23675 A polynomial has nonpositi...
deg1sclle 23676 A scalar polynomial has no...
deg1scl 23677 A nonzero scalar polynomia...
deg1mul2 23678 Degree of multiplication o...
deg1mul3 23679 Degree of multiplication o...
deg1mul3le 23680 Degree of multiplication o...
deg1tmle 23681 Limiting degree of a polyn...
deg1tm 23682 Exact degree of a polynomi...
deg1pwle 23683 Limiting degree of a varia...
deg1pw 23684 Exact degree of a variable...
ply1nz 23685 Univariate polynomials ove...
ply1nzb 23686 Univariate polynomials are...
ply1domn 23687 Corollary of ~ deg1mul2 : ...
ply1idom 23688 The ring of univariate pol...
ply1divmo 23699 Uniqueness of a quotient i...
ply1divex 23700 Lemma for ~ ply1divalg : e...
ply1divalg 23701 The division algorithm for...
ply1divalg2 23702 Reverse the order of multi...
uc1pval 23703 Value of the set of unitic...
isuc1p 23704 Being a unitic polynomial....
mon1pval 23705 Value of the set of monic ...
ismon1p 23706 Being a monic polynomial. ...
uc1pcl 23707 Unitic polynomials are pol...
mon1pcl 23708 Monic polynomials are poly...
uc1pn0 23709 Unitic polynomials are not...
mon1pn0 23710 Monic polynomials are not ...
uc1pdeg 23711 Unitic polynomials have no...
uc1pldg 23712 Unitic polynomials have un...
mon1pldg 23713 Unitic polynomials have on...
mon1puc1p 23714 Monic polynomials are unit...
uc1pmon1p 23715 Make a unitic polynomial m...
deg1submon1p 23716 The difference of two moni...
q1pval 23717 Value of the univariate po...
q1peqb 23718 Characterizing property of...
q1pcl 23719 Closure of the quotient by...
r1pval 23720 Value of the polynomial re...
r1pcl 23721 Closure of remainder follo...
r1pdeglt 23722 The remainder has a degree...
r1pid 23723 Express the original polyn...
dvdsq1p 23724 Divisibility in a polynomi...
dvdsr1p 23725 Divisibility in a polynomi...
ply1remlem 23726 A term of the form ` x - N...
ply1rem 23727 The polynomial remainder t...
facth1 23728 The factor theorem and its...
fta1glem1 23729 Lemma for ~ fta1g . (Cont...
fta1glem2 23730 Lemma for ~ fta1g . (Cont...
fta1g 23731 The one-sided fundamental ...
fta1blem 23732 Lemma for ~ fta1b . (Cont...
fta1b 23733 The assumption that ` R ` ...
drnguc1p 23734 Over a division ring, all ...
ig1peu 23735 There is a unique monic po...
ig1pval 23736 Substitutions for the poly...
ig1pval2 23737 Generator of the zero idea...
ig1pval3 23738 Characterizing properties ...
ig1pcl 23739 The monic generator of an ...
ig1pdvds 23740 The monic generator of an ...
ig1prsp 23741 Any ideal of polynomials o...
ply1lpir 23742 The ring of polynomials ov...
ply1pid 23743 The polynomials over a fie...
plyco0 23752 Two ways to say that a fun...
plyval 23753 Value of the polynomial se...
plybss 23754 Reverse closure of the par...
elply 23755 Definition of a polynomial...
elply2 23756 The coefficient function c...
plyun0 23757 The set of polynomials is ...
plyf 23758 The polynomial is a functi...
plyss 23759 The polynomial set functio...
plyssc 23760 Every polynomial ring is c...
elplyr 23761 Sufficient condition for e...
elplyd 23762 Sufficient condition for e...
ply1termlem 23763 Lemma for ~ ply1term . (C...
ply1term 23764 A one-term polynomial. (C...
plypow 23765 A power is a polynomial. ...
plyconst 23766 A constant function is a p...
ne0p 23767 A test to show that a poly...
ply0 23768 The zero function is a pol...
plyid 23769 The identity function is a...
plyeq0lem 23770 Lemma for ~ plyeq0 . If `...
plyeq0 23771 If a polynomial is zero at...
plypf1 23772 Write the set of complex p...
plyaddlem1 23773 Derive the coefficient fun...
plymullem1 23774 Derive the coefficient fun...
plyaddlem 23775 Lemma for ~ plyadd . (Con...
plymullem 23776 Lemma for ~ plymul . (Con...
plyadd 23777 The sum of two polynomials...
plymul 23778 The product of two polynom...
plysub 23779 The difference of two poly...
plyaddcl 23780 The sum of two polynomials...
plymulcl 23781 The product of two polynom...
plysubcl 23782 The difference of two poly...
coeval 23783 Value of the coefficient f...
coeeulem 23784 Lemma for ~ coeeu . (Cont...
coeeu 23785 Uniqueness of the coeffici...
coelem 23786 Lemma for properties of th...
coeeq 23787 If ` A ` satisfies the pro...
dgrval 23788 Value of the degree functi...
dgrlem 23789 Lemma for ~ dgrcl and simi...
coef 23790 The domain and range of th...
coef2 23791 The domain and range of th...
coef3 23792 The domain and range of th...
dgrcl 23793 The degree of any polynomi...
dgrub 23794 If the ` M ` -th coefficie...
dgrub2 23795 All the coefficients above...
dgrlb 23796 If all the coefficients ab...
coeidlem 23797 Lemma for ~ coeid . (Cont...
coeid 23798 Reconstruct a polynomial a...
coeid2 23799 Reconstruct a polynomial a...
coeid3 23800 Reconstruct a polynomial a...
plyco 23801 The composition of two pol...
coeeq2 23802 Compute the coefficient fu...
dgrle 23803 Given an explicit expressi...
dgreq 23804 If the highest term in a p...
0dgr 23805 A constant function has de...
0dgrb 23806 A function has degree zero...
dgrnznn 23807 A nonzero polynomial with ...
coefv0 23808 The result of evaluating a...
coeaddlem 23809 Lemma for ~ coeadd and ~ d...
coemullem 23810 Lemma for ~ coemul and ~ d...
coeadd 23811 The coefficient function o...
coemul 23812 A coefficient of a product...
coe11 23813 The coefficient function i...
coemulhi 23814 The leading coefficient of...
coemulc 23815 The coefficient function i...
coe0 23816 The coefficients of the ze...
coesub 23817 The coefficient function o...
coe1termlem 23818 The coefficient function o...
coe1term 23819 The coefficient function o...
dgr1term 23820 The degree of a monomial. ...
plycn 23821 A polynomial is a continuo...
dgr0 23822 The degree of the zero pol...
coeidp 23823 The coefficients of the id...
dgrid 23824 The degree of the identity...
dgreq0 23825 The leading coefficient of...
dgrlt 23826 Two ways to say that the d...
dgradd 23827 The degree of a sum of pol...
dgradd2 23828 The degree of a sum of pol...
dgrmul2 23829 The degree of a product of...
dgrmul 23830 The degree of a product of...
dgrmulc 23831 Scalar multiplication by a...
dgrsub 23832 The degree of a difference...
dgrcolem1 23833 The degree of a compositio...
dgrcolem2 23834 Lemma for ~ dgrco . (Cont...
dgrco 23835 The degree of a compositio...
plycjlem 23836 Lemma for ~ plycj and ~ co...
plycj 23837 The double conjugation of ...
coecj 23838 Double conjugation of a po...
plyrecj 23839 A polynomial with real coe...
plymul0or 23840 Polynomial multiplication ...
ofmulrt 23841 The set of roots of a prod...
plyreres 23842 Real-coefficient polynomia...
dvply1 23843 Derivative of a polynomial...
dvply2g 23844 The derivative of a polyno...
dvply2 23845 The derivative of a polyno...
dvnply2 23846 Polynomials have polynomia...
dvnply 23847 Polynomials have polynomia...
plycpn 23848 Polynomials are smooth. (...
quotval 23851 Value of the quotient func...
plydivlem1 23852 Lemma for ~ plydivalg . (...
plydivlem2 23853 Lemma for ~ plydivalg . (...
plydivlem3 23854 Lemma for ~ plydivex . Ba...
plydivlem4 23855 Lemma for ~ plydivex . In...
plydivex 23856 Lemma for ~ plydivalg . (...
plydiveu 23857 Lemma for ~ plydivalg . (...
plydivalg 23858 The division algorithm on ...
quotlem 23859 Lemma for properties of th...
quotcl 23860 The quotient of two polyno...
quotcl2 23861 Closure of the quotient fu...
quotdgr 23862 Remainder property of the ...
plyremlem 23863 Closure of a linear factor...
plyrem 23864 The polynomial remainder t...
facth 23865 The factor theorem. If a ...
fta1lem 23866 Lemma for ~ fta1 . (Contr...
fta1 23867 The easy direction of the ...
quotcan 23868 Exact division with a mult...
vieta1lem1 23869 Lemma for ~ vieta1 . (Con...
vieta1lem2 23870 Lemma for ~ vieta1 : induc...
vieta1 23871 The first-order Vieta's fo...
plyexmo 23872 An infinite set of values ...
elaa 23875 Elementhood in the set of ...
aacn 23876 An algebraic number is a c...
aasscn 23877 The algebraic numbers are ...
elqaalem1 23878 Lemma for ~ elqaa . The f...
elqaalem2 23879 Lemma for ~ elqaa . (Cont...
elqaalem3 23880 Lemma for ~ elqaa . (Cont...
elqaa 23881 The set of numbers generat...
qaa 23882 Every rational number is a...
qssaa 23883 The rational numbers are c...
iaa 23884 The imaginary unit is alge...
aareccl 23885 The reciprocal of an algeb...
aacjcl 23886 The conjugate of an algebr...
aannenlem1 23887 Lemma for ~ aannen . (Con...
aannenlem2 23888 Lemma for ~ aannen . (Con...
aannenlem3 23889 The algebraic numbers are ...
aannen 23890 The algebraic numbers are ...
aalioulem1 23891 Lemma for ~ aaliou . An i...
aalioulem2 23892 Lemma for ~ aaliou . (Con...
aalioulem3 23893 Lemma for ~ aaliou . (Con...
aalioulem4 23894 Lemma for ~ aaliou . (Con...
aalioulem5 23895 Lemma for ~ aaliou . (Con...
aalioulem6 23896 Lemma for ~ aaliou . (Con...
aaliou 23897 Liouville's theorem on dio...
geolim3 23898 Geometric series convergen...
aaliou2 23899 Liouville's approximation ...
aaliou2b 23900 Liouville's approximation ...
aaliou3lem1 23901 Lemma for ~ aaliou3 . (Co...
aaliou3lem2 23902 Lemma for ~ aaliou3 . (Co...
aaliou3lem3 23903 Lemma for ~ aaliou3 . (Co...
aaliou3lem8 23904 Lemma for ~ aaliou3 . (Co...
aaliou3lem4 23905 Lemma for ~ aaliou3 . (Co...
aaliou3lem5 23906 Lemma for ~ aaliou3 . (Co...
aaliou3lem6 23907 Lemma for ~ aaliou3 . (Co...
aaliou3lem7 23908 Lemma for ~ aaliou3 . (Co...
aaliou3lem9 23909 Example of a "Liouville nu...
aaliou3 23910 Example of a "Liouville nu...
taylfvallem1 23915 Lemma for ~ taylfval . (C...
taylfvallem 23916 Lemma for ~ taylfval . (C...
taylfval 23917 Define the Taylor polynomi...
eltayl 23918 Value of the Taylor series...
taylf 23919 The Taylor series defines ...
tayl0 23920 The Taylor series is alway...
taylplem1 23921 Lemma for ~ taylpfval and ...
taylplem2 23922 Lemma for ~ taylpfval and ...
taylpfval 23923 Define the Taylor polynomi...
taylpf 23924 The Taylor polynomial is a...
taylpval 23925 Value of the Taylor polyno...
taylply2 23926 The Taylor polynomial is a...
taylply 23927 The Taylor polynomial is a...
dvtaylp 23928 The derivative of the Tayl...
dvntaylp 23929 The ` M ` -th derivative o...
dvntaylp0 23930 The first ` N ` derivative...
taylthlem1 23931 Lemma for ~ taylth . This...
taylthlem2 23932 Lemma for ~ taylth . (Con...
taylth 23933 Taylor's theorem. The Tay...
ulmrel 23936 The uniform limit relation...
ulmscl 23937 Closure of the base set in...
ulmval 23938 Express the predicate: Th...
ulmcl 23939 Closure of a uniform limit...
ulmf 23940 Closure of a uniform limit...
ulmpm 23941 Closure of a uniform limit...
ulmf2 23942 Closure of a uniform limit...
ulm2 23943 Simplify ~ ulmval when ` F...
ulmi 23944 The uniform limit property...
ulmclm 23945 A uniform limit of functio...
ulmres 23946 A sequence of functions co...
ulmshftlem 23947 Lemma for ~ ulmshft . (Co...
ulmshft 23948 A sequence of functions co...
ulm0 23949 Every function converges u...
ulmuni 23950 An sequence of functions u...
ulmdm 23951 Two ways to express that a...
ulmcaulem 23952 Lemma for ~ ulmcau and ~ u...
ulmcau 23953 A sequence of functions co...
ulmcau2 23954 A sequence of functions co...
ulmss 23955 A uniform limit of functio...
ulmbdd 23956 A uniform limit of bounded...
ulmcn 23957 A uniform limit of continu...
ulmdvlem1 23958 Lemma for ~ ulmdv . (Cont...
ulmdvlem2 23959 Lemma for ~ ulmdv . (Cont...
ulmdvlem3 23960 Lemma for ~ ulmdv . (Cont...
ulmdv 23961 If ` F ` is a sequence of ...
mtest 23962 The Weierstrass M-test. I...
mtestbdd 23963 Given the hypotheses of th...
mbfulm 23964 A uniform limit of measura...
iblulm 23965 A uniform limit of integra...
itgulm 23966 A uniform limit of integra...
itgulm2 23967 A uniform limit of integra...
pserval 23968 Value of the function ` G ...
pserval2 23969 Value of the function ` G ...
psergf 23970 The sequence of terms in t...
radcnvlem1 23971 Lemma for ~ radcnvlt1 , ~ ...
radcnvlem2 23972 Lemma for ~ radcnvlt1 , ~ ...
radcnvlem3 23973 Lemma for ~ radcnvlt1 , ~ ...
radcnv0 23974 Zero is always a convergen...
radcnvcl 23975 The radius of convergence ...
radcnvlt1 23976 If ` X ` is within the ope...
radcnvlt2 23977 If ` X ` is within the ope...
radcnvle 23978 If ` X ` is a convergent p...
dvradcnv 23979 The radius of convergence ...
pserulm 23980 If ` S ` is a region conta...
psercn2 23981 Since by ~ pserulm the ser...
psercnlem2 23982 Lemma for ~ psercn . (Con...
psercnlem1 23983 Lemma for ~ psercn . (Con...
psercn 23984 An infinite series converg...
pserdvlem1 23985 Lemma for ~ pserdv . (Con...
pserdvlem2 23986 Lemma for ~ pserdv . (Con...
pserdv 23987 The derivative of a power ...
pserdv2 23988 The derivative of a power ...
abelthlem1 23989 Lemma for ~ abelth . (Con...
abelthlem2 23990 Lemma for ~ abelth . The ...
abelthlem3 23991 Lemma for ~ abelth . (Con...
abelthlem4 23992 Lemma for ~ abelth . (Con...
abelthlem5 23993 Lemma for ~ abelth . (Con...
abelthlem6 23994 Lemma for ~ abelth . (Con...
abelthlem7a 23995 Lemma for ~ abelth . (Con...
abelthlem7 23996 Lemma for ~ abelth . (Con...
abelthlem8 23997 Lemma for ~ abelth . (Con...
abelthlem9 23998 Lemma for ~ abelth . By a...
abelth 23999 Abel's theorem. If the po...
abelth2 24000 Abel's theorem, restricted...
efcn 24001 The exponential function i...
sincn 24002 Sine is continuous. (Cont...
coscn 24003 Cosine is continuous. (Co...
reeff1olem 24004 Lemma for ~ reeff1o . (Co...
reeff1o 24005 The real exponential funct...
reefiso 24006 The exponential function o...
efcvx 24007 The exponential function o...
reefgim 24008 The exponential function i...
pilem1 24009 Lemma for ~ pire , ~ pigt2...
pilem2 24010 Lemma for ~ pire , ~ pigt2...
pilem3 24011 Lemma for ~ pire , ~ pigt2...
pigt2lt4 24012 ` _pi ` is between 2 and 4...
sinpi 24013 The sine of ` _pi ` is 0. ...
pire 24014 ` _pi ` is a real number. ...
picn 24015 ` _pi ` is a complex numbe...
pipos 24016 ` _pi ` is positive. (Con...
pirp 24017 ` _pi ` is a positive real...
negpicn 24018 ` -u _pi ` is a real numbe...
sinhalfpilem 24019 Lemma for ~ sinhalfpi and ...
halfpire 24020 ` _pi / 2 ` is real. (Con...
neghalfpire 24021 ` -u _pi / 2 ` is real. (...
neghalfpirx 24022 ` -u _pi / 2 ` is an exten...
pidiv2halves 24023 Adding ` _pi / 2 ` to itse...
sinhalfpi 24024 The sine of ` _pi / 2 ` is...
coshalfpi 24025 The cosine of ` _pi / 2 ` ...
cosneghalfpi 24026 The cosine of ` -u _pi / 2...
efhalfpi 24027 The exponential of ` _i _p...
cospi 24028 The cosine of ` _pi ` is `...
efipi 24029 The exponential of ` _i x....
eulerid 24030 Euler's identity. (Contri...
sin2pi 24031 The sine of ` 2 _pi ` is 0...
cos2pi 24032 The cosine of ` 2 _pi ` is...
ef2pi 24033 The exponential of ` 2 _pi...
ef2kpi 24034 The exponential of ` 2 K _...
efper 24035 The exponential function i...
sinperlem 24036 Lemma for ~ sinper and ~ c...
sinper 24037 The sine function is perio...
cosper 24038 The cosine function is per...
sin2kpi 24039 If ` K ` is an integer, th...
cos2kpi 24040 If ` K ` is an integer, th...
sin2pim 24041 Sine of a number subtracte...
cos2pim 24042 Cosine of a number subtrac...
sinmpi 24043 Sine of a number less ` _p...
cosmpi 24044 Cosine of a number less ` ...
sinppi 24045 Sine of a number plus ` _p...
cosppi 24046 Cosine of a complex number...
efimpi 24047 The exponential function o...
sinhalfpip 24048 The sine of ` _pi / 2 ` pl...
sinhalfpim 24049 The sine of ` _pi / 2 ` mi...
coshalfpip 24050 The cosine of ` _pi / 2 ` ...
coshalfpim 24051 The cosine of ` _pi / 2 ` ...
ptolemy 24052 Ptolemy's Theorem. This t...
sincosq1lem 24053 Lemma for ~ sincosq1sgn . ...
sincosq1sgn 24054 The signs of the sine and ...
sincosq2sgn 24055 The signs of the sine and ...
sincosq3sgn 24056 The signs of the sine and ...
sincosq4sgn 24057 The signs of the sine and ...
coseq00topi 24058 Location of the zeroes of ...
coseq0negpitopi 24059 Location of the zeroes of ...
tanrpcl 24060 Positive real closure of t...
tangtx 24061 The tangent function is gr...
tanabsge 24062 The tangent function is gr...
sinq12gt0 24063 The sine of a number stric...
sinq12ge0 24064 The sine of a number betwe...
sinq34lt0t 24065 The sine of a number stric...
cosq14gt0 24066 The cosine of a number str...
cosq14ge0 24067 The cosine of a number bet...
sincosq1eq 24068 Complementarity of the sin...
sincos4thpi 24069 The sine and cosine of ` _...
tan4thpi 24070 The tangent of ` _pi / 4 `...
sincos6thpi 24071 The sine and cosine of ` _...
sincos3rdpi 24072 The sine and cosine of ` _...
pige3 24073 ` _pi ` is greater or equa...
abssinper 24074 The absolute value of sine...
sinkpi 24075 The sine of an integer mul...
coskpi 24076 The absolute value of the ...
sineq0 24077 A complex number whose sin...
coseq1 24078 A complex number whose cos...
efeq1 24079 A complex number whose exp...
cosne0 24080 The cosine function has no...
cosordlem 24081 Lemma for ~ cosord . (Con...
cosord 24082 Cosine is decreasing over ...
cos11 24083 Cosine is one-to-one over ...
sinord 24084 Sine is increasing over th...
recosf1o 24085 The cosine function is a b...
resinf1o 24086 The sine function is a bij...
tanord1 24087 The tangent function is st...
tanord 24088 The tangent function is st...
tanregt0 24089 The positivity of ` tan ( ...
negpitopissre 24090 ` ( -u _pi (,] _pi ) ` is ...
efgh 24091 The exponential function o...
efif1olem1 24092 Lemma for ~ efif1o . (Con...
efif1olem2 24093 Lemma for ~ efif1o . (Con...
efif1olem3 24094 Lemma for ~ efif1o . (Con...
efif1olem4 24095 The exponential function o...
efif1o 24096 The exponential function o...
efifo 24097 The exponential function o...
eff1olem 24098 The exponential function m...
eff1o 24099 The exponential function m...
efabl 24100 The image of a subgroup of...
efsubm 24101 The image of a subgroup of...
circgrp 24102 The circle group ` T ` is ...
circsubm 24103 The circle group ` T ` is ...
rzgrp 24104 The quotient group R/Z is ...
logrn 24109 The range of the natural l...
ellogrn 24110 Write out the property ` A...
dflog2 24111 The natural logarithm func...
relogrn 24112 The range of the natural l...
logrncn 24113 The range of the natural l...
eff1o2 24114 The exponential function r...
logf1o 24115 The natural logarithm func...
dfrelog 24116 The natural logarithm func...
relogf1o 24117 The natural logarithm func...
logrncl 24118 Closure of the natural log...
logcl 24119 Closure of the natural log...
logimcl 24120 Closure of the imaginary p...
logcld 24121 The logarithm of a nonzero...
logimcld 24122 The imaginary part of the ...
logimclad 24123 The imaginary part of the ...
abslogimle 24124 The imaginary part of the ...
logrnaddcl 24125 The range of the natural l...
relogcl 24126 Closure of the natural log...
eflog 24127 Relationship between the n...
logeq0im1 24128 If the logarithm of a numb...
logccne0 24129 The logarithm isn't 0 if i...
logne0 24130 Logarithm of a non-1 posit...
reeflog 24131 Relationship between the n...
logef 24132 Relationship between the n...
relogef 24133 Relationship between the n...
logeftb 24134 Relationship between the n...
relogeftb 24135 Relationship between the n...
log1 24136 The natural logarithm of `...
loge 24137 The natural logarithm of `...
logneg 24138 The natural logarithm of a...
logm1 24139 The natural logarithm of n...
lognegb 24140 If a number has imaginary ...
relogoprlem 24141 Lemma for ~ relogmul and ~...
relogmul 24142 The natural logarithm of t...
relogdiv 24143 The natural logarithm of t...
explog 24144 Exponentiation of a nonzer...
reexplog 24145 Exponentiation of a positi...
relogexp 24146 The natural logarithm of p...
relog 24147 Real part of a logarithm. ...
relogiso 24148 The natural logarithm func...
reloggim 24149 The natural logarithm is a...
logltb 24150 The natural logarithm func...
logfac 24151 The logarithm of a factori...
eflogeq 24152 Solve an equation involvin...
logleb 24153 Natural logarithm preserve...
rplogcl 24154 Closure of the logarithm f...
logge0 24155 The logarithm of a number ...
logcj 24156 The natural logarithm dist...
efiarg 24157 The exponential of the "ar...
cosargd 24158 The cosine of the argument...
cosarg0d 24159 The cosine of the argument...
argregt0 24160 Closure of the argument of...
argrege0 24161 Closure of the argument of...
argimgt0 24162 Closure of the argument of...
argimlt0 24163 Closure of the argument of...
logimul 24164 Multiplying a number by ` ...
logneg2 24165 The logarithm of the negat...
logmul2 24166 Generalization of ~ relogm...
logdiv2 24167 Generalization of ~ relogd...
abslogle 24168 Bound on the magnitude of ...
tanarg 24169 The basic relation between...
logdivlti 24170 The ` log x / x ` function...
logdivlt 24171 The ` log x / x ` function...
logdivle 24172 The ` log x / x ` function...
relogcld 24173 Closure of the natural log...
reeflogd 24174 Relationship between the n...
relogmuld 24175 The natural logarithm of t...
relogdivd 24176 The natural logarithm of t...
logled 24177 Natural logarithm preserve...
relogefd 24178 Relationship between the n...
rplogcld 24179 Closure of the logarithm f...
logge0d 24180 The logarithm of a number ...
divlogrlim 24181 The inverse logarithm func...
logno1 24182 The logarithm function is ...
dvrelog 24183 The derivative of the real...
relogcn 24184 The real logarithm functio...
ellogdm 24185 Elementhood in the "contin...
logdmn0 24186 A number in the continuous...
logdmnrp 24187 A number in the continuous...
logdmss 24188 The continuity domain of `...
logcnlem2 24189 Lemma for ~ logcn . (Cont...
logcnlem3 24190 Lemma for ~ logcn . (Cont...
logcnlem4 24191 Lemma for ~ logcn . (Cont...
logcnlem5 24192 Lemma for ~ logcn . (Cont...
logcn 24193 The logarithm function is ...
dvloglem 24194 Lemma for ~ dvlog . (Cont...
logdmopn 24195 The "continuous domain" of...
logf1o2 24196 The logarithm maps its con...
dvlog 24197 The derivative of the comp...
dvlog2lem 24198 Lemma for ~ dvlog2 . (Con...
dvlog2 24199 The derivative of the comp...
advlog 24200 The antiderivative of the ...
advlogexp 24201 The antiderivative of a po...
efopnlem1 24202 Lemma for ~ efopn . (Cont...
efopnlem2 24203 Lemma for ~ efopn . (Cont...
efopn 24204 The exponential map is an ...
logtayllem 24205 Lemma for ~ logtayl . (Co...
logtayl 24206 The Taylor series for ` -u...
logtaylsum 24207 The Taylor series for ` -u...
logtayl2 24208 Power series expression fo...
logccv 24209 The natural logarithm func...
cxpval 24210 Value of the complex power...
cxpef 24211 Value of the complex power...
0cxp 24212 Value of the complex power...
cxpexpz 24213 Relate the complex power f...
cxpexp 24214 Relate the complex power f...
logcxp 24215 Logarithm of a complex pow...
cxp0 24216 Value of the complex power...
cxp1 24217 Value of the complex power...
1cxp 24218 Value of the complex power...
ecxp 24219 Write the exponential func...
cxpcl 24220 Closure of the complex pow...
recxpcl 24221 Real closure of the comple...
rpcxpcl 24222 Positive real closure of t...
cxpne0 24223 Complex exponentiation is ...
cxpeq0 24224 Complex exponentiation is ...
cxpadd 24225 Sum of exponents law for c...
cxpp1 24226 Value of a nonzero complex...
cxpneg 24227 Value of a complex number ...
cxpsub 24228 Exponent subtraction law f...
cxpge0 24229 Nonnegative exponentiation...
mulcxplem 24230 Lemma for ~ mulcxp . (Con...
mulcxp 24231 Complex exponentiation of ...
cxprec 24232 Complex exponentiation of ...
divcxp 24233 Complex exponentiation of ...
cxpmul 24234 Product of exponents law f...
cxpmul2 24235 Product of exponents law f...
cxproot 24236 The complex power function...
cxpmul2z 24237 Generalize ~ cxpmul2 to ne...
abscxp 24238 Absolute value of a power,...
abscxp2 24239 Absolute value of a power,...
cxplt 24240 Ordering property for comp...
cxple 24241 Ordering property for comp...
cxplea 24242 Ordering property for comp...
cxple2 24243 Ordering property for comp...
cxplt2 24244 Ordering property for comp...
cxple2a 24245 Ordering property for comp...
cxplt3 24246 Ordering property for comp...
cxple3 24247 Ordering property for comp...
cxpsqrtlem 24248 Lemma for ~ cxpsqrt . (Co...
cxpsqrt 24249 The complex exponential fu...
logsqrt 24250 Logarithm of a square root...
cxp0d 24251 Value of the complex power...
cxp1d 24252 Value of the complex power...
1cxpd 24253 Value of the complex power...
cxpcld 24254 Closure of the complex pow...
cxpmul2d 24255 Product of exponents law f...
0cxpd 24256 Value of the complex power...
cxpexpzd 24257 Relate the complex power f...
cxpefd 24258 Value of the complex power...
cxpne0d 24259 Complex exponentiation is ...
cxpp1d 24260 Value of a nonzero complex...
cxpnegd 24261 Value of a complex number ...
cxpmul2zd 24262 Generalize ~ cxpmul2 to ne...
cxpaddd 24263 Sum of exponents law for c...
cxpsubd 24264 Exponent subtraction law f...
cxpltd 24265 Ordering property for comp...
cxpled 24266 Ordering property for comp...
cxplead 24267 Ordering property for comp...
divcxpd 24268 Complex exponentiation of ...
recxpcld 24269 Positive real closure of t...
cxpge0d 24270 Nonnegative exponentiation...
cxple2ad 24271 Ordering property for comp...
cxplt2d 24272 Ordering property for comp...
cxple2d 24273 Ordering property for comp...
mulcxpd 24274 Complex exponentiation of ...
cxprecd 24275 Complex exponentiation of ...
rpcxpcld 24276 Positive real closure of t...
logcxpd 24277 Logarithm of a complex pow...
cxplt3d 24278 Ordering property for comp...
cxple3d 24279 Ordering property for comp...
cxpmuld 24280 Product of exponents law f...
dvcxp1 24281 The derivative of a comple...
dvcxp2 24282 The derivative of a comple...
dvsqrt 24283 The derivative of the real...
dvcncxp1 24284 Derivative of complex powe...
dvcnsqrt 24285 Derivative of square root ...
cxpcn 24286 Domain of continuity of th...
cxpcn2 24287 Continuity of the complex ...
cxpcn3lem 24288 Lemma for ~ cxpcn3 . (Con...
cxpcn3 24289 Extend continuity of the c...
resqrtcn 24290 Continuity of the real squ...
sqrtcn 24291 Continuity of the square r...
cxpaddlelem 24292 Lemma for ~ cxpaddle . (C...
cxpaddle 24293 Ordering property for comp...
abscxpbnd 24294 Bound on the absolute valu...
root1id 24295 Property of an ` N ` -th r...
root1eq1 24296 The only powers of an ` N ...
root1cj 24297 Within the ` N ` -th roots...
cxpeq 24298 Solve an equation involvin...
loglesqrt 24299 An upper bound on the loga...
logreclem 24300 Symmetry of the natural lo...
logrec 24301 Logarithm of a reciprocal ...
logbval 24304 Define the value of the ` ...
logbcl 24305 General logarithm closure....
logbid1 24306 General logarithm is 1 whe...
logb1 24307 The logarithm of ` 1 ` to ...
elogb 24308 The general logarithm of a...
logbchbase 24309 Change of base for logarit...
relogbval 24310 Value of the general logar...
relogbcl 24311 Closure of the general log...
relogbzcl 24312 Closure of the general log...
relogbreexp 24313 Power law for the general ...
relogbzexp 24314 Power law for the general ...
relogbmul 24315 The logarithm of the produ...
relogbmulexp 24316 The logarithm of the produ...
relogbdiv 24317 The logarithm of the quoti...
relogbexp 24318 Identity law for general l...
nnlogbexp 24319 Identity law for general l...
logbrec 24320 Logarithm of a reciprocal ...
logbleb 24321 The general logarithm func...
logblt 24322 The general logarithm func...
relogbcxp 24323 Identity law for the gener...
cxplogb 24324 Identity law for the gener...
relogbcxpb 24325 The logarithm is the inver...
logbmpt 24326 The general logarithm to a...
logbf 24327 The general logarithm to a...
logbfval 24328 The general logarithm of a...
relogbf 24329 The general logarithm to a...
logblog 24330 The general logarithm to t...
angval 24331 Define the angle function,...
angcan 24332 Cancel a constant multipli...
angneg 24333 Cancel a negative sign in ...
angvald 24334 The (signed) angle between...
angcld 24335 The (signed) angle between...
angrteqvd 24336 Two vectors are at a right...
cosangneg2d 24337 The cosine of the angle be...
angrtmuld 24338 Perpendicularity of two ve...
ang180lem1 24339 Lemma for ~ ang180 . Show...
ang180lem2 24340 Lemma for ~ ang180 . Show...
ang180lem3 24341 Lemma for ~ ang180 . Sinc...
ang180lem4 24342 Lemma for ~ ang180 . Redu...
ang180lem5 24343 Lemma for ~ ang180 : Redu...
ang180 24344 The sum of angles ` m A B ...
lawcoslem1 24345 Lemma for ~ lawcos . Here...
lawcos 24346 Law of cosines (also known...
pythag 24347 Pythagorean theorem. Give...
isosctrlem1 24348 Lemma for ~ isosctr . (Co...
isosctrlem2 24349 Lemma for ~ isosctr . Cor...
isosctrlem3 24350 Lemma for ~ isosctr . Cor...
isosctr 24351 Isosceles triangle theorem...
ssscongptld 24352 If two triangles have equa...
affineequiv 24353 Equivalence between two wa...
affineequiv2 24354 Equivalence between two wa...
angpieqvdlem 24355 Equivalence used in the pr...
angpieqvdlem2 24356 Equivalence used in ~ angp...
angpined 24357 If the angle at ABC is ` _...
angpieqvd 24358 The angle ABC is ` _pi ` i...
chordthmlem 24359 If M is the midpoint of AB...
chordthmlem2 24360 If M is the midpoint of AB...
chordthmlem3 24361 If M is the midpoint of AB...
chordthmlem4 24362 If P is on the segment AB ...
chordthmlem5 24363 If P is on the segment AB ...
chordthm 24364 The intersecting chords th...
heron 24365 Heron's formula gives the ...
quad2 24366 The quadratic equation, wi...
quad 24367 The quadratic equation. (...
1cubrlem 24368 The cube roots of unity. ...
1cubr 24369 The cube roots of unity. ...
dcubic1lem 24370 Lemma for ~ dcubic1 and ~ ...
dcubic2 24371 Reverse direction of ~ dcu...
dcubic1 24372 Forward direction of ~ dcu...
dcubic 24373 Solutions to the depressed...
mcubic 24374 Solutions to a monic cubic...
cubic2 24375 The solution to the genera...
cubic 24376 The cubic equation, which ...
binom4 24377 Work out a quartic binomia...
dquartlem1 24378 Lemma for ~ dquart . (Con...
dquartlem2 24379 Lemma for ~ dquart . (Con...
dquart 24380 Solve a depressed quartic ...
quart1cl 24381 Closure lemmas for ~ quart...
quart1lem 24382 Lemma for ~ quart1 . (Con...
quart1 24383 Depress a quartic equation...
quartlem1 24384 Lemma for ~ quart . (Cont...
quartlem2 24385 Closure lemmas for ~ quart...
quartlem3 24386 Closure lemmas for ~ quart...
quartlem4 24387 Closure lemmas for ~ quart...
quart 24388 The quartic equation, writ...
asinlem 24395 The argument to the logari...
asinlem2 24396 The argument to the logari...
asinlem3a 24397 Lemma for ~ asinlem3 . (C...
asinlem3 24398 The argument to the logari...
asinf 24399 Domain and range of the ar...
asincl 24400 Closure for the arcsin fun...
acosf 24401 Domain and range of the ar...
acoscl 24402 Closure for the arccos fun...
atandm 24403 Since the property is a li...
atandm2 24404 This form of ~ atandm is a...
atandm3 24405 A compact form of ~ atandm...
atandm4 24406 A compact form of ~ atandm...
atanf 24407 Domain and range of the ar...
atancl 24408 Closure for the arctan fun...
asinval 24409 Value of the arcsin functi...
acosval 24410 Value of the arccos functi...
atanval 24411 Value of the arctan functi...
atanre 24412 A real number is in the do...
asinneg 24413 The arcsine function is od...
acosneg 24414 The negative symmetry rela...
efiasin 24415 The exponential of the arc...
sinasin 24416 The arcsine function is an...
cosacos 24417 The arccosine function is ...
asinsinlem 24418 Lemma for ~ asinsin . (Co...
asinsin 24419 The arcsine function compo...
acoscos 24420 The arccosine function is ...
asin1 24421 The arcsine of ` 1 ` is ` ...
acos1 24422 The arcsine of ` 1 ` is ` ...
reasinsin 24423 The arcsine function compo...
asinsinb 24424 Relationship between sine ...
acoscosb 24425 Relationship between sine ...
asinbnd 24426 The arcsine function has r...
acosbnd 24427 The arccosine function has...
asinrebnd 24428 Bounds on the arcsine func...
asinrecl 24429 The arcsine function is re...
acosrecl 24430 The arccosine function is ...
cosasin 24431 The cosine of the arcsine ...
sinacos 24432 The sine of the arccosine ...
atandmneg 24433 The domain of the arctange...
atanneg 24434 The arctangent function is...
atan0 24435 The arctangent of zero is ...
atandmcj 24436 The arctangent function di...
atancj 24437 The arctangent function di...
atanrecl 24438 The arctangent function is...
efiatan 24439 Value of the exponential o...
atanlogaddlem 24440 Lemma for ~ atanlogadd . ...
atanlogadd 24441 The rule ` sqrt ( z w ) = ...
atanlogsublem 24442 Lemma for ~ atanlogsub . ...
atanlogsub 24443 A variation on ~ atanlogad...
efiatan2 24444 Value of the exponential o...
2efiatan 24445 Value of the exponential o...
tanatan 24446 The arctangent function is...
atandmtan 24447 The tangent function has r...
cosatan 24448 The cosine of an arctangen...
cosatanne0 24449 The arctangent function ha...
atantan 24450 The arctangent function is...
atantanb 24451 Relationship between tange...
atanbndlem 24452 Lemma for ~ atanbnd . (Co...
atanbnd 24453 The arctangent function is...
atanord 24454 The arctangent function is...
atan1 24455 The arctangent of ` 1 ` is...
bndatandm 24456 A point in the open unit d...
atans 24457 The "domain of continuity"...
atans2 24458 It suffices to show that `...
atansopn 24459 The domain of continuity o...
atansssdm 24460 The domain of continuity o...
ressatans 24461 The real number line is a ...
dvatan 24462 The derivative of the arct...
atancn 24463 The arctangent is a contin...
atantayl 24464 The Taylor series for ` ar...
atantayl2 24465 The Taylor series for ` ar...
atantayl3 24466 The Taylor series for ` ar...
leibpilem1 24467 Lemma for ~ leibpi . (Con...
leibpilem2 24468 The Leibniz formula for ` ...
leibpi 24469 The Leibniz formula for ` ...
leibpisum 24470 The Leibniz formula for ` ...
log2cnv 24471 Using the Taylor series fo...
log2tlbnd 24472 Bound the error term in th...
log2ublem1 24473 Lemma for ~ log2ub . The ...
log2ublem2 24474 Lemma for ~ log2ub . (Con...
log2ublem3 24475 Lemma for ~ log2ub . In d...
log2ub 24476 ` log 2 ` is less than ` 2...
log2le1 24477 ` log 2 ` is less than ` 1...
birthdaylem1 24478 Lemma for ~ birthday . (C...
birthdaylem2 24479 For general ` N ` and ` K ...
birthdaylem3 24480 For general ` N ` and ` K ...
birthday 24481 The Birthday Problem. The...
dmarea 24484 The domain of the area fun...
areambl 24485 The fibers of a measurable...
areass 24486 A measurable region is a s...
dfarea 24487 Rewrite ~ df-area self-ref...
areaf 24488 Area measurement is a func...
areacl 24489 The area of a measurable r...
areage0 24490 The area of a measurable r...
areaval 24491 The area of a measurable r...
rlimcnp 24492 Relate a limit of a real-v...
rlimcnp2 24493 Relate a limit of a real-v...
rlimcnp3 24494 Relate a limit of a real-v...
xrlimcnp 24495 Relate a limit of a real-v...
efrlim 24496 The limit of the sequence ...
dfef2 24497 The limit of the sequence ...
cxplim 24498 A power to a negative expo...
sqrtlim 24499 The inverse square root fu...
rlimcxp 24500 Any power to a positive ex...
o1cxp 24501 An eventually bounded func...
cxp2limlem 24502 A linear factor grows slow...
cxp2lim 24503 Any power grows slower tha...
cxploglim 24504 The logarithm grows slower...
cxploglim2 24505 Every power of the logarit...
divsqrtsumlem 24506 Lemma for ~ divsqrsum and ...
divsqrsumf 24507 The function ` F ` used in...
divsqrsum 24508 The sum ` sum_ n <_ x ( 1 ...
divsqrtsum2 24509 A bound on the distance of...
divsqrtsumo1 24510 The sum ` sum_ n <_ x ( 1 ...
cvxcl 24511 Closure of a 0-1 linear co...
scvxcvx 24512 A strictly convex function...
jensenlem1 24513 Lemma for ~ jensen . (Con...
jensenlem2 24514 Lemma for ~ jensen . (Con...
jensen 24515 Jensen's inequality, a fin...
amgmlem 24516 Lemma for ~ amgm . (Contr...
amgm 24517 Inequality of arithmetic a...
logdifbnd 24520 Bound on the difference of...
logdiflbnd 24521 Lower bound on the differe...
emcllem1 24522 Lemma for ~ emcl . The se...
emcllem2 24523 Lemma for ~ emcl . ` F ` i...
emcllem3 24524 Lemma for ~ emcl . The fu...
emcllem4 24525 Lemma for ~ emcl . The di...
emcllem5 24526 Lemma for ~ emcl . The pa...
emcllem6 24527 Lemma for ~ emcl . By the...
emcllem7 24528 Lemma for ~ emcl and ~ har...
emcl 24529 Closure and bounds for the...
harmonicbnd 24530 A bound on the harmonic se...
harmonicbnd2 24531 A bound on the harmonic se...
emre 24532 The Euler-Mascheroni const...
emgt0 24533 The Euler-Mascheroni const...
harmonicbnd3 24534 A bound on the harmonic se...
harmoniclbnd 24535 A bound on the harmonic se...
harmonicubnd 24536 A bound on the harmonic se...
harmonicbnd4 24537 The asymptotic behavior of...
fsumharmonic 24538 Bound a finite sum based o...
zetacvg 24541 The zeta series is converg...
eldmgm 24548 Elementhood in the set of ...
dmgmaddn0 24549 If ` A ` is not a nonposit...
dmlogdmgm 24550 If ` A ` is in the continu...
rpdmgm 24551 A positive real number is ...
dmgmn0 24552 If ` A ` is not a nonposit...
dmgmaddnn0 24553 If ` A ` is not a nonposit...
dmgmdivn0 24554 Lemma for ~ lgamf . (Cont...
lgamgulmlem1 24555 Lemma for ~ lgamgulm . (C...
lgamgulmlem2 24556 Lemma for ~ lgamgulm . (C...
lgamgulmlem3 24557 Lemma for ~ lgamgulm . (C...
lgamgulmlem4 24558 Lemma for ~ lgamgulm . (C...
lgamgulmlem5 24559 Lemma for ~ lgamgulm . (C...
lgamgulmlem6 24560 The series ` G ` is unifor...
lgamgulm 24561 The series ` G ` is unifor...
lgamgulm2 24562 Rewrite the limit of the s...
lgambdd 24563 The log-Gamma function is ...
lgamucov 24564 The ` U ` regions used in ...
lgamucov2 24565 The ` U ` regions used in ...
lgamcvglem 24566 Lemma for ~ lgamf and ~ lg...
lgamcl 24567 The log-Gamma function is ...
lgamf 24568 The log-Gamma function is ...
gamf 24569 The Gamma function is a co...
gamcl 24570 The exponential of the log...
eflgam 24571 The exponential of the log...
gamne0 24572 The Gamma function is neve...
igamval 24573 Value of the inverse Gamma...
igamz 24574 Value of the inverse Gamma...
igamgam 24575 Value of the inverse Gamma...
igamlgam 24576 Value of the inverse Gamma...
igamf 24577 Closure of the inverse Gam...
igamcl 24578 Closure of the inverse Gam...
gamigam 24579 The Gamma function is the ...
lgamcvg 24580 The series ` G ` converges...
lgamcvg2 24581 The series ` G ` converges...
gamcvg 24582 The pointwise exponential ...
lgamp1 24583 The functional equation of...
gamp1 24584 The functional equation of...
gamcvg2lem 24585 Lemma for ~ gamcvg2 . (Co...
gamcvg2 24586 An infinite product expres...
regamcl 24587 The Gamma function is real...
relgamcl 24588 The log-Gamma function is ...
rpgamcl 24589 The log-Gamma function is ...
lgam1 24590 The log-Gamma function at ...
gam1 24591 The log-Gamma function at ...
facgam 24592 The Gamma function general...
gamfac 24593 The Gamma function general...
wilthlem1 24594 The only elements that are...
wilthlem2 24595 Lemma for ~ wilth : induct...
wilthlem3 24596 Lemma for ~ wilth . Here ...
wilth 24597 Wilson's theorem. A numbe...
wilthimp 24598 The forward implication of...
ftalem1 24599 Lemma for ~ fta : "growth...
ftalem2 24600 Lemma for ~ fta . There e...
ftalem3 24601 Lemma for ~ fta . There e...
ftalem4 24602 Lemma for ~ fta : Closure...
ftalem5 24603 Lemma for ~ fta : Main pr...
ftalem6 24604 Lemma for ~ fta : Dischar...
ftalem7 24605 Lemma for ~ fta . Shift t...
fta 24606 The Fundamental Theorem of...
basellem1 24607 Lemma for ~ basel . Closu...
basellem2 24608 Lemma for ~ basel . Show ...
basellem3 24609 Lemma for ~ basel . Using...
basellem4 24610 Lemma for ~ basel . By ~ ...
basellem5 24611 Lemma for ~ basel . Using...
basellem6 24612 Lemma for ~ basel . The f...
basellem7 24613 Lemma for ~ basel . The f...
basellem8 24614 Lemma for ~ basel . The f...
basellem9 24615 Lemma for ~ basel . Since...
basel 24616 The sum of the inverse squ...
efnnfsumcl 24629 Finite sum closure in the ...
ppisval 24630 The set of primes less tha...
ppisval2 24631 The set of primes less tha...
ppifi 24632 The set of primes less tha...
prmdvdsfi 24633 The set of prime divisors ...
chtf 24634 Domain and range of the Ch...
chtcl 24635 Real closure of the Chebys...
chtval 24636 Value of the Chebyshev fun...
efchtcl 24637 The Chebyshev function is ...
chtge0 24638 The Chebyshev function is ...
vmaval 24639 Value of the von Mangoldt ...
isppw 24640 Two ways to say that ` A `...
isppw2 24641 Two ways to say that ` A `...
vmappw 24642 Value of the von Mangoldt ...
vmaprm 24643 Value of the von Mangoldt ...
vmacl 24644 Closure for the von Mangol...
vmaf 24645 Functionality of the von M...
efvmacl 24646 The von Mangoldt is closed...
vmage0 24647 The von Mangoldt function ...
chpval 24648 Value of the second Chebys...
chpf 24649 Functionality of the secon...
chpcl 24650 Closure for the second Che...
efchpcl 24651 The second Chebyshev funct...
chpge0 24652 The second Chebyshev funct...
ppival 24653 Value of the prime-countin...
ppival2 24654 Value of the prime-countin...
ppival2g 24655 Value of the prime-countin...
ppif 24656 Domain and range of the pr...
ppicl 24657 Real closure of the prime-...
muval 24658 The value of the Möbi...
muval1 24659 The value of the Möbi...
muval2 24660 The value of the Möbi...
isnsqf 24661 Two ways to say that a num...
issqf 24662 Two ways to say that a num...
sqfpc 24663 The prime count of a squar...
dvdssqf 24664 A divisor of a squarefree ...
sqf11 24665 A squarefree number is com...
muf 24666 The Möbius function i...
mucl 24667 Closure of the Möbius...
sgmval 24668 The value of the divisor f...
sgmval2 24669 The value of the divisor f...
0sgm 24670 The value of the sum-of-di...
sgmf 24671 The divisor function is a ...
sgmcl 24672 Closure of the divisor fun...
sgmnncl 24673 Closure of the divisor fun...
mule1 24674 The Möbius function t...
chtfl 24675 The Chebyshev function doe...
chpfl 24676 The second Chebyshev funct...
ppiprm 24677 The prime-counting functio...
ppinprm 24678 The prime-counting functio...
chtprm 24679 The Chebyshev function at ...
chtnprm 24680 The Chebyshev function at ...
chpp1 24681 The second Chebyshev funct...
chtwordi 24682 The Chebyshev function is ...
chpwordi 24683 The second Chebyshev funct...
chtdif 24684 The difference of the Cheb...
efchtdvds 24685 The exponentiated Chebyshe...
ppifl 24686 The prime-counting functio...
ppip1le 24687 The prime-counting functio...
ppiwordi 24688 The prime-counting functio...
ppidif 24689 The difference of the prim...
ppi1 24690 The prime-counting functio...
cht1 24691 The Chebyshev function at ...
vma1 24692 The von Mangoldt function ...
chp1 24693 The second Chebyshev funct...
ppi1i 24694 Inference form of ~ ppiprm...
ppi2i 24695 Inference form of ~ ppinpr...
ppi2 24696 The prime-counting functio...
ppi3 24697 The prime-counting functio...
cht2 24698 The Chebyshev function at ...
cht3 24699 The Chebyshev function at ...
ppinncl 24700 Closure of the prime-count...
chtrpcl 24701 Closure of the Chebyshev f...
ppieq0 24702 The prime-counting functio...
ppiltx 24703 The prime-counting functio...
prmorcht 24704 Relate the primorial (prod...
mumullem1 24705 Lemma for ~ mumul . A mul...
mumullem2 24706 Lemma for ~ mumul . The p...
mumul 24707 The Möbius function i...
sqff1o 24708 There is a bijection from ...
fsumdvdsdiaglem 24709 A "diagonal commutation" o...
fsumdvdsdiag 24710 A "diagonal commutation" o...
fsumdvdscom 24711 A double commutation of di...
dvdsppwf1o 24712 A bijection from the divis...
dvdsflf1o 24713 A bijection from the numbe...
dvdsflsumcom 24714 A sum commutation from ` s...
fsumfldivdiaglem 24715 Lemma for ~ fsumfldivdiag ...
fsumfldivdiag 24716 The right-hand side of ~ d...
musum 24717 The sum of the Möbius...
musumsum 24718 Evaluate a collapsing sum ...
muinv 24719 The Möbius inversion ...
dvdsmulf1o 24720 If ` M ` and ` N ` are two...
fsumdvdsmul 24721 Product of two divisor sum...
sgmppw 24722 The value of the divisor f...
0sgmppw 24723 A prime power ` P ^ K ` ha...
1sgmprm 24724 The sum of divisors for a ...
1sgm2ppw 24725 The sum of the divisors of...
sgmmul 24726 The divisor function for f...
ppiublem1 24727 Lemma for ~ ppiub . (Cont...
ppiublem2 24728 A prime greater than ` 3 `...
ppiub 24729 An upper bound on the prim...
vmalelog 24730 The von Mangoldt function ...
chtlepsi 24731 The first Chebyshev functi...
chprpcl 24732 Closure of the second Cheb...
chpeq0 24733 The second Chebyshev funct...
chteq0 24734 The first Chebyshev functi...
chtleppi 24735 Upper bound on the ` theta...
chtublem 24736 Lemma for ~ chtub . (Cont...
chtub 24737 An upper bound on the Cheb...
fsumvma 24738 Rewrite a sum over the von...
fsumvma2 24739 Apply ~ fsumvma for the co...
pclogsum 24740 The logarithmic analogue o...
vmasum 24741 The sum of the von Mangold...
logfac2 24742 Another expression for the...
chpval2 24743 Express the second Chebysh...
chpchtsum 24744 The second Chebyshev funct...
chpub 24745 An upper bound on the seco...
logfacubnd 24746 A simple upper bound on th...
logfaclbnd 24747 A lower bound on the logar...
logfacbnd3 24748 Show the stronger statemen...
logfacrlim 24749 Combine the estimates ~ lo...
logexprlim 24750 The sum ` sum_ n <_ x , lo...
logfacrlim2 24751 Write out ~ logfacrlim as ...
mersenne 24752 A Mersenne prime is a prim...
perfect1 24753 Euclid's contribution to t...
perfectlem1 24754 Lemma for ~ perfect . (Co...
perfectlem2 24755 Lemma for ~ perfect . (Co...
perfect 24756 The Euclid-Euler theorem, ...
dchrval 24759 Value of the group of Diri...
dchrbas 24760 Base set of the group of D...
dchrelbas 24761 A Dirichlet character is a...
dchrelbas2 24762 A Dirichlet character is a...
dchrelbas3 24763 A Dirichlet character is a...
dchrelbasd 24764 A Dirichlet character is a...
dchrrcl 24765 Reverse closure for a Diri...
dchrmhm 24766 A Dirichlet character is a...
dchrf 24767 A Dirichlet character is a...
dchrelbas4 24768 A Dirichlet character is a...
dchrzrh1 24769 Value of a Dirichlet chara...
dchrzrhcl 24770 A Dirichlet character take...
dchrzrhmul 24771 A Dirichlet character is c...
dchrplusg 24772 Group operation on the gro...
dchrmul 24773 Group operation on the gro...
dchrmulcl 24774 Closure of the group opera...
dchrn0 24775 A Dirichlet character is n...
dchr1cl 24776 Closure of the principal D...
dchrmulid2 24777 Left identity for the prin...
dchrinvcl 24778 Closure of the group inver...
dchrabl 24779 The set of Dirichlet chara...
dchrfi 24780 The group of Dirichlet cha...
dchrghm 24781 A Dirichlet character rest...
dchr1 24782 Value of the principal Dir...
dchreq 24783 A Dirichlet character is d...
dchrresb 24784 A Dirichlet character is d...
dchrabs 24785 A Dirichlet character take...
dchrinv 24786 The inverse of a Dirichlet...
dchrabs2 24787 A Dirichlet character take...
dchr1re 24788 The principal Dirichlet ch...
dchrptlem1 24789 Lemma for ~ dchrpt . (Con...
dchrptlem2 24790 Lemma for ~ dchrpt . (Con...
dchrptlem3 24791 Lemma for ~ dchrpt . (Con...
dchrpt 24792 For any element other than...
dchrsum2 24793 An orthogonality relation ...
dchrsum 24794 An orthogonality relation ...
sumdchr2 24795 Lemma for ~ sumdchr . (Co...
dchrhash 24796 There are exactly ` phi ( ...
sumdchr 24797 An orthogonality relation ...
dchr2sum 24798 An orthogonality relation ...
sum2dchr 24799 An orthogonality relation ...
bcctr 24800 Value of the central binom...
pcbcctr 24801 Prime count of a central b...
bcmono 24802 The binomial coefficient i...
bcmax 24803 The binomial coefficient t...
bcp1ctr 24804 Ratio of two central binom...
bclbnd 24805 A bound on the binomial co...
efexple 24806 Convert a bound on a power...
bpos1lem 24807 Lemma for ~ bpos1 . (Cont...
bpos1 24808 Bertrand's postulate, chec...
bposlem1 24809 An upper bound on the prim...
bposlem2 24810 There are no odd primes in...
bposlem3 24811 Lemma for ~ bpos . Since ...
bposlem4 24812 Lemma for ~ bpos . (Contr...
bposlem5 24813 Lemma for ~ bpos . Bound ...
bposlem6 24814 Lemma for ~ bpos . By usi...
bposlem7 24815 Lemma for ~ bpos . The fu...
bposlem8 24816 Lemma for ~ bpos . Evalua...
bposlem9 24817 Lemma for ~ bpos . Derive...
bpos 24818 Bertrand's postulate: ther...
zabsle1 24821 ` { -u 1 , 0 , 1 } ` is th...
lgslem1 24822 When ` a ` is coprime to t...
lgslem2 24823 The set ` Z ` of all integ...
lgslem3 24824 The set ` Z ` of all integ...
lgslem4 24825 The function ` F ` is clos...
lgsval 24826 Value of the Legendre symb...
lgsfval 24827 Value of the function ` F ...
lgsfcl2 24828 The function ` F ` is clos...
lgscllem 24829 The Legendre symbol is an ...
lgsfcl 24830 Closure of the function ` ...
lgsfle1 24831 The function ` F ` has mag...
lgsval2lem 24832 Lemma for ~ lgsval2 . (Co...
lgsval4lem 24833 Lemma for ~ lgsval4 . (Co...
lgscl2 24834 The Legendre symbol is an ...
lgs0 24835 The Legendre symbol when t...
lgscl 24836 The Legendre symbol is an ...
lgsle1 24837 The Legendre symbol has ab...
lgsval2 24838 The Legendre symbol at a p...
lgs2 24839 The Legendre symbol at ` 2...
lgsval3 24840 The Legendre symbol at an ...
lgsvalmod 24841 The Legendre symbol is equ...
lgsval4 24842 Restate ~ lgsval for nonze...
lgsfcl3 24843 Closure of the function ` ...
lgsval4a 24844 Same as ~ lgsval4 for posi...
lgscl1 24845 The value of the Legendre ...
lgsneg 24846 The Legendre symbol is eit...
lgsneg1 24847 The Legendre symbol for no...
lgsmod 24848 The Legendre (Jacobi) symb...
lgsdilem 24849 Lemma for ~ lgsdi and ~ lg...
lgsdir2lem1 24850 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem2 24851 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem3 24852 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem4 24853 Lemma for ~ lgsdir2 . (Co...
lgsdir2lem5 24854 Lemma for ~ lgsdir2 . (Co...
lgsdir2 24855 The Legendre symbol is com...
lgsdirprm 24856 The Legendre symbol is com...
lgsdir 24857 The Legendre symbol is com...
lgsdilem2 24858 Lemma for ~ lgsdi . (Cont...
lgsdi 24859 The Legendre symbol is com...
lgsne0 24860 The Legendre symbol is non...
lgsabs1 24861 The Legendre symbol is non...
lgssq 24862 The Legendre symbol at a s...
lgssq2 24863 The Legendre symbol at a s...
lgsprme0 24864 The Legendre symbol at any...
1lgs 24865 The Legendre symbol at ` 1...
lgs1 24866 The Legendre symbol at ` 1...
lgsmodeq 24867 The Legendre (Jacobi) symb...
lgsmulsqcoprm 24868 The Legendre (Jacobi) symb...
lgsdirnn0 24869 Variation on ~ lgsdir vali...
lgsdinn0 24870 Variation on ~ lgsdi valid...
lgsqrlem1 24871 Lemma for ~ lgsqr . (Cont...
lgsqrlem2 24872 Lemma for ~ lgsqr . (Cont...
lgsqrlem3 24873 Lemma for ~ lgsqr . (Cont...
lgsqrlem4 24874 Lemma for ~ lgsqr . (Cont...
lgsqrlem5 24875 Lemma for ~ lgsqr . (Cont...
lgsqr 24876 The Legendre symbol for od...
lgsqrmod 24877 If the Legendre symbol of ...
lgsqrmodndvds 24878 If the Legendre symbol of ...
lgsdchrval 24879 The Legendre symbol functi...
lgsdchr 24880 The Legendre symbol functi...
gausslemma2dlem0a 24881 Auxiliary lemma 1 for ~ ga...
gausslemma2dlem0b 24882 Auxiliary lemma 2 for ~ ga...
gausslemma2dlem0c 24883 Auxiliary lemma 3 for ~ ga...
gausslemma2dlem0d 24884 Auxiliary lemma 4 for ~ ga...
gausslemma2dlem0e 24885 Auxiliary lemma 5 for ~ ga...
gausslemma2dlem0f 24886 Auxiliary lemma 6 for ~ ga...
gausslemma2dlem0g 24887 Auxiliary lemma 7 for ~ ga...
gausslemma2dlem0h 24888 Auxiliary lemma 8 for ~ ga...
gausslemma2dlem0i 24889 Auxiliary lemma 9 for ~ ga...
gausslemma2dlem1a 24890 Lemma for ~ gausslemma2dle...
gausslemma2dlem1 24891 Lemma 1 for ~ gausslemma2d...
gausslemma2dlem2 24892 Lemma 2 for ~ gausslemma2d...
gausslemma2dlem3 24893 Lemma 3 for ~ gausslemma2d...
gausslemma2dlem4 24894 Lemma 4 for ~ gausslemma2d...
gausslemma2dlem5a 24895 Lemma for ~ gausslemma2dle...
gausslemma2dlem5 24896 Lemma 5 for ~ gausslemma2d...
gausslemma2dlem6 24897 Lemma 6 for ~ gausslemma2d...
gausslemma2dlem7 24898 Lemma 7 for ~ gausslemma2d...
gausslemma2d 24899 Gauss' Lemma (see also the...
lgseisenlem1 24900 Lemma for ~ lgseisen . If...
lgseisenlem2 24901 Lemma for ~ lgseisen . Th...
lgseisenlem3 24902 Lemma for ~ lgseisen . (C...
lgseisenlem4 24903 Lemma for ~ lgseisen . Th...
lgseisen 24904 Eisenstein's lemma, an exp...
lgsquadlem1 24905 Lemma for ~ lgsquad . Cou...
lgsquadlem2 24906 Lemma for ~ lgsquad . Cou...
lgsquadlem3 24907 Lemma for ~ lgsquad . (Co...
lgsquad 24908 The Law of Quadratic Recip...
lgsquad2lem1 24909 Lemma for ~ lgsquad2 . (C...
lgsquad2lem2 24910 Lemma for ~ lgsquad2 . (C...
lgsquad2 24911 Extend ~ lgsquad to coprim...
lgsquad3 24912 Extend ~ lgsquad2 to integ...
m1lgs 24913 The first supplement to th...
2lgslem1a1 24914 Lemma 1 for ~ 2lgslem1a . ...
2lgslem1a2 24915 Lemma 2 for ~ 2lgslem1a . ...
2lgslem1a 24916 Lemma 1 for ~ 2lgslem1 . ...
2lgslem1b 24917 Lemma 2 for ~ 2lgslem1 . ...
2lgslem1c 24918 Lemma 3 for ~ 2lgslem1 . ...
2lgslem1 24919 Lemma 1 for ~ 2lgs . (Con...
2lgslem2 24920 Lemma 2 for ~ 2lgs . (Con...
2lgslem3a 24921 Lemma for ~ 2lgslem3a1 . ...
2lgslem3b 24922 Lemma for ~ 2lgslem3b1 . ...
2lgslem3c 24923 Lemma for ~ 2lgslem3c1 . ...
2lgslem3d 24924 Lemma for ~ 2lgslem3d1 . ...
2lgslem3a1 24925 Lemma 1 for ~ 2lgslem3 . ...
2lgslem3b1 24926 Lemma 2 for ~ 2lgslem3 . ...
2lgslem3c1 24927 Lemma 3 for ~ 2lgslem3 . ...
2lgslem3d1 24928 Lemma 4 for ~ 2lgslem3 . ...
2lgslem3 24929 Lemma 3 for ~ 2lgs . (Con...
2lgs2 24930 The Legendre symbol for ` ...
2lgslem4 24931 Lemma 4 for ~ 2lgs : speci...
2lgs 24932 The second supplement to t...
2lgsoddprmlem1 24933 Lemma 1 for ~ 2lgsoddprm ....
2lgsoddprmlem2 24934 Lemma 2 for ~ 2lgsoddprm ....
2lgsoddprmlem3a 24935 Lemma 1 for ~ 2lgsoddprmle...
2lgsoddprmlem3b 24936 Lemma 2 for ~ 2lgsoddprmle...
2lgsoddprmlem3c 24937 Lemma 3 for ~ 2lgsoddprmle...
2lgsoddprmlem3d 24938 Lemma 4 for ~ 2lgsoddprmle...
2lgsoddprmlem3 24939 Lemma 3 for ~ 2lgsoddprm ....
2lgsoddprmlem4 24940 Lemma 4 for ~ 2lgsoddprm ....
2lgsoddprm 24941 The second supplement to t...
2sqlem1 24942 Lemma for ~ 2sq . (Contri...
2sqlem2 24943 Lemma for ~ 2sq . (Contri...
mul2sq 24944 Fibonacci's identity (actu...
2sqlem3 24945 Lemma for ~ 2sqlem5 . (Co...
2sqlem4 24946 Lemma for ~ 2sqlem5 . (Co...
2sqlem5 24947 Lemma for ~ 2sq . If a nu...
2sqlem6 24948 Lemma for ~ 2sq . If a nu...
2sqlem7 24949 Lemma for ~ 2sq . (Contri...
2sqlem8a 24950 Lemma for ~ 2sqlem8 . (Co...
2sqlem8 24951 Lemma for ~ 2sq . (Contri...
2sqlem9 24952 Lemma for ~ 2sq . (Contri...
2sqlem10 24953 Lemma for ~ 2sq . Every f...
2sqlem11 24954 Lemma for ~ 2sq . (Contri...
2sq 24955 All primes of the form ` 4...
2sqblem 24956 The converse to ~ 2sq . (...
2sqb 24957 The converse to ~ 2sq . (...
chebbnd1lem1 24958 Lemma for ~ chebbnd1 : sho...
chebbnd1lem2 24959 Lemma for ~ chebbnd1 : Sh...
chebbnd1lem3 24960 Lemma for ~ chebbnd1 : get...
chebbnd1 24961 The Chebyshev bound: The ...
chtppilimlem1 24962 Lemma for ~ chtppilim . (...
chtppilimlem2 24963 Lemma for ~ chtppilim . (...
chtppilim 24964 The ` theta ` function is ...
chto1ub 24965 The ` theta ` function is ...
chebbnd2 24966 The Chebyshev bound, part ...
chto1lb 24967 The ` theta ` function is ...
chpchtlim 24968 The ` psi ` and ` theta ` ...
chpo1ub 24969 The ` psi ` function is up...
chpo1ubb 24970 The ` psi ` function is up...
vmadivsum 24971 The sum of the von Mangold...
vmadivsumb 24972 Give a total bound on the ...
rplogsumlem1 24973 Lemma for ~ rplogsum . (C...
rplogsumlem2 24974 Lemma for ~ rplogsum . Eq...
dchrisum0lem1a 24975 Lemma for ~ dchrisum0lem1 ...
rpvmasumlem 24976 Lemma for ~ rpvmasum . Ca...
dchrisumlema 24977 Lemma for ~ dchrisum . Le...
dchrisumlem1 24978 Lemma for ~ dchrisum . Le...
dchrisumlem2 24979 Lemma for ~ dchrisum . Le...
dchrisumlem3 24980 Lemma for ~ dchrisum . Le...
dchrisum 24981 If ` n e. [ M , +oo ) |-> ...
dchrmusumlema 24982 Lemma for ~ dchrmusum and ...
dchrmusum2 24983 The sum of the Möbius...
dchrvmasumlem1 24984 An alternative expression ...
dchrvmasum2lem 24985 Give an expression for ` l...
dchrvmasum2if 24986 Combine the results of ~ d...
dchrvmasumlem2 24987 Lemma for ~ dchrvmasum . ...
dchrvmasumlem3 24988 Lemma for ~ dchrvmasum . ...
dchrvmasumlema 24989 Lemma for ~ dchrvmasum and...
dchrvmasumiflem1 24990 Lemma for ~ dchrvmasumif ....
dchrvmasumiflem2 24991 Lemma for ~ dchrvmasum . ...
dchrvmasumif 24992 An asymptotic approximatio...
dchrvmaeq0 24993 The set ` W ` is the colle...
dchrisum0fval 24994 Value of the function ` F ...
dchrisum0fmul 24995 The function ` F ` , the d...
dchrisum0ff 24996 The function ` F ` is a re...
dchrisum0flblem1 24997 Lemma for ~ dchrisum0flb ....
dchrisum0flblem2 24998 Lemma for ~ dchrisum0flb ....
dchrisum0flb 24999 The divisor sum of a real ...
dchrisum0fno1 25000 The sum ` sum_ k <_ x , F ...
rpvmasum2 25001 A partial result along the...
dchrisum0re 25002 Suppose ` X ` is a non-pri...
dchrisum0lema 25003 Lemma for ~ dchrisum0 . A...
dchrisum0lem1b 25004 Lemma for ~ dchrisum0lem1 ...
dchrisum0lem1 25005 Lemma for ~ dchrisum0 . (...
dchrisum0lem2a 25006 Lemma for ~ dchrisum0 . (...
dchrisum0lem2 25007 Lemma for ~ dchrisum0 . (...
dchrisum0lem3 25008 Lemma for ~ dchrisum0 . (...
dchrisum0 25009 The sum ` sum_ n e. NN , X...
dchrisumn0 25010 The sum ` sum_ n e. NN , X...
dchrmusumlem 25011 The sum of the Möbius...
dchrvmasumlem 25012 The sum of the Möbius...
dchrmusum 25013 The sum of the Möbius...
dchrvmasum 25014 The sum of the von Mangold...
rpvmasum 25015 The sum of the von Mangold...
rplogsum 25016 The sum of ` log p / p ` o...
dirith2 25017 Dirichlet's theorem: there...
dirith 25018 Dirichlet's theorem: there...
mudivsum 25019 Asymptotic formula for ` s...
mulogsumlem 25020 Lemma for ~ mulogsum . (C...
mulogsum 25021 Asymptotic formula for ...
logdivsum 25022 Asymptotic analysis of ...
mulog2sumlem1 25023 Asymptotic formula for ...
mulog2sumlem2 25024 Lemma for ~ mulog2sum . (...
mulog2sumlem3 25025 Lemma for ~ mulog2sum . (...
mulog2sum 25026 Asymptotic formula for ...
vmalogdivsum2 25027 The sum ` sum_ n <_ x , La...
vmalogdivsum 25028 The sum ` sum_ n <_ x , La...
2vmadivsumlem 25029 Lemma for ~ 2vmadivsum . ...
2vmadivsum 25030 The sum ` sum_ m n <_ x , ...
logsqvma 25031 A formula for ` log ^ 2 ( ...
logsqvma2 25032 The Möbius inverse of...
log2sumbnd 25033 Bound on the difference be...
selberglem1 25034 Lemma for ~ selberg . Est...
selberglem2 25035 Lemma for ~ selberg . (Co...
selberglem3 25036 Lemma for ~ selberg . Est...
selberg 25037 Selberg's symmetry formula...
selbergb 25038 Convert eventual boundedne...
selberg2lem 25039 Lemma for ~ selberg2 . Eq...
selberg2 25040 Selberg's symmetry formula...
selberg2b 25041 Convert eventual boundedne...
chpdifbndlem1 25042 Lemma for ~ chpdifbnd . (...
chpdifbndlem2 25043 Lemma for ~ chpdifbnd . (...
chpdifbnd 25044 A bound on the difference ...
logdivbnd 25045 A bound on a sum of logs, ...
selberg3lem1 25046 Introduce a log weighting ...
selberg3lem2 25047 Lemma for ~ selberg3 . Eq...
selberg3 25048 Introduce a log weighting ...
selberg4lem1 25049 Lemma for ~ selberg4 . Eq...
selberg4 25050 The Selberg symmetry formu...
pntrval 25051 Define the residual of the...
pntrf 25052 Functionality of the resid...
pntrmax 25053 There is a bound on the re...
pntrsumo1 25054 A bound on a sum over ` R ...
pntrsumbnd 25055 A bound on a sum over ` R ...
pntrsumbnd2 25056 A bound on a sum over ` R ...
selbergr 25057 Selberg's symmetry formula...
selberg3r 25058 Selberg's symmetry formula...
selberg4r 25059 Selberg's symmetry formula...
selberg34r 25060 The sum of ~ selberg3r and...
pntsval 25061 Define the "Selberg functi...
pntsf 25062 Functionality of the Selbe...
selbergs 25063 Selberg's symmetry formula...
selbergsb 25064 Selberg's symmetry formula...
pntsval2 25065 The Selberg function can b...
pntrlog2bndlem1 25066 The sum of ~ selberg3r and...
pntrlog2bndlem2 25067 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem3 25068 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem4 25069 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem5 25070 Lemma for ~ pntrlog2bnd . ...
pntrlog2bndlem6a 25071 Lemma for ~ pntrlog2bndlem...
pntrlog2bndlem6 25072 Lemma for ~ pntrlog2bnd . ...
pntrlog2bnd 25073 A bound on ` R ( x ) log ^...
pntpbnd1a 25074 Lemma for ~ pntpbnd . (Co...
pntpbnd1 25075 Lemma for ~ pntpbnd . (Co...
pntpbnd2 25076 Lemma for ~ pntpbnd . (Co...
pntpbnd 25077 Lemma for ~ pnt . Establi...
pntibndlem1 25078 Lemma for ~ pntibnd . (Co...
pntibndlem2a 25079 Lemma for ~ pntibndlem2 . ...
pntibndlem2 25080 Lemma for ~ pntibnd . The...
pntibndlem3 25081 Lemma for ~ pntibnd . Pac...
pntibnd 25082 Lemma for ~ pnt . Establi...
pntlemd 25083 Lemma for ~ pnt . Closure...
pntlemc 25084 Lemma for ~ pnt . Closure...
pntlema 25085 Lemma for ~ pnt . Closure...
pntlemb 25086 Lemma for ~ pnt . Unpack ...
pntlemg 25087 Lemma for ~ pnt . Closure...
pntlemh 25088 Lemma for ~ pnt . Bounds ...
pntlemn 25089 Lemma for ~ pnt . The "na...
pntlemq 25090 Lemma for ~ pntlemj . (Co...
pntlemr 25091 Lemma for ~ pntlemj . (Co...
pntlemj 25092 Lemma for ~ pnt . The ind...
pntlemi 25093 Lemma for ~ pnt . Elimina...
pntlemf 25094 Lemma for ~ pnt . Add up ...
pntlemk 25095 Lemma for ~ pnt . Evaluat...
pntlemo 25096 Lemma for ~ pnt . Combine...
pntleme 25097 Lemma for ~ pnt . Package...
pntlem3 25098 Lemma for ~ pnt . Equatio...
pntlemp 25099 Lemma for ~ pnt . Wrappin...
pntleml 25100 Lemma for ~ pnt . Equatio...
pnt3 25101 The Prime Number Theorem, ...
pnt2 25102 The Prime Number Theorem, ...
pnt 25103 The Prime Number Theorem: ...
abvcxp 25104 Raising an absolute value ...
padicfval 25105 Value of the p-adic absolu...
padicval 25106 Value of the p-adic absolu...
ostth2lem1 25107 Lemma for ~ ostth2 , altho...
qrngbas 25108 The base set of the field ...
qdrng 25109 The rationals form a divis...
qrng0 25110 The zero element of the fi...
qrng1 25111 The unit element of the fi...
qrngneg 25112 The additive inverse in th...
qrngdiv 25113 The division operation in ...
qabvle 25114 By using induction on ` N ...
qabvexp 25115 Induct the product rule ~ ...
ostthlem1 25116 Lemma for ~ ostth . If tw...
ostthlem2 25117 Lemma for ~ ostth . Refin...
qabsabv 25118 The regular absolute value...
padicabv 25119 The p-adic absolute value ...
padicabvf 25120 The p-adic absolute value ...
padicabvcxp 25121 All positive powers of the...
ostth1 25122 - Lemma for ~ ostth : triv...
ostth2lem2 25123 Lemma for ~ ostth2 . (Con...
ostth2lem3 25124 Lemma for ~ ostth2 . (Con...
ostth2lem4 25125 Lemma for ~ ostth2 . (Con...
ostth2 25126 - Lemma for ~ ostth : regu...
ostth3 25127 - Lemma for ~ ostth : p-ad...
ostth 25128 Ostrowski's theorem, which...
itvndx 25139 Index value of the Interva...
lngndx 25140 Index value of the "line" ...
itvid 25141 Utility theorem: index-ind...
lngid 25142 Utility theorem: index-ind...
trkgstr 25143 Functionality of a Tarski ...
trkgbas 25144 The base set of a Tarski g...
trkgdist 25145 The measure of a distance ...
trkgitv 25146 The congruence relation in...
istrkgc 25153 Property of being a Tarski...
istrkgb 25154 Property of being a Tarski...
istrkgcb 25155 Property of being a Tarski...
istrkge 25156 Property of fulfilling Euc...
istrkgl 25157 Building lines from the se...
istrkgld 25158 Property of fulfilling the...
istrkg2ld 25159 Property of fulfilling the...
istrkg3ld 25160 Property of fulfilling the...
axtgcgrrflx 25161 Axiom of reflexivity of co...
axtgcgrid 25162 Axiom of identity of congr...
axtgsegcon 25163 Axiom of segment construct...
axtg5seg 25164 Five segments axiom, Axiom...
axtgbtwnid 25165 Identity of Betweenness. ...
axtgpasch 25166 Axiom of (Inner) Pasch, Ax...
axtgcont1 25167 Axiom of Continuity. Axio...
axtgcont 25168 Axiom of Continuity. Axio...
axtglowdim2 25169 Lower dimension axiom for ...
axtgupdim2 25170 Upper dimension axiom for ...
axtgeucl 25171 Euclid's Axiom. Axiom A10...
tgcgrcomimp 25172 Congruence commutes on the...
tgcgrcomr 25173 Congruence commutes on the...
tgcgrcoml 25174 Congruence commutes on the...
tgcgrcomlr 25175 Congruence commutes on bot...
tgcgreqb 25176 Congruence and equality. ...
tgcgreq 25177 Congruence and equality. ...
tgcgrneq 25178 Congruence and equality. ...
tgcgrtriv 25179 Degenerate segments are co...
tgcgrextend 25180 Link congruence over a pai...
tgsegconeq 25181 Two points that satisfy th...
tgbtwntriv2 25182 Betweenness always holds f...
tgbtwncom 25183 Betweenness commutes. The...
tgbtwncomb 25184 Betweenness commutes, bico...
tgbtwnne 25185 Betweenness and inequality...
tgbtwntriv1 25186 Betweenness always holds f...
tgbtwnswapid 25187 If you can swap the first ...
tgbtwnintr 25188 Inner transitivity law for...
tgbtwnexch3 25189 Exchange the first endpoin...
tgbtwnouttr2 25190 Outer transitivity law for...
tgbtwnexch2 25191 Exchange the outer point o...
tgbtwnouttr 25192 Outer transitivity law for...
tgbtwnexch 25193 Outer transitivity law for...
tgtrisegint 25194 A line segment between two...
tglowdim1 25195 Lower dimension axiom for ...
tglowdim1i 25196 Lower dimension axiom for ...
tgldimor 25197 Excluded-middle like state...
tgldim0eq 25198 In dimension zero, any two...
tgldim0itv 25199 In dimension zero, any two...
tgldim0cgr 25200 In dimension zero, any two...
tgbtwndiff 25201 There is always a ` c ` di...
tgdim01 25202 In geometries of dimension...
tgifscgr 25203 Inner five segment congrue...
tgcgrsub 25204 Removing identical parts f...
iscgrg 25207 The congruence property fo...
iscgrgd 25208 The property for two seque...
iscgrglt 25209 The property for two seque...
trgcgrg 25210 The property for two trian...
trgcgr 25211 Triangle congruence. (Con...
ercgrg 25212 The shape congruence relat...
tgcgrxfr 25213 A line segment can be divi...
cgr3id 25214 Reflexivity law for three-...
cgr3simp1 25215 Deduce segment congruence ...
cgr3simp2 25216 Deduce segment congruence ...
cgr3simp3 25217 Deduce segment congruence ...
cgr3swap12 25218 Permutation law for three-...
cgr3swap23 25219 Permutation law for three-...
cgr3swap13 25220 Permutation law for three-...
cgr3rotr 25221 Permutation law for three-...
cgr3rotl 25222 Permutation law for three-...
trgcgrcom 25223 Commutative law for three-...
cgr3tr 25224 Transitivity law for three...
tgbtwnxfr 25225 A condition for extending ...
tgcgr4 25226 Two quadrilaterals to be c...
isismt 25229 Property of being an isome...
ismot 25230 Property of being an isome...
motcgr 25231 Property of a motion: dist...
idmot 25232 The identity is a motion. ...
motf1o 25233 Motions are bijections. (...
motcl 25234 Closure of motions. (Cont...
motco 25235 The composition of two mot...
cnvmot 25236 The converse of a motion i...
motplusg 25237 The operation for motions ...
motgrp 25238 The motions of a geometry ...
motcgrg 25239 Property of a motion: dist...
motcgr3 25240 Property of a motion: dist...
tglng 25241 Lines of a Tarski Geometry...
tglnfn 25242 Lines as functions. (Cont...
tglnunirn 25243 Lines are sets of points. ...
tglnpt 25244 Lines are sets of points. ...
tglngne 25245 It takes two different poi...
tglngval 25246 The line going through poi...
tglnssp 25247 Lines are subset of the ge...
tgellng 25248 Property of lying on the l...
tgcolg 25249 We choose the notation ` (...
btwncolg1 25250 Betweenness implies coline...
btwncolg2 25251 Betweenness implies coline...
btwncolg3 25252 Betweenness implies coline...
colcom 25253 Swapping the points defini...
colrot1 25254 Rotating the points defini...
colrot2 25255 Rotating the points defini...
ncolcom 25256 Swapping non-colinear poin...
ncolrot1 25257 Rotating non-colinear poin...
ncolrot2 25258 Rotating non-colinear poin...
tgdim01ln 25259 In geometries of dimension...
ncoltgdim2 25260 If there are 3 non-colinea...
lnxfr 25261 Transfer law for colineari...
lnext 25262 Extend a line with a missi...
tgfscgr 25263 Congruence law for the gen...
lncgr 25264 Congruence rule for lines....
lnid 25265 Identity law for points on...
tgidinside 25266 Law for finding a point in...
tgbtwnconn1lem1 25267 Lemma for ~ tgbtwnconn1 . ...
tgbtwnconn1lem2 25268 Lemma for ~ tgbtwnconn1 . ...
tgbtwnconn1lem3 25269 Lemma for ~ tgbtwnconn1 . ...
tgbtwnconn1 25270 Connectivity law for betwe...
tgbtwnconn2 25271 Another connectivity law f...
tgbtwnconn3 25272 Inner connectivity law for...
tgbtwnconnln3 25273 Derive colinearity from be...
tgbtwnconn22 25274 Double connectivity law fo...
tgbtwnconnln1 25275 Derive colinearity from be...
tgbtwnconnln2 25276 Derive colinearity from be...
legval 25279 Value of the less-than rel...
legov 25280 Value of the less-than rel...
legov2 25281 An equivalent definition o...
legid 25282 Reflexivity of the less-th...
btwnleg 25283 Betweenness implies less-t...
legtrd 25284 Transitivity of the less-t...
legtri3 25285 Equality from the less-tha...
legtrid 25286 Trichotomy law for the les...
leg0 25287 Degenerated (zero-length) ...
legeq 25288 Deduce equality from "less...
legbtwn 25289 Deduce betweenness from "l...
tgcgrsub2 25290 Removing identical parts f...
ltgseg 25291 The set ` E ` denotes the ...
ltgov 25292 Strict "shorter than" geom...
legov3 25293 An equivalent definition o...
legso 25294 The shorter-than relations...
ishlg 25297 Rays : Definition 6.1 of ...
hlcomb 25298 The half-line relation com...
hlcomd 25299 The half-line relation com...
hlne1 25300 The half-line relation imp...
hlne2 25301 The half-line relation imp...
hlln 25302 The half-line relation imp...
hleqnid 25303 The endpoint does not belo...
hlid 25304 The half-line relation is ...
hltr 25305 The half-line relation is ...
hlbtwn 25306 Betweenness is a sufficien...
btwnhl1 25307 Deduce half-line from betw...
btwnhl2 25308 Deduce half-line from betw...
btwnhl 25309 Swap betweenness for a hal...
lnhl 25310 Either a point ` C ` on th...
hlcgrex 25311 Construct a point on a hal...
hlcgreulem 25312 Lemma for ~ hlcgreu . (Co...
hlcgreu 25313 The point constructed in ~...
btwnlng1 25314 Betweenness implies coline...
btwnlng2 25315 Betweenness implies coline...
btwnlng3 25316 Betweenness implies coline...
lncom 25317 Swapping the points defini...
lnrot1 25318 Rotating the points defini...
lnrot2 25319 Rotating the points defini...
ncolne1 25320 Non-colinear points are di...
ncolne2 25321 Non-colinear points are di...
tgisline 25322 The property of being a pr...
tglnne 25323 It takes two different poi...
tglndim0 25324 There are no lines in dime...
tgelrnln 25325 The property of being a pr...
tglineeltr 25326 Transitivity law for lines...
tglineelsb2 25327 If ` S ` lies on PQ , then...
tglinerflx1 25328 Reflexivity law for line m...
tglinerflx2 25329 Reflexivity law for line m...
tglinecom 25330 Commutativity law for line...
tglinethru 25331 If ` A ` is a line contain...
tghilberti1 25332 There is a line through an...
tghilberti2 25333 There is at most one line ...
tglinethrueu 25334 There is a unique line goi...
tglnne0 25335 A line ` A ` has at least ...
tglnpt2 25336 Find a second point on a l...
tglineintmo 25337 Two distinct lines interse...
tglineineq 25338 Two distinct lines interse...
tglineneq 25339 Given three non-colinear p...
tglineinteq 25340 Two distinct lines interse...
ncolncol 25341 Deduce non-colinearity fro...
coltr 25342 A transitivity law for col...
coltr3 25343 A transitivity law for col...
colline 25344 Three points are colinear ...
tglowdim2l 25345 Reformulation of the lower...
tglowdim2ln 25346 There is always one point ...
mirreu3 25349 Existential uniqueness of ...
mirval 25350 Value of the point inversi...
mirfv 25351 Value of the point inversi...
mircgr 25352 Property of the image by t...
mirbtwn 25353 Property of the image by t...
ismir 25354 Property of the image by t...
mirf 25355 Point inversion as functio...
mircl 25356 Closure of the point inver...
mirmir 25357 The point inversion functi...
mircom 25358 Variation on ~ mirmir . (...
mirreu 25359 Any point has a unique ant...
mireq 25360 Equality deduction for poi...
mirinv 25361 The only invariant point o...
mirne 25362 Mirror of non-center point...
mircinv 25363 The center point is invari...
mirf1o 25364 The point inversion functi...
miriso 25365 The point inversion functi...
mirbtwni 25366 Point inversion preserves ...
mirbtwnb 25367 Point inversion preserves ...
mircgrs 25368 Point inversion preserves ...
mirmir2 25369 Point inversion of a point...
mirmot 25370 Point investion is a motio...
mirln 25371 If two points are on the s...
mirln2 25372 If a point and its mirror ...
mirconn 25373 Point inversion of connect...
mirhl 25374 If two points ` X ` and ` ...
mirbtwnhl 25375 If the center of the point...
mirhl2 25376 Deduce half-line relation ...
mircgrextend 25377 Link congruence over a pai...
mirtrcgr 25378 Point inversion of one poi...
mirauto 25379 Point inversion preserves ...
miduniq 25380 Unicity of the middle poin...
miduniq1 25381 Unicity of the middle poin...
miduniq2 25382 If two point inversions co...
colmid 25383 Colinearity and equidistan...
symquadlem 25384 Lemma of the symetrial qua...
krippenlem 25385 Lemma for ~ krippen . We ...
krippen 25386 Krippenlemma (German for c...
midexlem 25387 Lemma for the existence of...
israg 25392 Property for 3 points A, B...
ragcom 25393 Commutative rule for right...
ragcol 25394 The right angle property i...
ragmir 25395 Right angle property is pr...
mirrag 25396 Right angle is conserved b...
ragtrivb 25397 Trivial right angle. Theo...
ragflat2 25398 Deduce equality from two r...
ragflat 25399 Deduce equality from two r...
ragtriva 25400 Trivial right angle. Theo...
ragflat3 25401 Right angle and colinearit...
ragcgr 25402 Right angle and colinearit...
motrag 25403 Right angles are preserved...
ragncol 25404 Right angle implies non-co...
perpln1 25405 Derive a line from perpend...
perpln2 25406 Derive a line from perpend...
isperp 25407 Property for 2 lines A, B ...
perpcom 25408 The "perpendicular" relati...
perpneq 25409 Two perpendicular lines ar...
isperp2 25410 Property for 2 lines A, B,...
isperp2d 25411 One direction of ~ isperp2...
ragperp 25412 Deduce that two lines are ...
footex 25413 Lemma for ~ foot : existen...
foot 25414 From a point ` C ` outside...
footne 25415 Uniqueness of the foot poi...
footeq 25416 Uniqueness of the foot poi...
hlperpnel 25417 A point on a half-line whi...
perprag 25418 Deduce a right angle from ...
perpdragALT 25419 Deduce a right angle from ...
perpdrag 25420 Deduce a right angle from ...
colperp 25421 Deduce a perpendicularity ...
colperpexlem1 25422 Lemma for ~ colperp . Fir...
colperpexlem2 25423 Lemma for ~ colperpex . S...
colperpexlem3 25424 Lemma for ~ colperpex . C...
colperpex 25425 In dimension 2 and above, ...
mideulem2 25426 Lemma for ~ opphllem , whi...
opphllem 25427 Lemma 8.24 of [Schwabhause...
mideulem 25428 Lemma for ~ mideu . We ca...
midex 25429 Existence of the midpoint,...
mideu 25430 Existence and uniqueness o...
islnopp 25431 The property for two point...
islnoppd 25432 Deduce that ` A ` and ` B ...
oppne1 25433 Points lying on opposite s...
oppne2 25434 Points lying on opposite s...
oppne3 25435 Points lying on opposite s...
oppcom 25436 Commutativity rule for "op...
opptgdim2 25437 If two points opposite to ...
oppnid 25438 The "opposite to a line" r...
opphllem1 25439 Lemma for ~ opphl . (Cont...
opphllem2 25440 Lemma for ~ opphl . Lemma...
opphllem3 25441 Lemma for ~ opphl : We as...
opphllem4 25442 Lemma for ~ opphl . (Cont...
opphllem5 25443 Second part of Lemma 9.4 o...
opphllem6 25444 First part of Lemma 9.4 of...
oppperpex 25445 Restating ~ colperpex usin...
opphl 25446 If two points ` A ` and ` ...
outpasch 25447 Axiom of Pasch, outer form...
hlpasch 25448 An application of the axio...
ishpg 25451 Value of the half-plane re...
hpgbr 25452 Half-planes : property for...
hpgne1 25453 Points on the open half pl...
hpgne2 25454 Points on the open half pl...
lnopp2hpgb 25455 Theorem 9.8 of [Schwabhaus...
lnoppnhpg 25456 If two points lie on the o...
hpgerlem 25457 Lemma for the proof that t...
hpgid 25458 The half-plane relation is...
hpgcom 25459 The half-plane relation co...
hpgtr 25460 The half-plane relation is...
colopp 25461 Opposite sides of a line f...
colhp 25462 Half-plane relation for co...
hphl 25463 If two points are on the s...
midf 25468 Midpoint as a function. (...
midcl 25469 Closure of the midpoint. ...
ismidb 25470 Property of the midpoint. ...
midbtwn 25471 Betweenness of midpoint. ...
midcgr 25472 Congruence of midpoint. (...
midid 25473 Midpoint of a null segment...
midcom 25474 Commutativity rule for the...
mirmid 25475 Point inversion preserves ...
lmieu 25476 Uniqueness of the line mir...
lmif 25477 Line mirror as a function....
lmicl 25478 Closure of the line mirror...
islmib 25479 Property of the line mirro...
lmicom 25480 The line mirroring functio...
lmilmi 25481 Line mirroring is an invol...
lmireu 25482 Any point has a unique ant...
lmieq 25483 Equality deduction for lin...
lmiinv 25484 The invariants of the line...
lmicinv 25485 The mirroring line is an i...
lmimid 25486 If we have a right angle, ...
lmif1o 25487 The line mirroring functio...
lmiisolem 25488 Lemma for ~ lmiiso . (Con...
lmiiso 25489 The line mirroring functio...
lmimot 25490 Line mirroring is a motion...
hypcgrlem1 25491 Lemma for ~ hypcgr , case ...
hypcgrlem2 25492 Lemma for ~ hypcgr , case ...
hypcgr 25493 If the catheti of two righ...
lmiopp 25494 Line mirroring produces po...
lnperpex 25495 Existence of a perpendicul...
trgcopy 25496 Triangle construction: a c...
trgcopyeulem 25497 Lemma for ~ trgcopyeu . (...
trgcopyeu 25498 Triangle construction: a c...
iscgra 25501 Property for two angles AB...
iscgra1 25502 A special version of ~ isc...
iscgrad 25503 Sufficient conditions for ...
cgrane1 25504 Angles imply inequality. ...
cgrane2 25505 Angles imply inequality. ...
cgrane3 25506 Angles imply inequality. ...
cgrane4 25507 Angles imply inequality. ...
cgrahl1 25508 Angle congruence is indepe...
cgrahl2 25509 Angle congruence is indepe...
cgracgr 25510 First direction of proposi...
cgraid 25511 Angle congruence is reflex...
cgraswap 25512 Swap rays in a congruence ...
cgrcgra 25513 Triangle congruence implie...
cgracom 25514 Angle congruence commutes....
cgratr 25515 Angle congruence is transi...
cgraswaplr 25516 Swap both side of angle co...
cgrabtwn 25517 Angle congruence preserves...
cgrahl 25518 Angle congruence preserves...
cgracol 25519 Angle congruence preserves...
cgrancol 25520 Angle congruence preserves...
dfcgra2 25521 This is the full statement...
sacgr 25522 Supplementary angles of co...
oacgr 25523 Vertical angle theorem. V...
acopy 25524 Angle construction. Theor...
acopyeu 25525 Angle construction. Theor...
isinag 25529 Property for point ` X ` t...
inagswap 25530 Swap the order of the half...
inaghl 25531 The "point lie in angle" r...
isleag 25533 Geometrical "less than" pr...
cgrg3col4 25534 Lemma 11.28 of [Schwabhaus...
tgsas1 25535 First congruence theorem: ...
tgsas 25536 First congruence theorem: ...
tgsas2 25537 First congruence theorem: ...
tgsas3 25538 First congruence theorem: ...
tgasa1 25539 Second congruence theorem:...
tgasa 25540 Second congruence theorem:...
tgsss1 25541 Third congruence theorem: ...
tgsss2 25542 Third congruence theorem: ...
tgsss3 25543 Third congruence theorem: ...
isoas 25544 Congruence theorem for iso...
iseqlg 25547 Property of a triangle bei...
iseqlgd 25548 Condition for a triangle t...
f1otrgds 25549 Convenient lemma for ~ f1o...
f1otrgitv 25550 Convenient lemma for ~ f1o...
f1otrg 25551 A bijection between bases ...
f1otrge 25552 A bijection between bases ...
ttgval 25555 Define a function to augme...
ttglem 25556 Lemma for ~ ttgbas and ~ t...
ttgbas 25557 The base set of a complex ...
ttgplusg 25558 The addition operation of ...
ttgsub 25559 The subtraction operation ...
ttgvsca 25560 The scalar product of a co...
ttgds 25561 The metric of a complex Hi...
ttgitvval 25562 Betweenness for a complex ...
ttgelitv 25563 Betweenness for a complex ...
ttgbtwnid 25564 Any complex module equippe...
ttgcontlem1 25565 Lemma for % ttgcont . (Co...
xmstrkgc 25566 Any metric space fulfills ...
cchhllem 25567 Lemma for chlbas and chlvs...
elee 25574 Membership in a Euclidean ...
mptelee 25575 A condition for a mapping ...
eleenn 25576 If ` A ` is in ` ( EE `` N...
eleei 25577 The forward direction of ~...
eedimeq 25578 A point belongs to at most...
brbtwn 25579 The binary relationship fo...
brcgr 25580 The binary relationship fo...
fveere 25581 The function value of a po...
fveecn 25582 The function value of a po...
eqeefv 25583 Two points are equal iff t...
eqeelen 25584 Two points are equal iff t...
brbtwn2 25585 Alternate characterization...
colinearalglem1 25586 Lemma for ~ colinearalg . ...
colinearalglem2 25587 Lemma for ~ colinearalg . ...
colinearalglem3 25588 Lemma for ~ colinearalg . ...
colinearalglem4 25589 Lemma for ~ colinearalg . ...
colinearalg 25590 An algebraic characterizat...
eleesub 25591 Membership of a subtractio...
eleesubd 25592 Membership of a subtractio...
axdimuniq 25593 The unique dimension axiom...
axcgrrflx 25594 ` A ` is as far from ` B `...
axcgrtr 25595 Congruence is transitive. ...
axcgrid 25596 If there is no distance be...
axsegconlem1 25597 Lemma for ~ axsegcon . Ha...
axsegconlem2 25598 Lemma for ~ axsegcon . Sh...
axsegconlem3 25599 Lemma for ~ axsegcon . Sh...
axsegconlem4 25600 Lemma for ~ axsegcon . Sh...
axsegconlem5 25601 Lemma for ~ axsegcon . Sh...
axsegconlem6 25602 Lemma for ~ axsegcon . Sh...
axsegconlem7 25603 Lemma for ~ axsegcon . Sh...
axsegconlem8 25604 Lemma for ~ axsegcon . Sh...
axsegconlem9 25605 Lemma for ~ axsegcon . Sh...
axsegconlem10 25606 Lemma for ~ axsegcon . Sh...
axsegcon 25607 Any segment ` A B ` can be...
ax5seglem1 25608 Lemma for ~ ax5seg . Rexp...
ax5seglem2 25609 Lemma for ~ ax5seg . Rexp...
ax5seglem3a 25610 Lemma for ~ ax5seg . (Con...
ax5seglem3 25611 Lemma for ~ ax5seg . Comb...
ax5seglem4 25612 Lemma for ~ ax5seg . Give...
ax5seglem5 25613 Lemma for ~ ax5seg . If `...
ax5seglem6 25614 Lemma for ~ ax5seg . Give...
ax5seglem7 25615 Lemma for ~ ax5seg . An a...
ax5seglem8 25616 Lemma for ~ ax5seg . Use ...
ax5seglem9 25617 Lemma for ~ ax5seg . Take...
ax5seg 25618 The five segment axiom. T...
axbtwnid 25619 Points are indivisible. T...
axpaschlem 25620 Lemma for ~ axpasch . Set...
axpasch 25621 The inner Pasch axiom. Ta...
axlowdimlem1 25622 Lemma for ~ axlowdim . Es...
axlowdimlem2 25623 Lemma for ~ axlowdim . Sh...
axlowdimlem3 25624 Lemma for ~ axlowdim . Se...
axlowdimlem4 25625 Lemma for ~ axlowdim . Se...
axlowdimlem5 25626 Lemma for ~ axlowdim . Sh...
axlowdimlem6 25627 Lemma for ~ axlowdim . Sh...
axlowdimlem7 25628 Lemma for ~ axlowdim . Se...
axlowdimlem8 25629 Lemma for ~ axlowdim . Ca...
axlowdimlem9 25630 Lemma for ~ axlowdim . Ca...
axlowdimlem10 25631 Lemma for ~ axlowdim . Se...
axlowdimlem11 25632 Lemma for ~ axlowdim . Ca...
axlowdimlem12 25633 Lemma for ~ axlowdim . Ca...
axlowdimlem13 25634 Lemma for ~ axlowdim . Es...
axlowdimlem14 25635 Lemma for ~ axlowdim . Ta...
axlowdimlem15 25636 Lemma for ~ axlowdim . Se...
axlowdimlem16 25637 Lemma for ~ axlowdim . Se...
axlowdimlem17 25638 Lemma for ~ axlowdim . Es...
axlowdim1 25639 The lower dimension axiom ...
axlowdim2 25640 The lower two-dimensional ...
axlowdim 25641 The general lower dimensio...
axeuclidlem 25642 Lemma for ~ axeuclid . Ha...
axeuclid 25643 Euclid's axiom. Take an a...
axcontlem1 25644 Lemma for ~ axcont . Chan...
axcontlem2 25645 Lemma for ~ axcont . The ...
axcontlem3 25646 Lemma for ~ axcont . Give...
axcontlem4 25647 Lemma for ~ axcont . Give...
axcontlem5 25648 Lemma for ~ axcont . Comp...
axcontlem6 25649 Lemma for ~ axcont . Stat...
axcontlem7 25650 Lemma for ~ axcont . Give...
axcontlem8 25651 Lemma for ~ axcont . A po...
axcontlem9 25652 Lemma for ~ axcont . Give...
axcontlem10 25653 Lemma for ~ axcont . Give...
axcontlem11 25654 Lemma for ~ axcont . Elim...
axcontlem12 25655 Lemma for ~ axcont . Elim...
axcont 25656 The axiom of continuity. ...
eengv 25659 The value of the Euclidean...
eengstr 25660 The Euclidean geometry as ...
eengbas 25661 The Base of the Euclidean ...
ebtwntg 25662 The betweenness relation u...
ecgrtg 25663 The congruence relation us...
elntg 25664 The line definition in the...
eengtrkg 25665 The geometry structure for...
eengtrkge 25666 The geometry structure for...
edgfndxnn 25669 The index value of the edg...
edgfndxid 25670 The value of the edge func...
baseltedgf 25671 The index value of the ` B...
slotsbaseefdif 25672 The slots ` Base ` and ` ....
vtxval 25677 The set of vertices of a g...
iedgval 25678 The set of indexed edges o...
1vgrex 25679 A graph with at least one ...
opvtxval 25680 The set of vertices of a g...
opvtxfv 25681 The set of vertices of a g...
opvtxov 25682 The set of vertices of a g...
opiedgval 25683 The set of indexed edges o...
opiedgfv 25684 The set of indexed edges o...
opiedgov 25685 The set of indexed edges o...
opvtxfvi 25686 The set of vertices of a g...
opiedgfvi 25687 The set of indexed edges o...
funvtxdm2val 25688 The set of vertices of an ...
funiedgdm2val 25689 The set of indexed edges o...
funvtxval0 25690 The set of vertices of an ...
funvtxdmge2val 25691 The set of vertices of an ...
funiedgdmge2val 25692 The set of indexed edges o...
basvtxval 25693 The set of vertices of a g...
edgfiedgval 25694 The set of indexed edges o...
funvtxval 25695 The set of vertices of a g...
funiedgval 25696 The set of indexed edges o...
structvtxvallem 25697 Lemma for ~ structvtxval a...
structvtxval 25698 The set of vertices of an ...
structiedg0val 25699 The set of indexed edges o...
structgrssvtxlem 25700 Lemma for ~ structgrssvtx ...
structgrssvtx 25701 The set of vertices of a g...
structgrssiedg 25702 The set of indexed edges o...
struct2grstr 25703 A graph represented as an ...
struct2grvtx 25704 The set of vertices of a g...
struct2griedg 25705 The set of indexed edges o...
graop 25706 Any representation of a gr...
grastruct 25707 Any representation of a gr...
gropd 25708 If any representation of a...
grstructd 25709 If any representation of a...
gropeld 25710 If any representation of a...
grstructeld 25711 If any representation of a...
snstrvtxval 25712 The set of vertices of a g...
snstriedgval 25713 The set of indexed edges o...
vtxval0 25714 Degenerated case 1 for ver...
iedgval0 25715 Degenerated case 1 for edg...
vtxvalsnop 25716 Degenerated case 2 for ver...
iedgvalsnop 25717 Degenerated case 2 for edg...
vtxval3sn 25718 Degenerated case 3 for ver...
iedgval3sn 25719 Degenerated case 3 for edg...
vtxvalprc 25720 Degenerated case 4 for ver...
iedgvalprc 25721 Degenerated case 4 for edg...
isuhgr 25726 The predicate "is an undir...
isushgr 25727 The predicate "is an undir...
uhgrf 25728 The edge function of an un...
ushgrf 25729 The edge function of an un...
uhgrss 25730 An edge is a subset of ver...
uhgreq12g 25731 If two sets have the same ...
uhgrfun 25732 The edge function of an un...
uhgrn0 25733 An edge is a nonempty subs...
lpvtx 25734 The endpoints of a loop (w...
ushgruhgr 25735 An undirected simple hyper...
isuhgrop 25736 The property of being an u...
uhgr0e 25737 The empty graph, with vert...
uhgr0vb 25738 The null graph, with no ve...
uhgr0 25739 The null graph represented...
uhgrun 25740 The union ` U ` of two (un...
uhgrunop 25741 The union of two (undirect...
ushgrun 25742 The union ` U ` of two (un...
ushgrunop 25743 The union of two (undirect...
uhgrstrrepelem 25744 Lemma for ~ uhgrstrrepe . ...
uhgrstrrepe 25745 Replacing (or adding) the ...
incistruhgr 25746 An _incident structure_ ` ...
isupgr 25751 The property of being an u...
wrdupgr 25752 The property of being an u...
upgrf 25753 The edge function of an un...
upgrfn 25754 The edge function of an un...
upgrss 25755 An edge is a subset of ver...
upgrn0 25756 An edge is a nonempty subs...
upgrle 25757 An edge of an undirected p...
upgrfi 25758 An edge is a finite subset...
upgrex 25759 An edge is an unordered pa...
upgrbi 25760 Show that an unordered pai...
isumgr 25761 The property of being an u...
isumgrs 25762 The simplified property of...
wrdumgr 25763 The property of being an u...
umgrf 25764 The edge function of an un...
umgrfn 25765 The edge function of an un...
umgredg2 25766 An edge of a multigraph ha...
umgrbi 25767 Show that an unordered pai...
upgruhgr 25768 An undirected pseudograph ...
umgrupgr 25769 An undirected multigraph i...
umgruhgr 25770 An undirected multigraph i...
upgrle2 25771 An edge of an undirected p...
umgrnloopv 25772 In a multigraph, there is ...
umgredgprv 25773 In a multigraph, an edge i...
umgrnloop 25774 In a multigraph, there is ...
umgrnloop0 25775 A multigraph has no loops....
umgr0e 25776 The empty graph, with vert...
upgr0e 25777 The empty graph, with vert...
upgr1elem 25778 Lemma for ~ upgr1e and ~ u...
upgr1e 25779 A pseudograph with one edg...
upgr0eop 25780 The empty graph, with vert...
upgr1eop 25781 A pseudograph with one edg...
upgr0eopALT 25782 Alternate proof of ~ upgr0...
upgr1eopALT 25783 Alternate proof of ~ upgr1...
upgrun 25784 The union ` U ` of two pse...
upgrunop 25785 The union of two pseudogra...
umgrun 25786 The union ` U ` of two mul...
umgrunop 25787 The union of two multigrap...
umgrislfupgrlem 25788 Lemma for ~ umgrislfupgr a...
umgrislfupgr 25789 A multigraph is a loop-fre...
lfgredgge2 25790 An edge of a loop-free gra...
lfgrnloop 25791 A loop-free graph has no l...
edgaval 25794 The edges of a graph. (Co...
edgaopval 25795 The edges of a graph repre...
edgaov 25796 The edges of a graph repre...
edgastruct 25797 The edges of a graph repre...
edgiedgb 25798 A set is an edge iff it is...
uhgredgiedgb 25799 In a hypergraph, a set is ...
edg0iedg0 25800 There is no edge in a grap...
uhgriedg0edg0 25801 A hypergraph has no edges ...
uhgredgn0 25802 An edge of a hypergraph is...
edguhgr 25803 An edge of a hypergraph is...
uhgredgrnv 25804 An edge of a hypergraph co...
uhgredgss 25805 The set of edges of a hype...
upgredgss 25806 The set of edges of a pseu...
umgredgss 25807 The set of edges of a mult...
edgupgr 25808 Properties of an edge of a...
edgumgr 25809 Properties of an edge of a...
uhgrvtxedgiedgb 25810 In a hypergraph, a vertex ...
upgredg 25811 For each edge in a pseudog...
umgredg 25812 For each edge in a multigr...
umgrpredgav 25813 An edge of a multigraph al...
upgredg2vtx 25814 For a vertex incident to a...
upgredgpr 25815 If a proper pair (of verti...
umgredgne 25816 An edge of a multigraph al...
umgrnloop2 25817 A multigraph has no loops....
umgredgnlp 25818 An edge of a multigraph is...
reluhgra 25823 The class of all undirecte...
relushgra 25824 The class of all undirecte...
uhgrav 25825 The classes of vertices an...
uhgraopelvv 25826 An undirected hypergraph i...
isuhgra 25827 The property of being an u...
uhgraf 25828 The edge function of an un...
uhgrafun 25829 The edge function of an un...
isushgra 25830 The property of being an u...
ushgraf 25831 The edge function of an un...
ushgrauhgra 25832 An undirected simple hyper...
uhgraop 25833 The property of being an u...
uhgrac 25834 The property of being an u...
uhgrass 25835 An edge is a subset of ver...
uhgraeq12d 25836 Equality of hypergraphs. ...
uhgrares 25837 A subgraph of a hypergraph...
uhgra0 25838 The empty graph, with vert...
uhgra0v 25839 The null graph, with no ve...
uhgraun 25840 The union of two (undirect...
relumgra 25843 The class of all undirecte...
isumgra 25844 The property of being an u...
wrdumgra 25845 The property of being an u...
umgraf2 25846 The edge function of an un...
umgraf 25847 The edge function of an un...
umgrass 25848 An edge is a subset of ver...
umgran0 25849 An edge is a nonempty subs...
umgrale 25850 An edge has at most two en...
umgrafi 25851 An edge is a finite subset...
umgraex 25852 An edge is an unordered pa...
umgrares 25853 A subgraph of a graph (for...
umgra0 25854 The empty graph, with vert...
umgra1 25855 The graph with one edge. ...
umisuhgra 25856 An undirected multigraph i...
umgraun 25857 The union of two (undirect...
reluslgra 25863 The class of all undirecte...
relusgra 25864 The class of all undirecte...
uslgrav 25866 The classes of vertices an...
usgrav 25867 The classes of vertices an...
edgval 25868 The edges of a graph. (Co...
edgopval 25869 The edges of a graph repre...
edgov 25870 The edges of a graph, show...
edguslgra 25871 The edges of an undirected...
isuslgra 25872 The property of being an u...
isusgra 25873 The property of being an u...
uslgraf 25874 The edge function of an un...
usgraf 25875 The edge function of an un...
isusgra0 25876 The property of being an u...
usgraf0 25877 The edge function of an un...
usgrafun 25878 The edge function of an un...
usgraop 25879 An undirected simple graph...
usgrac 25880 An undirected simple graph...
edgss 25881 The set of edges of an und...
edg 25882 An edge of an undirected s...
isausgra 25883 The property of an unorder...
ausisusgra 25884 The equivalence of the def...
ausisusgraedg 25885 The equivalence of the def...
usgraedgop 25886 An edge of an undirected s...
usgraf1o 25887 The edge function of an un...
uslgraf1oedg 25888 The edge function of an un...
usgraf1 25889 The edge function of an un...
usgrass 25890 An edge is a subset of ver...
usgraeq12d 25891 Equality of simple graphs ...
uslisushgra 25892 An undirected simple graph...
uslisumgra 25893 An undirected simple graph...
usisuslgra 25894 An undirected simple graph...
usisumgra 25895 An undirected simple graph...
usisuhgra 25896 An undirected simple graph...
elusuhgra 25897 An undirected simple graph...
usgrares 25898 A subgraph of a graph (for...
usgra0 25899 The empty graph, with vert...
usgra0v 25900 The empty graph with no ve...
uslgra1 25901 The graph with one edge, a...
usgra1 25902 The graph with one edge, a...
uslgraun 25903 The union of two simple gr...
usgraedg2 25904 The value of the "edge fun...
usgraedgprv 25905 In an undirected graph, an...
usgraedgrnv 25906 An edge of an undirected s...
usgranloopv 25907 In an undirected simple gr...
usgranloop 25908 In an undirected simple gr...
usgranloop0 25909 A simple undirected graph ...
usgraedgrn 25910 An edge of an undirected s...
usgra2edg 25911 If a vertex is adjacent to...
usgra2edg1 25912 If a vertex is adjacent to...
usgrarnedg 25913 For each edge in a simple ...
edgprvtx 25914 An edge of an undirected s...
usgraedg3 25915 The value of the "edge fun...
usgraedg4 25916 The value of the "edge fun...
usgraedgreu 25917 The value of the "edge fun...
usgrarnedg1 25918 For each edge in a simple ...
usgra1v 25919 A class with one (or no) v...
usgraidx2vlem1 25920 Lemma 1 for ~ usgraidx2v ....
usgraidx2vlem2 25921 Lemma 2 for ~ usgraidx2v ....
usgraidx2v 25922 The mapping of indices of ...
usgraedgleord 25923 In a graph the number of e...
usgraex0elv 25924 Lemma 0 for ~ usgraexmpl ....
usgraex1elv 25925 Lemma 1 for ~ usgraexmpl ....
usgraex2elv 25926 Lemma 2 for ~ usgraexmpl ....
usgraex3elv 25927 Lemma 3 for ~ usgraexmpl ....
usgraexmpldifpr 25928 Lemma for ~ usgraexmpl : a...
usgraexmplef 25929 Lemma for ~ usgraexmpl . ...
usgraexmpl 25930 ` <. V , E >. ` is a graph...
usgraexmplvtx 25931 The vertices ` 0 , 1 , 2 ,...
usgraexmpledg 25932 The edges ` { 0 , 1 } , { ...
usgraexmplc 25933 ` G = <. V , E >. ` is a g...
usgraexmplcvtx 25934 The vertices ` 0 , 1 , 2 ,...
usgraexmplcedg 25935 The edges ` { 0 , 1 } , { ...
fiusgraedgfi 25936 In a finite graph the numb...
usgrafisindb0 25937 The size of a finite simpl...
usgrafisindb1 25938 The size of a finite simpl...
usgrares1 25939 Restricting an undirected ...
usgrafilem1 25940 The domain of the edge fun...
usgrafilem2 25941 In a graph with a finite n...
usgrafisinds 25942 In a graph with a finite n...
usgrafisbase 25943 Induction base for ~ usgra...
usgrafis 25944 A simple undirected graph ...
usgrafiedg 25945 A simple undirected graph ...
nbgraop 25952 The set of neighbors of an...
nbgraopALT 25953 Alternate proof of ~ nbgra...
nbgraop1 25954 The set of neighbors of an...
nbgrael 25955 The set of neighbors of an...
nbgranv0 25956 There are no neighbors of ...
nbusgra 25957 The set of neighbors of a ...
nbgra0nb 25958 A vertex which is not endp...
nbgraeledg 25959 A class/vertex is a neighb...
nbgraisvtx 25960 Every neighbor of a class/...
nbgra0edg 25961 In a graph with no edges, ...
nbgrassvt 25962 The neighbors of a vertex ...
nbgranself 25963 A vertex in a graph (witho...
nbgrassovt 25964 The neighbors of a vertex ...
nbgranself2 25965 A class is not a neighbor ...
nbgrassvwo 25966 The neighbors of a vertex ...
nbgrassvwo2 25967 The neighbors of a vertex ...
nbgrasym 25968 A vertex in a graph is a n...
nbgracnvfv 25969 Applying the edge function...
nbgraf1olem1 25970 Lemma 1 for ~ nbgraf1o . ...
nbgraf1olem2 25971 Lemma 2 for ~ nbgraf1o . ...
nbgraf1olem3 25972 Lemma 3 for ~ nbgraf1o . ...
nbgraf1olem4 25973 Lemma 4 for ~ nbgraf1o . ...
nbgraf1olem5 25974 Lemma 5 for ~ nbgraf1o . ...
nbgraf1o0 25975 The set of neighbors of a ...
nbgraf1o 25976 The set of neighbors of a ...
nbusgrafi 25977 The class of neighbors of ...
nbfiusgrafi 25978 The class of neighbors of ...
edgusgranbfin 25979 The number of neighbors of...
nb3graprlem1 25980 Lemma 1 for ~ nb3grapr . ...
nb3graprlem2 25981 Lemma 2 for ~ nb3grapr . ...
nb3grapr 25982 The neighbors of a vertex ...
nb3grapr2 25983 The neighbors of a vertex ...
nb3gra2nb 25984 If the neighbors of two ve...
iscusgra 25985 The property of being a co...
iscusgra0 25986 The property of being a co...
cusisusgra 25987 A complete (undirected sim...
cusgrarn 25988 In a complete simple graph...
cusgra0v 25989 A graph with no vertices (...
cusgra1v 25990 A graph with one vertex (a...
cusgra2v 25991 A graph with two (differen...
nbcusgra 25992 In a complete (undirected ...
cusgra3v 25993 A graph with three (differ...
cusgra3vnbpr 25994 The neighbors of a vertex ...
cusgraexilem1 25995 Lemma 1 for ~ cusgraexi . ...
cusgraexilem2 25996 Lemma 2 for ~ cusgraexi . ...
cusgraexi 25997 For each set the identity ...
cusgraexg 25998 For each set there is an e...
cusgrasizeindb0 25999 Base case of the induction...
cusgrasizeindb1 26000 Base case of the induction...
cusgrares 26001 Restricting a complete sim...
cusgrasizeindslem1 26002 Lemma 1 for ~ cusgrasizein...
cusgrasizeindslem2 26003 Lemma 2 for ~ cusgrasizein...
cusgrasizeinds 26004 Part 1 of induction step i...
cusgrasize2inds 26005 Induction step in ~ cusgra...
cusgrasize 26006 The size of a finite compl...
cusgrafilem1 26007 Lemma 1 for ~ cusgrafi . ...
cusgrafilem2 26008 Lemma 2 for ~ cusgrafi . ...
cusgrafilem3 26009 Lemma 3 for ~ cusgrafi . ...
cusgrafi 26010 If the size of a complete ...
usgrasscusgra 26011 An undirected simple graph...
sizeusglecusglem1 26012 Lemma 1 for ~ sizeusglecus...
sizeusglecusglem2 26013 Lemma 2 for ~ sizeusglecus...
sizeusglecusg 26014 The size of an undirected ...
usgramaxsize 26015 The maximum size of an und...
isuvtx 26016 The set of all universal v...
uvtxel 26017 A universal vertex, i.e. a...
uvtxisvtx 26018 A universal vertex is a ve...
uvtx0 26019 There is no universal vert...
uvtx01vtx 26020 If a graph/class has no ed...
uvtxnbgra 26021 A universal vertex has all...
uvtxnm1nbgra 26022 A universal vertex has ` n...
uvtxnbgravtx 26023 A universal vertex is neig...
cusgrauvtxb 26024 An undirected simple graph...
uvtxnb 26025 A vertex in a undirected s...
relwlk 26046 The walks (in an undirecte...
wlks 26047 The set of walks (in an un...
iswlk 26048 Properties of a pair of fu...
2mwlk 26049 The two mappings determini...
wlkres 26050 Restrictions of walks (i.e...
wlkbprop 26051 Basic properties of a walk...
iswlkg 26052 Generalisation of ~ iswlk ...
wlkcomp 26053 A walk expressed by proper...
wlkcompim 26054 Implications for the prope...
wlkn0 26055 The set of vertices of a w...
wlkop 26056 A walk (in an undirected s...
wlkcpr 26057 A walk as class with two c...
wlkelwrd 26058 The components of a walk a...
edgwlk 26059 The (connected) edges of a...
wlklenvm1 26060 The number of edges of a w...
wlkon 26061 The set of walks between t...
iswlkon 26062 Properties of a pair of fu...
wlkonprop 26063 Properties of a walk betwe...
wlkoniswlk 26064 A walk between to vertices...
wlkonwlk 26065 A walk is a walk between i...
trls 26066 The set of trails (in an u...
istrl 26067 Properties of a pair of fu...
istrl2 26068 Properties of a pair of fu...
trliswlk 26069 A trail is a walk (in an u...
trlon 26070 The set of trails between ...
istrlon 26071 Properties of a pair of fu...
trlonprop 26072 Properties of a trail betw...
trlonistrl 26073 A trail between to vertice...
trlonwlkon 26074 A trail between two vertic...
0wlk 26075 A pair of an empty set (of...
0trl 26076 A pair of an empty set (of...
0wlkon 26077 A walk of length 0 from a ...
0trlon 26078 A trail of length 0 from a...
2trllemF 26079 Lemma 5 for ~ constr2trl ....
2trllemA 26080 Lemma 1 for ~ constr2trl ....
2trllemB 26081 Lemma 2 for ~ constr2trl ....
2trllemH 26082 Lemma 3 for ~ constr2trl ....
2trllemE 26083 Lemma 4 for ~ constr2trl ....
2wlklemA 26084 Lemma for ~ constr2wlk . ...
2wlklemB 26085 Lemma for ~ constr2wlk . ...
2wlklemC 26086 Lemma for ~ constr2wlk . ...
2trllemD 26087 Lemma 4 for ~ constr2trl ....
2trllemG 26088 Lemma 7 for ~ constr2trl ....
wlkntrllem1 26089 Lemma 1 for ~ wlkntrl : F...
wlkntrllem2 26090 Lemma 2 for ~ wlkntrl : T...
wlkntrllem3 26091 Lemma 3 for ~ wlkntrl : F...
wlkntrl 26092 A walk which is not a trai...
usgrwlknloop 26093 In an undirected simple gr...
2wlklem 26094 Lemma for ~ is2wlk and ~ 2...
is2wlk 26095 Properties of a pair of fu...
pths 26096 The set of paths (in an un...
spths 26097 The set of simple paths (i...
ispth 26098 Properties of a pair of fu...
isspth 26099 Properties of a pair of fu...
0pth 26100 A pair of an empty set (of...
0spth 26101 A pair of an empty set (of...
pthistrl 26102 A path is a trail (in an u...
spthispth 26103 A simple path is a path (i...
pthdepisspth 26104 A path with different star...
pthon 26105 The set of paths between t...
ispthon 26106 Properties of a pair of fu...
pthonprop 26107 Properties of a path betwe...
pthonispth 26108 A path between two vertice...
0pthon 26109 A path of length 0 from a ...
0pthon1 26110 A path of length 0 from a ...
0pthonv 26111 For each vertex there is a...
spthon 26112 The set of simple paths be...
isspthon 26113 Properties of a pair of fu...
isspthonpth 26114 Properties of a pair of fu...
spthonprp 26115 Properties of a simple pat...
spthonisspth 26116 A simple path between to v...
spthonepeq 26117 The endpoints of a simple ...
constr1trl 26118 Construction of a trail fr...
1pthonlem1 26119 Lemma 1 for ~ 1pthon . (C...
1pthonlem2 26120 Lemma 2 for ~ 1pthon . (C...
1pthon 26121 A path of length 1 from on...
1pthoncl 26122 A path of length 1 from on...
1pthon2v 26123 For each pair of adjacent ...
constr2spthlem1 26124 Lemma 1 for ~ constr2spth ...
2pthlem1 26125 Lemma 1 for ~ constr2pth ....
2pthlem2 26126 Lemma 2 for ~ constr2pth ....
2wlklem1 26127 Lemma 1 for ~ constr2wlk ....
constr2wlk 26128 Construction of a walk fro...
constr2trl 26129 Construction of a trail fr...
constr2spth 26130 A simple path of length 2 ...
constr2pth 26131 A path of length 2 from on...
2pthon 26132 A path of length 2 from on...
2pthoncl 26133 A path of length 2 from on...
2pthon3v 26134 For a vertex adjacent to t...
redwlklem 26135 Lemma for ~ redwlk . (Con...
redwlk 26136 A walk ending at the last ...
wlkdvspthlem 26137 Lemma for ~ wlkdvspth . (...
wlkdvspth 26138 A walk consisting of diffe...
usgra2adedgspthlem1 26139 Lemma 1 for ~ usgra2adedgs...
usgra2adedgspthlem2 26140 Lemma 2 for ~ usgra2adedgs...
usgra2adedgspth 26141 In an undirected simple gr...
usgra2adedgwlk 26142 In an undirected simple gr...
usgra2adedgwlkon 26143 In an undirected simple gr...
usgra2adedgwlkonALT 26144 Alternate proof for ~ usgr...
usg2wlk 26145 In an undirected simple gr...
usg2wlkon 26146 In an undirected simple gr...
usgra2wlkspthlem1 26147 Lemma 1 for ~ usgra2wlkspt...
usgra2wlkspthlem2 26148 Lemma 2 for ~ usgra2wlkspt...
usgra2wlkspth 26149 In a undirected simple gra...
crcts 26150 The set of circuits (in an...
cycls 26151 The set of cycles (in an u...
iscrct 26152 Properties of a pair of fu...
iscycl 26153 Properties of a pair of fu...
0crct 26154 A pair of an empty set (of...
0cycl 26155 A pair of an empty set (of...
crctistrl 26156 A circuit is a trail. (Co...
cyclispth 26157 A cycle is a path. (Contr...
cycliscrct 26158 A cycle is a circuit. (Co...
cyclnspth 26159 A (non trivial) cycle is n...
cycliswlk 26160 A cycle is a walk. (Contr...
cyclispthon 26161 A cycle is a path starting...
fargshiftlem 26162 If a class is a function, ...
fargshiftfv 26163 If a class is a function, ...
fargshiftf 26164 If a class is a function, ...
fargshiftf1 26165 If a function is 1-1, then...
fargshiftfo 26166 If a function is onto, the...
fargshiftfva 26167 The values of a shifted fu...
usgrcyclnl1 26168 In an undirected simple gr...
usgrcyclnl2 26169 In an undirected simple gr...
3cycl3dv 26170 In a simple graph, the ver...
nvnencycllem 26171 Lemma for ~ 3v3e3cycl1 and...
3v3e3cycl1 26172 If there is a cycle of len...
constr3lem1 26173 Lemma for ~ constr3trl etc...
constr3lem2 26174 Lemma for ~ constr3trl etc...
constr3lem4 26175 Lemma for ~ constr3trl etc...
constr3lem5 26176 Lemma for ~ constr3trl etc...
constr3lem6 26177 Lemma for ~ constr3pthlem3...
constr3trllem1 26178 Lemma for ~ constr3trl . ...
constr3trllem2 26179 Lemma for ~ constr3trl . ...
constr3trllem3 26180 Lemma for ~ constr3trl . ...
constr3trllem4 26181 Lemma for ~ constr3trl . ...
constr3trllem5 26182 Lemma for ~ constr3trl . ...
constr3pthlem1 26183 Lemma for ~ constr3pth . ...
constr3pthlem2 26184 Lemma for ~ constr3pth . ...
constr3pthlem3 26185 Lemma for ~ constr3pth . ...
constr3cycllem1 26186 Lemma for ~ constr3cycl . ...
constr3trl 26187 Construction of a trail fr...
constr3pth 26188 Construction of a path fro...
constr3cycl 26189 Construction of a 3-cycle ...
constr3cyclp 26190 Construction of a 3-cycle ...
constr3cyclpe 26191 If there are three (differ...
3v3e3cycl2 26192 If there are three (differ...
3v3e3cycl 26193 If and only if there is a ...
4cycl4v4e 26194 If there is a cycle of len...
4cycl4dv 26195 In a simple graph, the ver...
4cycl4dv4e 26196 If there is a cycle of len...
dfconngra1 26199 Alternative definition of ...
isconngra 26200 The property of being a co...
isconngra1 26201 The property of being a co...
0conngra 26202 A class/graph without vert...
1conngra 26203 A class/graph with (at mos...
cusconngra 26204 A complete (undirected sim...
wwlk 26209 The set of walks (in an un...
wwlkn 26210 The set of walks (in an un...
iswwlk 26211 Properties of a word to re...
iswwlkn 26212 Properties of a word to re...
wwlkprop 26213 Properties of a walk (in a...
wwlknprop 26214 Properties of a walk of a ...
wwlknimp 26215 Implications for a set bei...
wwlksswrd 26216 Walks (represented by word...
wwlkn0 26217 A walk of length 0 is repr...
wlkiswwlk1 26218 The sequence of vertices i...
wlkiswwlk2lem1 26219 Lemma 1 for ~ wlkiswwlk2 ....
wlkiswwlk2lem2 26220 Lemma 2 for ~ wlkiswwlk2 ....
wlkiswwlk2lem3 26221 Lemma 3 for ~ wlkiswwlk2 ....
wlkiswwlk2lem4 26222 Lemma 4 for ~ wlkiswwlk2 ....
wlkiswwlk2lem5 26223 Lemma 5 for ~ wlkiswwlk2 ....
wlkiswwlk2lem6 26224 Lemma 6 for ~ wlkiswwlk2 ....
wlkiswwlk2 26225 A walk as word corresponds...
wlkiswwlk 26226 A walk as word corresponds...
wlklniswwlkn1 26227 The sequence of vertices i...
wlklniswwlkn2 26228 A walk of length n as word...
wlklniswwlkn 26229 A walk of length n as word...
wwlkiswwlkn 26230 A walk of a fixed length a...
wwlksswwlkn 26231 The walks of a fixed lengt...
wwlknimpb 26232 Basic implications for a s...
wwlkn0s 26233 The set of all walks as wo...
vfwlkniswwlkn 26234 If the edge function of a ...
2wlkeq 26235 Conditions for two walks (...
usg2wlkeq 26236 Conditions for two walks w...
usg2wlkeq2 26237 Conditions for which two w...
wlknwwlknfun 26238 Lemma 1 for ~ wlknwwlknbij...
wlknwwlkninj 26239 Lemma 2 for ~ wlknwwlknbij...
wlknwwlknsur 26240 Lemma 3 for ~ wlknwwlknbij...
wlknwwlknbij 26241 Lemma 4 for ~ wlknwwlknbij...
wlknwwlknbij2 26242 There is a bijection betwe...
wlknwwlknen 26243 The set of walks of a fixe...
wlknwwlkneqs 26244 The set of walks of a fixe...
wlkiswwlkfun 26245 Lemma 1 for ~ wlkiswwlkbij...
wlkiswwlkinj 26246 Lemma 2 for ~ wlkiswwlkbij...
wlkiswwlksur 26247 Lemma 3 for ~ wlkiswwlkbij...
wlkiswwlkbij 26248 Lemma 4 for ~ wlkiswwlkbij...
wlkiswwlkbij2 26249 There is a bijection betwe...
wwlkeq 26250 Equality of two walks (as ...
wwlknred 26251 Reduction of a walk (as wo...
wwlknext 26252 Extension of a walk (as wo...
wwlknextbi 26253 Extension of a walk (as wo...
wwlknredwwlkn 26254 For each walk (as word) of...
wwlknredwwlkn0 26255 For each walk (as word) of...
wwlkextwrd 26256 Lemma 0 for ~ wwlkextbij ....
wwlkextfun 26257 Lemma 1 for ~ wwlkextbij ....
wwlkextinj 26258 Lemma 2 for ~ wwlkextbij ....
wwlkextsur 26259 Lemma 3 for ~ wwlkextbij ....
wwlkextbij0 26260 Lemma 4 for ~ wwlkextbij ....
wwlkextbij 26261 There is a bijection betwe...
wwlkexthasheq 26262 The number of the extensio...
wwlkm1edg 26263 Removing the trailing edge...
disjxwwlks 26264 Sets of walks (as words) e...
wwlknndef 26265 Conditions for ` WWalksN `...
wwlknfi 26266 The number of walks repres...
wlknfi 26267 The number of walks of fix...
wlknwwlknvbij 26268 There is a bijection betwe...
wwlkextproplem1 26269 Lemma 1 for ~ wwlkextprop ...
wwlkextproplem2 26270 Lemma 2 for ~ wwlkextprop ...
wwlkextproplem3 26271 Lemma 3 for ~ wwlkextprop ...
wwlkextprop 26272 Adding additional properti...
disjxwwlkn 26273 Sets of walks (as words) e...
hashwwlkext 26274 Number of walks (as words)...
clwlk 26281 The set of closed walks (i...
isclwlk0 26282 Properties of a pair of fu...
isclwlkg 26283 Generalisation of ~ isclwl...
isclwlk 26284 Properties of a pair of fu...
clwlkiswlk 26285 A closed walk is a walk (i...
clwlkswlks 26286 Closed walks are walks (in...
clwlksarewlks 26287 Closed walks are walks (in...
wlkv0 26288 If there is a walk in an e...
wlk0 26289 There is no walk in an emp...
clwlk0 26290 There is no closed walk in...
clwlkcomp 26291 A closed walk expressed by...
clwlkcompim 26292 Implications for the prope...
0clwlk 26293 A pair of an empty set (of...
clwwlk 26294 The set of closed walks (i...
clwwlkn 26295 The set of closed walks (i...
isclwwlk 26296 Properties of a word to re...
isclwwlkn 26297 Properties of a word to re...
clwwlkprop 26298 Properties of a closed wal...
clwwlkgt0 26299 A closed walk in an undire...
clwwlknprop 26300 Properties of a closed wal...
clwwlknndef 26301 Conditions for ` ClWWalksN...
clwwlkn0 26302 There is no closed walk of...
clwwlkn2 26303 In an undirected simple gr...
clwwlknimp 26304 Implications for a set bei...
clwwlksswrd 26305 Closed walks (represented ...
clwwlknfi 26306 If there is only a finite ...
clwlkisclwwlklem2a1 26307 Lemma 1 for ~ clwlkisclwwl...
clwlkisclwwlklem2a2 26308 Lemma 3 for ~ clwlkisclwwl...
clwlkisclwwlklem2a3 26309 Lemma 3 for ~ clwlkisclwwl...
clwlkisclwwlklem2fv1 26310 Lemma 4a for ~ clwlkisclww...
clwlkisclwwlklem2fv2 26311 Lemma 4b for ~ clwlkisclww...
clwlkisclwwlklem2a4 26312 Lemma 4 for ~ clwlkisclwwl...
clwlkisclwwlklem2a 26313 Lemma 2 for ~ clwlkisclwwl...
clwlkisclwwlklem2 26314 Lemma for ~ clwlkisclwwlk ...
clwlkisclwwlklem1 26315 Lemma for ~ clwlkisclwwlk ...
clwlkisclwwlklem0 26316 Lemma for ~ clwlkisclwwlk ...
clwlkisclwwlk 26317 A closed walk as word corr...
clwlkisclwwlk2 26318 A closed walk corresponds ...
clwwlkisclwwlkn 26319 A closed walk of a fixed l...
clwwlkssclwwlkn 26320 The closed walks of a fixe...
clwwlkel 26321 Obtaining a closed walk (a...
clwwlkf 26322 Lemma 1 for ~ clwwlkbij : ...
clwwlkfv 26323 Lemma 2 for ~ clwwlkbij : ...
clwwlkf1 26324 Lemma 3 for ~ clwwlkbij : ...
clwwlkfo 26325 Lemma 4 for ~ clwwlkbij : ...
clwwlkf1o 26326 Lemma 5 for ~ clwwlkbij : ...
clwwlkbij 26327 There is a bijection betwe...
clwwlknwwlkncl 26328 Obtaining a closed walk (a...
clwwlkvbij 26329 There is a bijection betwe...
clwwlkext2edg 26330 If a word concatenated wit...
wwlkext2clwwlk 26331 If a word represents a wal...
wwlksubclwwlk 26332 Any prefix of a word repre...
clwwisshclwwlem1 26333 Lemma 1 for ~ clwwisshclww...
clwwisshclwwlem 26334 Lemma for ~ clwwisshclww ....
clwwisshclww 26335 Cyclically shifting a clos...
clwwisshclwwn 26336 Cyclically shifting a clos...
clwwnisshclwwn 26337 Cyclically shifting a clos...
erclwwlkrel 26338 ` .~ ` is a relation. (Co...
erclwwlkeq 26339 Two classes are equivalent...
erclwwlkeqlen 26340 If two classes are equival...
erclwwlkref 26341 ` .~ ` is a reflexive rela...
erclwwlksym 26342 ` .~ ` is a symmetric rela...
erclwwlktr 26343 ` .~ ` is a transitive rel...
erclwwlk 26344 ` .~ ` is an equivalence r...
eleclclwwlknlem1 26345 Lemma 1 for ~ eleclclwwlkn...
eleclclwwlknlem2 26346 Lemma 2 for ~ eleclclwwlkn...
clwwlknscsh 26347 The set of cyclical shifts...
usg2cwwk2dif 26348 If a word represents a clo...
usg2cwwkdifex 26349 If a word represents a clo...
erclwwlknrel 26350 ` .~ ` is a relation. (Co...
erclwwlkneq 26351 Two classes are equivalent...
erclwwlkneqlen 26352 If two classes are equival...
erclwwlknref 26353 ` .~ ` is a reflexive rela...
erclwwlknsym 26354 ` .~ ` is a symmetric rela...
erclwwlkntr 26355 ` .~ ` is a transitive rel...
erclwwlkn 26356 ` .~ ` is an equivalence r...
qerclwwlknfi 26357 The quotient set of the se...
hashclwwlkn0 26358 The number of closed walks...
eclclwwlkn1 26359 An equivalence class accor...
eleclclwwlkn 26360 A member of an equivalence...
hashecclwwlkn1 26361 The size of every equivale...
usghashecclwwlk 26362 The size of every equivale...
hashclwwlkn 26363 The size of the set of clo...
clwwlkndivn 26364 The size of the set of clo...
wlklenvp1 26365 The number of vertices of ...
wlklenvclwlk 26366 The number of vertices in ...
clwlkfclwwlk2wrd 26367 The second component of a ...
clwlkfclwwlk1hashn 26368 The size of the first comp...
clwlkfclwwlk1hash 26369 The size of the first comp...
clwlkfclwwlk2sswd 26370 The size of a subword of t...
clwlkfclwwlk 26371 There is a function betwee...
clwlkfoclwwlk 26372 There is an onto function ...
clwlkf1clwwlklem1 26373 Lemma 1 for ~ clwlkf1clwwl...
clwlkf1clwwlklem2 26374 Lemma 2 for ~ clwlkf1clwwl...
clwlkf1clwwlklem3 26375 Lemma 3 for ~ clwlkf1clwwl...
clwlkf1clwwlklem 26376 Lemma for ~ clwlkf1clwwlk ...
clwlkf1clwwlk 26377 There is a one-to-one func...
clwlkf1oclwwlk 26378 There is a one-to-one onto...
clwlksizeeq 26379 The size of the set of clo...
clwlkndivn 26380 The size of the set of clo...
el2wlkonotlem 26389 Lemma for ~ el2wlkonot . ...
is2wlkonot 26390 The set of walks of length...
is2spthonot 26391 The set of simple paths of...
2wlkonot 26392 The set of walks of length...
2spthonot 26393 The set of simple paths of...
2wlksot 26394 The set of walks of length...
2spthsot 26395 The set of simple paths of...
el2wlkonot 26396 A walk of length 2 between...
el2spthonot 26397 A simple path of length 2 ...
el2spthonot0 26398 A simple path of length 2 ...
el2wlkonotot0 26399 A walk of length 2 between...
el2wlkonotot 26400 A walk of length 2 between...
el2wlkonotot1 26401 A walk of length 2 between...
2wlkonot3v 26402 If an ordered triple repre...
2spthonot3v 26403 If an ordered triple repre...
2wlkonotv 26404 If an ordered tripple repr...
el2wlksoton 26405 A walk of length 2 between...
el2spthsoton 26406 A simple path of length 2 ...
el2wlksot 26407 A walk of length 2 between...
el2pthsot 26408 A simple path of length 2 ...
el2wlksotot 26409 A walk of length 2 between...
usg2wlkonot 26410 A walk of length 2 between...
usg2wotspth 26411 A walk of length 2 between...
2pthwlkonot 26412 For two different vertices...
2wot2wont 26413 The set of (simple) paths ...
2spontn0vne 26414 If the set of simple paths...
usg2spthonot 26415 A simple path of length 2 ...
usg2spthonot0 26416 A simple path of length 2 ...
usg2spthonot1 26417 A simple path of length 2 ...
2spot2iun2spont 26418 The set of simple paths of...
2spotfi 26419 In a finite graph, the set...
vdgrfval 26422 The value of the vertex de...
vdgrval 26423 The value of the vertex de...
vdgrfival 26424 The value of the vertex de...
vdgrf 26425 The vertex degree function...
vdgrfif 26426 The vertex degree function...
vdgr0 26427 The degree of a vertex in ...
vdgrun 26428 The degree of a vertex in ...
vdgrfiun 26429 The degree of a vertex in ...
vdgr1d 26430 The vertex degree of a one...
vdgr1b 26431 The vertex degree of a one...
vdgr1c 26432 The vertex degree of a one...
vdgr1a 26433 The vertex degree of a one...
vdusgraval 26434 The value of the vertex de...
vdusgra0nedg 26435 If a vertex in a simple gr...
vdgrnn0pnf 26436 The degree of a vertex is ...
usgfidegfi 26437 In a finite graph, the deg...
usgfiregdegfi 26438 In a finite graph, the deg...
hashnbgravd 26439 The size of the set of the...
hashnbgravdg 26440 The size of the set of the...
nbhashnn0 26441 The number of the neighbor...
nbhashuvtx1 26442 If the number of the neigh...
nbhashuvtx 26443 If the number of the neigh...
uvtxhashnb 26444 A universal vertex has ` n...
usgravd0nedg 26445 If a vertex in a simple gr...
usgravd00 26446 If every vertex in a simpl...
usgrauvtxvdbi 26447 In a finite undirected sim...
vdiscusgra 26448 In a finite complete undir...
isrgra 26453 The property of being a k-...
isrusgra 26454 The property of being a k-...
rgraprop 26455 The properties of a k-regu...
rusgraprop 26456 The properties of a k-regu...
rusgrargra 26457 A k-regular undirected sim...
rusisusgra 26458 Any k-regular undirected s...
isrusgusrg 26459 A graph is a k-regular und...
isrusgusrgcl 26460 A graph represented by a c...
isrgrac 26461 The property of being a k-...
isrusgrac 26462 The property of being a k-...
0egra0rgra 26463 A graph is 0-regular if it...
0vgrargra 26464 A graph with no vertices i...
cusgraisrusgra 26465 A complete undirected simp...
0eusgraiff0rgra 26466 An undirected simple graph...
cusgraiffrusgra 26467 A finite undirected simple...
0eusgraiff0rgracl 26468 An undirected simple graph...
rusgraprop2 26469 The properties of a k-regu...
rusgraprop3 26470 The properties of a k-regu...
rusgraprop4 26471 The properties of a k-regu...
rusgrasn 26472 If a k-regular undirected ...
rusgranumwwlkl1 26473 In a k-regular graph, the ...
rusgranumwlkl1 26474 In a k-regular graph, ther...
rusgranumwlklem0 26475 Lemma 0 for ~ rusgranumwlk...
rusgranumwlklem1 26476 Lemma 1 for ~ rusgranumwlk...
rusgranumwlklem2 26477 Lemma 2 for ~ rusgranumwlk...
rusgranumwlklem3 26478 Lemma 3 for ~ rusgranumwlk...
rusgranumwlklem4 26479 Lemma 4 for ~ rusgranumwlk...
rusgranumwlkb0 26480 Induction base 0 for ~ rus...
rusgranumwlkb1 26481 Induction base 1 for ~ rus...
rusgra0edg 26482 Special case for graphs wi...
rusgranumwlks 26483 Induction step for ~ rusgr...
rusgranumwlk 26484 In a k-regular graph, the ...
rusgranumwlkg 26485 In a k-regular graph, the ...
rusgranumwwlkg 26486 In a k-regular graph, the ...
clwlknclwlkdifs 26487 The set of walks of length...
clwlknclwlkdifnum 26488 In a k-regular graph, the ...
releupa 26491 The set ` ( V EulPaths E )...
iseupa 26492 The property " ` <. F , P ...
eupagra 26493 If an eulerian path exists...
eupai 26494 Properties of an Eulerian ...
eupatrl 26495 An Eulerian path is a trai...
eupacl 26496 An Eulerian path has lengt...
eupaf1o 26497 The ` F ` function in an E...
eupafi 26498 Any graph with an Eulerian...
eupapf 26499 The ` P ` function in an E...
eupaseg 26500 The ` N ` -th edge in an e...
eupa0 26501 There is an Eulerian path ...
eupares 26502 The restriction of an Eule...
eupap1 26503 Append one path segment to...
eupath2lem1 26504 Lemma for ~ eupath2 . (Co...
eupath2lem2 26505 Lemma for ~ eupath2 . (Co...
eupath2lem3 26506 Lemma for ~ eupath2 . (Co...
eupath2 26507 The only vertices of odd d...
eupath 26508 A graph with an Eulerian p...
vdeg0i 26509 The base case for the indu...
umgrabi 26510 Show that an unordered pai...
vdegp1ai 26511 The induction step for a v...
vdegp1bi 26512 The induction step for a v...
vdegp1ci 26513 The induction step for a v...
konigsberg 26514 The Konigsberg Bridge prob...
isfrgra 26517 The property of being a fr...
frisusgrapr 26518 A friendship graph is an u...
frisusgra 26519 A friendship graph is an u...
frgra0v 26520 Any graph with no vertex i...
frgra0 26521 Any empty graph (graph wit...
frgraunss 26522 Any two (different) vertic...
frgraun 26523 Any two (different) vertic...
frisusgranb 26524 In a friendship graph, the...
frgra1v 26525 Any graph with only one ve...
frgra2v 26526 Any graph with two (differ...
frgra3vlem1 26527 Lemma 1 for ~ frgra3v . (...
frgra3vlem2 26528 Lemma 2 for ~ frgra3v . (...
frgra3v 26529 Any graph with three verti...
1vwmgra 26530 Every graph with one verte...
3vfriswmgralem 26531 Lemma for ~ 3vfriswmgra . ...
3vfriswmgra 26532 Every friendship graph wit...
1to2vfriswmgra 26533 Every friendship graph wit...
1to3vfriswmgra 26534 Every friendship graph wit...
1to3vfriendship 26535 The friendship theorem for...
2pthfrgrarn 26536 Between any two (different...
2pthfrgrarn2 26537 Between any two (different...
2pthfrgra 26538 Between any two (different...
3cyclfrgrarn1 26539 Every vertex in a friendsh...
3cyclfrgrarn 26540 Every vertex in a friendsh...
3cyclfrgrarn2 26541 Every vertex in a friendsh...
3cyclfrgra 26542 Every vertex in a friendsh...
4cycl2v2nb 26543 In a (maybe degenerated) 4...
4cycl2vnunb 26544 In a 4-cycle, two distinct...
n4cyclfrgra 26545 There is no 4-cycle in a f...
4cyclusnfrgra 26546 A graph with a 4-cycle is ...
frgranbnb 26547 If two neighbors of a spec...
frconngra 26548 A friendship graph is conn...
vdfrgra0 26549 A vertex in a friendship g...
vdn0frgrav2 26550 A vertex in a friendship g...
vdgn0frgrav2 26551 A vertex in a friendship g...
vdn1frgrav2 26552 Any vertex in a friendship...
vdgn1frgrav2 26553 Any vertex in a friendship...
vdgfrgragt2 26554 Any vertex in a friendship...
vdgn1frgrav3 26555 Any vertex in a friendship...
usgn0fidegnn0 26556 In a nonempty, finite grap...
frgrancvvdeqlem1 26557 Lemma 1 for ~ frgrancvvdeq...
frgrancvvdeqlem2 26558 Lemma 2 for ~ frgrancvvdeq...
frgrancvvdeqlem3 26559 Lemma 3 for ~ frgrancvvdeq...
frgrancvvdeqlem4 26560 Lemma 4 for ~ frgrancvvdeq...
frgrancvvdeqlem5 26561 Lemma 5 for ~ frgrancvvdeq...
frgrancvvdeqlem6 26562 Lemma 6 for ~ frgrancvvdeq...
frgrancvvdeqlem7 26563 Lemma 7 for ~ frgrancvvdeq...
frgrancvvdeqlemA 26564 Lemma A for ~ frgrancvvdeq...
frgrancvvdeqlemB 26565 Lemma B for ~ frgrancvvdeq...
frgrancvvdeqlemC 26566 Lemma C for ~ frgrancvvdeq...
frgrancvvdeqlem8 26567 Lemma 8 for ~ frgrancvvdeq...
frgrancvvdeqlem9 26568 Lemma 9 for ~ frgrancvvdeq...
frgrancvvdeq 26569 In a finite friendship gra...
frgrancvvdgeq 26570 In a friendship graph, two...
frgrawopreglem1 26571 Lemma 1 for ~ frgrawopreg ...
frgrawopreglem2 26572 Lemma 2 for ~ frgrawopreg ...
frgrawopreglem3 26573 Lemma 3 for ~ frgrawopreg ...
frgrawopreglem4 26574 Lemma 4 for ~ frgrawopreg ...
frgrawopreglem5 26575 Lemma 5 for ~ frgrawopreg ...
frgrawopreg 26576 In a friendship graph ther...
frgrawopreg1 26577 According to statement 5 i...
frgrawopreg2 26578 According to statement 5 i...
frgraregorufr0 26579 In a friendship graph ther...
frgraregorufr 26580 If there is a vertex havin...
frgraeu 26581 Any two (different) vertic...
frg2woteu 26582 For two different vertices...
frg2wotn0 26583 In a friendship graph, the...
frg2wot1 26584 In a friendship graph, the...
frg2spot1 26585 In a friendship graph, the...
frg2woteqm 26586 There is a (simple) path o...
frg2woteq 26587 There is a (simple) path o...
2spotdisj 26588 All simple paths of length...
2spotiundisj 26589 All simple paths of length...
frghash2spot 26590 The number of simple paths...
2spot0 26591 If there are no vertices, ...
usg2spot2nb 26592 The set of paths of length...
usgreghash2spotv 26593 According to statement 7 i...
usgreg2spot 26594 In a finite k-regular grap...
2spotmdisj 26595 The sets of paths of lengt...
usgreghash2spot 26596 In a finite k-regular grap...
frgregordn0 26597 If a nonempty friendship g...
frrusgraord 26598 If a nonempty finite frien...
frgraregorufrg 26599 If there is a vertex havin...
numclwlk3lem3 26600 Lemma 3 for ~ numclwwlk3 ....
extwwlkfablem1 26601 Lemma 1 for ~ extwwlkfab ....
extwwlkfablem2lem 26602 Lemma for ~ extwwlkfablem2...
clwwlkextfrlem1 26603 Lemma for ~ numclwwlk2lem1...
numclwwlkfvc 26604 Value of function ` C ` , ...
extwwlkfablem2 26605 Lemma 2 for ~ extwwlkfab ....
numclwwlkun 26606 The set of closed walks in...
numclwwlkdisj 26607 The sets of closed walks s...
numclwwlkovf 26608 Value of operation ` F ` ,...
numclwwlkffin 26609 In a finite graph, the val...
numclwwlkovfel2 26610 Properties of an element o...
numclwwlkovf2 26611 Value of operation ` F ` f...
numclwwlkovf2num 26612 In a k regular graph, ther...
numclwwlkovf2ex 26613 Extending a closed walk st...
numclwwlkovg 26614 Value of operation ` G ` ,...
numclwwlkovgel 26615 Properties of an element o...
numclwwlkovgelim 26616 Properties of an element o...
extwwlkfab 26617 The set of closed walks (h...
numclwlk1lem2foa 26618 Going forth and back form ...
numclwlk1lem2f 26619 T is a function. (Contrib...
numclwlk1lem2fv 26620 Value of the function T. (...
numclwlk1lem2f1 26621 T is a 1-1 function. (Con...
numclwlk1lem2fo 26622 T is an onto function. (C...
numclwlk1lem2f1o 26623 T is a 1-1 onto function. ...
numclwlk1lem2 26624 There is a bijection betwe...
numclwwlk1 26625 Statement 9 in [Huneke] p....
numclwwlkovq 26626 Value of operation Q, mapp...
numclwwlkqhash 26627 In a k-regular graph, the ...
numclwwlkovh 26628 Value of operation H, mapp...
numclwwlk2lem1 26629 In a friendship graph, for...
numclwlk2lem2f 26630 R is a function. (Contrib...
numclwlk2lem2fv 26631 Value of the function R. (...
numclwlk2lem2f1o 26632 R is a 1-1 onto function. ...
numclwwlk2lem3 26633 In a friendship graph, the...
numclwwlk2 26634 Statement 10 in [Huneke] p...
numclwwlk3lem 26635 Lemma for ~ numclwwlk3 . ...
numclwwlk3 26636 Statement 12 in [Huneke] p...
numclwwlk4 26637 The total number of closed...
numclwwlk5lem 26638 Lemma for ~ numclwwlk5 . ...
numclwwlk5 26639 Statement 13 in [Huneke] p...
numclwwlk6 26640 For a prime divisor p of k...
numclwwlk7 26641 Statement 14 in [Huneke] p...
numclwwlk8 26642 The size of the set of clo...
frgrareggt1 26643 If a finite friendship gra...
frgrareg 26644 If a finite friendship gra...
frgraregord013 26645 If a finite friendship gra...
frgraregord13 26646 If a nonempty finite frien...
frgraogt3nreg 26647 If a finite friendship gra...
friendshipgt3 26648 The friendship theorem for...
friendship 26649 The friendship theorem: I...
conventions 26650

...

conventions-label 26651

...

natded 26652 Here are typical n...
ex-natded5.2 26653 Theorem 5.2 of [Clemente] ...
ex-natded5.2-2 26654 A more efficient proof of ...
ex-natded5.2i 26655 The same as ~ ex-natded5.2...
ex-natded5.3 26656 Theorem 5.3 of [Clemente] ...
ex-natded5.3-2 26657 A more efficient proof of ...
ex-natded5.3i 26658 The same as ~ ex-natded5.3...
ex-natded5.5 26659 Theorem 5.5 of [Clemente] ...
ex-natded5.7 26660 Theorem 5.7 of [Clemente] ...
ex-natded5.7-2 26661 A more efficient proof of ...
ex-natded5.8 26662 Theorem 5.8 of [Clemente] ...
ex-natded5.8-2 26663 A more efficient proof of ...
ex-natded5.13 26664 Theorem 5.13 of [Clemente]...
ex-natded5.13-2 26665 A more efficient proof of ...
ex-natded9.20 26666 Theorem 9.20 of [Clemente]...
ex-natded9.20-2 26667 A more efficient proof of ...
ex-natded9.26 26668 Theorem 9.26 of [Clemente]...
ex-natded9.26-2 26669 A more efficient proof of ...
ex-or 26670 Example for ~ df-or . Exa...
ex-an 26671 Example for ~ df-an . Exa...
ex-dif 26672 Example for ~ df-dif . Ex...
ex-un 26673 Example for ~ df-un . Exa...
ex-in 26674 Example for ~ df-in . Exa...
ex-uni 26675 Example for ~ df-uni . Ex...
ex-ss 26676 Example for ~ df-ss . Exa...
ex-pss 26677 Example for ~ df-pss . Ex...
ex-pw 26678 Example for ~ df-pw . Exa...
ex-pr 26679 Example for ~ df-pr . (Co...
ex-br 26680 Example for ~ df-br . Exa...
ex-opab 26681 Example for ~ df-opab . E...
ex-eprel 26682 Example for ~ df-eprel . ...
ex-id 26683 Example for ~ df-id . Exa...
ex-po 26684 Example for ~ df-po . Exa...
ex-xp 26685 Example for ~ df-xp . Exa...
ex-cnv 26686 Example for ~ df-cnv . Ex...
ex-co 26687 Example for ~ df-co . Exa...
ex-dm 26688 Example for ~ df-dm . Exa...
ex-rn 26689 Example for ~ df-rn . Exa...
ex-res 26690 Example for ~ df-res . Ex...
ex-ima 26691 Example for ~ df-ima . Ex...
ex-fv 26692 Example for ~ df-fv . Exa...
ex-1st 26693 Example for ~ df-1st . Ex...
ex-2nd 26694 Example for ~ df-2nd . Ex...
1kp2ke3k 26695 Example for ~ df-dec , 100...
ex-fl 26696 Example for ~ df-fl . Exa...
ex-ceil 26697 Example for ~ df-ceil . (...
ex-mod 26698 Example for ~ df-mod . (C...
ex-exp 26699 Example for ~ df-exp . (C...
ex-fac 26700 Example for ~ df-fac . (C...
ex-bc 26701 Example for ~ df-bc . (Co...
ex-hash 26702 Example for ~ df-hash . (...
ex-sqrt 26703 Example for ~ df-sqrt . (...
ex-abs 26704 Example for ~ df-abs . (C...
ex-dvds 26705 Example for ~ df-dvds : 3 ...
ex-gcd 26706 Example for ~ df-gcd . (C...
ex-lcm 26707 Example for ~ df-lcm . (C...
ex-prmo 26708 Example for ~ df-prmo : ` ...
aevdemo 26709 Proof illustrating the com...
ex-ind-dvds 26710 Example of a proof by indu...
avril1 26711 Poisson d'Avril's Theorem....
2bornot2b 26712 The law of excluded middle...
helloworld 26713 The classic "Hello world" ...
1p1e2apr1 26714 One plus one equals two. ...
eqid1 26715 Law of identity (reflexivi...
1div0apr 26716 Division by zero is forbid...
topnfbey 26717 Nothing seems to be imposs...
isplig 26720 The predicate "is a planar...
tncp 26721 In any planar incidence ge...
lpni 26722 For any line in a planar i...
dummylink 26725 Alias for ~ a1ii that may ...
id1 26726 Alias for ~ idALT that may...
isgrpo 26735 The predicate "is a group ...
isgrpoi 26736 Properties that determine ...
grpofo 26737 A group operation maps ont...
grpocl 26738 Closure law for a group op...
grpolidinv 26739 A group has a left identit...
grpon0 26740 The base set of a group is...
grpoass 26741 A group operation is assoc...
grpoidinvlem1 26742 Lemma for ~ grpoidinv . (...
grpoidinvlem2 26743 Lemma for ~ grpoidinv . (...
grpoidinvlem3 26744 Lemma for ~ grpoidinv . (...
grpoidinvlem4 26745 Lemma for ~ grpoidinv . (...
grpoidinv 26746 A group has a left and rig...
grpoideu 26747 The left identity element ...
grporndm 26748 A group's range in terms o...
0ngrp 26749 The empty set is not a gro...
gidval 26750 The value of the identity ...
grpoidval 26751 Lemma for ~ grpoidcl and o...
grpoidcl 26752 The identity element of a ...
grpoidinv2 26753 A group's properties using...
grpolid 26754 The identity element of a ...
grporid 26755 The identity element of a ...
grporcan 26756 Right cancellation law for...
grpoinveu 26757 The left inverse element o...
grpoid 26758 Two ways of saying that an...
grporn 26759 The range of a group opera...
grpoinvfval 26760 The inverse function of a ...
grpoinvval 26761 The inverse of a group ele...
grpoinvcl 26762 A group element's inverse ...
grpoinv 26763 The properties of a group ...
grpolinv 26764 The left inverse of a grou...
grporinv 26765 The right inverse of a gro...
grpoinvid1 26766 The inverse of a group ele...
grpoinvid2 26767 The inverse of a group ele...
grpolcan 26768 Left cancellation law for ...
grpo2inv 26769 Double inverse law for gro...
grpoinvf 26770 Mapping of the inverse fun...
grpoinvop 26771 The inverse of the group o...
grpodivfval 26772 Group division (or subtrac...
grpodivval 26773 Group division (or subtrac...
grpodivinv 26774 Group division by an inver...
grpoinvdiv 26775 Inverse of a group divisio...
grpodivf 26776 Mapping for group division...
grpodivcl 26777 Closure of group division ...
grpodivdiv 26778 Double group division. (C...
grpomuldivass 26779 Associative-type law for m...
grpodivid 26780 Division of a group member...
grponpcan 26781 Cancellation law for group...
isablo 26784 The predicate "is an Abeli...
ablogrpo 26785 An Abelian group operation...
ablocom 26786 An Abelian group operation...
ablo32 26787 Commutative/associative la...
ablo4 26788 Commutative/associative la...
isabloi 26789 Properties that determine ...
ablomuldiv 26790 Law for group multiplicati...
ablodivdiv 26791 Law for double group divis...
ablodivdiv4 26792 Law for double group divis...
ablodiv32 26793 Swap the second and third ...
ablonnncan 26794 Cancellation law for group...
ablonncan 26795 Cancellation law for group...
ablonnncan1 26796 Cancellation law for group...
vcrel 26799 The class of all complex v...
vciOLD 26800 Obsolete version of ~ cvsi...
vcsm 26801 Functionality of th scalar...
vccl 26802 Closure of the scalar prod...
vcidOLD 26803 Identity element for the s...
vcdi 26804 Distributive law for the s...
vcdir 26805 Distributive law for the s...
vcass 26806 Associative law for the sc...
vc2OLD 26807 A vector plus itself is tw...
vcablo 26808 Vector addition is an Abel...
vcgrp 26809 Vector addition is a group...
vclcan 26810 Left cancellation law for ...
vczcl 26811 The zero vector is a vecto...
vc0rid 26812 The zero vector is a right...
vc0 26813 Zero times a vector is the...
vcz 26814 Anything times the zero ve...
vcm 26815 Minus 1 times a vector is ...
isvclem 26816 Lemma for ~ isvcOLD . (Co...
vcex 26817 The components of a comple...
isvcOLD 26818 The predicate "is a comple...
isvciOLD 26819 Properties that determine ...
cnaddabloOLD 26820 Obsolete as of 23-Jan-2020...
cnidOLD 26821 Obsolete as of 23-Jan-2020...
cncvcOLD 26822 Obsolete version of ~ cncv...
nvss 26832 Structure of the class of ...
nvvcop 26833 A normed complex vector sp...
nvrel 26841 The class of all normed co...
vafval 26842 Value of the function for ...
bafval 26843 Value of the function for ...
smfval 26844 Value of the function for ...
0vfval 26845 Value of the function for ...
nmcvfval 26846 Value of the norm function...
nvop2 26847 A normed complex vector sp...
nvvop 26848 The vector space component...
isnvlem 26849 Lemma for ~ isnv . (Contr...
nvex 26850 The components of a normed...
isnv 26851 The predicate "is a normed...
isnvi 26852 Properties that determine ...
nvi 26853 The properties of a normed...
nvvc 26854 The vector space component...
nvablo 26855 The vector addition operat...
nvgrp 26856 The vector addition operat...
nvgf 26857 Mapping for the vector add...
nvsf 26858 Mapping for the scalar mul...
nvgcl 26859 Closure law for the vector...
nvcom 26860 The vector addition (group...
nvass 26861 The vector addition (group...
nvadd32 26862 Commutative/associative la...
nvrcan 26863 Right cancellation law for...
nvadd4 26864 Rearrangement of 4 terms i...
nvscl 26865 Closure law for the scalar...
nvsid 26866 Identity element for the s...
nvsass 26867 Associative law for the sc...
nvscom 26868 Commutative law for the sc...
nvdi 26869 Distributive law for the s...
nvdir 26870 Distributive law for the s...
nv2 26871 A vector plus itself is tw...
vsfval 26872 Value of the function for ...
nvzcl 26873 Closure law for the zero v...
nv0rid 26874 The zero vector is a right...
nv0lid 26875 The zero vector is a left ...
nv0 26876 Zero times a vector is the...
nvsz 26877 Anything times the zero ve...
nvinv 26878 Minus 1 times a vector is ...
nvinvfval 26879 Function for the negative ...
nvm 26880 Vector subtraction in term...
nvmval 26881 Value of vector subtractio...
nvmval2 26882 Value of vector subtractio...
nvmfval 26883 Value of the function for ...
nvmf 26884 Mapping for the vector sub...
nvmcl 26885 Closure law for the vector...
nvnnncan1 26886 Cancellation law for vecto...
nvmdi 26887 Distributive law for scala...
nvnegneg 26888 Double negative of a vecto...
nvmul0or 26889 If a scalar product is zer...
nvrinv 26890 A vector minus itself. (C...
nvlinv 26891 Minus a vector plus itself...
nvpncan2 26892 Cancellation law for vecto...
nvpncan 26893 Cancellation law for vecto...
nvaddsub 26894 Commutative/associative la...
nvnpcan 26895 Cancellation law for a nor...
nvaddsub4 26896 Rearrangement of 4 terms i...
nvmeq0 26897 The difference between two...
nvmid 26898 A vector minus itself is t...
nvf 26899 Mapping for the norm funct...
nvcl 26900 The norm of a normed compl...
nvcli 26901 The norm of a normed compl...
nvs 26902 Proportionality property o...
nvsge0 26903 The norm of a scalar produ...
nvm1 26904 The norm of the negative o...
nvdif 26905 The norm of the difference...
nvpi 26906 The norm of a vector plus ...
nvz0 26907 The norm of a zero vector ...
nvz 26908 The norm of a vector is ze...
nvtri 26909 Triangle inequality for th...
nvmtri 26910 Triangle inequality for th...
nvabs 26911 Norm difference property o...
nvge0 26912 The norm of a normed compl...
nvgt0 26913 A nonzero norm is positive...
nv1 26914 From any nonzero vector, c...
nvop 26915 A complex inner product sp...
cnnv 26916 The set of complex numbers...
cnnvg 26917 The vector addition (group...
cnnvba 26918 The base set of the normed...
cnnvs 26919 The scalar product operati...
cnnvnm 26920 The norm operation of the ...
cnnvm 26921 The vector subtraction ope...
elimnv 26922 Hypothesis elimination lem...
elimnvu 26923 Hypothesis elimination lem...
imsval 26924 Value of the induced metri...
imsdval 26925 Value of the induced metri...
imsdval2 26926 Value of the distance func...
nvnd 26927 The norm of a normed compl...
imsdf 26928 Mapping for the induced me...
imsmetlem 26929 Lemma for ~ imsmet . (Con...
imsmet 26930 The induced metric of a no...
imsxmet 26931 The induced metric of a no...
cnims 26932 The metric induced on the ...
vacn 26933 Vector addition is jointly...
nmcvcn 26934 The norm of a normed compl...
nmcnc 26935 The norm of a normed compl...
smcnlem 26936 Lemma for ~ smcn . (Contr...
smcn 26937 Scalar multiplication is j...
vmcn 26938 Vector subtraction is join...
dipfval 26941 The inner product function...
ipval 26942 Value of the inner product...
ipval2lem2 26943 Lemma for ~ ipval3 . (Con...
ipval2lem3 26944 Lemma for ~ ipval3 . (Con...
ipval2lem4 26945 Lemma for ~ ipval3 . (Con...
ipval2 26946 Expansion of the inner pro...
4ipval2 26947 Four times the inner produ...
ipval3 26948 Expansion of the inner pro...
ipidsq 26949 The inner product of a vec...
ipnm 26950 Norm expressed in terms of...
dipcl 26951 An inner product is a comp...
ipf 26952 Mapping for the inner prod...
dipcj 26953 The complex conjugate of a...
ipipcj 26954 An inner product times its...
diporthcom 26955 Orthogonality (meaning inn...
dip0r 26956 Inner product with a zero ...
dip0l 26957 Inner product with a zero ...
ipz 26958 The inner product of a vec...
dipcn 26959 Inner product is jointly c...
sspval 26962 The set of all subspaces o...
isssp 26963 The predicate "is a subspa...
sspid 26964 A normed complex vector sp...
sspnv 26965 A subspace is a normed com...
sspba 26966 The base set of a subspace...
sspg 26967 Vector addition on a subsp...
sspgval 26968 Vector addition on a subsp...
ssps 26969 Scalar multiplication on a...
sspsval 26970 Scalar multiplication on a...
sspmlem 26971 Lemma for ~ sspm and other...
sspmval 26972 Vector addition on a subsp...
sspm 26973 Vector subtraction on a su...
sspz 26974 The zero vector of a subsp...
sspn 26975 The norm on a subspace is ...
sspnval 26976 The norm on a subspace in ...
sspimsval 26977 The induced metric on a su...
sspims 26978 The induced metric on a su...
lnoval 26991 The set of linear operator...
islno 26992 The predicate "is a linear...
lnolin 26993 Basic linearity property o...
lnof 26994 A linear operator is a map...
lno0 26995 The value of a linear oper...
lnocoi 26996 The composition of two lin...
lnoadd 26997 Addition property of a lin...
lnosub 26998 Subtraction property of a ...
lnomul 26999 Scalar multiplication prop...
nvo00 27000 Two ways to express a zero...
nmoofval 27001 The operator norm function...
nmooval 27002 The operator norm function...
nmosetre 27003 The set in the supremum of...
nmosetn0 27004 The set in the supremum of...
nmoxr 27005 The norm of an operator is...
nmooge0 27006 The norm of an operator is...
nmorepnf 27007 The norm of an operator is...
nmoreltpnf 27008 The norm of any operator i...
nmogtmnf 27009 The norm of an operator is...
nmoolb 27010 A lower bound for an opera...
nmoubi 27011 An upper bound for an oper...
nmoub3i 27012 An upper bound for an oper...
nmoub2i 27013 An upper bound for an oper...
nmobndi 27014 Two ways to express that a...
nmounbi 27015 Two ways two express that ...
nmounbseqi 27016 An unbounded operator dete...
nmounbseqiALT 27017 Alternate shorter proof of...
nmobndseqi 27018 A bounded sequence determi...
nmobndseqiALT 27019 Alternate shorter proof of...
bloval 27020 The class of bounded linea...
isblo 27021 The predicate "is a bounde...
isblo2 27022 The predicate "is a bounde...
bloln 27023 A bounded operator is a li...
blof 27024 A bounded operator is an o...
nmblore 27025 The norm of a bounded oper...
0ofval 27026 The zero operator between ...
0oval 27027 Value of the zero operator...
0oo 27028 The zero operator is an op...
0lno 27029 The zero operator is linea...
nmoo0 27030 The operator norm of the z...
0blo 27031 The zero operator is a bou...
nmlno0lem 27032 Lemma for ~ nmlno0i . (Co...
nmlno0i 27033 The norm of a linear opera...
nmlno0 27034 The norm of a linear opera...
nmlnoubi 27035 An upper bound for the ope...
nmlnogt0 27036 The norm of a nonzero line...
lnon0 27037 The domain of a nonzero li...
nmblolbii 27038 A lower bound for the norm...
nmblolbi 27039 A lower bound for the norm...
isblo3i 27040 The predicate "is a bounde...
blo3i 27041 Properties that determine ...
blometi 27042 Upper bound for the distan...
blocnilem 27043 Lemma for ~ blocni and ~ l...
blocni 27044 A linear operator is conti...
lnocni 27045 If a linear operator is co...
blocn 27046 A linear operator is conti...
blocn2 27047 A bounded linear operator ...
ajfval 27048 The adjoint function. (Co...
hmoval 27049 The set of Hermitian (self...
ishmo 27050 The predicate "is a hermit...
phnv 27053 Every complex inner produc...
phrel 27054 The class of all complex i...
phnvi 27055 Every complex inner produc...
isphg 27056 The predicate "is a comple...
phop 27057 A complex inner product sp...
cncph 27058 The set of complex numbers...
elimph 27059 Hypothesis elimination lem...
elimphu 27060 Hypothesis elimination lem...
isph 27061 The predicate "is an inner...
phpar2 27062 The parallelogram law for ...
phpar 27063 The parallelogram law for ...
ip0i 27064 A slight variant of Equati...
ip1ilem 27065 Lemma for ~ ip1i . (Contr...
ip1i 27066 Equation 6.47 of [Ponnusam...
ip2i 27067 Equation 6.48 of [Ponnusam...
ipdirilem 27068 Lemma for ~ ipdiri . (Con...
ipdiri 27069 Distributive law for inner...
ipasslem1 27070 Lemma for ~ ipassi . Show...
ipasslem2 27071 Lemma for ~ ipassi . Show...
ipasslem3 27072 Lemma for ~ ipassi . Show...
ipasslem4 27073 Lemma for ~ ipassi . Show...
ipasslem5 27074 Lemma for ~ ipassi . Show...
ipasslem7 27075 Lemma for ~ ipassi . Show...
ipasslem8 27076 Lemma for ~ ipassi . By ~...
ipasslem9 27077 Lemma for ~ ipassi . Conc...
ipasslem10 27078 Lemma for ~ ipassi . Show...
ipasslem11 27079 Lemma for ~ ipassi . Show...
ipassi 27080 Associative law for inner ...
dipdir 27081 Distributive law for inner...
dipdi 27082 Distributive law for inner...
ip2dii 27083 Inner product of two sums....
dipass 27084 Associative law for inner ...
dipassr 27085 "Associative" law for seco...
dipassr2 27086 "Associative" law for inne...
dipsubdir 27087 Distributive law for inner...
dipsubdi 27088 Distributive law for inner...
pythi 27089 The Pythagorean theorem fo...
siilem1 27090 Lemma for ~ sii . (Contri...
siilem2 27091 Lemma for ~ sii . (Contri...
siii 27092 Inference from ~ sii . (C...
sii 27093 Schwarz inequality. Part ...
sspph 27094 A subspace of an inner pro...
ipblnfi 27095 A function ` F ` generated...
ip2eqi 27096 Two vectors are equal iff ...
phoeqi 27097 A condition implying that ...
ajmoi 27098 Every operator has at most...
ajfuni 27099 The adjoint function is a ...
ajfun 27100 The adjoint function is a ...
ajval 27101 Value of the adjoint funct...
iscbn 27104 A complex Banach space is ...
cbncms 27105 The induced metric on comp...
bnnv 27106 Every complex Banach space...
bnrel 27107 The class of all complex B...
bnsscmcl 27108 A subspace of a Banach spa...
cnbn 27109 The set of complex numbers...
ubthlem1 27110 Lemma for ~ ubth . The fu...
ubthlem2 27111 Lemma for ~ ubth . Given ...
ubthlem3 27112 Lemma for ~ ubth . Prove ...
ubth 27113 Uniform Boundedness Theore...
minvecolem1 27114 Lemma for ~ minveco . The...
minvecolem2 27115 Lemma for ~ minveco . Any...
minvecolem3 27116 Lemma for ~ minveco . The...
minvecolem4a 27117 Lemma for ~ minveco . ` F ...
minvecolem4b 27118 Lemma for ~ minveco . The...
minvecolem4c 27119 Lemma for ~ minveco . The...
minvecolem4 27120 Lemma for ~ minveco . The...
minvecolem5 27121 Lemma for ~ minveco . Dis...
minvecolem6 27122 Lemma for ~ minveco . Any...
minvecolem7 27123 Lemma for ~ minveco . Sin...
minveco 27124 Minimizing vector theorem,...
ishlo 27127 The predicate "is a comple...
hlobn 27128 Every complex Hilbert spac...
hlph 27129 Every complex Hilbert spac...
hlrel 27130 The class of all complex H...
hlnv 27131 Every complex Hilbert spac...
hlnvi 27132 Every complex Hilbert spac...
hlvc 27133 Every complex Hilbert spac...
hlcmet 27134 The induced metric on a co...
hlmet 27135 The induced metric on a co...
hlpar2 27136 The parallelogram law sati...
hlpar 27137 The parallelogram law sati...
hlex 27138 The base set of a Hilbert ...
hladdf 27139 Mapping for Hilbert space ...
hlcom 27140 Hilbert space vector addit...
hlass 27141 Hilbert space vector addit...
hl0cl 27142 The Hilbert space zero vec...
hladdid 27143 Hilbert space addition wit...
hlmulf 27144 Mapping for Hilbert space ...
hlmulid 27145 Hilbert space scalar multi...
hlmulass 27146 Hilbert space scalar multi...
hldi 27147 Hilbert space scalar multi...
hldir 27148 Hilbert space scalar multi...
hlmul0 27149 Hilbert space scalar multi...
hlipf 27150 Mapping for Hilbert space ...
hlipcj 27151 Conjugate law for Hilbert ...
hlipdir 27152 Distributive law for Hilbe...
hlipass 27153 Associative law for Hilber...
hlipgt0 27154 The inner product of a Hil...
hlcompl 27155 Completeness of a Hilbert ...
cnchl 27156 The set of complex numbers...
ssphl 27157 A Banach subspace of an in...
htthlem 27158 Lemma for ~ htth . The co...
htth 27159 Hellinger-Toeplitz Theorem...
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here
h2hva 27215 The group (addition) opera...
h2hsm 27216 The scalar product operati...
h2hnm 27217 The norm function of Hilbe...
h2hvs 27218 The vector subtraction ope...
h2hmetdval 27219 Value of the distance func...
h2hcau 27220 The Cauchy sequences of Hi...
h2hlm 27221 The limit sequences of Hil...
axhilex-zf 27222 Derive axiom ~ ax-hilex fr...
axhfvadd-zf 27223 Derive axiom ~ ax-hfvadd f...
axhvcom-zf 27224 Derive axiom ~ ax-hvcom fr...
axhvass-zf 27225 Derive axiom ~ ax-hvass fr...
axhv0cl-zf 27226 Derive axiom ~ ax-hv0cl fr...
axhvaddid-zf 27227 Derive axiom ~ ax-hvaddid ...
axhfvmul-zf 27228 Derive axiom ~ ax-hfvmul f...
axhvmulid-zf 27229 Derive axiom ~ ax-hvmulid ...
axhvmulass-zf 27230 Derive axiom ~ ax-hvmulass...
axhvdistr1-zf 27231 Derive axiom ~ ax-hvdistr1...
axhvdistr2-zf 27232 Derive axiom ~ ax-hvdistr2...
axhvmul0-zf 27233 Derive axiom ~ ax-hvmul0 f...
axhfi-zf 27234 Derive axiom ~ ax-hfi from...
axhis1-zf 27235 Derive axiom ~ ax-his1 fro...
axhis2-zf 27236 Derive axiom ~ ax-his2 fro...
axhis3-zf 27237 Derive axiom ~ ax-his3 fro...
axhis4-zf 27238 Derive axiom ~ ax-his4 fro...
axhcompl-zf 27239 Derive axiom ~ ax-hcompl f...
hvmulex 27252 The Hilbert space scalar p...
hvaddcl 27253 Closure of vector addition...
hvmulcl 27254 Closure of scalar multipli...
hvmulcli 27255 Closure inference for scal...
hvsubf 27256 Mapping domain and codomai...
hvsubval 27257 Value of vector subtractio...
hvsubcl 27258 Closure of vector subtract...
hvaddcli 27259 Closure of vector addition...
hvcomi 27260 Commutation of vector addi...
hvsubvali 27261 Value of vector subtractio...
hvsubcli 27262 Closure of vector subtract...
ifhvhv0 27263 Prove ` if ( A e. ~H , A ,...
hvaddid2 27264 Addition with the zero vec...
hvmul0 27265 Scalar multiplication with...
hvmul0or 27266 If a scalar product is zer...
hvsubid 27267 Subtraction of a vector fr...
hvnegid 27268 Addition of negative of a ...
hv2neg 27269 Two ways to express the ne...
hvaddid2i 27270 Addition with the zero vec...
hvnegidi 27271 Addition of negative of a ...
hv2negi 27272 Two ways to express the ne...
hvm1neg 27273 Convert minus one times a ...
hvaddsubval 27274 Value of vector addition i...
hvadd32 27275 Commutative/associative la...
hvadd12 27276 Commutative/associative la...
hvadd4 27277 Hilbert vector space addit...
hvsub4 27278 Hilbert vector space addit...
hvaddsub12 27279 Commutative/associative la...
hvpncan 27280 Addition/subtraction cance...
hvpncan2 27281 Addition/subtraction cance...
hvaddsubass 27282 Associativity of sum and d...
hvpncan3 27283 Subtraction and addition o...
hvmulcom 27284 Scalar multiplication comm...
hvsubass 27285 Hilbert vector space assoc...
hvsub32 27286 Hilbert vector space commu...
hvmulassi 27287 Scalar multiplication asso...
hvmulcomi 27288 Scalar multiplication comm...
hvmul2negi 27289 Double negative in scalar ...
hvsubdistr1 27290 Scalar multiplication dist...
hvsubdistr2 27291 Scalar multiplication dist...
hvdistr1i 27292 Scalar multiplication dist...
hvsubdistr1i 27293 Scalar multiplication dist...
hvassi 27294 Hilbert vector space assoc...
hvadd32i 27295 Hilbert vector space commu...
hvsubassi 27296 Hilbert vector space assoc...
hvsub32i 27297 Hilbert vector space commu...
hvadd12i 27298 Hilbert vector space commu...
hvadd4i 27299 Hilbert vector space addit...
hvsubsub4i 27300 Hilbert vector space addit...
hvsubsub4 27301 Hilbert vector space addit...
hv2times 27302 Two times a vector. (Cont...
hvnegdii 27303 Distribution of negative o...
hvsubeq0i 27304 If the difference between ...
hvsubcan2i 27305 Vector cancellation law. ...
hvaddcani 27306 Cancellation law for vecto...
hvsubaddi 27307 Relationship between vecto...
hvnegdi 27308 Distribution of negative o...
hvsubeq0 27309 If the difference between ...
hvaddeq0 27310 If the sum of two vectors ...
hvaddcan 27311 Cancellation law for vecto...
hvaddcan2 27312 Cancellation law for vecto...
hvmulcan 27313 Cancellation law for scala...
hvmulcan2 27314 Cancellation law for scala...
hvsubcan 27315 Cancellation law for vecto...
hvsubcan2 27316 Cancellation law for vecto...
hvsub0 27317 Subtraction of a zero vect...
hvsubadd 27318 Relationship between vecto...
hvaddsub4 27319 Hilbert vector space addit...
hicl 27321 Closure of inner product. ...
hicli 27322 Closure inference for inne...
his5 27327 Associative law for inner ...
his52 27328 Associative law for inner ...
his35 27329 Move scalar multiplication...
his35i 27330 Move scalar multiplication...
his7 27331 Distributive law for inner...
hiassdi 27332 Distributive/associative l...
his2sub 27333 Distributive law for inner...
his2sub2 27334 Distributive law for inner...
hire 27335 A necessary and sufficient...
hiidrcl 27336 Real closure of inner prod...
hi01 27337 Inner product with the 0 v...
hi02 27338 Inner product with the 0 v...
hiidge0 27339 Inner product with self is...
his6 27340 Zero inner product with se...
his1i 27341 Conjugate law for inner pr...
abshicom 27342 Commuted inner products ha...
hial0 27343 A vector whose inner produ...
hial02 27344 A vector whose inner produ...
hisubcomi 27345 Two vector subtractions si...
hi2eq 27346 Lemma used to prove equali...
hial2eq 27347 Two vectors whose inner pr...
hial2eq2 27348 Two vectors whose inner pr...
orthcom 27349 Orthogonality commutes. (...
normlem0 27350 Lemma used to derive prope...
normlem1 27351 Lemma used to derive prope...
normlem2 27352 Lemma used to derive prope...
normlem3 27353 Lemma used to derive prope...
normlem4 27354 Lemma used to derive prope...
normlem5 27355 Lemma used to derive prope...
normlem6 27356 Lemma used to derive prope...
normlem7 27357 Lemma used to derive prope...
normlem8 27358 Lemma used to derive prope...
normlem9 27359 Lemma used to derive prope...
normlem7tALT 27360 Lemma used to derive prope...
bcseqi 27361 Equality case of Bunjakova...
normlem9at 27362 Lemma used to derive prope...
dfhnorm2 27363 Alternate definition of th...
normf 27364 The norm function maps fro...
normval 27365 The value of the norm of a...
normcl 27366 Real closure of the norm o...
normge0 27367 The norm of a vector is no...
normgt0 27368 The norm of nonzero vector...
norm0 27369 The norm of a zero vector....
norm-i 27370 Theorem 3.3(i) of [Beran] ...
normne0 27371 A norm is nonzero iff its ...
normcli 27372 Real closure of the norm o...
normsqi 27373 The square of a norm. (Co...
norm-i-i 27374 Theorem 3.3(i) of [Beran] ...
normsq 27375 The square of a norm. (Co...
normsub0i 27376 Two vectors are equal iff ...
normsub0 27377 Two vectors are equal iff ...
norm-ii-i 27378 Triangle inequality for no...
norm-ii 27379 Triangle inequality for no...
norm-iii-i 27380 Theorem 3.3(iii) of [Beran...
norm-iii 27381 Theorem 3.3(iii) of [Beran...
normsubi 27382 Negative doesn't change th...
normpythi 27383 Analogy to Pythagorean the...
normsub 27384 Swapping order of subtract...
normneg 27385 The norm of a vector equal...
normpyth 27386 Analogy to Pythagorean the...
normpyc 27387 Corollary to Pythagorean t...
norm3difi 27388 Norm of differences around...
norm3adifii 27389 Norm of differences around...
norm3lem 27390 Lemma involving norm of di...
norm3dif 27391 Norm of differences around...
norm3dif2 27392 Norm of differences around...
norm3lemt 27393 Lemma involving norm of di...
norm3adifi 27394 Norm of differences around...
normpari 27395 Parallelogram law for norm...
normpar 27396 Parallelogram law for norm...
normpar2i 27397 Corollary of parallelogram...
polid2i 27398 Generalized polarization i...
polidi 27399 Polarization identity. Re...
polid 27400 Polarization identity. Re...
hilablo 27401 Hilbert space vector addit...
hilid 27402 The group identity element...
hilvc 27403 Hilbert space is a complex...
hilnormi 27404 Hilbert space norm in term...
hilhhi 27405 Deduce the structure of Hi...
hhnv 27406 Hilbert space is a normed ...
hhva 27407 The group (addition) opera...
hhba 27408 The base set of Hilbert sp...
hh0v 27409 The zero vector of Hilbert...
hhsm 27410 The scalar product operati...
hhvs 27411 The vector subtraction ope...
hhnm 27412 The norm function of Hilbe...
hhims 27413 The induced metric of Hilb...
hhims2 27414 Hilbert space distance met...
hhmet 27415 The induced metric of Hilb...
hhxmet 27416 The induced metric of Hilb...
hhmetdval 27417 Value of the distance func...
hhip 27418 The inner product operatio...
hhph 27419 The Hilbert space of the H...
bcsiALT 27420 Bunjakovaskij-Cauchy-Schwa...
bcsiHIL 27421 Bunjakovaskij-Cauchy-Schwa...
bcs 27422 Bunjakovaskij-Cauchy-Schwa...
bcs2 27423 Corollary of the Bunjakova...
bcs3 27424 Corollary of the Bunjakova...
hcau 27425 Member of the set of Cauch...
hcauseq 27426 A Cauchy sequences on a Hi...
hcaucvg 27427 A Cauchy sequence on a Hil...
seq1hcau 27428 A sequence on a Hilbert sp...
hlimi 27429 Express the predicate: Th...
hlimseqi 27430 A sequence with a limit on...
hlimveci 27431 Closure of the limit of a ...
hlimconvi 27432 Convergence of a sequence ...
hlim2 27433 The limit of a sequence on...
hlimadd 27434 Limit of the sum of two se...
hilmet 27435 The Hilbert space norm det...
hilxmet 27436 The Hilbert space norm det...
hilmetdval 27437 Value of the distance func...
hilims 27438 Hilbert space distance met...
hhcau 27439 The Cauchy sequences of Hi...
hhlm 27440 The limit sequences of Hil...
hhcmpl 27441 Lemma used for derivation ...
hilcompl 27442 Lemma used for derivation ...
hhcms 27444 The Hilbert space induced ...
hhhl 27445 The Hilbert space structur...
hilcms 27446 The Hilbert space norm det...
hilhl 27447 The Hilbert space of the H...
issh 27449 Subspace ` H ` of a Hilber...
issh2 27450 Subspace ` H ` of a Hilber...
shss 27451 A subspace is a subset of ...
shel 27452 A member of a subspace of ...
shex 27453 The set of subspaces of a ...
shssii 27454 A closed subspace of a Hil...
sheli 27455 A member of a subspace of ...
shelii 27456 A member of a subspace of ...
sh0 27457 The zero vector belongs to...
shaddcl 27458 Closure of vector addition...
shmulcl 27459 Closure of vector scalar m...
issh3 27460 Subspace ` H ` of a Hilber...
shsubcl 27461 Closure of vector subtract...
isch 27463 Closed subspace ` H ` of a...
isch2 27464 Closed subspace ` H ` of a...
chsh 27465 A closed subspace is a sub...
chsssh 27466 Closed subspaces are subsp...
chex 27467 The set of closed subspace...
chshii 27468 A closed subspace is a sub...
ch0 27469 The zero vector belongs to...
chss 27470 A closed subspace of a Hil...
chel 27471 A member of a closed subsp...
chssii 27472 A closed subspace of a Hil...
cheli 27473 A member of a closed subsp...
chelii 27474 A member of a closed subsp...
chlimi 27475 The limit property of a cl...
hlim0 27476 The zero sequence in Hilbe...
hlimcaui 27477 If a sequence in Hilbert s...
hlimf 27478 Function-like behavior of ...
hlimuni 27479 A Hilbert space sequence c...
hlimreui 27480 The limit of a Hilbert spa...
hlimeui 27481 The limit of a Hilbert spa...
isch3 27482 A Hilbert subspace is clos...
chcompl 27483 Completeness of a closed s...
helch 27484 The unit Hilbert lattice e...
ifchhv 27485 Prove ` if ( A e. CH , A ,...
helsh 27486 Hilbert space is a subspac...
shsspwh 27487 Subspaces are subsets of H...
chsspwh 27488 Closed subspaces are subse...
hsn0elch 27489 The zero subspace belongs ...
norm1 27490 From any nonzero Hilbert s...
norm1exi 27491 A normalized vector exists...
norm1hex 27492 A normalized vector can ex...
elch0 27495 Membership in zero for clo...
h0elch 27496 The zero subspace is a clo...
h0elsh 27497 The zero subspace is a sub...
hhssva 27498 The vector addition operat...
hhsssm 27499 The scalar multiplication ...
hhssnm 27500 The norm operation on a su...
issubgoilem 27501 Lemma for ~ hhssabloilem ....
hhssabloilem 27502 Lemma for ~ hhssabloi . F...
hhssabloi 27503 Abelian group property of ...
hhssablo 27504 Abelian group property of ...
hhssnv 27505 Normed complex vector spac...
hhssnvt 27506 Normed complex vector spac...
hhsst 27507 A member of ` SH ` is a su...
hhshsslem1 27508 Lemma for ~ hhsssh . (Con...
hhshsslem2 27509 Lemma for ~ hhsssh . (Con...
hhsssh 27510 The predicate " ` H ` is a...
hhsssh2 27511 The predicate " ` H ` is a...
hhssba 27512 The base set of a subspace...
hhssvs 27513 The vector subtraction ope...
hhssvsf 27514 Mapping of the vector subt...
hhssph 27515 Inner product space proper...
hhssims 27516 Induced metric of a subspa...
hhssims2 27517 Induced metric of a subspa...
hhssmet 27518 Induced metric of a subspa...
hhssmetdval 27519 Value of the distance func...
hhsscms 27520 The induced metric of a cl...
hhssbn 27521 Banach space property of a...
hhsshl 27522 Hilbert space property of ...
ocval 27523 Value of orthogonal comple...
ocel 27524 Membership in orthogonal c...
shocel 27525 Membership in orthogonal c...
ocsh 27526 The orthogonal complement ...
shocsh 27527 The orthogonal complement ...
ocss 27528 An orthogonal complement i...
shocss 27529 An orthogonal complement i...
occon 27530 Contraposition law for ort...
occon2 27531 Double contraposition for ...
occon2i 27532 Double contraposition for ...
oc0 27533 The zero vector belongs to...
ocorth 27534 Members of a subset and it...
shocorth 27535 Members of a subspace and ...
ococss 27536 Inclusion in complement of...
shococss 27537 Inclusion in complement of...
shorth 27538 Members of orthogonal subs...
ocin 27539 Intersection of a Hilbert ...
occon3 27540 Hilbert lattice contraposi...
ocnel 27541 A nonzero vector in the co...
chocvali 27542 Value of the orthogonal co...
shuni 27543 Two subspaces with trivial...
chocunii 27544 Lemma for uniqueness part ...
pjhthmo 27545 Projection Theorem, unique...
occllem 27546 Lemma for ~ occl . (Contr...
occl 27547 Closure of complement of H...
shoccl 27548 Closure of complement of H...
choccl 27549 Closure of complement of H...
choccli 27550 Closure of ` CH ` orthocom...
shsval 27555 Value of subspace sum of t...
shsss 27556 The subspace sum is a subs...
shsel 27557 Membership in the subspace...
shsel3 27558 Membership in the subspace...
shseli 27559 Membership in subspace sum...
shscli 27560 Closure of subspace sum. ...
shscl 27561 Closure of subspace sum. ...
shscom 27562 Commutative law for subspa...
shsva 27563 Vector sum belongs to subs...
shsel1 27564 A subspace sum contains a ...
shsel2 27565 A subspace sum contains a ...
shsvs 27566 Vector subtraction belongs...
shsub1 27567 Subspace sum is an upper b...
shsub2 27568 Subspace sum is an upper b...
choc0 27569 The orthocomplement of the...
choc1 27570 The orthocomplement of the...
chocnul 27571 Orthogonal complement of t...
shintcli 27572 Closure of intersection of...
shintcl 27573 The intersection of a none...
chintcli 27574 The intersection of a none...
chintcl 27575 The intersection (infimum)...
spanval 27576 Value of the linear span o...
hsupval 27577 Value of supremum of set o...
chsupval 27578 The value of the supremum ...
spancl 27579 The span of a subset of Hi...
elspancl 27580 A member of a span is a ve...
shsupcl 27581 Closure of the subspace su...
hsupcl 27582 Closure of supremum of set...
chsupcl 27583 Closure of supremum of sub...
hsupss 27584 Subset relation for suprem...
chsupss 27585 Subset relation for suprem...
hsupunss 27586 The union of a set of Hilb...
chsupunss 27587 The union of a set of clos...
spanss2 27588 A subset of Hilbert space ...
shsupunss 27589 The union of a set of subs...
spanid 27590 A subspace of Hilbert spac...
spanss 27591 Ordering relationship for ...
spanssoc 27592 The span of a subset of Hi...
sshjval 27593 Value of join for subsets ...
shjval 27594 Value of join in ` SH ` . ...
chjval 27595 Value of join in ` CH ` . ...
chjvali 27596 Value of join in ` CH ` . ...
sshjval3 27597 Value of join for subsets ...
sshjcl 27598 Closure of join for subset...
shjcl 27599 Closure of join in ` SH ` ...
chjcl 27600 Closure of join in ` CH ` ...
shjcom 27601 Commutative law for Hilber...
shless 27602 Subset implies subset of s...
shlej1 27603 Add disjunct to both sides...
shlej2 27604 Add disjunct to both sides...
shincli 27605 Closure of intersection of...
shscomi 27606 Commutative law for subspa...
shsvai 27607 Vector sum belongs to subs...
shsel1i 27608 A subspace sum contains a ...
shsel2i 27609 A subspace sum contains a ...
shsvsi 27610 Vector subtraction belongs...
shunssi 27611 Union is smaller than subs...
shunssji 27612 Union is smaller than Hilb...
shsleji 27613 Subspace sum is smaller th...
shjcomi 27614 Commutative law for join i...
shsub1i 27615 Subspace sum is an upper b...
shsub2i 27616 Subspace sum is an upper b...
shub1i 27617 Hilbert lattice join is an...
shjcli 27618 Closure of ` CH ` join. (...
shjshcli 27619 ` SH ` closure of join. (...
shlessi 27620 Subset implies subset of s...
shlej1i 27621 Add disjunct to both sides...
shlej2i 27622 Add disjunct to both sides...
shslej 27623 Subspace sum is smaller th...
shincl 27624 Closure of intersection of...
shub1 27625 Hilbert lattice join is an...
shub2 27626 A subspace is a subset of ...
shsidmi 27627 Idempotent law for Hilbert...
shslubi 27628 The least upper bound law ...
shlesb1i 27629 Hilbert lattice ordering i...
shsval2i 27630 An alternate way to expres...
shsval3i 27631 An alternate way to expres...
shmodsi 27632 The modular law holds for ...
shmodi 27633 The modular law is implied...
pjhthlem1 27634 Lemma for ~ pjhth . (Cont...
pjhthlem2 27635 Lemma for ~ pjhth . (Cont...
pjhth 27636 Projection Theorem: Any H...
pjhtheu 27637 Projection Theorem: Any H...
pjhfval 27639 The value of the projectio...
pjhval 27640 Value of a projection. (C...
pjpreeq 27641 Equality with a projection...
pjeq 27642 Equality with a projection...
axpjcl 27643 Closure of a projection in...
pjhcl 27644 Closure of a projection in...
omlsilem 27645 Lemma for orthomodular law...
omlsii 27646 Subspace inference form of...
omlsi 27647 Subspace form of orthomodu...
ococi 27648 Complement of complement o...
ococ 27649 Complement of complement o...
dfch2 27650 Alternate definition of th...
ococin 27651 The double complement is t...
hsupval2 27652 Alternate definition of su...
chsupval2 27653 The value of the supremum ...
sshjval2 27654 Value of join in the set o...
chsupid 27655 A subspace is the supremum...
chsupsn 27656 Value of supremum of subse...
shlub 27657 Hilbert lattice join is th...
shlubi 27658 Hilbert lattice join is th...
pjhtheu2 27659 Uniqueness of ` y ` for th...
pjcli 27660 Closure of a projection in...
pjhcli 27661 Closure of a projection in...
pjpjpre 27662 Decomposition of a vector ...
axpjpj 27663 Decomposition of a vector ...
pjclii 27664 Closure of a projection in...
pjhclii 27665 Closure of a projection in...
pjpj0i 27666 Decomposition of a vector ...
pjpji 27667 Decomposition of a vector ...
pjpjhth 27668 Projection Theorem: Any H...
pjpjhthi 27669 Projection Theorem: Any H...
pjop 27670 Orthocomplement projection...
pjpo 27671 Projection in terms of ort...
pjopi 27672 Orthocomplement projection...
pjpoi 27673 Projection in terms of ort...
pjoc1i 27674 Projection of a vector in ...
pjchi 27675 Projection of a vector in ...
pjoccl 27676 The part of a vector that ...
pjoc1 27677 Projection of a vector in ...
pjomli 27678 Subspace form of orthomodu...
pjoml 27679 Subspace form of orthomodu...
pjococi 27680 Proof of orthocomplement t...
pjoc2i 27681 Projection of a vector in ...
pjoc2 27682 Projection of a vector in ...
sh0le 27683 The zero subspace is the s...
ch0le 27684 The zero subspace is the s...
shle0 27685 No subspace is smaller tha...
chle0 27686 No Hilbert lattice element...
chnlen0 27687 A Hilbert lattice element ...
ch0pss 27688 The zero subspace is a pro...
orthin 27689 The intersection of orthog...
ssjo 27690 The lattice join of a subs...
shne0i 27691 A nonzero subspace has a n...
shs0i 27692 Hilbert subspace sum with ...
shs00i 27693 Two subspaces are zero iff...
ch0lei 27694 The closed subspace zero i...
chle0i 27695 No Hilbert closed subspace...
chne0i 27696 A nonzero closed subspace ...
chocini 27697 Intersection of a closed s...
chj0i 27698 Join with lattice zero in ...
chm1i 27699 Meet with lattice one in `...
chjcli 27700 Closure of ` CH ` join. (...
chsleji 27701 Subspace sum is smaller th...
chseli 27702 Membership in subspace sum...
chincli 27703 Closure of Hilbert lattice...
chsscon3i 27704 Hilbert lattice contraposi...
chsscon1i 27705 Hilbert lattice contraposi...
chsscon2i 27706 Hilbert lattice contraposi...
chcon2i 27707 Hilbert lattice contraposi...
chcon1i 27708 Hilbert lattice contraposi...
chcon3i 27709 Hilbert lattice contraposi...
chunssji 27710 Union is smaller than ` CH...
chjcomi 27711 Commutative law for join i...
chub1i 27712 ` CH ` join is an upper bo...
chub2i 27713 ` CH ` join is an upper bo...
chlubi 27714 Hilbert lattice join is th...
chlubii 27715 Hilbert lattice join is th...
chlej1i 27716 Add join to both sides of ...
chlej2i 27717 Add join to both sides of ...
chlej12i 27718 Add join to both sides of ...
chlejb1i 27719 Hilbert lattice ordering i...
chdmm1i 27720 De Morgan's law for meet i...
chdmm2i 27721 De Morgan's law for meet i...
chdmm3i 27722 De Morgan's law for meet i...
chdmm4i 27723 De Morgan's law for meet i...
chdmj1i 27724 De Morgan's law for join i...
chdmj2i 27725 De Morgan's law for join i...
chdmj3i 27726 De Morgan's law for join i...
chdmj4i 27727 De Morgan's law for join i...
chnlei 27728 Equivalent expressions for...
chjassi 27729 Associative law for Hilber...
chj00i 27730 Two Hilbert lattice elemen...
chjoi 27731 The join of a closed subsp...
chj1i 27732 Join with Hilbert lattice ...
chm0i 27733 Meet with Hilbert lattice ...
chm0 27734 Meet with Hilbert lattice ...
shjshsi 27735 Hilbert lattice join equal...
shjshseli 27736 A closed subspace sum equa...
chne0 27737 A nonzero closed subspace ...
chocin 27738 Intersection of a closed s...
chssoc 27739 A closed subspace less tha...
chj0 27740 Join with Hilbert lattice ...
chslej 27741 Subspace sum is smaller th...
chincl 27742 Closure of Hilbert lattice...
chsscon3 27743 Hilbert lattice contraposi...
chsscon1 27744 Hilbert lattice contraposi...
chsscon2 27745 Hilbert lattice contraposi...
chpsscon3 27746 Hilbert lattice contraposi...
chpsscon1 27747 Hilbert lattice contraposi...
chpsscon2 27748 Hilbert lattice contraposi...
chjcom 27749 Commutative law for Hilber...
chub1 27750 Hilbert lattice join is gr...
chub2 27751 Hilbert lattice join is gr...
chlub 27752 Hilbert lattice join is th...
chlej1 27753 Add join to both sides of ...
chlej2 27754 Add join to both sides of ...
chlejb1 27755 Hilbert lattice ordering i...
chlejb2 27756 Hilbert lattice ordering i...
chnle 27757 Equivalent expressions for...
chjo 27758 The join of a closed subsp...
chabs1 27759 Hilbert lattice absorption...
chabs2 27760 Hilbert lattice absorption...
chabs1i 27761 Hilbert lattice absorption...
chabs2i 27762 Hilbert lattice absorption...
chjidm 27763 Idempotent law for Hilbert...
chjidmi 27764 Idempotent law for Hilbert...
chj12i 27765 A rearrangement of Hilbert...
chj4i 27766 Rearrangement of the join ...
chjjdiri 27767 Hilbert lattice join distr...
chdmm1 27768 De Morgan's law for meet i...
chdmm2 27769 De Morgan's law for meet i...
chdmm3 27770 De Morgan's law for meet i...
chdmm4 27771 De Morgan's law for meet i...
chdmj1 27772 De Morgan's law for join i...
chdmj2 27773 De Morgan's law for join i...
chdmj3 27774 De Morgan's law for join i...
chdmj4 27775 De Morgan's law for join i...
chjass 27776 Associative law for Hilber...
chj12 27777 A rearrangement of Hilbert...
chj4 27778 Rearrangement of the join ...
ledii 27779 An ortholattice is distrib...
lediri 27780 An ortholattice is distrib...
lejdii 27781 An ortholattice is distrib...
lejdiri 27782 An ortholattice is distrib...
ledi 27783 An ortholattice is distrib...
spansn0 27784 The span of the singleton ...
span0 27785 The span of the empty set ...
elspani 27786 Membership in the span of ...
spanuni 27787 The span of a union is the...
spanun 27788 The span of a union is the...
sshhococi 27789 The join of two Hilbert sp...
hne0 27790 Hilbert space has a nonzer...
chsup0 27791 The supremum of the empty ...
h1deoi 27792 Membership in orthocomplem...
h1dei 27793 Membership in 1-dimensiona...
h1did 27794 A generating vector belong...
h1dn0 27795 A nonzero vector generates...
h1de2i 27796 Membership in 1-dimensiona...
h1de2bi 27797 Membership in 1-dimensiona...
h1de2ctlem 27798 Lemma for ~ h1de2ci . (Co...
h1de2ci 27799 Membership in 1-dimensiona...
spansni 27800 The span of a singleton in...
elspansni 27801 Membership in the span of ...
spansn 27802 The span of a singleton in...
spansnch 27803 The span of a Hilbert spac...
spansnsh 27804 The span of a Hilbert spac...
spansnchi 27805 The span of a singleton in...
spansnid 27806 A vector belongs to the sp...
spansnmul 27807 A scalar product with a ve...
elspansncl 27808 A member of a span of a si...
elspansn 27809 Membership in the span of ...
elspansn2 27810 Membership in the span of ...
spansncol 27811 The singletons of collinea...
spansneleqi 27812 Membership relation implie...
spansneleq 27813 Membership relation that i...
spansnss 27814 The span of the singleton ...
elspansn3 27815 A member of the span of th...
elspansn4 27816 A span membership conditio...
elspansn5 27817 A vector belonging to both...
spansnss2 27818 The span of the singleton ...
normcan 27819 Cancellation-type law that...
pjspansn 27820 A projection on the span o...
spansnpji 27821 A subset of Hilbert space ...
spanunsni 27822 The span of the union of a...
spanpr 27823 The span of a pair of vect...
h1datomi 27824 A 1-dimensional subspace i...
h1datom 27825 A 1-dimensional subspace i...
cmbr 27827 Binary relation expressing...
pjoml2i 27828 Variation of orthomodular ...
pjoml3i 27829 Variation of orthomodular ...
pjoml4i 27830 Variation of orthomodular ...
pjoml5i 27831 The orthomodular law. Rem...
pjoml6i 27832 An equivalent of the ortho...
cmbri 27833 Binary relation expressing...
cmcmlem 27834 Commutation is symmetric. ...
cmcmi 27835 Commutation is symmetric. ...
cmcm2i 27836 Commutation with orthocomp...
cmcm3i 27837 Commutation with orthocomp...
cmcm4i 27838 Commutation with orthocomp...
cmbr2i 27839 Alternate definition of th...
cmcmii 27840 Commutation is symmetric. ...
cmcm2ii 27841 Commutation with orthocomp...
cmcm3ii 27842 Commutation with orthocomp...
cmbr3i 27843 Alternate definition for t...
cmbr4i 27844 Alternate definition for t...
lecmi 27845 Comparable Hilbert lattice...
lecmii 27846 Comparable Hilbert lattice...
cmj1i 27847 A Hilbert lattice element ...
cmj2i 27848 A Hilbert lattice element ...
cmm1i 27849 A Hilbert lattice element ...
cmm2i 27850 A Hilbert lattice element ...
cmbr3 27851 Alternate definition for t...
cm0 27852 The zero Hilbert lattice e...
cmidi 27853 The commutes relation is r...
pjoml2 27854 Variation of orthomodular ...
pjoml3 27855 Variation of orthomodular ...
pjoml5 27856 The orthomodular law. Rem...
cmcm 27857 Commutation is symmetric. ...
cmcm3 27858 Commutation with orthocomp...
cmcm2 27859 Commutation with orthocomp...
lecm 27860 Comparable Hilbert lattice...
fh1 27861 Foulis-Holland Theorem. I...
fh2 27862 Foulis-Holland Theorem. I...
cm2j 27863 A lattice element that com...
fh1i 27864 Foulis-Holland Theorem. I...
fh2i 27865 Foulis-Holland Theorem. I...
fh3i 27866 Variation of the Foulis-Ho...
fh4i 27867 Variation of the Foulis-Ho...
cm2ji 27868 A lattice element that com...
cm2mi 27869 A lattice element that com...
qlax1i 27870 One of the equations showi...
qlax2i 27871 One of the equations showi...
qlax3i 27872 One of the equations showi...
qlax4i 27873 One of the equations showi...
qlax5i 27874 One of the equations showi...
qlaxr1i 27875 One of the conditions show...
qlaxr2i 27876 One of the conditions show...
qlaxr4i 27877 One of the conditions show...
qlaxr5i 27878 One of the conditions show...
qlaxr3i 27879 A variation of the orthomo...
chscllem1 27880 Lemma for ~ chscl . (Cont...
chscllem2 27881 Lemma for ~ chscl . (Cont...
chscllem3 27882 Lemma for ~ chscl . (Cont...
chscllem4 27883 Lemma for ~ chscl . (Cont...
chscl 27884 The subspace sum of two cl...
osumi 27885 If two closed subspaces of...
osumcori 27886 Corollary of ~ osumi . (C...
osumcor2i 27887 Corollary of ~ osumi , sho...
osum 27888 If two closed subspaces of...
spansnji 27889 The subspace sum of a clos...
spansnj 27890 The subspace sum of a clos...
spansnscl 27891 The subspace sum of a clos...
sumspansn 27892 The sum of two vectors bel...
spansnm0i 27893 The meet of different one-...
nonbooli 27894 A Hilbert lattice with two...
spansncvi 27895 Hilbert space has the cove...
spansncv 27896 Hilbert space has the cove...
5oalem1 27897 Lemma for orthoarguesian l...
5oalem2 27898 Lemma for orthoarguesian l...
5oalem3 27899 Lemma for orthoarguesian l...
5oalem4 27900 Lemma for orthoarguesian l...
5oalem5 27901 Lemma for orthoarguesian l...
5oalem6 27902 Lemma for orthoarguesian l...
5oalem7 27903 Lemma for orthoarguesian l...
5oai 27904 Orthoarguesian law 5OA. Th...
3oalem1 27905 Lemma for 3OA (weak) ortho...
3oalem2 27906 Lemma for 3OA (weak) ortho...
3oalem3 27907 Lemma for 3OA (weak) ortho...
3oalem4 27908 Lemma for 3OA (weak) ortho...
3oalem5 27909 Lemma for 3OA (weak) ortho...
3oalem6 27910 Lemma for 3OA (weak) ortho...
3oai 27911 3OA (weak) orthoarguesian ...
pjorthi 27912 Projection components on o...
pjch1 27913 Property of identity proje...
pjo 27914 The orthogonal projection....
pjcompi 27915 Component of a projection....
pjidmi 27916 A projection is idempotent...
pjadjii 27917 A projection is self-adjoi...
pjaddii 27918 Projection of vector sum i...
pjinormii 27919 The inner product of a pro...
pjmulii 27920 Projection of (scalar) pro...
pjsubii 27921 Projection of vector diffe...
pjsslem 27922 Lemma for subset relations...
pjss2i 27923 Subset relationship for pr...
pjssmii 27924 Projection meet property. ...
pjssge0ii 27925 Theorem 4.5(iv)->(v) of [B...
pjdifnormii 27926 Theorem 4.5(v)<->(vi) of [...
pjcji 27927 The projection on a subspa...
pjadji 27928 A projection is self-adjoi...
pjaddi 27929 Projection of vector sum i...
pjinormi 27930 The inner product of a pro...
pjsubi 27931 Projection of vector diffe...
pjmuli 27932 Projection of scalar produ...
pjige0i 27933 The inner product of a pro...
pjige0 27934 The inner product of a pro...
pjcjt2 27935 The projection on a subspa...
pj0i 27936 The projection of the zero...
pjch 27937 Projection of a vector in ...
pjid 27938 The projection of a vector...
pjvec 27939 The set of vectors belongi...
pjocvec 27940 The set of vectors belongi...
pjocini 27941 Membership of projection i...
pjini 27942 Membership of projection i...
pjjsi 27943 A sufficient condition for...
pjfni 27944 Functionality of a project...
pjrni 27945 The range of a projection....
pjfoi 27946 A projection maps onto its...
pjfi 27947 The mapping of a projectio...
pjvi 27948 The value of a projection ...
pjhfo 27949 A projection maps onto its...
pjrn 27950 The range of a projection....
pjhf 27951 The mapping of a projectio...
pjfn 27952 Functionality of a project...
pjsumi 27953 The projection on a subspa...
pj11i 27954 One-to-one correspondence ...
pjdsi 27955 Vector decomposition into ...
pjds3i 27956 Vector decomposition into ...
pj11 27957 One-to-one correspondence ...
pjmfn 27958 Functionality of the proje...
pjmf1 27959 The projector function map...
pjoi0 27960 The inner product of proje...
pjoi0i 27961 The inner product of proje...
pjopythi 27962 Pythagorean theorem for pr...
pjopyth 27963 Pythagorean theorem for pr...
pjnormi 27964 The norm of the projection...
pjpythi 27965 Pythagorean theorem for pr...
pjneli 27966 If a vector does not belon...
pjnorm 27967 The norm of the projection...
pjpyth 27968 Pythagorean theorem for pr...
pjnel 27969 If a vector does not belon...
pjnorm2 27970 A vector belongs to the su...
mayete3i 27971 Mayet's equation E_3. Par...
mayetes3i 27972 Mayet's equation E^*_3, de...
hosmval 27978 Value of the sum of two Hi...
hommval 27979 Value of the scalar produc...
hodmval 27980 Value of the difference of...
hfsmval 27981 Value of the sum of two Hi...
hfmmval 27982 Value of the scalar produc...
hosval 27983 Value of the sum of two Hi...
homval 27984 Value of the scalar produc...
hodval 27985 Value of the difference of...
hfsval 27986 Value of the sum of two Hi...
hfmval 27987 Value of the scalar produc...
hoscl 27988 Closure of the sum of two ...
homcl 27989 Closure of the scalar prod...
hodcl 27990 Closure of the difference ...
ho0val 27993 Value of the zero Hilbert ...
ho0f 27994 Functionality of the zero ...
df0op2 27995 Alternate definition of Hi...
dfiop2 27996 Alternate definition of Hi...
hoif 27997 Functionality of the Hilbe...
hoival 27998 The value of the Hilbert s...
hoico1 27999 Composition with the Hilbe...
hoico2 28000 Composition with the Hilbe...
hoaddcl 28001 The sum of Hilbert space o...
homulcl 28002 The scalar product of a Hi...
hoeq 28003 Equality of Hilbert space ...
hoeqi 28004 Equality of Hilbert space ...
hoscli 28005 Closure of Hilbert space o...
hodcli 28006 Closure of Hilbert space o...
hocoi 28007 Composition of Hilbert spa...
hococli 28008 Closure of composition of ...
hocofi 28009 Mapping of composition of ...
hocofni 28010 Functionality of compositi...
hoaddcli 28011 Mapping of sum of Hilbert ...
hosubcli 28012 Mapping of difference of H...
hoaddfni 28013 Functionality of sum of Hi...
hosubfni 28014 Functionality of differenc...
hoaddcomi 28015 Commutativity of sum of Hi...
hosubcl 28016 Mapping of difference of H...
hoaddcom 28017 Commutativity of sum of Hi...
hodsi 28018 Relationship between Hilbe...
hoaddassi 28019 Associativity of sum of Hi...
hoadd12i 28020 Commutative/associative la...
hoadd32i 28021 Commutative/associative la...
hocadddiri 28022 Distributive law for Hilbe...
hocsubdiri 28023 Distributive law for Hilbe...
ho2coi 28024 Double composition of Hilb...
hoaddass 28025 Associativity of sum of Hi...
hoadd32 28026 Commutative/associative la...
hoadd4 28027 Rearrangement of 4 terms i...
hocsubdir 28028 Distributive law for Hilbe...
hoaddid1i 28029 Sum of a Hilbert space ope...
hodidi 28030 Difference of a Hilbert sp...
ho0coi 28031 Composition of the zero op...
hoid1i 28032 Composition of Hilbert spa...
hoid1ri 28033 Composition of Hilbert spa...
hoaddid1 28034 Sum of a Hilbert space ope...
hodid 28035 Difference of a Hilbert sp...
hon0 28036 A Hilbert space operator i...
hodseqi 28037 Subtraction and addition o...
ho0subi 28038 Subtraction of Hilbert spa...
honegsubi 28039 Relationship between Hilbe...
ho0sub 28040 Subtraction of Hilbert spa...
hosubid1 28041 The zero operator subtract...
honegsub 28042 Relationship between Hilbe...
homulid2 28043 An operator equals its sca...
homco1 28044 Associative law for scalar...
homulass 28045 Scalar product associative...
hoadddi 28046 Scalar product distributiv...
hoadddir 28047 Scalar product reverse dis...
homul12 28048 Swap first and second fact...
honegneg 28049 Double negative of a Hilbe...
hosubneg 28050 Relationship between opera...
hosubdi 28051 Scalar product distributiv...
honegdi 28052 Distribution of negative o...
honegsubdi 28053 Distribution of negative o...
honegsubdi2 28054 Distribution of negative o...
hosubsub2 28055 Law for double subtraction...
hosub4 28056 Rearrangement of 4 terms i...
hosubadd4 28057 Rearrangement of 4 terms i...
hoaddsubass 28058 Associative-type law for a...
hoaddsub 28059 Law for operator addition ...
hosubsub 28060 Law for double subtraction...
hosubsub4 28061 Law for double subtraction...
ho2times 28062 Two times a Hilbert space ...
hoaddsubassi 28063 Associativity of sum and d...
hoaddsubi 28064 Law for sum and difference...
hosd1i 28065 Hilbert space operator sum...
hosd2i 28066 Hilbert space operator sum...
hopncani 28067 Hilbert space operator can...
honpcani 28068 Hilbert space operator can...
hosubeq0i 28069 If the difference between ...
honpncani 28070 Hilbert space operator can...
ho01i 28071 A condition implying that ...
ho02i 28072 A condition implying that ...
hoeq1 28073 A condition implying that ...
hoeq2 28074 A condition implying that ...
adjmo 28075 Every Hilbert space operat...
adjsym 28076 Symmetry property of an ad...
eigrei 28077 A necessary and sufficient...
eigre 28078 A necessary and sufficient...
eigposi 28079 A sufficient condition (fi...
eigorthi 28080 A necessary and sufficient...
eigorth 28081 A necessary and sufficient...
nmopval 28099 Value of the norm of a Hil...
elcnop 28100 Property defining a contin...
ellnop 28101 Property defining a linear...
lnopf 28102 A linear Hilbert space ope...
elbdop 28103 Property defining a bounde...
bdopln 28104 A bounded linear Hilbert s...
bdopf 28105 A bounded linear Hilbert s...
nmopsetretALT 28106 The set in the supremum of...
nmopsetretHIL 28107 The set in the supremum of...
nmopsetn0 28108 The set in the supremum of...
nmopxr 28109 The norm of a Hilbert spac...
nmoprepnf 28110 The norm of a Hilbert spac...
nmopgtmnf 28111 The norm of a Hilbert spac...
nmopreltpnf 28112 The norm of a Hilbert spac...
nmopre 28113 The norm of a bounded oper...
elbdop2 28114 Property defining a bounde...
elunop 28115 Property defining a unitar...
elhmop 28116 Property defining a Hermit...
hmopf 28117 A Hermitian operator is a ...
hmopex 28118 The class of Hermitian ope...
nmfnval 28119 Value of the norm of a Hil...
nmfnsetre 28120 The set in the supremum of...
nmfnsetn0 28121 The set in the supremum of...
nmfnxr 28122 The norm of any Hilbert sp...
nmfnrepnf 28123 The norm of a Hilbert spac...
nlfnval 28124 Value of the null space of...
elcnfn 28125 Property defining a contin...
ellnfn 28126 Property defining a linear...
lnfnf 28127 A linear Hilbert space fun...
dfadj2 28128 Alternate definition of th...
funadj 28129 Functionality of the adjoi...
dmadjss 28130 The domain of the adjoint ...
dmadjop 28131 A member of the domain of ...
adjeu 28132 Elementhood in the domain ...
adjval 28133 Value of the adjoint funct...
adjval2 28134 Value of the adjoint funct...
cnvadj 28135 The adjoint function equal...
funcnvadj 28136 The converse of the adjoin...
adj1o 28137 The adjoint function maps ...
dmadjrn 28138 The adjoint of an operator...
eigvecval 28139 The set of eigenvectors of...
eigvalfval 28140 The eigenvalues of eigenve...
specval 28141 The value of the spectrum ...
speccl 28142 The spectrum of an operato...
hhlnoi 28143 The linear operators of Hi...
hhnmoi 28144 The norm of an operator in...
hhbloi 28145 A bounded linear operator ...
hh0oi 28146 The zero operator in Hilbe...
hhcno 28147 The continuous operators o...
hhcnf 28148 The continuous functionals...
dmadjrnb 28149 The adjoint of an operator...
nmoplb 28150 A lower bound for an opera...
nmopub 28151 An upper bound for an oper...
nmopub2tALT 28152 An upper bound for an oper...
nmopub2tHIL 28153 An upper bound for an oper...
nmopge0 28154 The norm of any Hilbert sp...
nmopgt0 28155 A linear Hilbert space ope...
cnopc 28156 Basic continuity property ...
lnopl 28157 Basic linearity property o...
unop 28158 Basic inner product proper...
unopf1o 28159 A unitary operator in Hilb...
unopnorm 28160 A unitary operator is idem...
cnvunop 28161 The inverse (converse) of ...
unopadj 28162 The inverse (converse) of ...
unoplin 28163 A unitary operator is line...
counop 28164 The composition of two uni...
hmop 28165 Basic inner product proper...
hmopre 28166 The inner product of the v...
nmfnlb 28167 A lower bound for a functi...
nmfnleub 28168 An upper bound for the nor...
nmfnleub2 28169 An upper bound for the nor...
nmfnge0 28170 The norm of any Hilbert sp...
elnlfn 28171 Membership in the null spa...
elnlfn2 28172 Membership in the null spa...
cnfnc 28173 Basic continuity property ...
lnfnl 28174 Basic linearity property o...
adjcl 28175 Closure of the adjoint of ...
adj1 28176 Property of an adjoint Hil...
adj2 28177 Property of an adjoint Hil...
adjeq 28178 A property that determines...
adjadj 28179 Double adjoint. Theorem 3...
adjvalval 28180 Value of the value of the ...
unopadj2 28181 The adjoint of a unitary o...
hmopadj 28182 A Hermitian operator is se...
hmdmadj 28183 Every Hermitian operator h...
hmopadj2 28184 An operator is Hermitian i...
hmoplin 28185 A Hermitian operator is li...
brafval 28186 The bra of a vector, expre...
braval 28187 A bra-ket juxtaposition, e...
braadd 28188 Linearity property of bra ...
bramul 28189 Linearity property of bra ...
brafn 28190 The bra function is a func...
bralnfn 28191 The Dirac bra function is ...
bracl 28192 Closure of the bra functio...
bra0 28193 The Dirac bra of the zero ...
brafnmul 28194 Anti-linearity property of...
kbfval 28195 The outer product of two v...
kbop 28196 The outer product of two v...
kbval 28197 The value of the operator ...
kbmul 28198 Multiplication property of...
kbpj 28199 If a vector ` A ` has norm...
eleigvec 28200 Membership in the set of e...
eleigvec2 28201 Membership in the set of e...
eleigveccl 28202 Closure of an eigenvector ...
eigvalval 28203 The eigenvalue of an eigen...
eigvalcl 28204 An eigenvalue is a complex...
eigvec1 28205 Property of an eigenvector...
eighmre 28206 The eigenvalues of a Hermi...
eighmorth 28207 Eigenvectors of a Hermitia...
nmopnegi 28208 Value of the norm of the n...
lnop0 28209 The value of a linear Hilb...
lnopmul 28210 Multiplicative property of...
lnopli 28211 Basic scalar product prope...
lnopfi 28212 A linear Hilbert space ope...
lnop0i 28213 The value of a linear Hilb...
lnopaddi 28214 Additive property of a lin...
lnopmuli 28215 Multiplicative property of...
lnopaddmuli 28216 Sum/product property of a ...
lnopsubi 28217 Subtraction property for a...
lnopsubmuli 28218 Subtraction/product proper...
lnopmulsubi 28219 Product/subtraction proper...
homco2 28220 Move a scalar product out ...
idunop 28221 The identity function (res...
0cnop 28222 The identically zero funct...
0cnfn 28223 The identically zero funct...
idcnop 28224 The identity function (res...
idhmop 28225 The Hilbert space identity...
0hmop 28226 The identically zero funct...
0lnop 28227 The identically zero funct...
0lnfn 28228 The identically zero funct...
nmop0 28229 The norm of the zero opera...
nmfn0 28230 The norm of the identicall...
hmopbdoptHIL 28231 A Hermitian operator is a ...
hoddii 28232 Distributive law for Hilbe...
hoddi 28233 Distributive law for Hilbe...
nmop0h 28234 The norm of any operator o...
idlnop 28235 The identity function (res...
0bdop 28236 The identically zero opera...
adj0 28237 Adjoint of the zero operat...
nmlnop0iALT 28238 A linear operator with a z...
nmlnop0iHIL 28239 A linear operator with a z...
nmlnopgt0i 28240 A linear Hilbert space ope...
nmlnop0 28241 A linear operator with a z...
nmlnopne0 28242 A linear operator with a n...
lnopmi 28243 The scalar product of a li...
lnophsi 28244 The sum of two linear oper...
lnophdi 28245 The difference of two line...
lnopcoi 28246 The composition of two lin...
lnopco0i 28247 The composition of a linea...
lnopeq0lem1 28248 Lemma for ~ lnopeq0i . Ap...
lnopeq0lem2 28249 Lemma for ~ lnopeq0i . (C...
lnopeq0i 28250 A condition implying that ...
lnopeqi 28251 Two linear Hilbert space o...
lnopeq 28252 Two linear Hilbert space o...
lnopunilem1 28253 Lemma for ~ lnopunii . (C...
lnopunilem2 28254 Lemma for ~ lnopunii . (C...
lnopunii 28255 If a linear operator (whos...
elunop2 28256 An operator is unitary iff...
nmopun 28257 Norm of a unitary Hilbert ...
unopbd 28258 A unitary operator is a bo...
lnophmlem1 28259 Lemma for ~ lnophmi . (Co...
lnophmlem2 28260 Lemma for ~ lnophmi . (Co...
lnophmi 28261 A linear operator is Hermi...
lnophm 28262 A linear operator is Hermi...
hmops 28263 The sum of two Hermitian o...
hmopm 28264 The scalar product of a He...
hmopd 28265 The difference of two Herm...
hmopco 28266 The composition of two com...
nmbdoplbi 28267 A lower bound for the norm...
nmbdoplb 28268 A lower bound for the norm...
nmcexi 28269 Lemma for ~ nmcopexi and ~...
nmcopexi 28270 The norm of a continuous l...
nmcoplbi 28271 A lower bound for the norm...
nmcopex 28272 The norm of a continuous l...
nmcoplb 28273 A lower bound for the norm...
nmophmi 28274 The norm of the scalar pro...
bdophmi 28275 The scalar product of a bo...
lnconi 28276 Lemma for ~ lnopconi and ~...
lnopconi 28277 A condition equivalent to ...
lnopcon 28278 A condition equivalent to ...
lnopcnbd 28279 A linear operator is conti...
lncnopbd 28280 A continuous linear operat...
lncnbd 28281 A continuous linear operat...
lnopcnre 28282 A linear operator is conti...
lnfnli 28283 Basic property of a linear...
lnfnfi 28284 A linear Hilbert space fun...
lnfn0i 28285 The value of a linear Hilb...
lnfnaddi 28286 Additive property of a lin...
lnfnmuli 28287 Multiplicative property of...
lnfnaddmuli 28288 Sum/product property of a ...
lnfnsubi 28289 Subtraction property for a...
lnfn0 28290 The value of a linear Hilb...
lnfnmul 28291 Multiplicative property of...
nmbdfnlbi 28292 A lower bound for the norm...
nmbdfnlb 28293 A lower bound for the norm...
nmcfnexi 28294 The norm of a continuous l...
nmcfnlbi 28295 A lower bound for the norm...
nmcfnex 28296 The norm of a continuous l...
nmcfnlb 28297 A lower bound of the norm ...
lnfnconi 28298 A condition equivalent to ...
lnfncon 28299 A condition equivalent to ...
lnfncnbd 28300 A linear functional is con...
imaelshi 28301 The image of a subspace un...
rnelshi 28302 The range of a linear oper...
nlelshi 28303 The null space of a linear...
nlelchi 28304 The null space of a contin...
riesz3i 28305 A continuous linear functi...
riesz4i 28306 A continuous linear functi...
riesz4 28307 A continuous linear functi...
riesz1 28308 Part 1 of the Riesz repres...
riesz2 28309 Part 2 of the Riesz repres...
cnlnadjlem1 28310 Lemma for ~ cnlnadji (Theo...
cnlnadjlem2 28311 Lemma for ~ cnlnadji . ` G...
cnlnadjlem3 28312 Lemma for ~ cnlnadji . By...
cnlnadjlem4 28313 Lemma for ~ cnlnadji . Th...
cnlnadjlem5 28314 Lemma for ~ cnlnadji . ` F...
cnlnadjlem6 28315 Lemma for ~ cnlnadji . ` F...
cnlnadjlem7 28316 Lemma for ~ cnlnadji . He...
cnlnadjlem8 28317 Lemma for ~ cnlnadji . ` F...
cnlnadjlem9 28318 Lemma for ~ cnlnadji . ` F...
cnlnadji 28319 Every continuous linear op...
cnlnadjeui 28320 Every continuous linear op...
cnlnadjeu 28321 Every continuous linear op...
cnlnadj 28322 Every continuous linear op...
cnlnssadj 28323 Every continuous linear Hi...
bdopssadj 28324 Every bounded linear Hilbe...
bdopadj 28325 Every bounded linear Hilbe...
adjbdln 28326 The adjoint of a bounded l...
adjbdlnb 28327 An operator is bounded and...
adjbd1o 28328 The mapping of adjoints of...
adjlnop 28329 The adjoint of an operator...
adjsslnop 28330 Every operator with an adj...
nmopadjlei 28331 Property of the norm of an...
nmopadjlem 28332 Lemma for ~ nmopadji . (C...
nmopadji 28333 Property of the norm of an...
adjeq0 28334 An operator is zero iff it...
adjmul 28335 The adjoint of the scalar ...
adjadd 28336 The adjoint of the sum of ...
nmoptrii 28337 Triangle inequality for th...
nmopcoi 28338 Upper bound for the norm o...
bdophsi 28339 The sum of two bounded lin...
bdophdi 28340 The difference between two...
bdopcoi 28341 The composition of two bou...
nmoptri2i 28342 Triangle-type inequality f...
adjcoi 28343 The adjoint of a compositi...
nmopcoadji 28344 The norm of an operator co...
nmopcoadj2i 28345 The norm of an operator co...
nmopcoadj0i 28346 An operator composed with ...
unierri 28347 If we approximate a chain ...
branmfn 28348 The norm of the bra functi...
brabn 28349 The bra of a vector is a b...
rnbra 28350 The set of bras equals the...
bra11 28351 The bra function maps vect...
bracnln 28352 A bra is a continuous line...
cnvbraval 28353 Value of the converse of t...
cnvbracl 28354 Closure of the converse of...
cnvbrabra 28355 The converse bra of the br...
bracnvbra 28356 The bra of the converse br...
bracnlnval 28357 The vector that a continuo...
cnvbramul 28358 Multiplication property of...
kbass1 28359 Dirac bra-ket associative ...
kbass2 28360 Dirac bra-ket associative ...
kbass3 28361 Dirac bra-ket associative ...
kbass4 28362 Dirac bra-ket associative ...
kbass5 28363 Dirac bra-ket associative ...
kbass6 28364 Dirac bra-ket associative ...
leopg 28365 Ordering relation for posi...
leop 28366 Ordering relation for oper...
leop2 28367 Ordering relation for oper...
leop3 28368 Operator ordering in terms...
leoppos 28369 Binary relation defining a...
leoprf2 28370 The ordering relation for ...
leoprf 28371 The ordering relation for ...
leopsq 28372 The square of a Hermitian ...
0leop 28373 The zero operator is a pos...
idleop 28374 The identity operator is a...
leopadd 28375 The sum of two positive op...
leopmuli 28376 The scalar product of a no...
leopmul 28377 The scalar product of a po...
leopmul2i 28378 Scalar product applied to ...
leoptri 28379 The positive operator orde...
leoptr 28380 The positive operator orde...
leopnmid 28381 A bounded Hermitian operat...
nmopleid 28382 A nonzero, bounded Hermiti...
opsqrlem1 28383 Lemma for opsqri . (Contr...
opsqrlem2 28384 Lemma for opsqri . ` F `` ...
opsqrlem3 28385 Lemma for opsqri . (Contr...
opsqrlem4 28386 Lemma for opsqri . (Contr...
opsqrlem5 28387 Lemma for opsqri . (Contr...
opsqrlem6 28388 Lemma for opsqri . (Contr...
pjhmopi 28389 A projector is a Hermitian...
pjlnopi 28390 A projector is a linear op...
pjnmopi 28391 The operator norm of a pro...
pjbdlni 28392 A projector is a bounded l...
pjhmop 28393 A projection is a Hermitia...
hmopidmchi 28394 An idempotent Hermitian op...
hmopidmpji 28395 An idempotent Hermitian op...
hmopidmch 28396 An idempotent Hermitian op...
hmopidmpj 28397 An idempotent Hermitian op...
pjsdii 28398 Distributive law for Hilbe...
pjddii 28399 Distributive law for Hilbe...
pjsdi2i 28400 Chained distributive law f...
pjcoi 28401 Composition of projections...
pjcocli 28402 Closure of composition of ...
pjcohcli 28403 Closure of composition of ...
pjadjcoi 28404 Adjoint of composition of ...
pjcofni 28405 Functionality of compositi...
pjss1coi 28406 Subset relationship for pr...
pjss2coi 28407 Subset relationship for pr...
pjssmi 28408 Projection meet property. ...
pjssge0i 28409 Theorem 4.5(iv)->(v) of [B...
pjdifnormi 28410 Theorem 4.5(v)<->(vi) of [...
pjnormssi 28411 Theorem 4.5(i)<->(vi) of [...
pjorthcoi 28412 Composition of projections...
pjscji 28413 The projection of orthogon...
pjssumi 28414 The projection on a subspa...
pjssposi 28415 Projector ordering can be ...
pjordi 28416 The definition of projecto...
pjssdif2i 28417 The projection subspace of...
pjssdif1i 28418 A necessary and sufficient...
pjimai 28419 The image of a projection....
pjidmcoi 28420 A projection is idempotent...
pjoccoi 28421 Composition of projections...
pjtoi 28422 Subspace sum of projection...
pjoci 28423 Projection of orthocomplem...
pjidmco 28424 A projection operator is i...
dfpjop 28425 Definition of projection o...
pjhmopidm 28426 Two ways to express the se...
elpjidm 28427 A projection operator is i...
elpjhmop 28428 A projection operator is H...
0leopj 28429 A projector is a positive ...
pjadj2 28430 A projector is self-adjoin...
pjadj3 28431 A projector is self-adjoin...
elpjch 28432 Reconstruction of the subs...
elpjrn 28433 Reconstruction of the subs...
pjinvari 28434 A closed subspace ` H ` wi...
pjin1i 28435 Lemma for Theorem 1.22 of ...
pjin2i 28436 Lemma for Theorem 1.22 of ...
pjin3i 28437 Lemma for Theorem 1.22 of ...
pjclem1 28438 Lemma for projection commu...
pjclem2 28439 Lemma for projection commu...
pjclem3 28440 Lemma for projection commu...
pjclem4a 28441 Lemma for projection commu...
pjclem4 28442 Lemma for projection commu...
pjci 28443 Two subspaces commute iff ...
pjcmul1i 28444 A necessary and sufficient...
pjcmul2i 28445 The projection subspace of...
pjcohocli 28446 Closure of composition of ...
pjadj2coi 28447 Adjoint of double composit...
pj2cocli 28448 Closure of double composit...
pj3lem1 28449 Lemma for projection tripl...
pj3si 28450 Stronger projection triple...
pj3i 28451 Projection triplet theorem...
pj3cor1i 28452 Projection triplet corolla...
pjs14i 28453 Theorem S-14 of Watanabe, ...
isst 28456 Property of a state. (Con...
ishst 28457 Property of a complex Hilb...
sticl 28458 ` [ 0 , 1 ] ` closure of t...
stcl 28459 Real closure of the value ...
hstcl 28460 Closure of the value of a ...
hst1a 28461 Unit value of a Hilbert-sp...
hstel2 28462 Properties of a Hilbert-sp...
hstorth 28463 Orthogonality property of ...
hstosum 28464 Orthogonal sum property of...
hstoc 28465 Sum of a Hilbert-space-val...
hstnmoc 28466 Sum of norms of a Hilbert-...
stge0 28467 The value of a state is no...
stle1 28468 The value of a state is le...
hstle1 28469 The norm of the value of a...
hst1h 28470 The norm of a Hilbert-spac...
hst0h 28471 The norm of a Hilbert-spac...
hstpyth 28472 Pythagorean property of a ...
hstle 28473 Ordering property of a Hil...
hstles 28474 Ordering property of a Hil...
hstoh 28475 A Hilbert-space-valued sta...
hst0 28476 A Hilbert-space-valued sta...
sthil 28477 The value of a state at th...
stj 28478 The value of a state on a ...
sto1i 28479 The state of a subspace pl...
sto2i 28480 The state of the orthocomp...
stge1i 28481 If a state is greater than...
stle0i 28482 If a state is less than or...
stlei 28483 Ordering law for states. ...
stlesi 28484 Ordering law for states. ...
stji1i 28485 Join of components of Sasa...
stm1i 28486 State of component of unit...
stm1ri 28487 State of component of unit...
stm1addi 28488 Sum of states whose meet i...
staddi 28489 If the sum of 2 states is ...
stm1add3i 28490 Sum of states whose meet i...
stadd3i 28491 If the sum of 3 states is ...
st0 28492 The state of the zero subs...
strlem1 28493 Lemma for strong state the...
strlem2 28494 Lemma for strong state the...
strlem3a 28495 Lemma for strong state the...
strlem3 28496 Lemma for strong state the...
strlem4 28497 Lemma for strong state the...
strlem5 28498 Lemma for strong state the...
strlem6 28499 Lemma for strong state the...
stri 28500 Strong state theorem. The...
strb 28501 Strong state theorem (bidi...
hstrlem2 28502 Lemma for strong set of CH...
hstrlem3a 28503 Lemma for strong set of CH...
hstrlem3 28504 Lemma for strong set of CH...
hstrlem4 28505 Lemma for strong set of CH...
hstrlem5 28506 Lemma for strong set of CH...
hstrlem6 28507 Lemma for strong set of CH...
hstri 28508 Hilbert space admits a str...
hstrbi 28509 Strong CH-state theorem (b...
largei 28510 A Hilbert lattice admits a...
jplem1 28511 Lemma for Jauch-Piron theo...
jplem2 28512 Lemma for Jauch-Piron theo...
jpi 28513 The function ` S ` , that ...
golem1 28514 Lemma for Godowski's equat...
golem2 28515 Lemma for Godowski's equat...
goeqi 28516 Godowski's equation, shown...
stcltr1i 28517 Property of a strong class...
stcltr2i 28518 Property of a strong class...
stcltrlem1 28519 Lemma for strong classical...
stcltrlem2 28520 Lemma for strong classical...
stcltrthi 28521 Theorem for classically st...
cvbr 28525 Binary relation expressing...
cvbr2 28526 Binary relation expressing...
cvcon3 28527 Contraposition law for the...
cvpss 28528 The covers relation implie...
cvnbtwn 28529 The covers relation implie...
cvnbtwn2 28530 The covers relation implie...
cvnbtwn3 28531 The covers relation implie...
cvnbtwn4 28532 The covers relation implie...
cvnsym 28533 The covers relation is not...
cvnref 28534 The covers relation is not...
cvntr 28535 The covers relation is not...
spansncv2 28536 Hilbert space has the cove...
mdbr 28537 Binary relation expressing...
mdi 28538 Consequence of the modular...
mdbr2 28539 Binary relation expressing...
mdbr3 28540 Binary relation expressing...
mdbr4 28541 Binary relation expressing...
dmdbr 28542 Binary relation expressing...
dmdmd 28543 The dual modular pair prop...
mddmd 28544 The modular pair property ...
dmdi 28545 Consequence of the dual mo...
dmdbr2 28546 Binary relation expressing...
dmdi2 28547 Consequence of the dual mo...
dmdbr3 28548 Binary relation expressing...
dmdbr4 28549 Binary relation expressing...
dmdi4 28550 Consequence of the dual mo...
dmdbr5 28551 Binary relation expressing...
mddmd2 28552 Relationship between modul...
mdsl0 28553 A sublattice condition tha...
ssmd1 28554 Ordering implies the modul...
ssmd2 28555 Ordering implies the modul...
ssdmd1 28556 Ordering implies the dual ...
ssdmd2 28557 Ordering implies the dual ...
dmdsl3 28558 Sublattice mapping for a d...
mdsl3 28559 Sublattice mapping for a m...
mdslle1i 28560 Order preservation of the ...
mdslle2i 28561 Order preservation of the ...
mdslj1i 28562 Join preservation of the o...
mdslj2i 28563 Meet preservation of the r...
mdsl1i 28564 If the modular pair proper...
mdsl2i 28565 If the modular pair proper...
mdsl2bi 28566 If the modular pair proper...
cvmdi 28567 The covering property impl...
mdslmd1lem1 28568 Lemma for ~ mdslmd1i . (C...
mdslmd1lem2 28569 Lemma for ~ mdslmd1i . (C...
mdslmd1lem3 28570 Lemma for ~ mdslmd1i . (C...
mdslmd1lem4 28571 Lemma for ~ mdslmd1i . (C...
mdslmd1i 28572 Preservation of the modula...
mdslmd2i 28573 Preservation of the modula...
mdsldmd1i 28574 Preservation of the dual m...
mdslmd3i 28575 Modular pair conditions th...
mdslmd4i 28576 Modular pair condition tha...
csmdsymi 28577 Cross-symmetry implies M-s...
mdexchi 28578 An exchange lemma for modu...
cvmd 28579 The covering property impl...
cvdmd 28580 The covering property impl...
ela 28582 Atoms in a Hilbert lattice...
elat2 28583 Expanded membership relati...
elatcv0 28584 A Hilbert lattice element ...
atcv0 28585 An atom covers the zero su...
atssch 28586 Atoms are a subset of the ...
atelch 28587 An atom is a Hilbert latti...
atne0 28588 An atom is not the Hilbert...
atss 28589 A lattice element smaller ...
atsseq 28590 Two atoms in a subset rela...
atcveq0 28591 A Hilbert lattice element ...
h1da 28592 A 1-dimensional subspace i...
spansna 28593 The span of the singleton ...
sh1dle 28594 A 1-dimensional subspace i...
ch1dle 28595 A 1-dimensional subspace i...
atom1d 28596 The 1-dimensional subspace...
superpos 28597 Superposition Principle. ...
chcv1 28598 The Hilbert lattice has th...
chcv2 28599 The Hilbert lattice has th...
chjatom 28600 The join of a closed subsp...
shatomici 28601 The lattice of Hilbert sub...
hatomici 28602 The Hilbert lattice is ato...
hatomic 28603 A Hilbert lattice is atomi...
shatomistici 28604 The lattice of Hilbert sub...
hatomistici 28605 ` CH ` is atomistic, i.e. ...
chpssati 28606 Two Hilbert lattice elemen...
chrelati 28607 The Hilbert lattice is rel...
chrelat2i 28608 A consequence of relative ...
cvati 28609 If a Hilbert lattice eleme...
cvbr4i 28610 An alternate way to expres...
cvexchlem 28611 Lemma for ~ cvexchi . (Co...
cvexchi 28612 The Hilbert lattice satisf...
chrelat2 28613 A consequence of relative ...
chrelat3 28614 A consequence of relative ...
chrelat3i 28615 A consequence of the relat...
chrelat4i 28616 A consequence of relative ...
cvexch 28617 The Hilbert lattice satisf...
cvp 28618 The Hilbert lattice satisf...
atnssm0 28619 The meet of a Hilbert latt...
atnemeq0 28620 The meet of distinct atoms...
atssma 28621 The meet with an atom's su...
atcv0eq 28622 Two atoms covering the zer...
atcv1 28623 Two atoms covering the zer...
atexch 28624 The Hilbert lattice satisf...
atomli 28625 An assertion holding in at...
atoml2i 28626 An assertion holding in at...
atordi 28627 An ordering law for a Hilb...
atcvatlem 28628 Lemma for ~ atcvati . (Co...
atcvati 28629 A nonzero Hilbert lattice ...
atcvat2i 28630 A Hilbert lattice element ...
atord 28631 An ordering law for a Hilb...
atcvat2 28632 A Hilbert lattice element ...
chirredlem1 28633 Lemma for ~ chirredi . (C...
chirredlem2 28634 Lemma for ~ chirredi . (C...
chirredlem3 28635 Lemma for ~ chirredi . (C...
chirredlem4 28636 Lemma for ~ chirredi . (C...
chirredi 28637 The Hilbert lattice is irr...
chirred 28638 The Hilbert lattice is irr...
atcvat3i 28639 A condition implying that ...
atcvat4i 28640 A condition implying exist...
atdmd 28641 Two Hilbert lattice elemen...
atmd 28642 Two Hilbert lattice elemen...
atmd2 28643 Two Hilbert lattice elemen...
atabsi 28644 Absorption of an incompara...
atabs2i 28645 Absorption of an incompara...
mdsymlem1 28646 Lemma for ~ mdsymi . (Con...
mdsymlem2 28647 Lemma for ~ mdsymi . (Con...
mdsymlem3 28648 Lemma for ~ mdsymi . (Con...
mdsymlem4 28649 Lemma for ~ mdsymi . This...
mdsymlem5 28650 Lemma for ~ mdsymi . (Con...
mdsymlem6 28651 Lemma for ~ mdsymi . This...
mdsymlem7 28652 Lemma for ~ mdsymi . Lemm...
mdsymlem8 28653 Lemma for ~ mdsymi . Lemm...
mdsymi 28654 M-symmetry of the Hilbert ...
mdsym 28655 M-symmetry of the Hilbert ...
dmdsym 28656 Dual M-symmetry of the Hil...
atdmd2 28657 Two Hilbert lattice elemen...
sumdmdii 28658 If the subspace sum of two...
cmmdi 28659 Commuting subspaces form a...
cmdmdi 28660 Commuting subspaces form a...
sumdmdlem 28661 Lemma for ~ sumdmdi . The...
sumdmdlem2 28662 Lemma for ~ sumdmdi . (Co...
sumdmdi 28663 The subspace sum of two Hi...
dmdbr4ati 28664 Dual modular pair property...
dmdbr5ati 28665 Dual modular pair property...
dmdbr6ati 28666 Dual modular pair property...
dmdbr7ati 28667 Dual modular pair property...
mdoc1i 28668 Orthocomplements form a mo...
mdoc2i 28669 Orthocomplements form a mo...
dmdoc1i 28670 Orthocomplements form a du...
dmdoc2i 28671 Orthocomplements form a du...
mdcompli 28672 A condition equivalent to ...
dmdcompli 28673 A condition equivalent to ...
mddmdin0i 28674 If dual modular implies mo...
cdjreui 28675 A member of the sum of dis...
cdj1i 28676 Two ways to express " ` A ...
cdj3lem1 28677 A property of " ` A ` and ...
cdj3lem2 28678 Lemma for ~ cdj3i . Value...
cdj3lem2a 28679 Lemma for ~ cdj3i . Closu...
cdj3lem2b 28680 Lemma for ~ cdj3i . The f...
cdj3lem3 28681 Lemma for ~ cdj3i . Value...
cdj3lem3a 28682 Lemma for ~ cdj3i . Closu...
cdj3lem3b 28683 Lemma for ~ cdj3i . The s...
cdj3i 28684 Two ways to express " ` A ...
The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here
mathbox 28685 (_This theorem is a dummy ...
foo3 28686 A theorem about the univer...
xfree 28687 A partial converse to ~ 19...
xfree2 28688 A partial converse to ~ 19...
addltmulALT 28689 A proof readability experi...
bian1d 28690 Adding a superfluous conju...
or3di 28691 Distributive law for disju...
or3dir 28692 Distributive law for disju...
3o1cs 28693 Deduction eliminating disj...
3o2cs 28694 Deduction eliminating disj...
3o3cs 28695 Deduction eliminating disj...
spc2ed 28696 Existential specialization...
spc2d 28697 Specialization with 2 quan...
eqvincg 28698 A variable introduction la...
vtocl2d 28699 Implicit substitution of t...
ralcom4f 28700 Commutation of restricted ...
rexcom4f 28701 Commutation of restricted ...
19.9d2rf 28702 A deduction version of one...
19.9d2r 28703 A deduction version of one...
clelsb3f 28704 Substitution applied to an...
sbceqbidf 28705 Equality theorem for class...
sbcies 28706 A special version of class...
moel 28707 "At most one" element in a...
mo5f 28708 Alternate definition of "a...
nmo 28709 Negation of "at most one"....
moimd 28710 "At most one" is preserved...
rmoeqALT 28711 Equality's restricted exis...
2reuswap2 28712 A condition allowing swap ...
reuxfr3d 28713 Transfer existential uniqu...
reuxfr4d 28714 Transfer existential uniqu...
rexunirn 28715 Restricted existential qua...
rmoxfrdOLD 28716 Transfer "at most one" res...
rmoxfrd 28717 Transfer "at most one" res...
ssrmo 28718 "At most one" existential ...
rmo3f 28719 Restricted "at most one" u...
rmo4fOLD 28720 Restricted "at most one" u...
rmo4f 28721 Restricted "at most one" u...
rabrab 28722 Abstract builder restricte...
rabtru 28723 Abtract builder using the ...
rabid2f 28724 An "identity" law for rest...
rabexgfGS 28725 Separation Scheme in terms...
rabsnel 28726 Truth implied by equality ...
foresf1o 28727 From a surjective function...
rabfodom 28728 Domination relation for re...
abrexdomjm 28729 An indexed set is dominate...
abrexdom2jm 28730 An indexed set is dominate...
abrexexd 28731 Existence of a class abstr...
elabreximd 28732 Class substitution in an i...
elabreximdv 28733 Class substitution in an i...
abrexss 28734 A necessary condition for ...
eqri 28735 Infer equality of classes ...
rabss3d 28736 Subclass law for restricte...
inin 28737 Intersection with an inter...
inindif 28738 See ~ inundif . (Contribu...
difeq 28739 Rewriting an equation with...
indifundif 28740 A remarkable equation with...
elpwincl1 28741 Closure of intersection wi...
elpwdifcl 28742 Closure of class differenc...
elpwiuncl 28743 Closure of indexed union w...
elpreq 28744 Equality wihin a pair. (C...
ifeqeqx 28745 An equality theorem tailor...
elimifd 28746 Elimination of a condition...
elim2if 28747 Elimination of two conditi...
elim2ifim 28748 Elimination of two conditi...
uniinn0 28749 Sufficient and necessary c...
uniin1 28750 Union of intersection. Ge...
uniin2 28751 Union of intersection. Ge...
difuncomp 28752 Express a class difference...
pwuniss 28753 Condition for a class unio...
elpwunicl 28754 Closure of a set union wit...
cbviunf 28755 Rule used to change the bo...
iuneq12daf 28756 Equality deduction for ind...
iunin1f 28757 Indexed union of intersect...
iunxsngf 28758 A singleton index picks ou...
ssiun3 28759 Subset equivalence for an ...
ssiun2sf 28760 Subset relationship for an...
iuninc 28761 The union of an increasing...
iundifdifd 28762 The intersection of a set ...
iundifdif 28763 The intersection of a set ...
iunrdx 28764 Re-index an indexed union....
iunpreima 28765 Preimage of an indexed uni...
disjnf 28766 In case ` x ` is not free ...
cbvdisjf 28767 Change bound variables in ...
disjss1f 28768 A subset of a disjoint col...
disjeq1f 28769 Equality theorem for disjo...
disjdifprg 28770 A trivial partition into a...
disjdifprg2 28771 A trivial partition of a s...
disji2f 28772 Property of a disjoint col...
disjif 28773 Property of a disjoint col...
disjorf 28774 Two ways to say that a col...
disjorsf 28775 Two ways to say that a col...
disjif2 28776 Property of a disjoint col...
disjabrex 28777 Rewriting a disjoint colle...
disjabrexf 28778 Rewriting a disjoint colle...
disjpreima 28779 A preimage of a disjoint s...
disjrnmpt 28780 Rewriting a disjoint colle...
disjin 28781 If a collection is disjoin...
disjin2 28782 If a collection is disjoin...
disjxpin 28783 Derive a disjunction over ...
iundisjf 28784 Rewrite a countable union ...
iundisj2f 28785 A disjoint union is disjoi...
disjrdx 28786 Re-index a disjunct collec...
disjex 28787 Two ways to say that two c...
disjexc 28788 A variant of ~ disjex , ap...
disjunsn 28789 Append an element to a dis...
disjun0 28790 Adding the empty element p...
disjiunel 28791 A set of elements B of a d...
disjuniel 28792 A set of elements B of a d...
xpdisjres 28793 Restriction of a constant ...
opeldifid 28794 Ordered pair elementhood o...
difres 28795 Case when class difference...
imadifxp 28796 Image of the difference wi...
relfi 28797 A relation (set) is finite...
fcoinver 28798 Build an equivalence relat...
fcoinvbr 28799 Binary relation for the eq...
brabgaf 28800 The law of concretion for ...
brelg 28801 Two things in a binary rel...
br8d 28802 Substitution for an eight-...
opabdm 28803 Domain of an ordered-pair ...
opabrn 28804 Range of an ordered-pair c...
ssrelf 28805 A subclass relationship de...
eqrelrd2 28806 A version of ~ eqrelrdv2 w...
erbr3b 28807 Biconditional for equivale...
iunsnima 28808 Image of a singleton by an...
mptexgf 28809 If the domain of a functio...
ac6sf2 28810 Alternate version of ~ ac6...
idssxp 28811 A diagonal set as a subset...
fnresin 28812 Restriction of a function ...
f1o3d 28813 Describe an implicit one-t...
rinvf1o 28814 Sufficient conditions for ...
fresf1o 28815 Conditions for a restricti...
f1mptrn 28816 Express injection for a ma...
dfimafnf 28817 Alternate definition of th...
funimass4f 28818 Membership relation for th...
suppss2f 28819 Show that the support of a...
fovcld 28820 Closure law for an operati...
ofrn 28821 The range of the function ...
ofrn2 28822 The range of the function ...
off2 28823 The function operation pro...
ofresid 28824 Applying an operation rest...
fimarab 28825 Expressing the image of a ...
unipreima 28826 Preimage of a class union....
sspreima 28827 The preimage of a subset i...
opfv 28828 Value of a function produc...
xppreima 28829 The preimage of a Cartesia...
xppreima2 28830 The preimage of a Cartesia...
elunirn2 28831 Condition for the membersh...
abfmpunirn 28832 Membership in a union of a...
rabfmpunirn 28833 Membership in a union of a...
abfmpeld 28834 Membership in an element o...
abfmpel 28835 Membership in an element o...
fmptdF 28836 Domain and co-domain of th...
mpteq12df 28837 An equality theorem for th...
resmptf 28838 Restriction of the mapping...
fmptcof2 28839 Composition of two functio...
fcomptf 28840 Express composition of two...
acunirnmpt 28841 Axiom of choice for the un...
acunirnmpt2 28842 Axiom of choice for the un...
acunirnmpt2f 28843 Axiom of choice for the un...
aciunf1lem 28844 Choice in an index union. ...
aciunf1 28845 Choice in an index union. ...
cofmpt 28846 Express composition of a m...
ofoprabco 28847 Function operation as a co...
ofpreima 28848 Express the preimage of a ...
ofpreima2 28849 Express the preimage of a ...
funcnvmptOLD 28850 Condition for a function i...
funcnvmpt 28851 Condition for a function i...
funcnv5mpt 28852 Two ways to say that a fun...
funcnv4mpt 28853 Two ways to say that a fun...
fgreu 28854 Exactly one point of a fun...
fcnvgreu 28855 If the converse of a relat...
rnmpt2ss 28856 The range of an operation ...
mptssALT 28857 Deduce subset relation of ...
partfun 28858 Rewrite a function defined...
dfcnv2 28859 Alternative definition of ...
mpt2mptxf 28860 Express a two-argument fun...
gtiso 28861 Two ways to write a strict...
isoun 28862 Infer an isomorphism from ...
disjdsct 28863 A disjoint collection is d...
df1stres 28864 Definition for a restricti...
df2ndres 28865 Definition for a restricti...
1stpreimas 28866 The preimage of a singleto...
1stpreima 28867 The preimage by ` 1st ` is...
2ndpreima 28868 The preimage by ` 2nd ` is...
curry2ima 28869 The image of a curried fun...
supssd 28870 Inequality deduction for s...
infssd 28871 Inequality deduction for i...
imafi2 28872 The image by a finite set ...
unifi3 28873 If a union is finite, then...
snct 28874 A singleton is countable. ...
prct 28875 An unordered pair is count...
fnct 28876 If the domain of a functio...
dmct 28877 The domain of a countable ...
cnvct 28878 If a set is countable, so ...
rnct 28879 The range of a countable s...
mptct 28880 A countable mapping set is...
mpt2cti 28881 An operation is countable ...
abrexct 28882 An image set of a countabl...
mptctf 28883 A countable mapping set is...
abrexctf 28884 An image set of a countabl...
padct 28885 Index a countable set with...
cnvoprab 28886 The converse of a class ab...
f1od2 28887 Describe an implicit one-t...
fcobij 28888 Composing functions with a...
fcobijfs 28889 Composing finitely support...
suppss3 28890 Deduce a function's suppor...
ffs2 28891 Rewrite a function's suppo...
ffsrn 28892 The range of a finitely su...
resf1o 28893 Restriction of functions t...
maprnin 28894 Restricting the range of t...
fpwrelmapffslem 28895 Lemma for ~ fpwrelmapffs ....
fpwrelmap 28896 Define a canonical mapping...
fpwrelmapffs 28897 Define a canonical mapping...
addeq0 28898 Two complex which add up t...
subeqxfrd 28899 Transfer two terms of a su...
znsqcld 28900 Squaring of nonzero relati...
nn0sqeq1 28901 Integer square one. (Cont...
1neg1t1neg1 28902 An integer unit times itse...
lt2addrd 28903 If the right-hand side of ...
xgepnf 28904 An extended real which is ...
xlemnf 28905 An extended real which is ...
xrlelttric 28906 Trichotomy law for extende...
xaddeq0 28907 Two extended reals which a...
infxrmnf 28908 The infinimum of a set of ...
xrinfm 28909 The extended real numbers ...
le2halvesd 28910 A sum is less than the who...
xraddge02 28911 A number is less than or e...
xrge0addge 28912 A number is less than or e...
xlt2addrd 28913 If the right-hand side of ...
xrsupssd 28914 Inequality deduction for s...
xrge0infss 28915 Any subset of nonnegative ...
xrge0infssd 28916 Inequality deduction for i...
xrge0addcld 28917 Nonnegative extended reals...
xrge0subcld 28918 Condition for closure of n...
infxrge0lb 28919 A member of a set of nonne...
infxrge0glb 28920 The infimum of a set of no...
infxrge0gelb 28921 The infimum of a set of no...
dfrp2 28922 Alternate definition of th...
xrofsup 28923 The supremum is preserved ...
supxrnemnf 28924 The supremum of a nonempty...
xrhaus 28925 The topology of the extend...
joiniooico 28926 Disjoint joining an open i...
ubico 28927 A right-open interval does...
xeqlelt 28928 Equality in terms of 'less...
eliccelico 28929 Relate elementhood to a cl...
elicoelioo 28930 Relate elementhood to a cl...
iocinioc2 28931 Intersection between two o...
xrdifh 28932 Class difference of a half...
iocinif 28933 Relate intersection of two...
difioo 28934 The difference between two...
difico 28935 The difference between two...
nndiffz1 28936 Upper set of the positive ...
ssnnssfz 28937 For any finite subset of `...
fzspl 28938 Split the last element of ...
fzdif2 28939 Split the last element of ...
fzsplit3 28940 Split a finite interval of...
bcm1n 28941 The proportion of one bino...
iundisjfi 28942 Rewrite a countable union ...
iundisj2fi 28943 A disjoint union is disjoi...
iundisjcnt 28944 Rewrite a countable union ...
iundisj2cnt 28945 A countable disjoint union...
f1ocnt 28946 Given a countable set ` A ...
fz1nnct 28947 NN and integer ranges star...
fz1nntr 28948 NN and integer ranges star...
hashunif 28949 The cardinality of a disjo...
numdenneg 28950 Numerator and denominator ...
divnumden2 28951 Calculate the reduced form...
nnindf 28952 Principle of Mathematical ...
nnindd 28953 Principle of Mathematical ...
nn0min 28954 Extracting the minimum pos...
ltesubnnd 28955 Subtracting an integer num...
xdivval 28958 Value of division: the (un...
xrecex 28959 Existence of reciprocal of...
xmulcand 28960 Cancellation law for exten...
xreceu 28961 Existential uniqueness of ...
xdivcld 28962 Closure law for the extend...
xdivcl 28963 Closure law for the extend...
xdivmul 28964 Relationship between divis...
rexdiv 28965 The extended real division...
xdivrec 28966 Relationship between divis...
xdivid 28967 A number divided by itself...
xdiv0 28968 Division into zero is zero...
xdiv0rp 28969 Division into zero is zero...
eliccioo 28970 Membership in a closed int...
elxrge02 28971 Elementhood in the set of ...
xdivpnfrp 28972 Plus infinity divided by a...
rpxdivcld 28973 Closure law for extended d...
xrpxdivcld 28974 Closure law for extended d...
bhmafibid1 28975 The Brahmagupta-Fibonacci ...
bhmafibid2 28976 The Brahmagupta-Fibonacci ...
2sqn0 28977 If the sum of two squares ...
2sqcoprm 28978 If the sum of two squares ...
2sqmod 28979 Given two decompositions o...
2sqmo 28980 There exists at most one d...
ressplusf 28981 The group operation functi...
ressnm 28982 The norm in a restricted s...
abvpropd2 28983 Weaker version of ~ abvpro...
oppgle 28984 less-than relation of an o...
oppglt 28985 less-than relation of an o...
ressprs 28986 The restriction of a preor...
oduprs 28987 Being a preset is a self-d...
posrasymb 28988 A poset ordering is asymet...
tospos 28989 A Toset is a Poset. (Cont...
resspos 28990 The restriction of a Poset...
resstos 28991 The restriction of a Toset...
tleile 28992 In a Toset, two elements m...
tltnle 28993 In a Toset, less-than is e...
odutos 28994 Being a toset is a self-du...
tlt2 28995 In a Toset, two elements m...
tlt3 28996 In a Toset, two elements m...
trleile 28997 In a Toset, two elements m...
toslublem 28998 Lemma for ~ toslub and ~ x...
toslub 28999 In a toset, the lowest upp...
tosglblem 29000 Lemma for ~ tosglb and ~ x...
tosglb 29001 Same theorem as ~ toslub ,...
clatp0cl 29002 The poset zero of a comple...
clatp1cl 29003 The poset one of a complet...
xrs0 29006 The zero of the extended r...
xrslt 29007 The "strictly less than" r...
xrsinvgval 29008 The inversion operation in...
xrsmulgzz 29009 The "multiple" function in...
xrstos 29010 The extended real numbers ...
xrsclat 29011 The extended real numbers ...
xrsp0 29012 The poset 0 of the extende...
xrsp1 29013 The poset 1 of the extende...
ressmulgnn 29014 Values for the group multi...
ressmulgnn0 29015 Values for the group multi...
xrge0base 29016 The base of the extended n...
xrge00 29017 The zero of the extended n...
xrge0plusg 29018 The additive law of the ex...
xrge0le 29019 The lower-or-equal relatio...
xrge0mulgnn0 29020 The group multiple functio...
xrge0addass 29021 Associativity of extended ...
xrge0addgt0 29022 The sum of nonnegative and...
xrge0adddir 29023 Right-distributivity of ex...
xrge0adddi 29024 Left-distributivity of ext...
xrge0npcan 29025 Extended nonnegative real ...
fsumrp0cl 29026 Closure of a finite sum of...
abliso 29027 The image of an Abelian gr...
isomnd 29032 A (left) ordered monoid is...
isogrp 29033 A (left) ordered group is ...
ogrpgrp 29034 An left ordered group is a...
omndmnd 29035 A left ordered monoid is a...
omndtos 29036 A left ordered monoid is a...
omndadd 29037 In an ordered monoid, the ...
omndaddr 29038 In a right ordered monoid,...
omndadd2d 29039 In a commutative left orde...
omndadd2rd 29040 In a left- and right- orde...
submomnd 29041 A submonoid of an ordered ...
xrge0omnd 29042 The nonnegative extended r...
omndmul2 29043 In an ordered monoid, the ...
omndmul3 29044 In an ordered monoid, the ...
omndmul 29045 In a commutative ordered m...
ogrpinvOLD 29046 In an ordered group, the o...
ogrpinv0le 29047 In an ordered group, the o...
ogrpsub 29048 In an ordered group, the o...
ogrpaddlt 29049 In an ordered group, stric...
ogrpaddltbi 29050 In a right ordered group, ...
ogrpaddltrd 29051 In a right ordered group, ...
ogrpaddltrbid 29052 In a right ordered group, ...
ogrpsublt 29053 In an ordered group, stric...
ogrpinv0lt 29054 In an ordered group, the o...
ogrpinvlt 29055 In an ordered group, the o...
sgnsv 29058 The sign mapping. (Contri...
sgnsval 29059 The sign value. (Contribu...
sgnsf 29060 The sign function. (Contr...
inftmrel 29065 The infinitesimal relation...
isinftm 29066 Express ` x ` is infinites...
isarchi 29067 Express the predicate " ` ...
pnfinf 29068 Plus infinity is an infini...
xrnarchi 29069 The completed real line is...
isarchi2 29070 Alternative way to express...
submarchi 29071 A submonoid is archimedean...
isarchi3 29072 This is the usual definiti...
archirng 29073 Property of Archimedean or...
archirngz 29074 Property of Archimedean le...
archiexdiv 29075 In an Archimedean group, g...
archiabllem1a 29076 Lemma for ~ archiabl : In...
archiabllem1b 29077 Lemma for ~ archiabl . (C...
archiabllem1 29078 Archimedean ordered groups...
archiabllem2a 29079 Lemma for ~ archiabl , whi...
archiabllem2c 29080 Lemma for ~ archiabl . (C...
archiabllem2b 29081 Lemma for ~ archiabl . (C...
archiabllem2 29082 Archimedean ordered groups...
archiabl 29083 Archimedean left- and righ...
isslmd 29086 The predicate "is a semimo...
slmdlema 29087 Lemma for properties of a ...
lmodslmd 29088 Left semimodules generaliz...
slmdcmn 29089 A semimodule is a commutat...
slmdmnd 29090 A semimodule is a monoid. ...
slmdsrg 29091 The scalar component of a ...
slmdbn0 29092 The base set of a semimodu...
slmdacl 29093 Closure of ring addition f...
slmdmcl 29094 Closure of ring multiplica...
slmdsn0 29095 The set of scalars in a se...
slmdvacl 29096 Closure of vector addition...
slmdass 29097 Semiring left module vecto...
slmdvscl 29098 Closure of scalar product ...
slmdvsdi 29099 Distributive law for scala...
slmdvsdir 29100 Distributive law for scala...
slmdvsass 29101 Associative law for scalar...
slmd0cl 29102 The ring zero in a semimod...
slmd1cl 29103 The ring unit in a semirin...
slmdvs1 29104 Scalar product with ring u...
slmd0vcl 29105 The zero vector is a vecto...
slmd0vlid 29106 Left identity law for the ...
slmd0vrid 29107 Right identity law for the...
slmd0vs 29108 Zero times a vector is the...
slmdvs0 29109 Anything times the zero ve...
gsumle 29110 A finite sum in an ordered...
gsummpt2co 29111 Split a finite sum into a ...
gsummpt2d 29112 Express a finite sum over ...
gsumvsca1 29113 Scalar product of a finite...
gsumvsca2 29114 Scalar product of a finite...
gsummptres 29115 Extend a finite group sum ...
xrge0tsmsd 29116 Any finite or infinite sum...
xrge0tsmsbi 29117 Any limit of a finite or i...
xrge0tsmseq 29118 Any limit of a finite or i...
rngurd 29119 Deduce the unit of a ring ...
ress1r 29120 ` 1r ` is unaffected by re...
dvrdir 29121 Distributive law for the d...
rdivmuldivd 29122 Multiplication of two rati...
ringinvval 29123 The ring inverse expressed...
dvrcan5 29124 Cancellation law for commo...
subrgchr 29125 If ` A ` is a subring of `...
isorng 29130 An ordered ring is a ring ...
orngring 29131 An ordered ring is a ring....
orngogrp 29132 An ordered ring is an orde...
isofld 29133 An ordered field is a fiel...
orngmul 29134 In an ordered ring, the or...
orngsqr 29135 In an ordered ring, all sq...
ornglmulle 29136 In an ordered ring, multip...
orngrmulle 29137 In an ordered ring, multip...
ornglmullt 29138 In an ordered ring, multip...
orngrmullt 29139 In an ordered ring, multip...
orngmullt 29140 In an ordered ring, the st...
ofldfld 29141 An ordered field is a fiel...
ofldtos 29142 An ordered field is a tota...
orng0le1 29143 In an ordered ring, the ri...
ofldlt1 29144 In an ordered field, the r...
ofldchr 29145 The characteristic of an o...
suborng 29146 Every subring of an ordere...
subofld 29147 Every subfield of an order...
isarchiofld 29148 Axiom of Archimedes : a ch...
rhmdvdsr 29149 A ring homomorphism preser...
rhmopp 29150 A ring homomorphism is als...
elrhmunit 29151 Ring homomorphisms preserv...
rhmdvd 29152 A ring homomorphism preser...
rhmunitinv 29153 Ring homomorphisms preserv...
kerunit 29154 If a unit element lies in ...
reldmresv 29157 The scalar restriction is ...
resvval 29158 Value of structure restric...
resvid2 29159 General behavior of trivia...
resvval2 29160 Value of nontrivial struct...
resvsca 29161 Base set of a structure re...
resvlem 29162 Other elements of a struct...
resvbas 29163 ` Base ` is unaffected by ...
resvplusg 29164 ` +g ` is unaffected by sc...
resvvsca 29165 ` .s ` is unaffected by sc...
resvmulr 29166 ` .s ` is unaffected by sc...
resv0g 29167 ` 0g ` is unaffected by sc...
resv1r 29168 ` 1r ` is unaffected by sc...
resvcmn 29169 Scalar restriction preserv...
gzcrng 29170 The gaussian integers form...
reofld 29171 The real numbers form an o...
nn0omnd 29172 The nonnegative integers f...
rearchi 29173 The field of the real numb...
nn0archi 29174 The monoid of the nonnegat...
xrge0slmod 29175 The extended nonnegative r...
symgfcoeu 29176 Uniqueness property of per...
psgndmfi 29177 For a finite base set, the...
psgnid 29178 Permutation sign of the id...
pmtrprfv2 29179 In a transposition of two ...
pmtrto1cl 29180 Useful lemma for the follo...
psgnfzto1stlem 29181 Lemma for ~ psgnfzto1st . ...
fzto1stfv1 29182 Value of our permutation `...
fzto1st1 29183 Special case where the per...
fzto1st 29184 The function moving one el...
fzto1stinvn 29185 Value of the inverse of ou...
psgnfzto1st 29186 The permutation sign for m...
smatfval 29189 Value of the submatrix. (...
smatrcl 29190 Closure of the rectangular...
smatlem 29191 Lemma for the next theorem...
smattl 29192 Entries of a submatrix, to...
smattr 29193 Entries of a submatrix, to...
smatbl 29194 Entries of a submatrix, bo...
smatbr 29195 Entries of a submatrix, bo...
smatcl 29196 Closure of the square subm...
matmpt2 29197 Write a square matrix as a...
1smat1 29198 The submatrix of the ident...
submat1n 29199 One case where the submatr...
submatres 29200 Special case where the sub...
submateqlem1 29201 Lemma for ~ submateq . (C...
submateqlem2 29202 Lemma for ~ submateq . (C...
submateq 29203 Sufficient condition for t...
submatminr1 29204 If we take a submatrix by ...
lmatval 29207 Value of the literal matri...
lmatfval 29208 Entries of a literal matri...
lmatfvlem 29209 Useful lemma to extract li...
lmatcl 29210 Closure of the literal mat...
lmat22lem 29211 Lemma for ~ lmat22e11 and ...
lmat22e11 29212 Entry of a 2x2 literal mat...
lmat22e12 29213 Entry of a 2x2 literal mat...
lmat22e21 29214 Entry of a 2x2 literal mat...
lmat22e22 29215 Entry of a 2x2 literal mat...
lmat22det 29216 The determinant of a liter...
mdetpmtr1 29217 The determinant of a matri...
mdetpmtr2 29218 The determinant of a matri...
mdetpmtr12 29219 The determinant of a matri...
mdetlap1 29220 A Laplace expansion of the...
madjusmdetlem1 29221 Lemma for ~ madjusmdet . ...
madjusmdetlem2 29222 Lemma for ~ madjusmdet . ...
madjusmdetlem3 29223 Lemma for ~ madjusmdet . ...
madjusmdetlem4 29224 Lemma for ~ madjusmdet . ...
madjusmdet 29225 Express the cofactor of th...
mdetlap 29226 Laplace expansion of the d...
fvproj 29227 Value of a function on pai...
fimaproj 29228 Image of a cartesian produ...
txomap 29229 Given two open maps ` F ` ...
qtopt1 29230 If every equivalence class...
qtophaus 29231 If an open map's graph in ...
circtopn 29232 The topology of the unit c...
circcn 29233 The function gluing the re...
reff 29234 For any cover refinement, ...
locfinreflem 29235 A locally finite refinemen...
locfinref 29236 A locally finite refinemen...
iscref 29239 The property that every op...
crefeq 29240 Equality theorem for the "...
creftop 29241 A space where every open c...
crefi 29242 The property that every op...
crefdf 29243 A formulation of ~ crefi e...
crefss 29244 The "every open cover has ...
cmpcref 29245 Equivalent definition of c...
cmpfiref 29246 Every open cover of a Comp...
ldlfcntref 29249 Every open cover of a Lind...
ispcmp 29252 The predicate "is a paraco...
cmppcmp 29253 Every compact space is par...
dispcmp 29254 Every discrete space is pa...
pcmplfin 29255 Given a paracompact topolo...
pcmplfinf 29256 Given a paracompact topolo...
metidval 29261 Value of the metric identi...
metidss 29262 As a relation, the metric ...
metidv 29263 ` A ` and ` B ` identify b...
metideq 29264 Basic property of the metr...
metider 29265 The metric identification ...
pstmval 29266 Value of the metric induce...
pstmfval 29267 Function value of the metr...
pstmxmet 29268 The metric induced by a ps...
hauseqcn 29269 In a Hausdorff topology, t...
unitsscn 29270 The closed unit is a subse...
elunitrn 29271 The closed unit is a subse...
elunitcn 29272 The closed unit is a subse...
elunitge0 29273 An element of the closed u...
unitssxrge0 29274 The closed unit is a subse...
unitdivcld 29275 Necessary conditions for a...
iistmd 29276 The closed unit forms a to...
unicls 29277 The union of the closed se...
tpr2tp 29278 The usual topology on ` ( ...
tpr2uni 29279 The usual topology on ` ( ...
xpinpreima 29280 Rewrite the cartesian prod...
xpinpreima2 29281 Rewrite the cartesian prod...
sqsscirc1 29282 The complex square of side...
sqsscirc2 29283 The complex square of side...
cnre2csqlem 29284 Lemma for ~ cnre2csqima . ...
cnre2csqima 29285 Image of a centered square...
tpr2rico 29286 For any point of an open s...
cnvordtrestixx 29287 The restriction of the 'gr...
prsdm 29288 Domain of the relation of ...
prsrn 29289 Range of the relation of a...
prsss 29290 Relation of a subpreset. ...
prsssdm 29291 Domain of a subpreset rela...
ordtprsval 29292 Value of the order topolog...
ordtprsuni 29293 Value of the order topolog...
ordtcnvNEW 29294 The order dual generates t...
ordtrestNEW 29295 The subspace topology of a...
ordtrest2NEWlem 29296 Lemma for ~ ordtrest2NEW ....
ordtrest2NEW 29297 An interval-closed set ` A...
ordtconlem1 29298 Connectedness in the order...
ordtcon 29299 Connectedness in the order...
mndpluscn 29300 A mapping that is both a h...
mhmhmeotmd 29301 Deduce a Topological Monoi...
rmulccn 29302 Multiplication by a real c...
raddcn 29303 Addition in the real numbe...
xrmulc1cn 29304 The operation multiplying ...
fmcncfil 29305 The image of a Cauchy filt...
xrge0hmph 29306 The extended nonnegative r...
xrge0iifcnv 29307 Define a bijection from ` ...
xrge0iifcv 29308 The defined function's val...
xrge0iifiso 29309 The defined bijection from...
xrge0iifhmeo 29310 Expose a homeomorphism fro...
xrge0iifhom 29311 The defined function from ...
xrge0iif1 29312 Condition for the defined ...
xrge0iifmhm 29313 The defined function from ...
xrge0pluscn 29314 The addition operation of ...
xrge0mulc1cn 29315 The operation multiplying ...
xrge0tps 29316 The extended nonnegative r...
xrge0topn 29317 The topology of the extend...
xrge0haus 29318 The topology of the extend...
xrge0tmdOLD 29319 The extended nonnegative r...
xrge0tmd 29320 The extended nonnegative r...
lmlim 29321 Relate a limit in a given ...
lmlimxrge0 29322 Relate a limit in the nonn...
rge0scvg 29323 Implication of convergence...
fsumcvg4 29324 A serie with finite suppor...
pnfneige0 29325 A neighborhood of ` +oo ` ...
lmxrge0 29326 Express "sequence ` F ` co...
lmdvg 29327 If a monotonic sequence of...
lmdvglim 29328 If a monotonic real number...
pl1cn 29329 A univariate polynomial is...
zringnm 29332 The norm (function) for a ...
zzsnm 29333 The norm of the ring of th...
zlm0 29334 Zero of a ` ZZ ` -module. ...
zlm1 29335 Unit of a ` ZZ ` -module (...
zlmds 29336 Distance in a ` ZZ ` -modu...
zlmtset 29337 Topology in a ` ZZ ` -modu...
zlmnm 29338 Norm of a ` ZZ ` -module (...
zhmnrg 29339 The ` ZZ ` -module built f...
nmmulg 29340 The norm of a group produc...
zrhnm 29341 The norm of the image by `...
cnzh 29342 The ` ZZ ` -module of ` CC...
rezh 29343 The ` ZZ ` -module of ` RR...
qqhval 29346 Value of the canonical hom...
zrhf1ker 29347 The kernel of the homomorp...
zrhchr 29348 The kernel of the homomorp...
zrhker 29349 The kernel of the homomorp...
zrhunitpreima 29350 The preimage by ` ZRHom ` ...
elzrhunit 29351 Condition for the image by...
elzdif0 29352 Lemma for ~ qqhval2 . (Co...
qqhval2lem 29353 Lemma for ~ qqhval2 . (Co...
qqhval2 29354 Value of the canonical hom...
qqhvval 29355 Value of the canonical hom...
qqh0 29356 The image of ` 0 ` by the ...
qqh1 29357 The image of ` 1 ` by the ...
qqhf 29358 ` QQHom ` as a function. ...
qqhvq 29359 The image of a quotient by...
qqhghm 29360 The ` QQHom ` homomorphism...
qqhrhm 29361 The ` QQHom ` homomorphism...
qqhnm 29362 The norm of the image by `...
qqhcn 29363 The ` QQHom ` homomorphism...
qqhucn 29364 The ` QQHom ` homomorphism...
rrhval 29368 Value of the canonical hom...
rrhcn 29369 If the topology of ` R ` i...
rrhf 29370 If the topology of ` R ` i...
isrrext 29372 Express the property " ` R...
rrextnrg 29373 An extension of ` RR ` is ...
rrextdrg 29374 An extension of ` RR ` is ...
rrextnlm 29375 The norm of an extension o...
rrextchr 29376 The ring characteristic of...
rrextcusp 29377 An extension of ` RR ` is ...
rrexttps 29378 An extension of ` RR ` is ...
rrexthaus 29379 The topology of an extensi...
rrextust 29380 The uniformity of an exten...
rerrext 29381 The field of the real numb...
cnrrext 29382 The field of the complex n...
qqtopn 29383 The topology of the field ...
rrhfe 29384 If ` R ` is an extension o...
rrhcne 29385 If ` R ` is an extension o...
rrhqima 29386 The ` RRHom ` homomorphism...
rrh0 29387 The image of ` 0 ` by the ...
xrhval 29390 The value of the embedding...
zrhre 29391 The ` ZRHom ` homomorphism...
qqhre 29392 The ` QQHom ` homomorphism...
rrhre 29393 The ` RRHom ` homomorphism...
relmntop 29396 Manifold is a relation. (...
ismntoplly 29397 Property of being a manifo...
ismntop 29398 Property of being a manifo...
nexple 29399 A lower bound for an expon...
indv 29402 Value of the indicator fun...
indval 29403 Value of the indicator fun...
indval2 29404 Alternate value of the ind...
indf 29405 An indicator function as a...
indfval 29406 Value of the indicator fun...
pr01ssre 29407 The range of the indicator...
ind1 29408 Value of the indicator fun...
ind0 29409 Value of the indicator fun...
ind1a 29410 Value of the indicator fun...
indpi1 29411 Preimage of the singleton ...
indsum 29412 Finite sum of a product wi...
indf1o 29413 The bijection between a po...
indpreima 29414 A function with range ` { ...
indf1ofs 29415 The bijection between fini...
esumex 29418 An extended sum is a set b...
esumcl 29419 Closure for extended sum i...
esumeq12dvaf 29420 Equality deduction for ext...
esumeq12dva 29421 Equality deduction for ext...
esumeq12d 29422 Equality deduction for ext...
esumeq1 29423 Equality theorem for an ex...
esumeq1d 29424 Equality theorem for an ex...
esumeq2 29425 Equality theorem for exten...
esumeq2d 29426 Equality deduction for ext...
esumeq2dv 29427 Equality deduction for ext...
esumeq2sdv 29428 Equality deduction for ext...
nfesum1 29429 Bound-variable hypothesis ...
nfesum2 29430 Bound-variable hypothesis ...
cbvesum 29431 Change bound variable in a...
cbvesumv 29432 Change bound variable in a...
esumid 29433 Identify the extended sum ...
esumgsum 29434 A finite extended sum is t...
esumval 29435 Develop the value of the e...
esumel 29436 The extended sum is a limi...
esumnul 29437 Extended sum over the empt...
esum0 29438 Extended sum of zero. (Co...
esumf1o 29439 Re-index an extended sum u...
esumc 29440 Convert from the collectio...
esumrnmpt 29441 Rewrite an extended sum in...
esumsplit 29442 Split an extended sum into...
esummono 29443 Extended sum is monotonic....
esumpad 29444 Extend an extended sum by ...
esumpad2 29445 Remove zeroes from an exte...
esumadd 29446 Addition of infinite sums....
esumle 29447 If all of the terms of an ...
gsumesum 29448 Relate a group sum on ` ( ...
esumlub 29449 The extended sum is the lo...
esumaddf 29450 Addition of infinite sums....
esumlef 29451 If all of the terms of an ...
esumcst 29452 The extended sum of a cons...
esumsnf 29453 The extended sum of a sing...
esumsn 29454 The extended sum of a sing...
esumpr 29455 Extended sum over a pair. ...
esumpr2 29456 Extended sum over a pair, ...
esumrnmpt2 29457 Rewrite an extended sum in...
esumfzf 29458 Formulating a partial exte...
esumfsup 29459 Formulating an extended su...
esumfsupre 29460 Formulating an extended su...
esumss 29461 Change the index set to a ...
esumpinfval 29462 The value of the extended ...
esumpfinvallem 29463 Lemma for ~ esumpfinval . ...
esumpfinval 29464 The value of the extended ...
esumpfinvalf 29465 Same as ~ esumpfinval , mi...
esumpinfsum 29466 The value of the extended ...
esumpcvgval 29467 The value of the extended ...
esumpmono 29468 The partial sums in an ext...
esumcocn 29469 Lemma for ~ esummulc2 and ...
esummulc1 29470 An extended sum multiplied...
esummulc2 29471 An extended sum multiplied...
esumdivc 29472 An extended sum divided by...
hashf2 29473 Lemma for ~ hasheuni . (C...
hasheuni 29474 The cardinality of a disjo...
esumcvg 29475 The sequence of partial su...
esumcvg2 29476 Simpler version of ~ esumc...
esumcvgsum 29477 The value of the extended ...
esumsup 29478 Express an extended sum as...
esumgect 29479 "Send ` n ` to ` +oo ` " i...
esumcvgre 29480 All terms of a converging ...
esum2dlem 29481 Lemma for ~ esum2d (finite...
esum2d 29482 Write a double extended su...
esumiun 29483 Sum over a non necessarily...
ofceq 29486 Equality theorem for funct...
ofcfval 29487 Value of an operation appl...
ofcval 29488 Evaluate a function/consta...
ofcfn 29489 The function operation pro...
ofcfeqd2 29490 Equality theorem for funct...
ofcfval3 29491 General value of ` ( F oFC...
ofcf 29492 The function/constant oper...
ofcfval2 29493 The function operation exp...
ofcfval4 29494 The function/constant oper...
ofcc 29495 Left operation by a consta...
ofcof 29496 Relate function operation ...
sigaex 29499 Lemma for ~ issiga and ~ i...
sigaval 29500 The set of sigma-algebra w...
issiga 29501 An alternative definition ...
isrnsigaOLD 29502 The property of being a si...
isrnsiga 29503 The property of being a si...
0elsiga 29504 A sigma-algebra contains t...
baselsiga 29505 A sigma-algebra contains i...
sigasspw 29506 A sigma-algebra is a set o...
sigaclcu 29507 A sigma-algebra is closed ...
sigaclcuni 29508 A sigma-algebra is closed ...
sigaclfu 29509 A sigma-algebra is closed ...
sigaclcu2 29510 A sigma-algebra is closed ...
sigaclfu2 29511 A sigma-algebra is closed ...
sigaclcu3 29512 A sigma-algebra is closed ...
issgon 29513 Property of being a sigma-...
sgon 29514 A sigma-algebra is a sigma...
elsigass 29515 An element of a sigma-alge...
elrnsiga 29516 Dropping the base informat...
isrnsigau 29517 The property of being a si...
unielsiga 29518 A sigma-algebra contains i...
dmvlsiga 29519 Lebesgue-measurable subset...
pwsiga 29520 Any power set forms a sigm...
prsiga 29521 The smallest possible sigm...
sigaclci 29522 A sigma-algebra is closed ...
difelsiga 29523 A sigma-algebra is closed ...
unelsiga 29524 A sigma-algebra is closed ...
inelsiga 29525 A sigma-algebra is closed ...
sigainb 29526 Building a sigma-algebra f...
insiga 29527 The intersection of a coll...
sigagenval 29530 Value of the generated sig...
sigagensiga 29531 A generated sigma-algebra ...
sgsiga 29532 A generated sigma-algebra ...
unisg 29533 The sigma-algebra generate...
dmsigagen 29534 A sigma-algebra can be gen...
sssigagen 29535 A set is a subset of the s...
sssigagen2 29536 A subset of the generating...
elsigagen 29537 Any element of a set is al...
elsigagen2 29538 Any countable union of ele...
sigagenss 29539 The generated sigma-algebr...
sigagenss2 29540 Sufficient condition for i...
sigagenid 29541 The sigma-algebra generate...
ispisys 29542 The property of being a pi...
ispisys2 29543 The property of being a pi...
inelpisys 29544 Pi-systems are closed unde...
sigapisys 29545 All sigma-algebras are pi-...
isldsys 29546 The property of being a la...
pwldsys 29547 The power set of the unive...
unelldsys 29548 Lambda-systems are closed ...
sigaldsys 29549 All sigma-algebras are lam...
ldsysgenld 29550 The intersection of all la...
sigapildsyslem 29551 Lemma for ~ sigapildsys . ...
sigapildsys 29552 Sigma-algebra are exactly ...
ldgenpisyslem1 29553 Lemma for ~ ldgenpisys . ...
ldgenpisyslem2 29554 Lemma for ~ ldgenpisys . ...
ldgenpisyslem3 29555 Lemma for ~ ldgenpisys . ...
ldgenpisys 29556 The lambda system ` E ` ge...
dynkin 29557 Dynkin's lambda-pi theorem...
isros 29558 The property of being a ri...
rossspw 29559 A ring of sets is a collec...
0elros 29560 A ring of sets contains th...
unelros 29561 A ring of sets is closed u...
difelros 29562 A ring of sets is closed u...
inelros 29563 A ring of sets is closed u...
fiunelros 29564 A ring of sets is closed u...
issros 29565 The property of being a se...
srossspw 29566 A semi-ring of sets is a c...
0elsros 29567 A semi-ring of sets contai...
inelsros 29568 A semi-ring of sets is clo...
diffiunisros 29569 In semiring of sets, compl...
rossros 29570 Rings of sets are semi-rin...
brsiga 29573 The Borel Algebra on real ...
brsigarn 29574 The Borel Algebra is a sig...
brsigasspwrn 29575 The Borel Algebra is a set...
unibrsiga 29576 The union of the Borel Alg...
cldssbrsiga 29577 A Borel Algebra contains a...
sxval 29580 Value of the product sigma...
sxsiga 29581 A product sigma-algebra is...
sxsigon 29582 A product sigma-algebra is...
sxuni 29583 The base set of a product ...
elsx 29584 The cartesian product of t...
measbase 29587 The base set of a measure ...
measval 29588 The value of the ` measure...
ismeas 29589 The property of being a me...
isrnmeas 29590 The property of being a me...
dmmeas 29591 The domain of a measure is...
measbasedom 29592 The base set of a measure ...
measfrge0 29593 A measure is a function ov...
measfn 29594 A measure is a function on...
measvxrge0 29595 The values of a measure ar...
measvnul 29596 The measure of the empty s...
measge0 29597 A measure is nonnegative. ...
measle0 29598 If the measure of a given ...
measvun 29599 The measure of a countable...
measxun2 29600 The measure the union of t...
measun 29601 The measure the union of t...
measvunilem 29602 Lemma for ~ measvuni . (C...
measvunilem0 29603 Lemma for ~ measvuni . (C...
measvuni 29604 The measure of a countable...
measssd 29605 A measure is monotone with...
measunl 29606 A measure is sub-additive ...
measiuns 29607 The measure of the union o...
measiun 29608 A measure is sub-additive....
meascnbl 29609 A measure is continuous fr...
measinblem 29610 Lemma for ~ measinb . (Co...
measinb 29611 Building a measure restric...
measres 29612 Building a measure restric...
measinb2 29613 Building a measure restric...
measdivcstOLD 29614 Division of a measure by a...
measdivcst 29615 Division of a measure by a...
cntmeas 29616 The Counting measure is a ...
pwcntmeas 29617 The counting measure is a ...
cntnevol 29618 Counting and Lebesgue meas...
voliune 29619 The Lebesgue measure funct...
volfiniune 29620 The Lebesgue measure funct...
volmeas 29621 The Lebesgue measure is a ...
ddeval1 29624 Value of the delta measure...
ddeval0 29625 Value of the delta measure...
ddemeas 29626 The Dirac delta measure is...
relae 29630 'almost everywhere' is a r...
brae 29631 'almost everywhere' relati...
braew 29632 'almost everywhere' relati...
truae 29633 A truth holds almost every...
aean 29634 A conjunction holds almost...
faeval 29636 Value of the 'almost every...
relfae 29637 The 'almost everywhere' bu...
brfae 29638 'almost everywhere' relati...
ismbfm 29641 The predicate " ` F ` is a...
elunirnmbfm 29642 The property of being a me...
mbfmfun 29643 A measurable function is a...
mbfmf 29644 A measurable function as a...
isanmbfm 29645 The predicate to be a meas...
mbfmcnvima 29646 The preimage by a measurab...
mbfmbfm 29647 A measurable function to a...
mbfmcst 29648 A constant function is mea...
1stmbfm 29649 The first projection map i...
2ndmbfm 29650 The second projection map ...
imambfm 29651 If the sigma-algebra in th...
cnmbfm 29652 A continuous function is m...
mbfmco 29653 The composition of two mea...
mbfmco2 29654 The pair building of two m...
mbfmvolf 29655 Measurable functions with ...
elmbfmvol2 29656 Measurable functions with ...
mbfmcnt 29657 All functions are measurab...
br2base 29658 The base set for the gener...
dya2ub 29659 An upper bound for a dyadi...
sxbrsigalem0 29660 The closed half-spaces of ...
sxbrsigalem3 29661 The sigma-algebra generate...
dya2iocival 29662 The function ` I ` returns...
dya2iocress 29663 Dyadic intervals are subse...
dya2iocbrsiga 29664 Dyadic intervals are Borel...
dya2icobrsiga 29665 Dyadic intervals are Borel...
dya2icoseg 29666 For any point and any clos...
dya2icoseg2 29667 For any point and any open...
dya2iocrfn 29668 The function returning dya...
dya2iocct 29669 The dyadic rectangle set i...
dya2iocnrect 29670 For any point of an open r...
dya2iocnei 29671 For any point of an open s...
dya2iocuni 29672 Every open set of ` ( RR X...
dya2iocucvr 29673 The dyadic rectangular set...
sxbrsigalem1 29674 The Borel algebra on ` ( R...
sxbrsigalem2 29675 The sigma-algebra generate...
sxbrsigalem4 29676 The Borel algebra on ` ( R...
sxbrsigalem5 29677 First direction for ~ sxbr...
sxbrsigalem6 29678 First direction for ~ sxbr...
sxbrsiga 29679 The product sigma-algebra ...
omsval 29682 Value of the function mapp...
omsfval 29683 Value of the outer measure...
omscl 29684 A closure lemma for the co...
omsf 29685 A constructed outer measur...
oms0 29686 A constructed outer measur...
omsmon 29687 A constructed outer measur...
omssubaddlem 29688 For any small margin ` E `...
omssubadd 29689 A constructed outer measur...
carsgval 29692 Value of the Caratheodory ...
carsgcl 29693 Closure of the Caratheodor...
elcarsg 29694 Property of being a Catath...
baselcarsg 29695 The universe set, ` O ` , ...
0elcarsg 29696 The empty set is Caratheod...
carsguni 29697 The union of all Caratheod...
elcarsgss 29698 Caratheodory measurable se...
difelcarsg 29699 The Caratheodory measurabl...
inelcarsg 29700 The Caratheodory measurabl...
unelcarsg 29701 The Caratheodory-measurabl...
difelcarsg2 29702 The Caratheodory-measurabl...
carsgmon 29703 Utility lemma: Apply mono...
carsgsigalem 29704 Lemma for the following th...
fiunelcarsg 29705 The Caratheodory measurabl...
carsgclctunlem1 29706 Lemma for ~ carsgclctun . ...
carsggect 29707 The outer measure is count...
carsgclctunlem2 29708 Lemma for ~ carsgclctun . ...
carsgclctunlem3 29709 Lemma for ~ carsgclctun . ...
carsgclctun 29710 The Caratheodory measurabl...
carsgsiga 29711 The Caratheodory measurabl...
omsmeas 29712 The restriction of a const...
pmeasmono 29713 This theorem's hypotheses ...
pmeasadd 29714 A premeasure on a ring of ...
itgeq12dv 29715 Equality theorem for an in...
sitgval 29721 Value of the simple functi...
issibf 29722 The predicate " ` F ` is a...
sibf0 29723 The constant zero function...
sibfmbl 29724 A simple function is measu...
sibff 29725 A simple function is a fun...
sibfrn 29726 A simple function has fini...
sibfima 29727 Any preimage of a singleto...
sibfinima 29728 The measure of the interse...
sibfof 29729 Applying function operatio...
sitgfval 29730 Value of the Bochner integ...
sitgclg 29731 Closure of the Bochner int...
sitgclbn 29732 Closure of the Bochner int...
sitgclcn 29733 Closure of the Bochner int...
sitgclre 29734 Closure of the Bochner int...
sitg0 29735 The integral of the consta...
sitgf 29736 The integral for simple fu...
sitgaddlemb 29737 Lemma for * sitgadd . (Co...
sitmval 29738 Value of the simple functi...
sitmfval 29739 Value of the integral dist...
sitmcl 29740 Closure of the integral di...
sitmf 29741 The integral metric as a f...
oddpwdc 29743 Lemma for ~ eulerpart . T...
oddpwdcv 29744 Lemma for ~ eulerpart : va...
eulerpartlemsv1 29745 Lemma for ~ eulerpart . V...
eulerpartlemelr 29746 Lemma for ~ eulerpart . (...
eulerpartlemsv2 29747 Lemma for ~ eulerpart . V...
eulerpartlemsf 29748 Lemma for ~ eulerpart . (...
eulerpartlems 29749 Lemma for ~ eulerpart . (...
eulerpartlemsv3 29750 Lemma for ~ eulerpart . V...
eulerpartlemgc 29751 Lemma for ~ eulerpart . (...
eulerpartleme 29752 Lemma for ~ eulerpart . (...
eulerpartlemv 29753 Lemma for ~ eulerpart . (...
eulerpartlemo 29754 Lemma for ~ eulerpart : ` ...
eulerpartlemd 29755 Lemma for ~ eulerpart : ` ...
eulerpartlem1 29756 Lemma for ~ eulerpart . (...
eulerpartlemb 29757 Lemma for ~ eulerpart . T...
eulerpartlemt0 29758 Lemma for ~ eulerpart . (...
eulerpartlemf 29759 Lemma for ~ eulerpart : O...
eulerpartlemt 29760 Lemma for ~ eulerpart . (...
eulerpartgbij 29761 Lemma for ~ eulerpart : T...
eulerpartlemgv 29762 Lemma for ~ eulerpart : va...
eulerpartlemr 29763 Lemma for ~ eulerpart . (...
eulerpartlemmf 29764 Lemma for ~ eulerpart . (...
eulerpartlemgvv 29765 Lemma for ~ eulerpart : va...
eulerpartlemgu 29766 Lemma for ~ eulerpart : R...
eulerpartlemgh 29767 Lemma for ~ eulerpart : T...
eulerpartlemgf 29768 Lemma for ~ eulerpart : I...
eulerpartlemgs2 29769 Lemma for ~ eulerpart : T...
eulerpartlemn 29770 Lemma for ~ eulerpart . (...
eulerpart 29771 Euler's theorem on partiti...
subiwrd 29774 Lemma for ~ sseqp1 . (Con...
subiwrdlen 29775 Length of a subword of an ...
iwrdsplit 29776 Lemma for ~ sseqp1 . (Con...
sseqval 29777 Value of the strong sequen...
sseqfv1 29778 Value of the strong sequen...
sseqfn 29779 A strong recursive sequenc...
sseqmw 29780 Lemma for ~ sseqf amd ~ ss...
sseqf 29781 A strong recursive sequenc...
sseqfres 29782 The first elements in the ...
sseqfv2 29783 Value of the strong sequen...
sseqp1 29784 Value of the strong sequen...
fiblem 29787 Lemma for ~ fib0 , ~ fib1 ...
fib0 29788 Value of the Fibonacci seq...
fib1 29789 Value of the Fibonacci seq...
fibp1 29790 Value of the Fibonacci seq...
fib2 29791 Value of the Fibonacci seq...
fib3 29792 Value of the Fibonacci seq...
fib4 29793 Value of the Fibonacci seq...
fib5 29794 Value of the Fibonacci seq...
fib6 29795 Value of the Fibonacci seq...
elprob 29798 The property of being a pr...
domprobmeas 29799 A probability measure is a...
domprobsiga 29800 The domain of a probabilit...
probtot 29801 The probability of the uni...
prob01 29802 A probability is an elemen...
probnul 29803 The probability of the emp...
unveldomd 29804 The universe is an element...
unveldom 29805 The universe is an element...
nuleldmp 29806 The empty set is an elemen...
probcun 29807 The probability of the uni...
probun 29808 The probability of the uni...
probdif 29809 The probability of the dif...
probinc 29810 A probability law is incre...
probdsb 29811 The probability of the com...
probmeasd 29812 A probability measure is a...
probvalrnd 29813 The value of a probability...
probtotrnd 29814 The probability of the uni...
totprobd 29815 Law of total probability, ...
totprob 29816 Law of total probability. ...
probfinmeasbOLD 29817 Build a probability measur...
probfinmeasb 29818 Build a probability measur...
probmeasb 29819 Build a probability from a...
cndprobval 29822 The value of the condition...
cndprobin 29823 An identity linking condit...
cndprob01 29824 The conditional probabilit...
cndprobtot 29825 The conditional probabilit...
cndprobnul 29826 The conditional probabilit...
cndprobprob 29827 The conditional probabilit...
bayesth 29828 Bayes Theorem. (Contribut...
rrvmbfm 29831 A real-valued random varia...
isrrvv 29832 Elementhood to the set of ...
rrvvf 29833 A real-valued random varia...
rrvfn 29834 A real-valued random varia...
rrvdm 29835 The domain of a random var...
rrvrnss 29836 The range of a random vari...
rrvf2 29837 A real-valued random varia...
rrvdmss 29838 The domain of a random var...
rrvfinvima 29839 For a real-value random va...
0rrv 29840 The constant function equa...
rrvadd 29841 The sum of two random vari...
rrvmulc 29842 A random variable multipli...
rrvsum 29843 An indexed sum of random v...
orvcval 29846 Value of the preimage mapp...
orvcval2 29847 Another way to express the...
elorvc 29848 Elementhood of a preimage....
orvcval4 29849 The value of the preimage ...
orvcoel 29850 If the relation produces o...
orvccel 29851 If the relation produces c...
elorrvc 29852 Elementhood of a preimage ...
orrvcval4 29853 The value of the preimage ...
orrvcoel 29854 If the relation produces o...
orrvccel 29855 If the relation produces c...
orvcgteel 29856 Preimage maps produced by ...
orvcelval 29857 Preimage maps produced by ...
orvcelel 29858 Preimage maps produced by ...
dstrvval 29859 The value of the distribut...
dstrvprob 29860 The distribution of a rand...
orvclteel 29861 Preimage maps produced by ...
dstfrvel 29862 Elementhood of preimage ma...
dstfrvunirn 29863 The limit of all preimage ...
orvclteinc 29864 Preimage maps produced by ...
dstfrvinc 29865 A cumulative distribution ...
dstfrvclim1 29866 The limit of the cumulativ...
coinfliplem 29867 Division in the extended r...
coinflipprob 29868 The ` P ` we defined for c...
coinflipspace 29869 The space of our coin-flip...
coinflipuniv 29870 The universe of our coin-f...
coinfliprv 29871 The ` X ` we defined for c...
coinflippv 29872 The probability of heads i...
coinflippvt 29873 The probability of tails i...
ballotlemoex 29874 ` O ` is a set. (Contribu...
ballotlem1 29875 The size of the universe i...
ballotlemelo 29876 Elementhood in ` O ` . (C...
ballotlem2 29877 The probability that the f...
ballotlemfval 29878 The value of F. (Contribut...
ballotlemfelz 29879 ` ( F `` C ) ` has values ...
ballotlemfp1 29880 If the ` J ` th ballot is ...
ballotlemfc0 29881 ` F ` takes value 0 betwee...
ballotlemfcc 29882 ` F ` takes value 0 betwee...
ballotlemfmpn 29883 ` ( F `` C ) ` finishes co...
ballotlemfval0 29884 ` ( F `` C ) ` always star...
ballotleme 29885 Elements of ` E ` . (Cont...
ballotlemodife 29886 Elements of ` ( O \ E ) ` ...
ballotlem4 29887 If the first pick is a vot...
ballotlem5 29888 If A is not ahead througho...
ballotlemi 29889 Value of ` I ` for a given...
ballotlemiex 29890 Properties of ` ( I `` C )...
ballotlemi1 29891 The first tie cannot be re...
ballotlemii 29892 The first tie cannot be re...
ballotlemsup 29893 The set of zeroes of ` F `...
ballotlemimin 29894 ` ( I `` C ) ` is the firs...
ballotlemic 29895 If the first vote is for B...
ballotlem1c 29896 If the first vote is for A...
ballotlemsval 29897 Value of ` S ` . (Contrib...
ballotlemsv 29898 Value of ` S ` evaluated a...
ballotlemsgt1 29899 ` S ` maps values less tha...
ballotlemsdom 29900 Domain of ` S ` for a give...
ballotlemsel1i 29901 The range ` ( 1 ... ( I ``...
ballotlemsf1o 29902 The defined ` S ` is a bij...
ballotlemsi 29903 The image by ` S ` of the ...
ballotlemsima 29904 The image by ` S ` of an i...
ballotlemieq 29905 If two countings share the...
ballotlemrval 29906 Value of ` R ` . (Contrib...
ballotlemscr 29907 The image of ` ( R `` C ) ...
ballotlemrv 29908 Value of ` R ` evaluated a...
ballotlemrv1 29909 Value of ` R ` before the ...
ballotlemrv2 29910 Value of ` R ` after the t...
ballotlemro 29911 Range of ` R ` is included...
ballotlemgval 29912 Expand the value of ` .^ `...
ballotlemgun 29913 A property of the defined ...
ballotlemfg 29914 Express the value of ` ( F...
ballotlemfrc 29915 Express the value of ` ( F...
ballotlemfrci 29916 Reverse counting preserves...
ballotlemfrceq 29917 Value of ` F ` for a rever...
ballotlemfrcn0 29918 Value of ` F ` for a rever...
ballotlemrc 29919 Range of ` R ` . (Contrib...
ballotlemirc 29920 Applying ` R ` does not ch...
ballotlemrinv0 29921 Lemma for ~ ballotlemrinv ...
ballotlemrinv 29922 ` R ` is its own inverse :...
ballotlem1ri 29923 When the vote on the first...
ballotlem7 29924 ` R ` is a bijection betwe...
ballotlem8 29925 There are as many counting...
ballotth 29926 Bertrand's ballot problem ...
sgncl 29927 Closure of the signum. (C...
sgnclre 29928 Closure of the signum. (C...
sgnneg 29929 Negation of the signum. (...
sgn3da 29930 A conditional containing a...
sgnmul 29931 Signum of a product. (Con...
sgnmulrp2 29932 Multiplication by a positi...
sgnsub 29933 Subtraction of a number of...
sgnnbi 29934 Negative signum. (Contrib...
sgnpbi 29935 Positive signum. (Contrib...
sgn0bi 29936 Zero signum. (Contributed...
sgnsgn 29937 Signum is idempotent. (Co...
sgnmulsgn 29938 If two real numbers are of...
sgnmulsgp 29939 If two real numbers are of...
fzssfzo 29940 Condition for an integer i...
gsumncl 29941 Closure of a group sum in ...
gsumnunsn 29942 Closure of a group sum in ...
wrdres 29943 Condition for the restrict...
wrdsplex 29944 Existence of a split of a ...
ccatmulgnn0dir 29945 Concatenation of words fol...
ofcccat 29946 Letterwise operations on w...
ofcs1 29947 Letterwise operations on a...
ofcs2 29948 Letterwise operations on a...
plymul02 29949 Product of a polynomial wi...
plymulx0 29950 Coefficients of a polynomi...
plymulx 29951 Coefficients of a polynomi...
plyrecld 29952 Closure of a polynomial wi...
signsplypnf 29953 The quotient of a polynomi...
signsply0 29954 Lemma for the rule of sign...
signspval 29955 The value of the skipping ...
signsw0glem 29956 Neutral element property o...
signswbase 29957 The base of ` W ` is the t...
signswplusg 29958 The operation of ` W ` . ...
signsw0g 29959 The neutral element of ` W...
signswmnd 29960 ` W ` is a monoid structur...
signswrid 29961 The zero-skipping operatio...
signswlid 29962 The zero-skipping operatio...
signswn0 29963 The zero-skipping operatio...
signswch 29964 The zero-skipping operatio...
signslema 29965 Computational part of sign...
signstfv 29966 Value of the zero-skipping...
signstfval 29967 Value of the zero-skipping...
signstcl 29968 Closure of the zero skippi...
signstf 29969 The zero skipping sign wor...
signstlen 29970 Length of the zero skippin...
signstf0 29971 Sign of a single letter wo...
signstfvn 29972 Zero-skipping sign in a wo...
signsvtn0 29973 If the last letter is non ...
signstfvp 29974 Zero-skipping sign in a wo...
signstfvneq0 29975 In case the first letter i...
signstfvcl 29976 Closure of the zero skippi...
signstfvc 29977 Zero-skipping sign in a wo...
signstres 29978 Restriction of a zero skip...
signstfveq0a 29979 Lemma for ~ signstfveq0 . ...
signstfveq0 29980 In case the last letter is...
signsvvfval 29981 The value of ` V ` , which...
signsvvf 29982 ` V ` is a function. (Con...
signsvf0 29983 There is no change of sign...
signsvf1 29984 In a single-letter word, w...
signsvfn 29985 Number of changes in a wor...
signsvtp 29986 Adding a letter of the sam...
signsvtn 29987 Adding a letter of a diffe...
signsvfpn 29988 Adding a letter of the sam...
signsvfnn 29989 Adding a letter of a diffe...
signlem0 29990 Adding a zero as the highe...
signshf 29991 ` H ` , corresponding to t...
signshwrd 29992 ` H ` , corresponding to t...
signshlen 29993 Length of ` H ` , correspo...
signshnz 29994 ` H ` is not the empty wor...
istrkg2d 29997 Property of fulfilling dim...
axtglowdim2OLD 29998 Lower dimension axiom for ...
axtgupdim2OLD 29999 Upper dimension axiom for ...
afsval 30002 Value of the AFS relation ...
brafs 30003 Binary relationship form o...
tg5segofs 30004 Rephrase ~ axtg5seg using ...
bnj170 30017 ` /\ ` -manipulation. (Co...
bnj240 30018 ` /\ ` -manipulation. (Co...
bnj248 30019 ` /\ ` -manipulation. (Co...
bnj250 30020 ` /\ ` -manipulation. (Co...
bnj251 30021 ` /\ ` -manipulation. (Co...
bnj252 30022 ` /\ ` -manipulation. (Co...
bnj253 30023 ` /\ ` -manipulation. (Co...
bnj255 30024 ` /\ ` -manipulation. (Co...
bnj256 30025 ` /\ ` -manipulation. (Co...
bnj257 30026 ` /\ ` -manipulation. (Co...
bnj258 30027 ` /\ ` -manipulation. (Co...
bnj268 30028 ` /\ ` -manipulation. (Co...
bnj290 30029 ` /\ ` -manipulation. (Co...
bnj291 30030 ` /\ ` -manipulation. (Co...
bnj312 30031 ` /\ ` -manipulation. (Co...
bnj334 30032 ` /\ ` -manipulation. (Co...
bnj345 30033 ` /\ ` -manipulation. (Co...
bnj422 30034 ` /\ ` -manipulation. (Co...
bnj432 30035 ` /\ ` -manipulation. (Co...
bnj446 30036 ` /\ ` -manipulation. (Co...
bnj21 30037 First-order logic and set ...
bnj23 30038 First-order logic and set ...
bnj31 30039 First-order logic and set ...
bnj62 30040 First-order logic and set ...
bnj89 30041 First-order logic and set ...
bnj90 30042 First-order logic and set ...
bnj101 30043 First-order logic and set ...
bnj105 30044 First-order logic and set ...
bnj115 30045 First-order logic and set ...
bnj132 30046 First-order logic and set ...
bnj133 30047 First-order logic and set ...
bnj142OLD 30048 First-order logic and set ...
bnj145OLD 30049 First-order logic and set ...
bnj156 30050 First-order logic and set ...
bnj158 30051 First-order logic and set ...
bnj168 30052 First-order logic and set ...
bnj206 30053 First-order logic and set ...
bnj216 30054 First-order logic and set ...
bnj219 30055 First-order logic and set ...
bnj226 30056 First-order logic and set ...
bnj228 30057 First-order logic and set ...
bnj519 30058 First-order logic and set ...
bnj521 30059 First-order logic and set ...
bnj524 30060 First-order logic and set ...
bnj525 30061 First-order logic and set ...
bnj534 30062 First-order logic and set ...
bnj538 30063 First-order logic and set ...
bnj538OLD 30064 First-order logic and set ...
bnj529 30065 First-order logic and set ...
bnj551 30066 First-order logic and set ...
bnj563 30067 First-order logic and set ...
bnj564 30068 First-order logic and set ...
bnj593 30069 First-order logic and set ...
bnj596 30070 First-order logic and set ...
bnj610 30071 Pass from equality ( ` x =...
bnj642 30072 ` /\ ` -manipulation. (Co...
bnj643 30073 ` /\ ` -manipulation. (Co...
bnj645 30074 ` /\ ` -manipulation. (Co...
bnj658 30075 ` /\ ` -manipulation. (Co...
bnj667 30076 ` /\ ` -manipulation. (Co...
bnj705 30077 ` /\ ` -manipulation. (Co...
bnj706 30078 ` /\ ` -manipulation. (Co...
bnj707 30079 ` /\ ` -manipulation. (Co...
bnj708 30080 ` /\ ` -manipulation. (Co...
bnj721 30081 ` /\ ` -manipulation. (Co...
bnj832 30082 ` /\ ` -manipulation. (Co...
bnj835 30083 ` /\ ` -manipulation. (Co...
bnj836 30084 ` /\ ` -manipulation. (Co...
bnj837 30085 ` /\ ` -manipulation. (Co...
bnj769 30086 ` /\ ` -manipulation. (Co...
bnj770 30087 ` /\ ` -manipulation. (Co...
bnj771 30088 ` /\ ` -manipulation. (Co...
bnj887 30089 ` /\ ` -manipulation. (Co...
bnj918 30090 First-order logic and set ...
bnj919 30091 First-order logic and set ...
bnj923 30092 First-order logic and set ...
bnj927 30093 First-order logic and set ...
bnj930 30094 First-order logic and set ...
bnj931 30095 First-order logic and set ...
bnj937 30096 First-order logic and set ...
bnj941 30097 First-order logic and set ...
bnj945 30098 Technical lemma for ~ bnj6...
bnj946 30099 First-order logic and set ...
bnj951 30100 ` /\ ` -manipulation. (Co...
bnj956 30101 First-order logic and set ...
bnj976 30102 First-order logic and set ...
bnj982 30103 First-order logic and set ...
bnj1019 30104 First-order logic and set ...
bnj1023 30105 First-order logic and set ...
bnj1095 30106 First-order logic and set ...
bnj1096 30107 First-order logic and set ...
bnj1098 30108 First-order logic and set ...
bnj1101 30109 First-order logic and set ...
bnj1113 30110 First-order logic and set ...
bnj1109 30111 First-order logic and set ...
bnj1131 30112 First-order logic and set ...
bnj1138 30113 First-order logic and set ...
bnj1142 30114 First-order logic and set ...
bnj1143 30115 First-order logic and set ...
bnj1146 30116 First-order logic and set ...
bnj1149 30117 First-order logic and set ...
bnj1185 30118 First-order logic and set ...
bnj1196 30119 First-order logic and set ...
bnj1198 30120 First-order logic and set ...
bnj1209 30121 First-order logic and set ...
bnj1211 30122 First-order logic and set ...
bnj1213 30123 First-order logic and set ...
bnj1212 30124 First-order logic and set ...
bnj1219 30125 First-order logic and set ...
bnj1224 30126 First-order logic and set ...
bnj1230 30127 First-order logic and set ...
bnj1232 30128 First-order logic and set ...
bnj1235 30129 First-order logic and set ...
bnj1239 30130 First-order logic and set ...
bnj1238 30131 First-order logic and set ...
bnj1241 30132 First-order logic and set ...
bnj1247 30133 First-order logic and set ...
bnj1254 30134 First-order logic and set ...
bnj1262 30135 First-order logic and set ...
bnj1266 30136 First-order logic and set ...
bnj1265 30137 First-order logic and set ...
bnj1275 30138 First-order logic and set ...
bnj1276 30139 First-order logic and set ...
bnj1292 30140 First-order logic and set ...
bnj1293 30141 First-order logic and set ...
bnj1294 30142 First-order logic and set ...
bnj1299 30143 First-order logic and set ...
bnj1304 30144 First-order logic and set ...
bnj1316 30145 First-order logic and set ...
bnj1317 30146 First-order logic and set ...
bnj1322 30147 First-order logic and set ...
bnj1340 30148 First-order logic and set ...
bnj1345 30149 First-order logic and set ...
bnj1350 30150 First-order logic and set ...
bnj1351 30151 First-order logic and set ...
bnj1352 30152 First-order logic and set ...
bnj1361 30153 First-order logic and set ...
bnj1366 30154 First-order logic and set ...
bnj1379 30155 First-order logic and set ...
bnj1383 30156 First-order logic and set ...
bnj1385 30157 First-order logic and set ...
bnj1386 30158 First-order logic and set ...
bnj1397 30159 First-order logic and set ...
bnj1400 30160 First-order logic and set ...
bnj1405 30161 First-order logic and set ...
bnj1422 30162 First-order logic and set ...
bnj1424 30163 First-order logic and set ...
bnj1436 30164 First-order logic and set ...
bnj1441 30165 First-order logic and set ...
bnj1454 30166 First-order logic and set ...
bnj1459 30167 First-order logic and set ...
bnj1464 30168 Conversion of implicit sub...
bnj1465 30169 First-order logic and set ...
bnj1468 30170 Conversion of implicit sub...
bnj1476 30171 First-order logic and set ...
bnj1502 30172 First-order logic and set ...
bnj1503 30173 First-order logic and set ...
bnj1517 30174 First-order logic and set ...
bnj1521 30175 First-order logic and set ...
bnj1533 30176 First-order logic and set ...
bnj1534 30177 First-order logic and set ...
bnj1536 30178 First-order logic and set ...
bnj1538 30179 First-order logic and set ...
bnj1541 30180 First-order logic and set ...
bnj1542 30181 First-order logic and set ...
bnj110 30182 Well-founded induction res...
bnj157 30183 Well-founded induction res...
bnj66 30184 Technical lemma for ~ bnj6...
bnj91 30185 First-order logic and set ...
bnj92 30186 First-order logic and set ...
bnj93 30187 Technical lemma for ~ bnj9...
bnj95 30188 Technical lemma for ~ bnj1...
bnj96 30189 Technical lemma for ~ bnj1...
bnj97 30190 Technical lemma for ~ bnj1...
bnj98 30191 Technical lemma for ~ bnj1...
bnj106 30192 First-order logic and set ...
bnj118 30193 First-order logic and set ...
bnj121 30194 First-order logic and set ...
bnj124 30195 Technical lemma for ~ bnj1...
bnj125 30196 Technical lemma for ~ bnj1...
bnj126 30197 Technical lemma for ~ bnj1...
bnj130 30198 Technical lemma for ~ bnj1...
bnj149 30199 Technical lemma for ~ bnj1...
bnj150 30200 Technical lemma for ~ bnj1...
bnj151 30201 Technical lemma for ~ bnj1...
bnj154 30202 Technical lemma for ~ bnj1...
bnj155 30203 Technical lemma for ~ bnj1...
bnj153 30204 Technical lemma for ~ bnj8...
bnj207 30205 Technical lemma for ~ bnj8...
bnj213 30206 First-order logic and set ...
bnj222 30207 Technical lemma for ~ bnj2...
bnj229 30208 Technical lemma for ~ bnj5...
bnj517 30209 Technical lemma for ~ bnj5...
bnj518 30210 Technical lemma for ~ bnj8...
bnj523 30211 Technical lemma for ~ bnj8...
bnj526 30212 Technical lemma for ~ bnj8...
bnj528 30213 Technical lemma for ~ bnj8...
bnj535 30214 Technical lemma for ~ bnj8...
bnj539 30215 Technical lemma for ~ bnj8...
bnj540 30216 Technical lemma for ~ bnj8...
bnj543 30217 Technical lemma for ~ bnj8...
bnj544 30218 Technical lemma for ~ bnj8...
bnj545 30219 Technical lemma for ~ bnj8...
bnj546 30220 Technical lemma for ~ bnj8...
bnj548 30221 Technical lemma for ~ bnj8...
bnj553 30222 Technical lemma for ~ bnj8...
bnj554 30223 Technical lemma for ~ bnj8...
bnj556 30224 Technical lemma for ~ bnj8...
bnj557 30225 Technical lemma for ~ bnj8...
bnj558 30226 Technical lemma for ~ bnj8...
bnj561 30227 Technical lemma for ~ bnj8...
bnj562 30228 Technical lemma for ~ bnj8...
bnj570 30229 Technical lemma for ~ bnj8...
bnj571 30230 Technical lemma for ~ bnj8...
bnj605 30231 Technical lemma. This lem...
bnj581 30232 Technical lemma for ~ bnj5...
bnj589 30233 Technical lemma for ~ bnj8...
bnj590 30234 Technical lemma for ~ bnj8...
bnj591 30235 Technical lemma for ~ bnj8...
bnj594 30236 Technical lemma for ~ bnj8...
bnj580 30237 Technical lemma for ~ bnj5...
bnj579 30238 Technical lemma for ~ bnj8...
bnj602 30239 Equality theorem for the `...
bnj607 30240 Technical lemma for ~ bnj8...
bnj609 30241 Technical lemma for ~ bnj8...
bnj611 30242 Technical lemma for ~ bnj8...
bnj600 30243 Technical lemma for ~ bnj8...
bnj601 30244 Technical lemma for ~ bnj8...
bnj852 30245 Technical lemma for ~ bnj6...
bnj864 30246 Technical lemma for ~ bnj6...
bnj865 30247 Technical lemma for ~ bnj6...
bnj873 30248 Technical lemma for ~ bnj6...
bnj849 30249 Technical lemma for ~ bnj6...
bnj882 30250 Definition (using hypothes...
bnj18eq1 30251 Equality theorem for trans...
bnj893 30252 Property of ` _trCl ` . U...
bnj900 30253 Technical lemma for ~ bnj6...
bnj906 30254 Property of ` _trCl ` . (...
bnj908 30255 Technical lemma for ~ bnj6...
bnj911 30256 Technical lemma for ~ bnj6...
bnj916 30257 Technical lemma for ~ bnj6...
bnj917 30258 Technical lemma for ~ bnj6...
bnj934 30259 Technical lemma for ~ bnj6...
bnj929 30260 Technical lemma for ~ bnj6...
bnj938 30261 Technical lemma for ~ bnj6...
bnj944 30262 Technical lemma for ~ bnj6...
bnj953 30263 Technical lemma for ~ bnj6...
bnj958 30264 Technical lemma for ~ bnj6...
bnj1000 30265 Technical lemma for ~ bnj8...
bnj965 30266 Technical lemma for ~ bnj8...
bnj964 30267 Technical lemma for ~ bnj6...
bnj966 30268 Technical lemma for ~ bnj6...
bnj967 30269 Technical lemma for ~ bnj6...
bnj969 30270 Technical lemma for ~ bnj6...
bnj970 30271 Technical lemma for ~ bnj6...
bnj910 30272 Technical lemma for ~ bnj6...
bnj978 30273 Technical lemma for ~ bnj6...
bnj981 30274 Technical lemma for ~ bnj6...
bnj983 30275 Technical lemma for ~ bnj6...
bnj984 30276 Technical lemma for ~ bnj6...
bnj985 30277 Technical lemma for ~ bnj6...
bnj986 30278 Technical lemma for ~ bnj6...
bnj996 30279 Technical lemma for ~ bnj6...
bnj998 30280 Technical lemma for ~ bnj6...
bnj999 30281 Technical lemma for ~ bnj6...
bnj1001 30282 Technical lemma for ~ bnj6...
bnj1006 30283 Technical lemma for ~ bnj6...
bnj1014 30284 Technical lemma for ~ bnj6...
bnj1015 30285 Technical lemma for ~ bnj6...
bnj1018 30286 Technical lemma for ~ bnj6...
bnj1020 30287 Technical lemma for ~ bnj6...
bnj1021 30288 Technical lemma for ~ bnj6...
bnj907 30289 Technical lemma for ~ bnj6...
bnj1029 30290 Property of ` _trCl ` . (...
bnj1033 30291 Technical lemma for ~ bnj6...
bnj1034 30292 Technical lemma for ~ bnj6...
bnj1039 30293 Technical lemma for ~ bnj6...
bnj1040 30294 Technical lemma for ~ bnj6...
bnj1047 30295 Technical lemma for ~ bnj6...
bnj1049 30296 Technical lemma for ~ bnj6...
bnj1052 30297 Technical lemma for ~ bnj6...
bnj1053 30298 Technical lemma for ~ bnj6...
bnj1071 30299 Technical lemma for ~ bnj6...
bnj1083 30300 Technical lemma for ~ bnj6...
bnj1090 30301 Technical lemma for ~ bnj6...
bnj1093 30302 Technical lemma for ~ bnj6...
bnj1097 30303 Technical lemma for ~ bnj6...
bnj1110 30304 Technical lemma for ~ bnj6...
bnj1112 30305 Technical lemma for ~ bnj6...
bnj1118 30306 Technical lemma for ~ bnj6...
bnj1121 30307 Technical lemma for ~ bnj6...
bnj1123 30308 Technical lemma for ~ bnj6...
bnj1030 30309 Technical lemma for ~ bnj6...
bnj1124 30310 Property of ` _trCl ` . (...
bnj1133 30311 Technical lemma for ~ bnj6...
bnj1128 30312 Technical lemma for ~ bnj6...
bnj1127 30313 Property of ` _trCl ` . (...
bnj1125 30314 Property of ` _trCl ` . (...
bnj1145 30315 Technical lemma for ~ bnj6...
bnj1147 30316 Property of ` _trCl ` . (...
bnj1137 30317 Property of ` _trCl ` . (...
bnj1148 30318 Property of ` _pred ` . (...
bnj1136 30319 Technical lemma for ~ bnj6...
bnj1152 30320 Technical lemma for ~ bnj6...
bnj1154 30321 Property of ` Fr ` . (Con...
bnj1171 30322 Technical lemma for ~ bnj6...
bnj1172 30323 Technical lemma for ~ bnj6...
bnj1173 30324 Technical lemma for ~ bnj6...
bnj1174 30325 Technical lemma for ~ bnj6...
bnj1175 30326 Technical lemma for ~ bnj6...
bnj1176 30327 Technical lemma for ~ bnj6...
bnj1177 30328 Technical lemma for ~ bnj6...
bnj1186 30329 Technical lemma for ~ bnj6...
bnj1190 30330 Technical lemma for ~ bnj6...
bnj1189 30331 Technical lemma for ~ bnj6...
bnj69 30332 Existence of a minimal ele...
bnj1228 30333 Existence of a minimal ele...
bnj1204 30334 Well-founded induction. T...
bnj1234 30335 Technical lemma for ~ bnj6...
bnj1245 30336 Technical lemma for ~ bnj6...
bnj1256 30337 Technical lemma for ~ bnj6...
bnj1259 30338 Technical lemma for ~ bnj6...
bnj1253 30339 Technical lemma for ~ bnj6...
bnj1279 30340 Technical lemma for ~ bnj6...
bnj1286 30341 Technical lemma for ~ bnj6...
bnj1280 30342 Technical lemma for ~ bnj6...
bnj1296 30343 Technical lemma for ~ bnj6...
bnj1309 30344 Technical lemma for ~ bnj6...
bnj1307 30345 Technical lemma for ~ bnj6...
bnj1311 30346 Technical lemma for ~ bnj6...
bnj1318 30347 Technical lemma for ~ bnj6...
bnj1326 30348 Technical lemma for ~ bnj6...
bnj1321 30349 Technical lemma for ~ bnj6...
bnj1364 30350 Property of ` _FrSe ` . (...
bnj1371 30351 Technical lemma for ~ bnj6...
bnj1373 30352 Technical lemma for ~ bnj6...
bnj1374 30353 Technical lemma for ~ bnj6...
bnj1384 30354 Technical lemma for ~ bnj6...
bnj1388 30355 Technical lemma for ~ bnj6...
bnj1398 30356 Technical lemma for ~ bnj6...
bnj1413 30357 Property of ` _trCl ` . (...
bnj1408 30358 Technical lemma for ~ bnj1...
bnj1414 30359 Property of ` _trCl ` . (...
bnj1415 30360 Technical lemma for ~ bnj6...
bnj1416 30361 Technical lemma for ~ bnj6...
bnj1418 30362 Property of ` _pred ` . (...
bnj1417 30363 Technical lemma for ~ bnj6...
bnj1421 30364 Technical lemma for ~ bnj6...
bnj1444 30365 Technical lemma for ~ bnj6...
bnj1445 30366 Technical lemma for ~ bnj6...
bnj1446 30367 Technical lemma for ~ bnj6...
bnj1447 30368 Technical lemma for ~ bnj6...
bnj1448 30369 Technical lemma for ~ bnj6...
bnj1449 30370 Technical lemma for ~ bnj6...
bnj1442 30371 Technical lemma for ~ bnj6...
bnj1450 30372 Technical lemma for ~ bnj6...
bnj1423 30373 Technical lemma for ~ bnj6...
bnj1452 30374 Technical lemma for ~ bnj6...
bnj1466 30375 Technical lemma for ~ bnj6...
bnj1467 30376 Technical lemma for ~ bnj6...
bnj1463 30377 Technical lemma for ~ bnj6...
bnj1489 30378 Technical lemma for ~ bnj6...
bnj1491 30379 Technical lemma for ~ bnj6...
bnj1312 30380 Technical lemma for ~ bnj6...
bnj1493 30381 Technical lemma for ~ bnj6...
bnj1497 30382 Technical lemma for ~ bnj6...
bnj1498 30383 Technical lemma for ~ bnj6...
bnj60 30384 Well-founded recursion, pa...
bnj1514 30385 Technical lemma for ~ bnj1...
bnj1518 30386 Technical lemma for ~ bnj1...
bnj1519 30387 Technical lemma for ~ bnj1...
bnj1520 30388 Technical lemma for ~ bnj1...
bnj1501 30389 Technical lemma for ~ bnj1...
bnj1500 30390 Well-founded recursion, pa...
bnj1525 30391 Technical lemma for ~ bnj1...
bnj1529 30392 Technical lemma for ~ bnj1...
bnj1523 30393 Technical lemma for ~ bnj1...
bnj1522 30394 Well-founded recursion, pa...
quartfull 30401 The quartic equation, writ...
deranglem 30402 Lemma for derangements. (...
derangval 30403 Define the derangement fun...
derangf 30404 The derangement number is ...
derang0 30405 The derangement number of ...
derangsn 30406 The derangement number of ...
derangenlem 30407 One half of ~ derangen . ...
derangen 30408 The derangement number is ...
subfacval 30409 The subfactorial is define...
derangen2 30410 Write the derangement numb...
subfacf 30411 The subfactorial is a func...
subfaclefac 30412 The subfactorial is less t...
subfac0 30413 The subfactorial at zero. ...
subfac1 30414 The subfactorial at one. ...
subfacp1lem1 30415 Lemma for ~ subfacp1 . Th...
subfacp1lem2a 30416 Lemma for ~ subfacp1 . Pr...
subfacp1lem2b 30417 Lemma for ~ subfacp1 . Pr...
subfacp1lem3 30418 Lemma for ~ subfacp1 . In...
subfacp1lem4 30419 Lemma for ~ subfacp1 . Th...
subfacp1lem5 30420 Lemma for ~ subfacp1 . In...
subfacp1lem6 30421 Lemma for ~ subfacp1 . By...
subfacp1 30422 A two-term recurrence for ...
subfacval2 30423 A closed-form expression f...
subfaclim 30424 The subfactorial converges...
subfacval3 30425 Another closed form expres...
derangfmla 30426 The derangements formula, ...
erdszelem1 30427 Lemma for ~ erdsze . (Con...
erdszelem2 30428 Lemma for ~ erdsze . (Con...
erdszelem3 30429 Lemma for ~ erdsze . (Con...
erdszelem4 30430 Lemma for ~ erdsze . (Con...
erdszelem5 30431 Lemma for ~ erdsze . (Con...
erdszelem6 30432 Lemma for ~ erdsze . (Con...
erdszelem7 30433 Lemma for ~ erdsze . (Con...
erdszelem8 30434 Lemma for ~ erdsze . (Con...
erdszelem9 30435 Lemma for ~ erdsze . (Con...
erdszelem10 30436 Lemma for ~ erdsze . (Con...
erdszelem11 30437 Lemma for ~ erdsze . (Con...
erdsze 30438 The Erdős-Szekeres th...
erdsze2lem1 30439 Lemma for ~ erdsze2 . (Co...
erdsze2lem2 30440 Lemma for ~ erdsze2 . (Co...
erdsze2 30441 Generalize the statement o...
kur14lem1 30442 Lemma for ~ kur14 . (Cont...
kur14lem2 30443 Lemma for ~ kur14 . Write...
kur14lem3 30444 Lemma for ~ kur14 . A clo...
kur14lem4 30445 Lemma for ~ kur14 . Compl...
kur14lem5 30446 Lemma for ~ kur14 . Closu...
kur14lem6 30447 Lemma for ~ kur14 . If ` ...
kur14lem7 30448 Lemma for ~ kur14 : main p...
kur14lem8 30449 Lemma for ~ kur14 . Show ...
kur14lem9 30450 Lemma for ~ kur14 . Since...
kur14lem10 30451 Lemma for ~ kur14 . Disch...
kur14 30452 Kuratowski's closure-compl...
ispcon 30459 The property of being a pa...
pconcn 30460 The property of being a pa...
pcontop 30461 A simply connected space i...
isscon 30462 The property of being a si...
sconpcon 30463 A simply connected space i...
scontop 30464 A simply connected space i...
sconpht 30465 A closed path in a simply ...
cnpcon 30466 An image of a path-connect...
pconcon 30467 A path-connected space is ...
txpcon 30468 The topological product of...
ptpcon 30469 The topological product of...
indispcon 30470 The indiscrete topology (o...
conpcon 30471 A connected and locally pa...
qtoppcon 30472 A quotient of a path-conne...
pconpi1 30473 All fundamental groups in ...
sconpht2 30474 Any two paths in a simply ...
sconpi1 30475 A path-connected topologic...
txsconlem 30476 Lemma for ~ txscon . (Con...
txscon 30477 The topological product of...
cvxpcon 30478 A convex subset of the com...
cvxscon 30479 A convex subset of the com...
blscon 30480 An open ball in the comple...
cnllyscon 30481 The topology of the comple...
rescon 30482 A subset of ` RR ` is simp...
iooscon 30483 An open interval is simply...
iccscon 30484 A closed interval is simpl...
retopscon 30485 The real numbers are simpl...
iccllyscon 30486 A closed interval is local...
rellyscon 30487 The real numbers are local...
iiscon 30488 The unit interval is simpl...
iillyscon 30489 The unit interval is local...
iinllycon 30490 The unit interval is local...
fncvm 30493 Lemma for covering maps. ...
cvmscbv 30494 Change bound variables in ...
iscvm 30495 The property of being a co...
cvmtop1 30496 Reverse closure for a cove...
cvmtop2 30497 Reverse closure for a cove...
cvmcn 30498 A covering map is a contin...
cvmcov 30499 Property of a covering map...
cvmsrcl 30500 Reverse closure for an eve...
cvmsi 30501 One direction of ~ cvmsval...
cvmsval 30502 Elementhood in the set ` S...
cvmsss 30503 An even covering is a subs...
cvmsn0 30504 An even covering is nonemp...
cvmsuni 30505 An even covering of ` U ` ...
cvmsdisj 30506 An even covering of ` U ` ...
cvmshmeo 30507 Every element of an even c...
cvmsf1o 30508 ` F ` , localized to an el...
cvmscld 30509 The sets of an even coveri...
cvmsss2 30510 An open subset of an evenl...
cvmcov2 30511 The covering map property ...
cvmseu 30512 Every element in ` U. T ` ...
cvmsiota 30513 Identify the unique elemen...
cvmopnlem 30514 Lemma for ~ cvmopn . (Con...
cvmfolem 30515 Lemma for ~ cvmfo . (Cont...
cvmopn 30516 A covering map is an open ...
cvmliftmolem1 30517 Lemma for ~ cvmliftmo . (...
cvmliftmolem2 30518 Lemma for ~ cvmliftmo . (...
cvmliftmoi 30519 A lift of a continuous fun...
cvmliftmo 30520 A lift of a continuous fun...
cvmliftlem1 30521 Lemma for ~ cvmlift . In ...
cvmliftlem2 30522 Lemma for ~ cvmlift . ` W ...
cvmliftlem3 30523 Lemma for ~ cvmlift . Sin...
cvmliftlem4 30524 Lemma for ~ cvmlift . The...
cvmliftlem5 30525 Lemma for ~ cvmlift . Def...
cvmliftlem6 30526 Lemma for ~ cvmlift . Ind...
cvmliftlem7 30527 Lemma for ~ cvmlift . Pro...
cvmliftlem8 30528 Lemma for ~ cvmlift . The...
cvmliftlem9 30529 Lemma for ~ cvmlift . The...
cvmliftlem10 30530 Lemma for ~ cvmlift . The...
cvmliftlem11 30531 Lemma for ~ cvmlift . (Co...
cvmliftlem13 30532 Lemma for ~ cvmlift . The...
cvmliftlem14 30533 Lemma for ~ cvmlift . Put...
cvmliftlem15 30534 Lemma for ~ cvmlift . Dis...
cvmlift 30535 One of the important prope...
cvmfo 30536 A covering map is an onto ...
cvmliftiota 30537 Write out a function ` H `...
cvmlift2lem1 30538 Lemma for ~ cvmlift2 . (C...
cvmlift2lem9a 30539 Lemma for ~ cvmlift2 and ~...
cvmlift2lem2 30540 Lemma for ~ cvmlift2 . (C...
cvmlift2lem3 30541 Lemma for ~ cvmlift2 . (C...
cvmlift2lem4 30542 Lemma for ~ cvmlift2 . (C...
cvmlift2lem5 30543 Lemma for ~ cvmlift2 . (C...
cvmlift2lem6 30544 Lemma for ~ cvmlift2 . (C...
cvmlift2lem7 30545 Lemma for ~ cvmlift2 . (C...
cvmlift2lem8 30546 Lemma for ~ cvmlift2 . (C...
cvmlift2lem9 30547 Lemma for ~ cvmlift2 . (C...
cvmlift2lem10 30548 Lemma for ~ cvmlift2 . (C...
cvmlift2lem11 30549 Lemma for ~ cvmlift2 . (C...
cvmlift2lem12 30550 Lemma for ~ cvmlift2 . (C...
cvmlift2lem13 30551 Lemma for ~ cvmlift2 . (C...
cvmlift2 30552 A two-dimensional version ...
cvmliftphtlem 30553 Lemma for ~ cvmliftpht . ...
cvmliftpht 30554 If ` G ` and ` H ` are pat...
cvmlift3lem1 30555 Lemma for ~ cvmlift3 . (C...
cvmlift3lem2 30556 Lemma for ~ cvmlift2 . (C...
cvmlift3lem3 30557 Lemma for ~ cvmlift2 . (C...
cvmlift3lem4 30558 Lemma for ~ cvmlift2 . (C...
cvmlift3lem5 30559 Lemma for ~ cvmlift2 . (C...
cvmlift3lem6 30560 Lemma for ~ cvmlift3 . (C...
cvmlift3lem7 30561 Lemma for ~ cvmlift3 . (C...
cvmlift3lem8 30562 Lemma for ~ cvmlift2 . (C...
cvmlift3lem9 30563 Lemma for ~ cvmlift2 . (C...
cvmlift3 30564 A general version of ~ cvm...
snmlff 30565 The function ` F ` from ~ ...
snmlfval 30566 The function ` F ` from ~ ...
snmlval 30567 The property " ` A ` is si...
snmlflim 30568 If ` A ` is simply normal,...
mvtval 30651 The set of variable typeco...
mrexval 30652 The set of "raw expression...
mexval 30653 The set of expressions, wh...
mexval2 30654 The set of expressions, wh...
mdvval 30655 The set of disjoint variab...
mvrsval 30656 The set of variables in an...
mvrsfpw 30657 The set of variables in an...
mrsubffval 30658 The substitution of some v...
mrsubfval 30659 The substitution of some v...
mrsubval 30660 The substitution of some v...
mrsubcv 30661 The value of a substituted...
mrsubvr 30662 The value of a substituted...
mrsubff 30663 A substitution is a functi...
mrsubrn 30664 Although it is defined for...
mrsubff1 30665 When restricted to complet...
mrsubff1o 30666 When restricted to complet...
mrsub0 30667 The value of the substitut...
mrsubf 30668 A substitution is a functi...
mrsubccat 30669 Substitution distributes o...
mrsubcn 30670 A substitution does not ch...
elmrsubrn 30671 Characterization of the su...
mrsubco 30672 The composition of two sub...
mrsubvrs 30673 The set of variables in a ...
msubffval 30674 A substitution applied to ...
msubfval 30675 A substitution applied to ...
msubval 30676 A substitution applied to ...
msubrsub 30677 A substitution applied to ...
msubty 30678 The type of a substituted ...
elmsubrn 30679 Characterization of substi...
msubrn 30680 Although it is defined for...
msubff 30681 A substitution is a functi...
msubco 30682 The composition of two sub...
msubf 30683 A substitution is a functi...
mvhfval 30684 Value of the function mapp...
mvhval 30685 Value of the function mapp...
mpstval 30686 A pre-statement is an orde...
elmpst 30687 Property of being a pre-st...
msrfval 30688 Value of the reduct of a p...
msrval 30689 Value of the reduct of a p...
mpstssv 30690 A pre-statement is an orde...
mpst123 30691 Decompose a pre-statement ...
mpstrcl 30692 The elements of a pre-stat...
msrf 30693 The reduct of a pre-statem...
msrrcl 30694 If ` X ` and ` Y ` have th...
mstaval 30695 Value of the set of statem...
msrid 30696 The reduct of a statement ...
msrfo 30697 The reduct of a pre-statem...
mstapst 30698 A statement is a pre-state...
elmsta 30699 Property of being a statem...
ismfs 30700 A formal system is a tuple...
mfsdisj 30701 The constants and variable...
mtyf2 30702 The type function maps var...
mtyf 30703 The type function maps var...
mvtss 30704 The set of variable typeco...
maxsta 30705 An axiom is a statement. ...
mvtinf 30706 Each variable typecode has...
msubff1 30707 When restricted to complet...
msubff1o 30708 When restricted to complet...
mvhf 30709 The function mapping varia...
mvhf1 30710 The function mapping varia...
msubvrs 30711 The set of variables in a ...
mclsrcl 30712 Reverse closure for the cl...
mclsssvlem 30713 Lemma for ~ mclsssv . (Co...
mclsval 30714 The function mapping varia...
mclsssv 30715 The closure of a set of ex...
ssmclslem 30716 Lemma for ~ ssmcls . (Con...
vhmcls 30717 All variable hypotheses ar...
ssmcls 30718 The original expressions a...
ss2mcls 30719 The closure is monotonic u...
mclsax 30720 The closure is closed unde...
mclsind 30721 Induction theorem for clos...
mppspstlem 30722 Lemma for ~ mppspst . (Co...
mppsval 30723 Definition of a provable p...
elmpps 30724 Definition of a provable p...
mppspst 30725 A provable pre-statement i...
mthmval 30726 A theorem is a pre-stateme...
elmthm 30727 A theorem is a pre-stateme...
mthmi 30728 A statement whose reduct i...
mthmsta 30729 A theorem is a pre-stateme...
mppsthm 30730 A provable pre-statement i...
mthmblem 30731 Lemma for ~ mthmb . (Cont...
mthmb 30732 If two statements have the...
mthmpps 30733 Given a theorem, there is ...
mclsppslem 30734 The closure is closed unde...
mclspps 30735 The closure is closed unde...
problem1 30812 Practice problem 1. Clues...
problem2 30813 Practice problem 2. Clues...
problem2OLD 30814 Practice problem 2. Clues...
problem3 30815 Practice problem 3. Clues...
problem4 30816 Practice problem 4. Clues...
problem5 30817 Practice problem 5. Clues...
quad3 30818 Variant of quadratic equat...
climuzcnv 30819 Utility lemma to convert b...
sinccvglem 30820 ` ( ( sin `` x ) / x ) ~~>...
sinccvg 30821 ` ( ( sin `` x ) / x ) ~~>...
circum 30822 The circumference of a cir...
elfzm12 30823 Membership in a curtailed ...
nn0seqcvg 30824 A strictly-decreasing nonn...
lediv2aALT 30825 Division of both sides of ...
abs2sqlei 30826 The absolute values of two...
abs2sqlti 30827 The absolute values of two...
abs2sqle 30828 The absolute values of two...
abs2sqlt 30829 The absolute values of two...
abs2difi 30830 Difference of absolute val...
abs2difabsi 30831 Absolute value of differen...
axextprim 30832 ~ ax-ext without distinct ...
axrepprim 30833 ~ ax-rep without distinct ...
axunprim 30834 ~ ax-un without distinct v...
axpowprim 30835 ~ ax-pow without distinct ...
axregprim 30836 ~ ax-reg without distinct ...
axinfprim 30837 ~ ax-inf without distinct ...
axacprim 30838 ~ ax-ac without distinct v...
untelirr 30839 We call a class "untanged"...
untuni 30840 The union of a class is un...
untsucf 30841 If a class is untangled, t...
unt0 30842 The null set is untangled....
untint 30843 If there is an untangled e...
efrunt 30844 If ` A ` is well-founded b...
untangtr 30845 A transitive class is unta...
3orel1 30846 Partial elimination of a t...
3orel2 30847 Partial elimination of a t...
3orel3 30848 Partial elimination of a t...
3pm3.2ni 30849 Triple negated disjunction...
3jaodd 30850 Double deduction form of ~...
3orit 30851 Closed form of ~ 3ori . (...
biimpexp 30852 A biconditional in the ant...
3orel13 30853 Elimination of two disjunc...
nepss 30854 Two classes are inequal if...
3ccased 30855 Triple disjunction form of...
dfso3 30856 Expansion of the definitio...
brtpid1 30857 A binary relationship invo...
brtpid2 30858 A binary relationship invo...
brtpid3 30859 A binary relationship invo...
ceqsrexv2 30860 Alternate elimitation of a...
iota5f 30861 A method for computing iot...
ceqsralv2 30862 Alternate elimination of a...
sqdivzi 30863 Distribution of square ove...
subdivcomb1 30864 Bring a term in a subtract...
subdivcomb2 30865 Bring a term in a subtract...
supfz 30866 The supremum of a finite s...
inffz 30867 The infimum of a finite se...
inffzOLD 30868 The infimum of a finite se...
fz0n 30869 The sequence ` ( 0 ... ( N...
shftvalg 30870 Value of a sequence shifte...
divcnvlin 30871 Limit of the ratio of two ...
climlec3 30872 Comparison of a constant t...
logi 30873 Calculate the logarithm of...
iexpire 30874 ` _i ` raised to itself is...
bcneg1 30875 The binomial coefficent ov...
bcm1nt 30876 The proportion of one bion...
bcprod 30877 A product identity for bin...
bccolsum 30878 A column-sum rule for bino...
iprodefisumlem 30879 Lemma for ~ iprodefisum . ...
iprodefisum 30880 Applying the exponential f...
iprodgam 30881 An infinite product versio...
faclimlem1 30882 Lemma for ~ faclim . Clos...
faclimlem2 30883 Lemma for ~ faclim . Show...
faclimlem3 30884 Lemma for ~ faclim . Alge...
faclim 30885 An infinite product expres...
iprodfac 30886 An infinite product expres...
faclim2 30887 Another factorial limit du...
pdivsq 30888 Condition for a prime divi...
dvdspw 30889 Exponentiation law for div...
gcd32 30890 Swap the second and third ...
gcdabsorb 30891 Absorption law for gcd. (...
brtp 30892 A condition for a binary r...
dftr6 30893 A potential definition of ...
coep 30894 Composition with epsilon. ...
coepr 30895 Composition with the conve...
dffr5 30896 A quantifier free definiti...
dfso2 30897 Quantifier free definition...
dfpo2 30898 Quantifier free definition...
br8 30899 Substitution for an eight-...
br6 30900 Substitution for a six-pla...
br4 30901 Substitution for a four-pl...
dfres3 30902 Alternate definition of re...
cnvco1 30903 Another distributive law o...
cnvco2 30904 Another distributive law o...
eldm3 30905 Quantifier-free definition...
elrn3 30906 Quantifier-free definition...
pocnv 30907 The converse of a partial ...
socnv 30908 The converse of a strict o...
funpsstri 30909 A condition for subset tri...
fundmpss 30910 If a class ` F ` is a prop...
fvresval 30911 The value of a function at...
funsseq 30912 Given two functions with e...
fununiq 30913 The uniqueness condition o...
funbreq 30914 An equality condition for ...
mpteq12d 30915 An equality inference for ...
fprb 30916 A condition for functionho...
br1steq 30917 Uniqueness condition for b...
br2ndeq 30918 Uniqueness condition for b...
br1steqg 30919 Uniqueness condition for b...
br2ndeqg 30920 Uniqueness condition for b...
dfdm5 30921 Definition of domain in te...
dfrn5 30922 Definition of range in ter...
opelco3 30923 Alternate way of saying th...
elima4 30924 Quantifier-free expression...
fv1stcnv 30925 The value of the converse ...
fv2ndcnv 30926 The value of the converse ...
setinds 30927 Principle of ` _E ` induct...
setinds2f 30928 ` _E ` induction schema, u...
setinds2 30929 ` _E ` induction schema, u...
elpotr 30930 A class of transitive sets...
dford5reg 30931 Given ~ ax-reg , an ordina...
dfon2lem1 30932 Lemma for ~ dfon2 . (Cont...
dfon2lem2 30933 Lemma for ~ dfon2 . (Cont...
dfon2lem3 30934 Lemma for ~ dfon2 . All s...
dfon2lem4 30935 Lemma for ~ dfon2 . If tw...
dfon2lem5 30936 Lemma for ~ dfon2 . Two s...
dfon2lem6 30937 Lemma for ~ dfon2 . A tra...
dfon2lem7 30938 Lemma for ~ dfon2 . All e...
dfon2lem8 30939 Lemma for ~ dfon2 . The i...
dfon2lem9 30940 Lemma for ~ dfon2 . A cla...
dfon2 30941 ` On ` consists of all set...
domep 30942 The domain of the epsilon ...
rdgprc0 30943 The value of the recursive...
rdgprc 30944 The value of the recursive...
dfrdg2 30945 Alternate definition of th...
dfrdg3 30946 Generalization of ~ dfrdg2...
axextdfeq 30947 A version of ~ ax-ext for ...
ax8dfeq 30948 A version of ~ ax-8 for us...
axextdist 30949 ~ ax-ext with distinctors ...
axext4dist 30950 ~ axext4 with distinctors ...
19.12b 30951 Version of ~ 19.12vv with ...
exnel 30952 There is always a set not ...
distel 30953 Distinctors in terms of me...
axextndbi 30954 ~ axextnd as a bicondition...
hbntg 30955 A more general form of ~ h...
hbimtg 30956 A more general and closed ...
hbaltg 30957 A more general and closed ...
hbng 30958 A more general form of ~ h...
hbimg 30959 A more general form of ~ h...
tfisg 30960 A closed form of ~ tfis . ...
dftrpred2 30963 A definition of the transi...
trpredeq1 30964 Equality theorem for trans...
trpredeq2 30965 Equality theorem for trans...
trpredeq3 30966 Equality theorem for trans...
trpredeq1d 30967 Equality deduction for tra...
trpredeq2d 30968 Equality deduction for tra...
trpredeq3d 30969 Equality deduction for tra...
eltrpred 30970 A class is a transitive pr...
trpredlem1 30971 Technical lemma for transi...
trpredpred 30972 Assuming it exists, the pr...
trpredss 30973 The transitive predecessor...
trpredtr 30974 The transitive predecessor...
trpredmintr 30975 The transitive predecessor...
trpredelss 30976 Given a transitive predece...
dftrpred3g 30977 The transitive predecessor...
dftrpred4g 30978 Another recursive expressi...
trpredpo 30979 If ` R ` partially orders ...
trpred0 30980 The class of transitive pr...
trpredex 30981 The transitive predecessor...
trpredrec 30982 If ` Y ` is an ` R ` , ` A...
frmin 30983 Every (possibly proper) su...
frind 30984 The principle of founded i...
frindi 30985 The principle of founded i...
frinsg 30986 Founded Induction Schema. ...
frins 30987 Founded Induction Schema. ...
frins2fg 30988 Founded Induction schema, ...
frins2f 30989 Founded Induction schema, ...
frins2g 30990 Founded Induction schema, ...
frins2 30991 Founded Induction schema, ...
frins3 30992 Founded Induction schema, ...
orderseqlem 30993 Lemma for ~ poseq and ~ so...
poseq 30994 A partial ordering of sequ...
soseq 30995 A linear ordering of seque...
wsuceq123 31004 Equality theorem for well-...
wsuceq1 31005 Equality theorem for well-...
wsuceq2 31006 Equality theorem for well-...
wsuceq3 31007 Equality theorem for well-...
nfwsuc 31008 Bound-variable hypothesis ...
wlimeq12 31009 Equality theorem for the l...
wlimeq1 31010 Equality theorem for the l...
wlimeq2 31011 Equality theorem for the l...
nfwlim 31012 Bound-variable hypothesis ...
elwlim 31013 Membership in the limit cl...
elwlimOLD 31014 Membership in the limit cl...
wzel 31015 The zero of a well-founded...
wzelOLD 31016 The zero of a well-founded...
wsuclem 31017 Lemma for the supremum pro...
wsuclemOLD 31018 Obsolete version of ~ wsuc...
wsucex 31019 Existence theorem for well...
wsuccl 31020 If ` X ` is a set with an ...
wsuclb 31021 A well-founded successor i...
wlimss 31022 The class of limit points ...
frr3g 31023 Functions defined by found...
frrlem1 31024 Lemma for founded recursio...
frrlem2 31025 Lemma for founded recursio...
frrlem3 31026 Lemma for founded recursio...
frrlem4 31027 Lemma for founded recursio...
frrlem5 31028 Lemma for founded recursio...
frrlem5b 31029 Lemma for founded recursio...
frrlem5c 31030 Lemma for founded recursio...
frrlem5d 31031 Lemma for founded recursio...
frrlem5e 31032 Lemma for founded recursio...
frrlem6 31033 Lemma for founded recursio...
frrlem7 31034 Lemma for founded recursio...
frrlem10 31035 Lemma for founded recursio...
frrlem11 31036 Lemma for founded recursio...
elno 31043 Membership in the surreals...
sltval 31044 The value of the surreal l...
bdayval 31045 The value of the birthday ...
nofun 31046 A surreal is a function. ...
nodmon 31047 The domain of a surreal is...
norn 31048 The range of a surreal is ...
nofnbday 31049 A surreal is a function ov...
nodmord 31050 The domain of a surreal ha...
elno2 31051 An alternative condition f...
elno3 31052 Another condition for memb...
sltval2 31053 Alternate expression for s...
nofv 31054 The function value of a su...
nosgnn0 31055 ` (/) ` is not a surreal s...
nosgnn0i 31056 If ` X ` is a surreal sign...
noreson 31057 The restriction of a surre...
sltsgn1 31058 If ` A
sltsgn2 31059 If ` A
sltintdifex 31060 If ` A
sltres 31061 If the restrictions of two...
noxpsgn 31062 The Cartesian product of a...
noxp1o 31063 The Cartesian product of a...
noxp2o 31064 The Cartesian product of a...
noseponlem 31065 Lemma for ~ nosepon . Con...
nosepon 31066 Given two unequal surreals...
sltsolem1 31067 Lemma for ~ sltso . The s...
sltso 31068 Surreal less than totally ...
sltirr 31069 Surreal less than is irref...
slttr 31070 Surreal less than is trans...
sltasym 31071 Surreal less than is asymm...
slttri 31072 Surreal less than obeys tr...
slttrieq2 31073 Trichotomy law for surreal...
bdayfo 31074 The birthday function maps...
bdayfun 31075 The birthday function is a...
bdayrn 31076 The birthday function's ra...
bdaydm 31077 The birthday function's do...
bdayfn 31078 The birthday function is a...
bdayelon 31079 The value of the birthday ...
noprc 31080 The surreal numbers are a ...
fvnobday 31081 The value of a surreal at ...
nodenselem3 31082 Lemma for ~ nodense . If ...
nodenselem4 31083 Lemma for ~ nodense . Sho...
nodenselem5 31084 Lemma for ~ nodense . If ...
nodenselem6 31085 The restriction of a surre...
nodenselem7 31086 Lemma for ~ nodense . ` A ...
nodenselem8 31087 Lemma for ~ nodense . Giv...
nodense 31088 Given two distinct surreal...
nocvxminlem 31089 Lemma for ~ nocvxmin . Gi...
nocvxmin 31090 Given a nonempty convex cl...
nobndlem1 31091 Lemma for ~ nobndup and ~ ...
nobndlem2 31092 Lemma for ~ nobndup and ~ ...
nobndlem3 31093 Lemma for ~ nobndup and ~ ...
nobndlem4 31094 Lemma for ~ nobndup and ~ ...
nobndlem5 31095 Lemma for ~ nobndup and ~ ...
nobndlem6 31096 Lemma for ~ nobndup and ~ ...
nobndlem7 31097 Lemma for ~ nobndup and ~ ...
nobndlem8 31098 Lemma for ~ nobndup and ~ ...
nobndup 31099 Any set of surreals is bou...
nobnddown 31100 Any set of surreals is bou...
nofulllem1 31101 Lemma for nofull (future) ...
nofulllem2 31102 Lemma for nofull (future) ...
nofulllem3 31103 Lemma for nofull (future) ...
nofulllem4 31104 Lemma for nofull (future) ...
nofulllem5 31105 Lemma for nofull (future) ...
brv 31154 The binary relationship ov...
txpss3v 31155 A tail Cartesian product i...
txprel 31156 A tail Cartesian product i...
brtxp 31157 Characterize a trinary rel...
brtxp2 31158 The binary relationship ov...
dfpprod2 31159 Expanded definition of par...
pprodcnveq 31160 A converse law for paralle...
pprodss4v 31161 The parallel product is a ...
brpprod 31162 Characterize a quatary rel...
brpprod3a 31163 Condition for parallel pro...
brpprod3b 31164 Condition for parallel pro...
relsset 31165 The subset class is a rela...
brsset 31166 For sets, the ` SSet ` bin...
idsset 31167 ` _I ` is equal to ` SSet ...
eltrans 31168 Membership in the class of...
dfon3 31169 A quantifier-free definiti...
dfon4 31170 Another quantifier-free de...
brtxpsd 31171 Expansion of a common form...
brtxpsd2 31172 Another common abbreviatio...
brtxpsd3 31173 A third common abbreviatio...
relbigcup 31174 The ` Bigcup ` relationshi...
brbigcup 31175 Binary relationship over `...
dfbigcup2 31176 ` Bigcup ` using maps-to n...
fobigcup 31177 ` Bigcup ` maps the univer...
fnbigcup 31178 ` Bigcup ` is a function o...
fvbigcup 31179 For sets, ` Bigcup ` yield...
elfix 31180 Membership in the fixpoint...
elfix2 31181 Alternative membership in ...
dffix2 31182 The fixpoints of a class i...
fixssdm 31183 The fixpoints of a class a...
fixssrn 31184 The fixpoints of a class a...
fixcnv 31185 The fixpoints of a class a...
fixun 31186 The fixpoint operator dist...
ellimits 31187 Membership in the class of...
limitssson 31188 The class of all limit ord...
dfom5b 31189 A quantifier-free definiti...
sscoid 31190 A condition for subset and...
dffun10 31191 Another potential definiti...
elfuns 31192 Membership in the class of...
elfunsg 31193 Closed form of ~ elfuns . ...
brsingle 31194 The binary relationship fo...
elsingles 31195 Membership in the class of...
fnsingle 31196 The singleton relationship...
fvsingle 31197 The value of the singleton...
dfsingles2 31198 Alternate definition of th...
snelsingles 31199 A singleton is a member of...
dfiota3 31200 A definiton of iota using ...
dffv5 31201 Another quantifier free de...
unisnif 31202 Express union of singleton...
brimage 31203 Binary relationship form o...
brimageg 31204 Closed form of ~ brimage ....
funimage 31205 ` Image A ` is a function....
fnimage 31206 ` Image R ` is a function ...
imageval 31207 The image functor in maps-...
fvimage 31208 The value of the image fun...
brcart 31209 Binary relationship form o...
brdomain 31210 The binary relationship fo...
brrange 31211 The binary relationship fo...
brdomaing 31212 Closed form of ~ brdomain ...
brrangeg 31213 Closed form of ~ brrange ....
brimg 31214 The binary relationship fo...
brapply 31215 The binary relationship fo...
brcup 31216 Binary relationship form o...
brcap 31217 Binary relationship form o...
brsuccf 31218 Binary relationship form o...
funpartlem 31219 Lemma for ~ funpartfun . ...
funpartfun 31220 The functional part of ` F...
funpartss 31221 The functional part of ` F...
funpartfv 31222 The function value of the ...
fullfunfnv 31223 The full functional part o...
fullfunfv 31224 The function value of the ...
brfullfun 31225 A binary relationship form...
brrestrict 31226 The binary relationship fo...
dfrecs2 31227 A quantifier-free definiti...
dfrdg4 31228 A quantifier-free definiti...
dfint3 31229 Quantifier-free definition...
imagesset 31230 The Image functor applied ...
brub 31231 Binary relationship form o...
brlb 31232 Binary relationship form o...
altopex 31237 Alternative ordered pairs ...
altopthsn 31238 Two alternate ordered pair...
altopeq12 31239 Equality for alternate ord...
altopeq1 31240 Equality for alternate ord...
altopeq2 31241 Equality for alternate ord...
altopth1 31242 Equality of the first memb...
altopth2 31243 Equality of the second mem...
altopthg 31244 Alternate ordered pair the...
altopthbg 31245 Alternate ordered pair the...
altopth 31246 The alternate ordered pair...
altopthb 31247 Alternate ordered pair the...
altopthc 31248 Alternate ordered pair the...
altopthd 31249 Alternate ordered pair the...
altxpeq1 31250 Equality for alternate Car...
altxpeq2 31251 Equality for alternate Car...
elaltxp 31252 Membership in alternate Ca...
altopelaltxp 31253 Alternate ordered pair mem...
altxpsspw 31254 An inclusion rule for alte...
altxpexg 31255 The alternate Cartesian pr...
rankaltopb 31256 Compute the rank of an alt...
nfaltop 31257 Bound-variable hypothesis ...
sbcaltop 31258 Distribution of class subs...
cgrrflx2d 31261 Deduction form of ~ axcgrr...
cgrtr4d 31262 Deduction form of ~ axcgrt...
cgrtr4and 31263 Deduction form of ~ axcgrt...
cgrrflx 31264 Reflexivity law for congru...
cgrrflxd 31265 Deduction form of ~ cgrrfl...
cgrcomim 31266 Congruence commutes on the...
cgrcom 31267 Congruence commutes betwee...
cgrcomand 31268 Deduction form of ~ cgrcom...
cgrtr 31269 Transitivity law for congr...
cgrtrand 31270 Deduction form of ~ cgrtr ...
cgrtr3 31271 Transitivity law for congr...
cgrtr3and 31272 Deduction form of ~ cgrtr3...
cgrcoml 31273 Congruence commutes on the...
cgrcomr 31274 Congruence commutes on the...
cgrcomlr 31275 Congruence commutes on bot...
cgrcomland 31276 Deduction form of ~ cgrcom...
cgrcomrand 31277 Deduction form of ~ cgrcom...
cgrcomlrand 31278 Deduction form of ~ cgrcom...
cgrtriv 31279 Degenerate segments are co...
cgrid2 31280 Identity law for congruenc...
cgrdegen 31281 Two congruent segments are...
brofs 31282 Binary relationship form o...
5segofs 31283 Rephrase ~ ax5seg using th...
ofscom 31284 The outer five segment pre...
cgrextend 31285 Link congruence over a pai...
cgrextendand 31286 Deduction form of ~ cgrext...
segconeq 31287 Two points that satsify th...
segconeu 31288 Existential uniqueness ver...
btwntriv2 31289 Betweenness always holds f...
btwncomim 31290 Betweenness commutes. Imp...
btwncom 31291 Betweenness commutes. (Co...
btwncomand 31292 Deduction form of ~ btwnco...
btwntriv1 31293 Betweenness always holds f...
btwnswapid 31294 If you can swap the first ...
btwnswapid2 31295 If you can swap arguments ...
btwnintr 31296 Inner transitivity law for...
btwnexch3 31297 Exchange the first endpoin...
btwnexch3and 31298 Deduction form of ~ btwnex...
btwnouttr2 31299 Outer transitivity law for...
btwnexch2 31300 Exchange the outer point o...
btwnouttr 31301 Outer transitivity law for...
btwnexch 31302 Outer transitivity law for...
btwnexchand 31303 Deduction form of ~ btwnex...
btwndiff 31304 There is always a ` c ` di...
trisegint 31305 A line segment between two...
funtransport 31308 The ` TransportTo ` relati...
fvtransport 31309 Calculate the value of the...
transportcl 31310 Closure law for segment tr...
transportprops 31311 Calculate the defining pro...
brifs 31320 Binary relationship form o...
ifscgr 31321 Inner five segment congrue...
cgrsub 31322 Removing identical parts f...
brcgr3 31323 Binary relationship form o...
cgr3permute3 31324 Permutation law for three-...
cgr3permute1 31325 Permutation law for three-...
cgr3permute2 31326 Permutation law for three-...
cgr3permute4 31327 Permutation law for three-...
cgr3permute5 31328 Permutation law for three-...
cgr3tr4 31329 Transitivity law for three...
cgr3com 31330 Commutativity law for thre...
cgr3rflx 31331 Identity law for three-pla...
cgrxfr 31332 A line segment can be divi...
btwnxfr 31333 A condition for extending ...
colinrel 31334 Colinearity is a relations...
brcolinear2 31335 Alternate colinearity bina...
brcolinear 31336 The binary relationship fo...
colinearex 31337 The colinear predicate exi...
colineardim1 31338 If ` A ` is colinear with ...
colinearperm1 31339 Permutation law for coline...
colinearperm3 31340 Permutation law for coline...
colinearperm2 31341 Permutation law for coline...
colinearperm4 31342 Permutation law for coline...
colinearperm5 31343 Permutation law for coline...
colineartriv1 31344 Trivial case of colinearit...
colineartriv2 31345 Trivial case of colinearit...
btwncolinear1 31346 Betweenness implies coline...
btwncolinear2 31347 Betweenness implies coline...
btwncolinear3 31348 Betweenness implies coline...
btwncolinear4 31349 Betweenness implies coline...
btwncolinear5 31350 Betweenness implies coline...
btwncolinear6 31351 Betweenness implies coline...
colinearxfr 31352 Transfer law for colineari...
lineext 31353 Extend a line with a missi...
brofs2 31354 Change some conditions for...
brifs2 31355 Change some conditions for...
brfs 31356 Binary relationship form o...
fscgr 31357 Congruence law for the gen...
linecgr 31358 Congruence rule for lines....
linecgrand 31359 Deduction form of ~ linecg...
lineid 31360 Identity law for points on...
idinside 31361 Law for finding a point in...
endofsegid 31362 If ` A ` , ` B ` , and ` C...
endofsegidand 31363 Deduction form of ~ endofs...
btwnconn1lem1 31364 Lemma for ~ btwnconn1 . T...
btwnconn1lem2 31365 Lemma for ~ btwnconn1 . N...
btwnconn1lem3 31366 Lemma for ~ btwnconn1 . E...
btwnconn1lem4 31367 Lemma for ~ btwnconn1 . A...
btwnconn1lem5 31368 Lemma for ~ btwnconn1 . N...
btwnconn1lem6 31369 Lemma for ~ btwnconn1 . N...
btwnconn1lem7 31370 Lemma for ~ btwnconn1 . U...
btwnconn1lem8 31371 Lemma for ~ btwnconn1 . N...
btwnconn1lem9 31372 Lemma for ~ btwnconn1 . N...
btwnconn1lem10 31373 Lemma for ~ btwnconn1 . N...
btwnconn1lem11 31374 Lemma for ~ btwnconn1 . N...
btwnconn1lem12 31375 Lemma for ~ btwnconn1 . U...
btwnconn1lem13 31376 Lemma for ~ btwnconn1 . B...
btwnconn1lem14 31377 Lemma for ~ btwnconn1 . F...
btwnconn1 31378 Connectitivy law for betwe...
btwnconn2 31379 Another connectivity law f...
btwnconn3 31380 Inner connectivity law for...
midofsegid 31381 If two points fall in the ...
segcon2 31382 Generalization of ~ axsegc...
brsegle 31385 Binary relationship form o...
brsegle2 31386 Alternate characterization...
seglecgr12im 31387 Substitution law for segme...
seglecgr12 31388 Substitution law for segme...
seglerflx 31389 Segment comparison is refl...
seglemin 31390 Any segment is at least as...
segletr 31391 Segment less than is trans...
segleantisym 31392 Antisymmetry law for segme...
seglelin 31393 Linearity law for segment ...
btwnsegle 31394 If ` B ` falls between ` A...
colinbtwnle 31395 Given three colinear point...
broutsideof 31398 Binary relationship form o...
broutsideof2 31399 Alternate form of ` Outsid...
outsidene1 31400 Outsideness implies inequa...
outsidene2 31401 Outsideness implies inequa...
btwnoutside 31402 A principle linking outsid...
broutsideof3 31403 Characterization of outsid...
outsideofrflx 31404 Reflexitivity of outsidene...
outsideofcom 31405 Commutitivity law for outs...
outsideoftr 31406 Transitivity law for outsi...
outsideofeq 31407 Uniqueness law for ` Outsi...
outsideofeu 31408 Given a non-degenerate ray...
outsidele 31409 Relate ` OutsideOf ` to ` ...
outsideofcol 31410 Outside of implies colinea...
funray 31417 Show that the ` Ray ` rela...
fvray 31418 Calculate the value of the...
funline 31419 Show that the ` Line ` rel...
linedegen 31420 When ` Line ` is applied w...
fvline 31421 Calculate the value of the...
liness 31422 A line is a subset of the ...
fvline2 31423 Alternate definition of a ...
lineunray 31424 A line is composed of a po...
lineelsb2 31425 If ` S ` lies on ` P Q ` ,...
linerflx1 31426 Reflexivity law for line m...
linecom 31427 Commutativity law for line...
linerflx2 31428 Reflexivity law for line m...
ellines 31429 Membership in the set of a...
linethru 31430 If ` A ` is a line contain...
hilbert1.1 31431 There is a line through an...
hilbert1.2 31432 There is at most one line ...
linethrueu 31433 There is a unique line goi...
lineintmo 31434 Two distinct lines interse...
fwddifval 31439 Calculate the value of the...
fwddifnval 31440 The value of the forward d...
fwddifn0 31441 The value of the n-iterate...
fwddifnp1 31442 The value of the n-iterate...
rankung 31443 The rank of the union of t...
ranksng 31444 The rank of a singleton. ...
rankelg 31445 The membership relation is...
rankpwg 31446 The rank of a power set. ...
rank0 31447 The rank of the empty set ...
rankeq1o 31448 The only set with rank ` 1...
elhf 31451 Membership in the heredita...
elhf2 31452 Alternate form of membersh...
elhf2g 31453 Hereditarily finiteness vi...
0hf 31454 The empty set is a heredit...
hfun 31455 The union of two HF sets i...
hfsn 31456 The singleton of an HF set...
hfadj 31457 Adjoining one HF element t...
hfelhf 31458 Any member of an HF set is...
hftr 31459 The class of all hereditar...
hfext 31460 Extensionality for HF sets...
hfuni 31461 The union of an HF set is ...
hfpw 31462 The power class of an HF s...
hfninf 31463 ` _om ` is not hereditaril...
a1i14 31464 Add two antecedents to a w...
a1i24 31465 Add two antecedents to a w...
exp5d 31466 An exportation inference. ...
exp5g 31467 An exportation inference. ...
exp5k 31468 An exportation inference. ...
exp56 31469 An exportation inference. ...
exp58 31470 An exportation inference. ...
exp510 31471 An exportation inference. ...
exp511 31472 An exportation inference. ...
exp512 31473 An exportation inference. ...
3com12d 31474 Commutation in consequent....
imp5p 31475 A triple importation infer...
imp5q 31476 A triple importation infer...
ecase13d 31477 Deduction for elimination ...
subtr 31478 Transitivity of implicit s...
subtr2 31479 Transitivity of implicit s...
trer 31480 A relation intersected wit...
elicc3 31481 An equivalent membership c...
finminlem 31482 A useful lemma about finit...
gtinf 31483 Any number greater than an...
gtinfOLD 31484 Any number greater than an...
opnrebl 31485 A set is open in the stand...
opnrebl2 31486 A set is open in the stand...
nn0prpwlem 31487 Lemma for ~ nn0prpw . Use...
nn0prpw 31488 Two nonnegative integers a...
topbnd 31489 Two equivalent expressions...
opnbnd 31490 A set is open iff it is di...
cldbnd 31491 A set is closed iff it con...
ntruni 31492 A union of interiors is a ...
clsun 31493 A pairwise union of closur...
clsint2 31494 The closure of an intersec...
opnregcld 31495 A set is regularly closed ...
cldregopn 31496 A set if regularly open if...
neiin 31497 Two neighborhoods intersec...
hmeoclda 31498 Homeomorphisms preserve cl...
hmeocldb 31499 Homeomorphisms preserve cl...
ivthALT 31500 An alternate proof of the ...
fnerel 31503 Fineness is a relation. (...
isfne 31504 The predicate " ` B ` is f...
isfne4 31505 The predicate " ` B ` is f...
isfne4b 31506 A condition for a topology...
isfne2 31507 The predicate " ` B ` is f...
isfne3 31508 The predicate " ` B ` is f...
fnebas 31509 A finer cover covers the s...
fnetg 31510 A finer cover generates a ...
fnessex 31511 If ` B ` is finer than ` A...
fneuni 31512 If ` B ` is finer than ` A...
fneint 31513 If a cover is finer than a...
fness 31514 A cover is finer than its ...
fneref 31515 Reflexivity of the finenes...
fnetr 31516 Transitivity of the finene...
fneval 31517 Two covers are finer than ...
fneer 31518 Fineness intersected with ...
topfne 31519 Fineness for covers corres...
topfneec 31520 A cover is equivalent to a...
topfneec2 31521 A topology is precisely id...
fnessref 31522 A cover is finer iff it ha...
refssfne 31523 A cover is a refinement if...
neibastop1 31524 A collection of neighborho...
neibastop2lem 31525 Lemma for ~ neibastop2 . ...
neibastop2 31526 In the topology generated ...
neibastop3 31527 The topology generated by ...
topmtcl 31528 The meet of a collection o...
topmeet 31529 Two equivalent formulation...
topjoin 31530 Two equivalent formulation...
fnemeet1 31531 The meet of a collection o...
fnemeet2 31532 The meet of equivalence cl...
fnejoin1 31533 Join of equivalence classe...
fnejoin2 31534 Join of equivalence classe...
fgmin 31535 Minimality property of a g...
neifg 31536 The neighborhood filter of...
tailfval 31537 The tail function for a di...
tailval 31538 The tail of an element in ...
eltail 31539 An element of a tail. (Co...
tailf 31540 The tail function of a dir...
tailini 31541 A tail contains its initia...
tailfb 31542 The collection of tails of...
filnetlem1 31543 Lemma for ~ filnet . Chan...
filnetlem2 31544 Lemma for ~ filnet . The ...
filnetlem3 31545 Lemma for ~ filnet . (Con...
filnetlem4 31546 Lemma for ~ filnet . (Con...
filnet 31547 A filter has the same conv...
tb-ax1 31548 The first of three axioms ...
tb-ax2 31549 The second of three axioms...
tb-ax3 31550 The third of three axioms ...
tbsyl 31551 The weak syllogism from Ta...
re1ax2lem 31552 Lemma for ~ re1ax2 . (Con...
re1ax2 31553 ~ ax-2 rederived from the ...
naim1 31554 Constructor theorem for ` ...
naim2 31555 Constructor theorem for ` ...
naim1i 31556 Constructor rule for ` -/\...
naim2i 31557 Constructor rule for ` -/\...
naim12i 31558 Constructor rule for ` -/\...
nabi1 31559 Constructor theorem for ` ...
nabi2 31560 Constructor theorem for ` ...
nabi1i 31561 Constructor rule for ` -/\...
nabi2i 31562 Constructor rule for ` -/\...
nabi12i 31563 Constructor rule for ` -/\...
df3nandALT1 31566 The double nand expressed ...
df3nandALT2 31567 The double nand expressed ...
andnand1 31568 Double and in terms of dou...
imnand2 31569 An ` -> ` nand relation. ...
allt 31570 For all sets, ` T. ` is tr...
alnof 31571 For all sets, ` F. ` is no...
nalf 31572 Not all sets hold ` F. ` a...
extt 31573 There exists a set that ho...
nextnt 31574 There does not exist a set...
nextf 31575 There does not exist a set...
unnf 31576 There does not exist exact...
unnt 31577 There does not exist exact...
mont 31578 There does not exist at mo...
mof 31579 There exist at most one se...
meran1 31580 A single axiom for proposi...
meran2 31581 A single axiom for proposi...
meran3 31582 A single axiom for proposi...
waj-ax 31583 A single axiom for proposi...
lukshef-ax2 31584 A single axiom for proposi...
arg-ax 31585 ? (Contributed by Anthony...
negsym1 31586 In the paper "On Variable ...
imsym1 31587 A symmetry with ` -> ` . ...
bisym1 31588 A symmetry with ` <-> ` . ...
consym1 31589 A symmetry with ` /\ ` . ...
dissym1 31590 A symmetry with ` \/ ` . ...
nandsym1 31591 A symmetry with ` -/\ ` . ...
unisym1 31592 A symmetry with ` A. ` . ...
exisym1 31593 A symmetry with ` E. ` . ...
unqsym1 31594 A symmetry with ` E! ` . ...
amosym1 31595 A symmetry with ` E* ` . ...
subsym1 31596 A symmetry with ` [ x / y ...
ontopbas 31597 An ordinal number is a top...
onsstopbas 31598 The class of ordinal numbe...
onpsstopbas 31599 The class of ordinal numbe...
ontgval 31600 The topology generated fro...
ontgsucval 31601 The topology generated fro...
onsuctop 31602 A successor ordinal number...
onsuctopon 31603 One of the topologies on a...
ordtoplem 31604 Membership of the class of...
ordtop 31605 An ordinal is a topology i...
onsucconi 31606 A successor ordinal number...
onsuccon 31607 A successor ordinal number...
ordtopcon 31608 An ordinal topology is con...
onintopsscon 31609 An ordinal topology is con...
onsuct0 31610 A successor ordinal number...
ordtopt0 31611 An ordinal topology is T_0...
onsucsuccmpi 31612 The successor of a success...
onsucsuccmp 31613 The successor of a success...
limsucncmpi 31614 The successor of a limit o...
limsucncmp 31615 The successor of a limit o...
ordcmp 31616 An ordinal topology is com...
ssoninhaus 31617 The ordinal topologies ` 1...
onint1 31618 The ordinal T_1 spaces are...
oninhaus 31619 The ordinal Hausdorff spac...
fveleq 31620 Please add description her...
findfvcl 31621 Please add description her...
findreccl 31622 Please add description her...
findabrcl 31623 Please add description her...
nnssi2 31624 Convert a theorem for real...
nnssi3 31625 Convert a theorem for real...
nndivsub 31626 Please add description her...
nndivlub 31627 A factor of a positive int...
ee7.2aOLD 31630 Lemma for Euclid's Element...
dnival 31631 Value of the "distance to ...
dnicld1 31632 Closure theorem for the "d...
dnicld2 31633 Closure theorem for the "d...
dnif 31634 The "distance to nearest i...
dnizeq0 31635 The distance to nearest in...
dnizphlfeqhlf 31636 The distance to nearest in...
rddif2 31637 Variant of ~ rddif . (Con...
dnibndlem1 31638 Lemma for ~ dnibnd . (Con...
dnibndlem2 31639 Lemma for ~ dnibnd . (Con...
dnibndlem3 31640 Lemma for ~ dnibnd . (Con...
dnibndlem4 31641 Lemma for ~ dnibnd . (Con...
dnibndlem5 31642 Lemma for ~ dnibnd . (Con...
dnibndlem6 31643 Lemma for ~ dnibnd . (Con...
dnibndlem7 31644 Lemma for ~ dnibnd . (Con...
dnibndlem8 31645 Lemma for ~ dnibnd . (Con...
dnibndlem9 31646 Lemma for ~ dnibnd . (Con...
dnibndlem10 31647 Lemma for ~ dnibnd . (Con...
dnibndlem11 31648 Lemma for ~ dnibnd . (Con...
dnibndlem12 31649 Lemma for ~ dnibnd . (Con...
dnibndlem13 31650 Lemma for ~ dnibnd . (Con...
dnibnd 31651 The "distance to nearest i...
dnicn 31652 The "distance to nearest i...
knoppcnlem1 31653 Lemma for ~ knoppcn . (Co...
knoppcnlem2 31654 Lemma for ~ knoppcn . (Co...
knoppcnlem3 31655 Lemma for ~ knoppcn . (Co...
knoppcnlem4 31656 Lemma for ~ knoppcn . (Co...
knoppcnlem5 31657 Lemma for ~ knoppcn . (Co...
knoppcnlem6 31658 Lemma for ~ knoppcn . (Co...
knoppcnlem7 31659 Lemma for ~ knoppcn . (Co...
knoppcnlem8 31660 Lemma for ~ knoppcn . (Co...
knoppcnlem9 31661 Lemma for ~ knoppcn . (Co...
knoppcnlem10 31662 Lemma for ~ knoppcn . (Co...
knoppcnlem11 31663 Lemma for ~ knoppcn . (Co...
knoppcn 31664 The continuous nowhere dif...
knoppcld 31665 Closure theorem for Knopp'...
addgtge0d 31666 Addition of positive and n...
unblimceq0lem 31667 Lemma for ~ unblimceq0 . ...
unblimceq0 31668 If ` F ` is unbounded near...
unbdqndv1 31669 If the difference quotient...
unbdqndv2lem1 31670 Lemma for ~ unbdqndv2 . (...
unbdqndv2lem2 31671 Lemma for ~ unbdqndv2 . (...
unbdqndv2 31672 Variant of ~ unbdqndv1 wit...
knoppndvlem1 31673 Lemma for ~ knoppndv . (C...
knoppndvlem2 31674 Lemma for ~ knoppndv . (C...
knoppndvlem3 31675 Lemma for ~ knoppndv . (C...
knoppndvlem4 31676 Lemma for ~ knoppndv . (C...
knoppndvlem5 31677 Lemma for ~ knoppndv . (C...
knoppndvlem6 31678 Lemma for ~ knoppndv . (C...
knoppndvlem7 31679 Lemma for ~ knoppndv . (C...
knoppndvlem8 31680 Lemma for ~ knoppndv . (C...
knoppndvlem9 31681 Lemma for ~ knoppndv . (C...
knoppndvlem10 31682 Lemma for ~ knoppndv . (C...
knoppndvlem11 31683 Lemma for ~ knoppndv . (C...
knoppndvlem12 31684 Lemma for ~ knoppndv . (C...
knoppndvlem13 31685 Lemma for ~ knoppndv . (C...
knoppndvlem14 31686 Lemma for ~ knoppndv . (C...
knoppndvlem15 31687 Lemma for ~ knoppndv . (C...
knoppndvlem16 31688 Lemma for ~ knoppndv . (C...
knoppndvlem17 31689 Lemma for ~ knoppndv . (C...
knoppndvlem18 31690 Lemma for ~ knoppndv . (C...
knoppndvlem19 31691 Lemma for ~ knoppndv . (C...
knoppndvlem20 31692 Lemma for ~ knoppndv . (C...
knoppndvlem21 31693 Lemma for ~ knoppndv . (C...
knoppndvlem22 31694 Lemma for ~ knoppndv . (C...
knoppndv 31695 The continuous nowhere dif...
knoppf 31696 Knopp's function is a func...
knoppcn2 31697 Variant of ~ knoppcn with ...
cnndvlem1 31698 Lemma for ~ cnndv . (Cont...
cnndvlem2 31699 Lemma for ~ cnndv . (Cont...
cnndv 31700 There exists a continuous ...
bj-mp2c 31701 A double modus ponens infe...
bj-mp2d 31702 A double modus ponens infe...
bj-0 31703 A syntactic theorem. See ...
bj-1 31704 In this proof, the use of ...
bj-a1k 31705 Weakening of ~ ax-1 . Thi...
bj-jarri 31706 Inference associated with ...
bj-jarrii 31707 Inference associated with ...
bj-imim2ALT 31708 More direct proof of ~ imi...
bj-imim21 31709 The propositional function...
bj-imim21i 31710 Inference associated with ...
bj-orim2 31711 Proof of ~ orim2 from the ...
bj-curry 31712 A non-intuitionistic posit...
bj-peirce 31713 Proof of ~ peirce from min...
bj-currypeirce 31714 Curry's axiom (a non-intui...
bj-peircecurry 31715 Peirce's axiom ~ peirce im...
pm4.81ALT 31716 Alternate proof of ~ pm4.8...
bj-con4iALT 31717 Alternate proof of ~ con4i...
bj-con2com 31718 A commuted form of the con...
bj-con2comi 31719 Inference associated with ...
bj-pm2.01i 31720 Inference associated with ...
bj-nimn 31721 If a formula is true, then...
bj-nimni 31722 Inference associated with ...
bj-peircei 31723 Inference associated with ...
bj-looinvi 31724 Inference associated with ...
bj-looinvii 31725 Inference associated with ...
bj-jaoi1 31726 Shortens 11 proofs by a to...
bj-jaoi2 31727 Shortens 9 proofs by a tot...
bj-dfbi4 31728 Alternate definition of th...
bj-dfbi5 31729 Alternate definition of th...
bj-dfbi6 31730 Alternate definition of th...
bj-bijust0 31731 The general statement that...
bj-consensus 31732 Version of ~ consensus exp...
bj-consensusALT 31733 Alternate proof of ~ bj-co...
bj-dfifc2 31734 This should be the alterna...
bj-df-ifc 31735 The definition of "ifc" if...
bj-ififc 31736 A theorem linking ` if- ` ...
sylancl2 31737 Shortens 5 proofs. (Contr...
sylancl3 31738 Shortens 11 proofs by a to...
bj-imbi12 31739 Imported form (uncurried f...
bj-trut 31740 A proposition is equivalen...
bj-biorfi 31741 This should be labeled "bi...
bj-falor 31742 Dual of ~ truan (which has...
bj-falor2 31743 Dual of ~ truan . (Contri...
bj-bibibi 31744 A property of the bicondit...
bj-imn3ani 31745 Duplication of ~ bnj1224 ....
bj-andnotim 31746 Two ways of expressing a c...
bj-bi3ant 31747 This used to be in the mai...
bj-bisym 31748 This used to be in the mai...
bj-axdd2 31749 This implication, proved u...
bj-axd2d 31750 This implication, proved u...
bj-axtd 31751 This implication, proved f...
bj-gl4lem 31752 Lemma for ~ bj-gl4 . Note...
bj-gl4 31753 In a normal modal logic, t...
bj-axc4 31754 Over minimal calculus, the...
prvlem1 31759 An elementary property of ...
prvlem2 31760 An elementary property of ...
bj-babygodel 31761 See the section header com...
bj-babylob 31762 See the section header com...
bj-godellob 31763 Proof of Gödel's theo...
bj-nf2 31766 Alternate definition of ~ ...
bj-nf3 31767 Alternate definition of ~ ...
bj-nf4 31768 Alternate definition of ~ ...
bj-nftht 31769 Closed form of ~ nfth . (...
bj-nfntht 31770 Closed form of ~ nfnth . ...
bj-nfntht2 31771 Closed form of ~ nfnth . ...
bj-nfth 31772 Any variable is not free i...
bj-nftru 31773 The true constant has no f...
bj-nfnth 31774 Any variable is not free i...
bj-nffal 31775 The false constant has no ...
bj-genr 31776 Generalization rule on the...
bj-genl 31777 Generalization rule on the...
bj-genan 31778 Generalization rule on a c...
bj-2alim 31779 Closed form of ~ 2alimi . ...
bj-2exim 31780 Closed form of ~ 2eximi . ...
bj-alanim 31781 Closed form of ~ alanimi ....
bj-2albi 31782 Closed form of ~ 2albii . ...
bj-notalbii 31783 Equivalence of universal q...
bj-2exbi 31784 Closed form of ~ 2exbii . ...
bj-3exbi 31785 Closed form of ~ 3exbii . ...
bj-sylgt2 31786 Uncurried form of ~ sylgt ...
bj-exlimh 31787 Closed form of close to ~ ...
bj-exlimh2 31788 Uncurried form of ~ bj-exl...
bj-alrimhi 31789 An inference associated wi...
bj-nexdh 31790 Closed form of ~ nexdh (an...
bj-nexdh2 31791 Uncurried form of ~ bj-nex...
bj-hbxfrbi 31792 Closed form of ~ hbxfrbi ....
bj-nfbi 31793 Closed form of ~ nfbii (wi...
bj-nfxfr 31794 Proof of ~ nfxfr from ~ bj...
bj-nfn 31795 A variable is non-free in ...
bj-exlime 31796 Variant of ~ exlimih where...
bj-exnalimn 31797 A transformation of quanti...
bj-nalnaleximiOLD 31798 An inference for distribut...
bj-nalnalimiOLD 31799 An inference for distribut...
bj-exaleximi 31800 An inference for distribut...
bj-exalimi 31801 An inference for distribut...
bj-ax12ig 31802 A lemma used to prove a we...
bj-ax12i 31803 A weakening of ~ bj-ax12ig...
bj-ax12iOLD 31804 Old proof of ~ bj-ax12i . ...
bj-ax5ea 31805 If a formula holds for som...
bj-nfv 31806 A non-occurring variable i...
bj-ax12wlem 31807 A lemma used to prove a we...
bj-ssbim 31810 Distribute substitution ov...
bj-ssbbi 31811 Biconditional property for...
bj-ssbimi 31812 Distribute substitution ov...
bj-ssbbii 31813 Biconditional property for...
bj-alsb 31814 If a proposition is true f...
bj-sbex 31815 If a proposition is true f...
bj-ssbeq 31816 Substitution in an equalit...
bj-ssb0 31817 Substitution for a variabl...
bj-ssbequ 31818 Equality property for subs...
bj-ssblem1 31819 A lemma for the definiens ...
bj-ssblem2 31820 The converse may not be pr...
bj-ssb1a 31821 One direction of a simplif...
bj-ssb1 31822 A simplified definition of...
bj-ax12 31823 A weaker form of ~ ax-12 a...
bj-ax12ssb 31824 The axiom ~ bj-ax12 expres...
bj-modal5e 31825 Dual statement of ~ hbe1 (...
bj-19.41al 31826 Special case of ~ 19.41 pr...
bj-equsexval 31827 Special case of ~ equsexv ...
bj-sb56 31828 Proof of ~ sb56 from Tarsk...
bj-dfssb2 31829 An alternate definition of...
bj-ssbn 31830 The result of a substituti...
bj-ssbft 31831 See ~ sbft . This proof i...
bj-ssbequ2 31832 Note that ~ ax-12 is used ...
bj-ssbequ1 31833 This uses ~ ax-12 with a d...
bj-ssbid2 31834 A special case of ~ bj-ssb...
bj-ssbid2ALT 31835 Alternate proof of ~ bj-ss...
bj-ssbid1 31836 A special case of ~ bj-ssb...
bj-ssbid1ALT 31837 Alternate proof of ~ bj-ss...
bj-ssbssblem 31838 Composition of two substit...
bj-ssbcom3lem 31839 Lemma for bj-ssbcom3 when ...
bj-ax6elem1 31840 Lemma for ~ bj-ax6e . (Co...
bj-ax6elem2 31841 Lemma for ~ bj-ax6e . (Co...
bj-ax6e 31842 Proof of ~ ax6e (hence ~ a...
bj-extru 31843 There exists a variable su...
bj-spimevw 31844 Existential introduction, ...
bj-spnfw 31845 Theorem close to a closed ...
bj-cbvexiw 31846 Change bound variable. Th...
bj-cbvexivw 31847 Change bound variable. Th...
bj-modald 31848 A short form of the axiom ...
bj-denot 31849 A weakening of ~ ax-6 and ...
bj-eqs 31850 A lemma for substitutions,...
bj-cbvexw 31851 Change bound variable. Th...
bj-ax12w 31852 The general statement that...
bj-elequ2g 31853 A form of ~ elequ2 with a ...
bj-ax89 31854 A theorem which could be u...
bj-elequ12 31855 An identity law for the no...
bj-cleljusti 31856 One direction of ~ cleljus...
bj-alcomexcom 31857 Commutation of universal q...
bj-hbalt 31858 Closed form of ~ hbal . W...
axc11n11 31859 Proof of ~ axc11n from { ~...
axc11n11r 31860 Proof of ~ axc11n from { ~...
bj-axc16g16 31861 Proof of ~ axc16g from { ~...
bj-ax12v3 31862 A weak version of ~ ax-12 ...
bj-ax12v3ALT 31863 Alternate proof of ~ bj-ax...
bj-sb 31864 A weak variant of ~ sbid2 ...
bj-modalbe 31865 The predicate-calculus ver...
bj-spst 31866 Closed form of ~ sps . On...
bj-19.21bit 31867 Closed form of ~ 19.21bi ....
bj-19.23bit 31868 Closed form of ~ 19.23bi ....
bj-nexrt 31869 Closed form of ~ nexr . C...
bj-alrim 31870 Closed form of ~ alrimi . ...
bj-alrim2 31871 Imported form (uncurried f...
bj-nfdt0 31872 A theorem close to a close...
bj-nfdt 31873 Closed form of ~ nf5d and ...
bj-nexdt 31874 Closed form of ~ nexd . (...
bj-nexdvt 31875 Closed form of ~ nexdv . ...
bj-19.3t 31876 Closed form of ~ 19.3 . (...
bj-alexbiex 31877 Adding a second quantifier...
bj-exexbiex 31878 Adding a second quantifier...
bj-alalbial 31879 Adding a second quantifier...
bj-exalbial 31880 Adding a second quantifier...
bj-19.9htbi 31881 Strengthening ~ 19.9ht by ...
bj-hbntbi 31882 Strengthening ~ hbnt by re...
bj-biexal1 31883 A general FOL biconditiona...
bj-biexal2 31884 A general FOL biconditiona...
bj-biexal3 31885 A general FOL biconditiona...
bj-bialal 31886 A general FOL biconditiona...
bj-biexex 31887 A general FOL biconditiona...
bj-hbext 31888 Closed form of ~ hbex . (...
bj-nfalt 31889 Closed form of ~ nfal . (...
bj-nfext 31890 Closed form of ~ nfex . (...
bj-eeanvw 31891 Version of ~ eeanv with a ...
bj-modal4e 31892 Dual statement of ~ hba1 (...
bj-modalb 31893 A short form of the axiom ...
bj-axc10 31894 Alternate (shorter) proof ...
bj-alequex 31895 A fol lemma. Can be used ...
bj-spimt2 31896 A step in the proof of ~ s...
bj-cbv3ta 31897 Closed form of ~ cbv3 . (...
bj-cbv3tb 31898 Closed form of ~ cbv3 . (...
bj-hbsb3t 31899 A theorem close to a close...
bj-hbsb3 31900 Shorter proof of ~ hbsb3 ....
bj-nfs1t 31901 A theorem close to a close...
bj-nfs1t2 31902 A theorem close to a close...
bj-nfs1 31903 Shorter proof of ~ nfs1 (t...
bj-axc10v 31904 Version of ~ axc10 with a ...
bj-spimtv 31905 Version of ~ spimt with a ...
bj-spimedv 31906 Version of ~ spimed with a...
bj-spimev 31907 Version of ~ spime with a ...
bj-spimvv 31908 Version of ~ spimv and ~ s...
bj-spimevv 31909 Version of ~ spimev with a...
bj-spvv 31910 Version of ~ spv with a dv...
bj-speiv 31911 Version of ~ spei with a d...
bj-chvarv 31912 Version of ~ chvar with a ...
bj-chvarvv 31913 Version of ~ chvarv with a...
bj-cbv3v2 31914 Version of ~ cbv3 with two...
bj-cbv3hv2 31915 Version of ~ cbv3h with tw...
bj-cbv1v 31916 Version of ~ cbv1 with a d...
bj-cbv1hv 31917 Version of ~ cbv1h with a ...
bj-cbv2hv 31918 Version of ~ cbv2h with a ...
bj-cbv2v 31919 Version of ~ cbv2 with a d...
bj-cbvalvv 31920 Version of ~ cbvalv with a...
bj-cbvexvv 31921 Version of ~ cbvexv with a...
bj-cbvaldv 31922 Version of ~ cbvald with a...
bj-cbvexdv 31923 Version of ~ cbvexd with a...
bj-cbval2v 31924 Version of ~ cbval2 with a...
bj-cbvex2v 31925 Version of ~ cbvex2 with a...
bj-cbval2vv 31926 Version of ~ cbval2v with ...
bj-cbvex2vv 31927 Version of ~ cbvex2v with ...
bj-cbvaldvav 31928 Version of ~ cbvaldva with...
bj-cbvexdvav 31929 Version of ~ cbvexdva with...
bj-cbvex4vv 31930 Version of ~ cbvex4v with ...
bj-equsalv 31931 Version of ~ equsal with a...
bj-equsalhv 31932 Version of ~ equsalh with ...
bj-axc11nv 31933 Version of ~ axc11n with a...
bj-aecomsv 31934 Version of ~ aecoms with a...
bj-axc11v 31935 Version of ~ axc11 with a ...
bj-dral1v 31936 Version of ~ dral1 with a ...
bj-drex1v 31937 Version of ~ drex1 with a ...
bj-drnf1v 31938 Version of ~ drnf1 with a ...
bj-drnf2v 31939 Version of ~ drnf2 with a ...
bj-equs45fv 31940 Version of ~ equs45f with ...
bj-sb2v 31941 Version of ~ sb2 with a dv...
bj-stdpc4v 31942 Version of ~ stdpc4 with a...
bj-2stdpc4v 31943 Version of ~ 2stdpc4 with ...
bj-sb3v 31944 Version of ~ sb3 with a dv...
bj-sb4v 31945 Version of ~ sb4 with a dv...
bj-hbs1 31946 Version of ~ hbsb2 with a ...
bj-nfs1v 31947 Version of ~ nfsb2 with a ...
bj-hbsb2av 31948 Version of ~ hbsb2a with a...
bj-hbsb3v 31949 Version of ~ hbsb3 with a ...
bj-equsb1v 31950 Version of ~ equsb1 with a...
bj-sbftv 31951 Version of ~ sbft with a d...
bj-sbfv 31952 Version of ~ sbf with a dv...
bj-sbfvv 31953 Version of ~ sbf with two ...
bj-sbtv 31954 Version of ~ sbt with a dv...
bj-sb6 31955 Remove dependency on ~ ax-...
bj-sb5 31956 Remove dependency on ~ ax-...
bj-axext3 31957 Remove dependency on ~ ax-...
bj-axext4 31958 Remove dependency on ~ ax-...
bj-hbab1 31959 Remove dependency on ~ ax-...
bj-nfsab1 31960 Remove dependency on ~ ax-...
bj-abeq2 31961 Remove dependency on ~ ax-...
bj-abeq1 31962 Remove dependency on ~ ax-...
bj-abbi 31963 Remove dependency on ~ ax-...
bj-abbi2i 31964 Remove dependency on ~ ax-...
bj-abbii 31965 Remove dependency on ~ ax-...
bj-abbid 31966 Remove dependency on ~ ax-...
bj-abbidv 31967 Remove dependency on ~ ax-...
bj-abbi2dv 31968 Remove dependency on ~ ax-...
bj-abbi1dv 31969 Remove dependency on ~ ax-...
bj-abid2 31970 Remove dependency on ~ ax-...
bj-clabel 31971 Remove dependency on ~ ax-...
bj-sbab 31972 Remove dependency on ~ ax-...
bj-nfab1 31973 Remove dependency on ~ ax-...
bj-vjust 31974 Remove dependency on ~ ax-...
bj-cdeqab 31975 Remove dependency on ~ ax-...
bj-axrep1 31976 Remove dependency on ~ ax-...
bj-axrep2 31977 Remove dependency on ~ ax-...
bj-axrep3 31978 Remove dependency on ~ ax-...
bj-axrep4 31979 Remove dependency on ~ ax-...
bj-axrep5 31980 Remove dependency on ~ ax-...
bj-axsep 31981 Remove dependency on ~ ax-...
bj-nalset 31982 Remove dependency on ~ ax-...
bj-zfpow 31983 Remove dependency on ~ ax-...
bj-el 31984 Remove dependency on ~ ax-...
bj-dtru 31985 Remove dependency on ~ ax-...
bj-axc16b 31986 Remove dependency on ~ ax-...
bj-eunex 31987 Remove dependency on ~ ax-...
bj-dtrucor 31988 Remove dependency on ~ ax-...
bj-dtrucor2v 31989 Version of ~ dtrucor2 with...
bj-dvdemo1 31990 Remove dependency on ~ ax-...
bj-dvdemo2 31991 Remove dependency on ~ ax-...
bj-sb3b 31992 Simplified definition of s...
bj-hbaeb2 31993 Biconditional version of a...
bj-hbaeb 31994 Biconditional version of ~...
bj-hbnaeb 31995 Biconditional version of ~...
bj-dvv 31996 A special instance of ~ bj...
bj-equsal1t 31997 Duplication of ~ wl-equsal...
bj-equsal1ti 31998 Inference associated with ...
bj-equsal1 31999 One direction of ~ equsal ...
bj-equsal2 32000 One direction of ~ equsal ...
bj-equsal 32001 Shorter proof of ~ equsal ...
stdpc5t 32002 Closed form of ~ stdpc5 . ...
bj-stdpc5 32003 More direct proof of ~ std...
2stdpc5 32004 A double ~ stdpc5 (one dir...
bj-19.21t 32005 Proof of ~ 19.21t from ~ s...
exlimii 32006 Inference associated with ...
ax11-pm 32007 Proof of ~ ax-11 similar t...
ax6er 32008 Another form of ~ ax6e . ...
exlimiieq1 32009 Inferring a theorem when i...
exlimiieq2 32010 Inferring a theorem when i...
ax11-pm2 32011 Proof of ~ ax-11 from the ...
bj-sbsb 32012 Biconditional showing two ...
bj-dfsb2 32013 Alternate (dual) definitio...
bj-sbf3 32014 Substitution has no effect...
bj-sbf4 32015 Substitution has no effect...
bj-sbnf 32016 Move non-free predicate in...
bj-eu3f 32017 Version of ~ eu3v where th...
bj-eumo0 32018 Existential uniqueness imp...
bj-nfdiOLD 32019 Obsolete proof temporarily...
bj-sbieOLD 32020 Obsolete proof temporarily...
bj-sbidmOLD 32021 Obsolete proof temporarily...
bj-mo3OLD 32022 Obsolete proof temporarily...
bj-syl66ib 32023 A mixed syllogism inferenc...
bj-nfbiit 32024 Closed form of ~ nfbii (th...
bj-nfimt 32025 Closed form of ~ nfim . (...
bj-nfimt2 32026 Uncurried form of ~ bj-nfi...
bj-dvelimdv 32027 Deduction form of ~ dvelim...
bj-dvelimdv1 32028 Curried form (exported for...
bj-dvelimv 32029 A version of ~ dvelim usin...
bj-nfeel2 32030 Non-freeness in an equalit...
bj-axc14nf 32031 Proof of a version of ~ ax...
bj-axc14 32032 Alternate proof of ~ axc14...
eliminable1 32033 A theorem used to prove th...
eliminable2a 32034 A theorem used to prove th...
eliminable2b 32035 A theorem used to prove th...
eliminable2c 32036 A theorem used to prove th...
eliminable3a 32037 A theorem used to prove th...
eliminable3b 32038 A theorem used to prove th...
bj-termab 32039 Every class can be written...
bj-eleq1w 32040 Weaker version of ~ eleq1 ...
bj-eleq2w 32041 Weaker version of ~ eleq2 ...
bj-clelsb3 32042 Remove dependency on ~ ax-...
bj-hblem 32043 Remove dependency on ~ ax-...
bj-nfcjust 32044 Remove dependency on ~ ax-...
bj-nfcrii 32045 Remove dependency on ~ ax-...
bj-nfcri 32046 Remove dependency on ~ ax-...
bj-nfnfc 32047 Remove dependency on ~ ax-...
bj-vexwt 32048 Closed form of ~ bj-vexw ....
bj-vexw 32049 If ` ph ` is a theorem, th...
bj-vexwvt 32050 Closed form of ~ bj-vexwv ...
bj-vexwv 32051 Version of ~ bj-vexw with ...
bj-denotes 32052 This would be the justific...
bj-issetwt 32053 Closed form of ~ bj-issetw...
bj-issetw 32054 The closest one can get to...
bj-elissetv 32055 Version of ~ bj-elisset wi...
bj-elisset 32056 Remove from ~ elisset depe...
bj-issetiv 32057 Version of ~ bj-isseti wit...
bj-isseti 32058 Remove from ~ isseti depen...
bj-ralvw 32059 A weak version of ~ ralv n...
bj-rexvwv 32060 A weak version of ~ rexv n...
bj-rababwv 32061 A weak version of ~ rabab ...
bj-ralcom4 32062 Remove from ~ ralcom4 depe...
bj-rexcom4 32063 Remove from ~ rexcom4 depe...
bj-rexcom4a 32064 Remove from ~ rexcom4a dep...
bj-rexcom4bv 32065 Version of ~ bj-rexcom4b w...
bj-rexcom4b 32066 Remove from ~ rexcom4b dep...
bj-ceqsalt0 32067 The FOL content of ~ ceqsa...
bj-ceqsalt1 32068 The FOL content of ~ ceqsa...
bj-ceqsalt 32069 Remove from ~ ceqsalt depe...
bj-ceqsaltv 32070 Version of ~ bj-ceqsalt wi...
bj-ceqsalg0 32071 The FOL content of ~ ceqsa...
bj-ceqsalg 32072 Remove from ~ ceqsalg depe...
bj-ceqsalgALT 32073 Alternate proof of ~ bj-ce...
bj-ceqsalgv 32074 Version of ~ bj-ceqsalg wi...
bj-ceqsalgvALT 32075 Alternate proof of ~ bj-ce...
bj-ceqsal 32076 Remove from ~ ceqsal depen...
bj-ceqsalv 32077 Remove from ~ ceqsalv depe...
bj-spcimdv 32078 Remove from ~ spcimdv depe...
bj-nfcsym 32079 The class-form not-free pr...
bj-ax8 32080 Proof of ~ ax-8 from ~ df-...
bj-df-clel 32081 Candidate definition for ~...
bj-dfclel 32082 Characterization of the el...
bj-ax9 32083 Proof of ~ ax-9 from ~ ax-...
bj-cleqhyp 32084 The hypothesis of ~ bj-df-...
bj-df-cleq 32085 Candidate definition for ~...
bj-dfcleq 32086 Proof of class extensional...
bj-sbeqALT 32087 Substitution in an equalit...
bj-sbeq 32088 Distribute proper substitu...
bj-sbceqgALT 32089 Distribute proper substitu...
bj-csbsnlem 32090 Lemma for ~ bj-csbsn (in t...
bj-csbsn 32091 Substitution in a singleto...
bj-sbel1 32092 Version of ~ sbcel1g when ...
bj-abv 32093 The class of sets verifyin...
bj-ab0 32094 The class of sets verifyin...
bj-abf 32095 Shorter proof of ~ abf (wh...
bj-csbprc 32096 More direct proof of ~ csb...
bj-exlimmpi 32097 Lemma for ~ bj-vtoclg1f1 (...
bj-exlimmpbi 32098 Lemma for theorems of the ...
bj-exlimmpbir 32099 Lemma for theorems of the ...
bj-vtoclf 32100 Remove dependency on ~ ax-...
bj-vtocl 32101 Remove dependency on ~ ax-...
bj-vtoclg1f1 32102 The FOL content of ~ vtocl...
bj-vtoclg1f 32103 Reprove ~ vtoclg1f from ~ ...
bj-vtoclg1fv 32104 Version of ~ bj-vtoclg1f w...
bj-rabbida2 32105 Version of ~ rabbidva2 wit...
bj-rabbida 32106 Version of ~ rabbidva with...
bj-rabbid 32107 Version of ~ rabbidv with ...
bj-rabeqd 32108 Deduction form of ~ rabeq ...
bj-rabeqbid 32109 Version of ~ rabeqbidv wit...
bj-rabeqbida 32110 Version of ~ rabeqbidva wi...
bj-seex 32111 Version of ~ seex with a d...
bj-nfcf 32112 Version of ~ df-nfc with a...
bj-axsep2 32113 Remove dependency on ~ ax-...
bj-unrab 32114 Generalization of ~ unrab ...
bj-inrab 32115 Generalization of ~ inrab ...
bj-inrab2 32116 Shorter proof of ~ inrab ....
bj-inrab3 32117 Generalization of ~ dfrab3...
bj-rabtr 32118 Restricted class abstracti...
bj-rabtrALT 32119 Alternate proof of ~ bj-ra...
bj-rabtrALTALT 32120 Alternate proof of ~ bj-ra...
bj-rabtrAUTO 32121 Proof of ~ bj-rabtr found ...
bj-ru0 32124 The FOL part of Russell's ...
bj-ru1 32125 A version of Russell's par...
bj-ru 32126 Remove dependency on ~ ax-...
bj-n0i 32127 Inference associated with ...
bj-nel0 32128 From the general negation ...
bj-disjcsn 32129 A class is disjoint from i...
bj-disjsn01 32130 Disjointness of the single...
bj-1ex 32131 ` 1o ` is a set. (Contrib...
bj-2ex 32132 ` 2o ` is a set. (Contrib...
bj-0nel1 32133 The empty set does not bel...
bj-1nel0 32134 ` 1o ` does not belong to ...
bj-xpimasn 32135 The image of a singleton, ...
bj-xpima1sn 32136 The image of a singleton b...
bj-xpima1snALT 32137 Alternate proof of ~ bj-xp...
bj-xpima2sn 32138 The image of a singleton b...
bj-xpnzex 32139 If the first factor of a p...
bj-xpexg2 32140 Exported form (curried for...
bj-xpnzexb 32141 If the first factor of a p...
bj-cleq 32142 Substitution property for ...
bj-sels 32143 If a class is a set, then ...
bj-snsetex 32144 The class of sets "whose s...
bj-clex 32145 Sethood of certain classes...
bj-sngleq 32148 Substitution property for ...
bj-elsngl 32149 Characterization of the el...
bj-snglc 32150 Characterization of the el...
bj-snglss 32151 The singletonization of a ...
bj-0nelsngl 32152 The empty set is not a mem...
bj-snglinv 32153 Inverse of singletonizatio...
bj-snglex 32154 A class is a set if and on...
bj-tageq 32157 Substitution property for ...
bj-eltag 32158 Characterization of the el...
bj-0eltag 32159 The empty set belongs to t...
bj-tagn0 32160 The tagging of a class is ...
bj-tagss 32161 The tagging of a class is ...
bj-snglsstag 32162 The singletonization is in...
bj-sngltagi 32163 The singletonization is in...
bj-sngltag 32164 The singletonization and t...
bj-tagci 32165 Characterization of the el...
bj-tagcg 32166 Characterization of the el...
bj-taginv 32167 Inverse of tagging. (Cont...
bj-tagex 32168 A class is a set if and on...
bj-xtageq 32169 The products of a given cl...
bj-xtagex 32170 The product of a set and t...
bj-projeq 32173 Substitution property for ...
bj-projeq2 32174 Substitution property for ...
bj-projun 32175 The class projection on a ...
bj-projex 32176 Sethood of the class proje...
bj-projval 32177 Value of the class project...
bj-1upleq 32180 Substitution property for ...
bj-pr1eq 32183 Substitution property for ...
bj-pr1un 32184 The first projection prese...
bj-pr1val 32185 Value of the first project...
bj-pr11val 32186 Value of the first project...
bj-pr1ex 32187 Sethood of the first proje...
bj-1uplth 32188 The characteristic propert...
bj-1uplex 32189 A monuple is a set if and ...
bj-1upln0 32190 A monuple is nonempty. (C...
bj-2upleq 32193 Substitution property for ...
bj-pr21val 32194 Value of the first project...
bj-pr2eq 32197 Substitution property for ...
bj-pr2un 32198 The second projection pres...
bj-pr2val 32199 Value of the second projec...
bj-pr22val 32200 Value of the second projec...
bj-pr2ex 32201 Sethood of the second proj...
bj-2uplth 32202 The characteristic propert...
bj-2uplex 32203 A couple is a set if and o...
bj-2upln0 32204 A couple is nonempty. (Co...
bj-2upln1upl 32205 A couple is never equal to...
bj-vjust2 32206 Justification theorem for ...
bj-df-v 32207 Alternate definition of th...
bj-df-nul 32208 Alternate definition of th...
bj-nul 32209 Two formulations of the ax...
bj-nuliota 32210 Definition of the empty se...
bj-nuliotaALT 32211 Alternate proof of ~ bj-nu...
bj-vtoclgfALT 32212 Alternate proof of ~ vtocl...
bj-pwcfsdom 32213 Remove hypothesis from ~ p...
bj-grur1 32214 Remove hypothesis from ~ g...
bj-rest00 32215 An elementwise intersectio...
bj-restsn 32216 An elementwise intersectio...
bj-restsnss 32217 Special case of ~ bj-rests...
bj-restsnss2 32218 Special case of ~ bj-rests...
bj-restsn0 32219 An elementwise intersectio...
bj-restsn10 32220 Special case of ~ bj-rests...
bj-restsnid 32221 The elementwise intersecti...
bj-rest10 32222 An elementwise intersectio...
bj-rest10b 32223 Alternate version of ~ bj-...
bj-restn0 32224 An elementwise intersectio...
bj-restn0b 32225 Alternate version of ~ bj-...
bj-restpw 32226 The elementwise intersecti...
bj-rest0 32227 An elementwise intersectio...
bj-restb 32228 An elementwise intersectio...
bj-restv 32229 An elementwise intersectio...
bj-resta 32230 An elementwise intersectio...
bj-restuni 32231 The union of an elementwis...
bj-restuni2 32232 The union of an elementwis...
bj-restreg 32233 A reformulation of the axi...
bj-toptopon2 32234 A topology is the same thi...
bj-topontopon 32235 A topology on a set is a t...
bj-funtopon 32236 ` TopOn ` is a function. ...
bj-elpw3 32237 A variant of ~ elpwg . (C...
bj-sspwpw 32238 The union of a set is incl...
bj-sspwpwab 32239 The class of families whos...
bj-sspwpweq 32240 The class of families whos...
bj-toponss 32241 The set of topologies on a...
bj-dmtopon 32242 The domain of ` TopOn ` is...
bj-fntopon 32243 ` TopOn ` is a function wi...
bj-toprntopon 32244 A topology is the same thi...
bj-xnex 32245 Lemma for ~ snnex and ~ bj...
bj-pwnex 32246 The class of all power set...
bj-topnex 32247 The class of all topologie...
bj-0nelmpt 32250 The empty set is not an el...
bj-mptval 32251 Value of a function given ...
bj-dfmpt2a 32252 An equivalent definition o...
bj-mpt2mptALT 32253 Alternate proof of ~ mpt2m...
bj-elid 32262 Characterization of the el...
bj-elid2 32263 Characterization of the el...
bj-elid3 32264 Characterization of the el...
bj-diagval 32267 Value of the diagonal. (C...
bj-eldiag 32268 Characterization of the el...
bj-eldiag2 32269 Characterization of the el...
bj-inftyexpiinv 32272 Utility theorem for the in...
bj-inftyexpiinj 32273 Injectivity of the paramet...
bj-inftyexpidisj 32274 An element of the circle a...
bj-ccinftydisj 32277 The circle at infinity is ...
bj-elccinfty 32278 A lemma for infinite exten...
bj-ccssccbar 32281 Complex numbers are extend...
bj-ccinftyssccbar 32282 Infinite extended complex ...
bj-pinftyccb 32285 The class ` pinfty ` is an...
bj-pinftynrr 32286 The extended complex numbe...
bj-minftyccb 32289 The class ` minfty ` is an...
bj-minftynrr 32290 The extended complex numbe...
bj-pinftynminfty 32291 The extended complex numbe...
bj-rrhatsscchat 32300 The real projective line i...
bj-cmnssmnd 32313 Commutative monoids are mo...
bj-cmnssmndel 32314 Commutative monoids are mo...
bj-ablssgrp 32315 Abelian groups are groups....
bj-ablssgrpel 32316 Abelian groups are groups ...
bj-ablsscmn 32317 Abelian groups are commuta...
bj-ablsscmnel 32318 Abelian groups are commuta...
bj-modssabl 32319 (The additive groups of) m...
bj-vecssmod 32320 Vector spaces are modules....
bj-vecssmodel 32321 Vector spaces are modules ...
bj-finsumval0 32324 Value of a finite sum. (C...
bj-rrvecssvec 32327 Real vector spaces are vec...
bj-rrvecssvecel 32328 Real vector spaces are vec...
bj-rrvecsscmn 32329 (The additive groups of) r...
bj-rrvecsscmnel 32330 (The additive groups of) r...
bj-subcom 32331 A consequence of commutati...
bj-ldiv 32332 Left-division. (Contribut...
bj-rdiv 32333 Right-division. (Contribu...
bj-mdiv 32334 A division law. (Contribu...
bj-lineq 32335 Solution of a (scalar) lin...
bj-lineqi 32336 Solution of a (scalar) lin...
bj-bary1lem 32337 A lemma for barycentric co...
bj-bary1lem1 32338 Existence and uniqueness (...
bj-bary1 32339 Barycentric coordinates in...
taupilem3 32342 Lemma for tau-related theo...
taupilemrplb 32343 A set of positive reals ha...
taupilem1 32344 Lemma for ~ taupi . A pos...
taupilem2 32345 Lemma for ~ taupi . The s...
taupi 32346 Relationship between ` _ta...
csbdif 32347 Distribution of class subs...
csbpredg 32348 Move class substitution in...
csbwrecsg 32349 Move class substitution in...
csbrecsg 32350 Move class substitution in...
csbrdgg 32351 Move class substitution in...
csboprabg 32352 Move class substitution in...
csbmpt22g 32353 Move class substitution in...
mpnanrd 32354 Eliminate the right side o...
con1bii2 32355 A contraposition inference...
con2bii2 32356 A contraposition inference...
vtoclefex 32357 Implicit substitution of a...
rnmptsn 32358 The range of a function ma...
f1omptsnlem 32359 This is the core of the pr...
f1omptsn 32360 A function mapping to sing...
mptsnunlem 32361 This is the core of the pr...
mptsnun 32362 A class ` B ` is equal to ...
dissneqlem 32363 This is the core of the pr...
dissneq 32364 Any topology that contains...
exlimim 32365 Closed form of ~ exlimimd ...
exlimimd 32366 Existential elimination ru...
exlimimdd 32367 Existential elimination ru...
exellim 32368 Closed form of ~ exellimdd...
exellimddv 32369 Eliminate an antecedent wh...
topdifinfindis 32370 Part of Exercise 3 of [Mun...
topdifinffinlem 32371 This is the core of the pr...
topdifinffin 32372 Part of Exercise 3 of [Mun...
topdifinf 32373 Part of Exercise 3 of [Mun...
topdifinfeq 32374 Two different ways of defi...
icorempt2 32375 Closed-below, open-above i...
icoreresf 32376 Closed-below, open-above i...
icoreval 32377 Value of the closed-below,...
icoreelrnab 32378 Elementhood in the set of ...
isbasisrelowllem1 32379 Lemma for ~ isbasisrelowl ...
isbasisrelowllem2 32380 Lemma for ~ isbasisrelowl ...
icoreclin 32381 The set of closed-below, o...
isbasisrelowl 32382 The set of all closed-belo...
icoreunrn 32383 The union of all closed-be...
istoprelowl 32384 The set of all closed-belo...
icoreelrn 32385 A class abstraction which ...
iooelexlt 32386 An element of an open inte...
relowlssretop 32387 The lower limit topology o...
relowlpssretop 32388 The lower limit topology o...
sucneqond 32389 Inequality of an ordinal s...
sucneqoni 32390 Inequality of an ordinal s...
onsucuni3 32391 If an ordinal number has a...
1oequni2o 32392 The ordinal number ` 1o ` ...
rdgsucuni 32393 If an ordinal number has a...
rdgeqoa 32394 If a recursive function wi...
elxp8 32395 Membership in a Cartesian ...
dffinxpf 32398 This theorem is the same a...
finxpeq1 32399 Equality theorem for Carte...
finxpeq2 32400 Equality theorem for Carte...
csbfinxpg 32401 Distribute proper substitu...
finxpreclem1 32402 Lemma for ` ^^ ` recursion...
finxpreclem2 32403 Lemma for ` ^^ ` recursion...
finxp0 32404 The value of Cartesian exp...
finxp1o 32405 The value of Cartesian exp...
finxpreclem3 32406 Lemma for ` ^^ ` recursion...
finxpreclem4 32407 Lemma for ` ^^ ` recursion...
finxpreclem5 32408 Lemma for ` ^^ ` recursion...
finxpreclem6 32409 Lemma for ` ^^ ` recursion...
finxpsuclem 32410 Lemma for ~ finxpsuc . (C...
finxpsuc 32411 The value of Cartesian exp...
finxp2o 32412 The value of Cartesian exp...
finxp3o 32413 The value of Cartesian exp...
finxpnom 32414 Cartesian exponentiation w...
finxp00 32415 Cartesian exponentiation o...
wl-section-prop 32416 Intuitionistic logic is no...
wl-section-boot 32420 In this section, I provide...
wl-imim1i 32421 Inference adding common co...
wl-syl 32422 An inference version of th...
wl-syl5 32423 A syllogism rule of infere...
wl-pm2.18d 32424 Deduction based on reducti...
wl-con4i 32425 Inference rule. Copy of ~...
wl-pm2.24i 32426 Inference rule. Copy of ~...
wl-a1i 32427 Inference rule. Copy of ~...
wl-mpi 32428 A nested modus ponens infe...
wl-imim2i 32429 Inference adding common an...
wl-syl6 32430 A syllogism rule of infere...
wl-ax3 32431 ~ ax-3 proved from Lukasie...
wl-ax1 32432 ~ ax-1 proved from Lukasie...
wl-pm2.27 32433 This theorem, called "Asse...
wl-com12 32434 Inference that swaps (comm...
wl-pm2.21 32435 From a wff and its negatio...
wl-con1i 32436 A contraposition inference...
wl-ja 32437 Inference joining the ante...
wl-imim2 32438 A closed form of syllogism...
wl-a1d 32439 Deduction introducing an e...
wl-ax2 32440 ~ ax-2 proved from Lukasie...
wl-id 32441 Principle of identity. Th...
wl-notnotr 32442 Converse of double negatio...
wl-pm2.04 32443 Swap antecedents. Theorem...
wl-section-impchain 32444 An implication like ` ( ps...
wl-impchain-mp-x 32445 This series of theorems pr...
wl-impchain-mp-0 32446 This theorem is the start ...
wl-impchain-mp-1 32447 This theorem is in fact a ...
wl-impchain-mp-2 32448 This theorem is in fact a ...
wl-impchain-com-1.x 32449 It is often convenient to ...
wl-impchain-com-1.1 32450 A degenerate form of antec...
wl-impchain-com-1.2 32451 This theorem is in fact a ...
wl-impchain-com-1.3 32452 This theorem is in fact a ...
wl-impchain-com-1.4 32453 This theorem is in fact a ...
wl-impchain-com-n.m 32454 This series of theorems al...
wl-impchain-com-2.3 32455 This theorem is in fact a ...
wl-impchain-com-2.4 32456 This theorem is in fact a ...
wl-impchain-com-3.2.1 32457 This theorem is in fact a ...
wl-impchain-a1-x 32458 If an implication chain is...
wl-impchain-a1-1 32459 Inference rule, a copy of ...
wl-impchain-a1-2 32460 Inference rule, a copy of ...
wl-impchain-a1-3 32461 Inference rule, a copy of ...
wl-section-nf 32462 The current definition of ...
wl-nf-nf2 32463 By ~ ax-10 the definition ...
wl-nf2-nf 32464 ~ hba1 is sufficient to le...
wl-ax13lem1 32466 A version of ~ ax-wl-13v w...
wl-jarri 32467 Dropping a nested antecede...
wl-jarli 32468 Dropping a nested conseque...
wl-mps 32469 Replacing a nested consequ...
wl-syls1 32470 Replacing a nested consequ...
wl-syls2 32471 Replacing a nested anteced...
wl-embant 32472 A true wff can always be a...
wl-orel12 32473 In a conjunctive normal fo...
wl-cases2-dnf 32474 A particular instance of ~...
wl-dfnan2 32475 An alternative definition ...
wl-nancom 32476 The 'nand' operator commut...
wl-nannan 32477 Lemma for handling nested ...
wl-nannot 32478 Show equivalence between n...
wl-nanbi1 32479 Introduce a right anti-con...
wl-nanbi2 32480 Introduce a left anti-conj...
wl-naev 32481 If some set variables can ...
wl-hbae1 32482 This specialization of ~ h...
wl-naevhba1v 32483 An instance of ~ hbn1w app...
wl-hbnaev 32484 Any variable is free in ` ...
wl-spae 32485 Prove an instance of ~ sp ...
wl-cbv3vv 32486 Avoiding ~ ax-11 . (Contr...
wl-speqv 32487 Under the assumption ` -. ...
wl-19.8eqv 32488 Under the assumption ` -. ...
wl-19.2reqv 32489 Under the assumption ` -. ...
wl-dveeq12 32490 The current form of ~ ax-1...
wl-nfalv 32491 If ` x ` is not present in...
wl-nfimf1 32492 An antecedent is irrelevan...
wl-nfnbi 32493 Being free does not depend...
wl-nfae1 32494 Unlike ~ nfae , this speci...
wl-nfnae1 32495 Unlike ~ nfnae , this spec...
wl-aetr 32496 A transitive law for varia...
wl-dral1d 32497 A version of ~ dral1 with ...
wl-cbvalnaed 32498 ~ wl-cbvalnae with a conte...
wl-cbvalnae 32499 A more general version of ...
wl-exeq 32500 The semantics of ` E. x y ...
wl-aleq 32501 The semantics of ` A. x y ...
wl-nfeqfb 32502 Extend ~ nfeqf to an equiv...
wl-nfs1t 32503 If ` y ` is not free in ` ...
wl-equsald 32504 Deduction version of ~ equ...
wl-equsal 32505 A useful equivalence relat...
wl-equsal1t 32506 The expression ` x = y ` i...
wl-equsalcom 32507 This simple equivalence ea...
wl-equsal1i 32508 The antecedent ` x = y ` i...
wl-sb6rft 32509 A specialization of ~ wl-e...
wl-sbrimt 32510 Substitution with a variab...
wl-sblimt 32511 Substitution with a variab...
wl-sb8t 32512 Substitution of variable i...
wl-sb8et 32513 Substitution of variable i...
wl-sbhbt 32514 Closed form of ~ sbhb . C...
wl-sbnf1 32515 Two ways expressing that `...
wl-equsb3 32516 ~ equsb3 with a distinctor...
wl-equsb4 32517 Substitution applied to an...
wl-sb6nae 32518 Version of ~ sb6 suitable ...
wl-sb5nae 32519 Version of ~ sb5 suitable ...
wl-2sb6d 32520 Version of ~ 2sb6 with a c...
wl-sbcom2d-lem1 32521 Lemma used to prove ~ wl-s...
wl-sbcom2d-lem2 32522 Lemma used to prove ~ wl-s...
wl-sbcom2d 32523 Version of ~ sbcom2 with a...
wl-sbalnae 32524 A theorem used in eliminat...
wl-sbal1 32525 A theorem used in eliminat...
wl-sbal2 32526 Move quantifier in and out...
wl-lem-exsb 32527 This theorem provides a ba...
wl-lem-nexmo 32528 This theorem provides a ba...
wl-lem-moexsb 32529 The antecedent ` A. x ( ph...
wl-alanbii 32530 This theorem extends ~ ala...
wl-mo2df 32531 Version of ~ mo2 with a co...
wl-mo2tf 32532 Closed form of ~ mo2 with ...
wl-eudf 32533 Version of ~ df-eu with a ...
wl-eutf 32534 Closed form of ~ df-eu wit...
wl-euequ1f 32535 ~ euequ1 proved with a dis...
wl-mo2t 32536 Closed form of ~ mo2 . (C...
wl-mo3t 32537 Closed form of ~ mo3 . (C...
wl-sb8eut 32538 Substitution of variable i...
wl-sb8mot 32539 Substitution of variable i...
wl-ax11-lem1 32541 A transitive law for varia...
wl-ax11-lem2 32542 Lemma. (Contributed by Wo...
wl-ax11-lem3 32543 Lemma. (Contributed by Wo...
wl-ax11-lem4 32544 Lemma. (Contributed by Wo...
wl-ax11-lem5 32545 Lemma. (Contributed by Wo...
wl-ax11-lem6 32546 Lemma. (Contributed by Wo...
wl-ax11-lem7 32547 Lemma. (Contributed by Wo...
wl-ax11-lem8 32548 Lemma. (Contributed by Wo...
wl-ax11-lem9 32549 The easy part when ` x ` c...
wl-ax11-lem10 32550 We now have prepared every...
wl-sbcom3 32551 Substituting ` y ` for ` x...
rabiun 32552 Abstraction restricted to ...
iundif1 32553 Indexed union of class dif...
imadifss 32554 The difference of images i...
cureq 32555 Equality theorem for curry...
unceq 32556 Equality theorem for uncur...
curf 32557 Functional property of cur...
uncf 32558 Functional property of unc...
curfv 32559 Value of currying. (Contr...
uncov 32560 Value of uncurrying. (Con...
curunc 32561 Currying of uncurrying. (...
unccur 32562 Uncurrying of currying. (...
phpreu 32563 Theorem related to pigeonh...
finixpnum 32564 A finite Cartesian product...
fin2solem 32565 Lemma for ~ fin2so . (Con...
fin2so 32566 Any totally ordered Tarski...
ltflcei 32567 Theorem to move the floor ...
leceifl 32568 Theorem to move the floor ...
sin2h 32569 Half-angle rule for sine. ...
cos2h 32570 Half-angle rule for cosine...
tan2h 32571 Half-angle rule for tangen...
pigt3 32572 ` _pi ` is greater than 3....
lindsdom 32573 A linearly independent set...
lindsenlbs 32574 A maximal linearly indepen...
matunitlindflem1 32575 One direction of ~ matunit...
matunitlindflem2 32576 One direction of ~ matunit...
matunitlindf 32577 A matrix over a field is i...
ptrest 32578 Expressing a restriction o...
ptrecube 32579 Any point in an open set o...
poimirlem1 32580 Lemma for ~ poimir - the v...
poimirlem2 32581 Lemma for ~ poimir - conse...
poimirlem3 32582 Lemma for ~ poimir to add ...
poimirlem4 32583 Lemma for ~ poimir connect...
poimirlem5 32584 Lemma for ~ poimir to esta...
poimirlem6 32585 Lemma for ~ poimir establi...
poimirlem7 32586 Lemma for ~ poimir , simil...
poimirlem8 32587 Lemma for ~ poimir , estab...
poimirlem9 32588 Lemma for ~ poimir , estab...
poimirlem10 32589 Lemma for ~ poimir establi...
poimirlem11 32590 Lemma for ~ poimir connect...
poimirlem12 32591 Lemma for ~ poimir connect...
poimirlem13 32592 Lemma for ~ poimir - for a...
poimirlem14 32593 Lemma for ~ poimir - for a...
poimirlem15 32594 Lemma for ~ poimir , that ...
poimirlem16 32595 Lemma for ~ poimir establi...
poimirlem17 32596 Lemma for ~ poimir establi...
poimirlem18 32597 Lemma for ~ poimir stating...
poimirlem19 32598 Lemma for ~ poimir establi...
poimirlem20 32599 Lemma for ~ poimir establi...
poimirlem21 32600 Lemma for ~ poimir stating...
poimirlem22 32601 Lemma for ~ poimir , that ...
poimirlem23 32602 Lemma for ~ poimir , two w...
poimirlem24 32603 Lemma for ~ poimir , two w...
poimirlem25 32604 Lemma for ~ poimir stating...
poimirlem26 32605 Lemma for ~ poimir showing...
poimirlem27 32606 Lemma for ~ poimir showing...
poimirlem28 32607 Lemma for ~ poimir , a var...
poimirlem29 32608 Lemma for ~ poimir connect...
poimirlem30 32609 Lemma for ~ poimir combini...
poimirlem31 32610 Lemma for ~ poimir , assig...
poimirlem32 32611 Lemma for ~ poimir , combi...
poimir 32612 Poincare-Miranda theorem. ...
broucube 32613 Brouwer - or as Kulpa call...
heicant 32614 Heine-Cantor theorem: a co...
opnmbllem0 32615 Lemma for ~ ismblfin ; cou...
mblfinlem1 32616 Lemma for ~ ismblfin , ord...
mblfinlem2 32617 Lemma for ~ ismblfin , eff...
mblfinlem3 32618 The difference between two...
mblfinlem4 32619 Backward direction of ~ is...
ismblfin 32620 Measurability in terms of ...
ovoliunnfl 32621 ~ ovoliun is incompatible ...
ex-ovoliunnfl 32622 Demonstration of ~ ovoliun...
voliunnfl 32623 ~ voliun is incompatible w...
volsupnfl 32624 ~ volsup is incompatible w...
0mbf 32625 The empty function is meas...
mbfresfi 32626 Measurability of a piecewi...
mbfposadd 32627 If the sum of two measurab...
cnambfre 32628 A real-valued, a.e. contin...
dvtanlem 32629 Lemma for ~ dvtan - the do...
dvtan 32630 Derivative of tangent. (C...
itg2addnclem 32631 An alternate expression fo...
itg2addnclem2 32632 Lemma for ~ itg2addnc . T...
itg2addnclem3 32633 Lemma incomprehensible in ...
itg2addnc 32634 Alternate proof of ~ itg2a...
itg2gt0cn 32635 ~ itg2gt0 holds on functio...
ibladdnclem 32636 Lemma for ~ ibladdnc ; cf ...
ibladdnc 32637 Choice-free analogue of ~ ...
itgaddnclem1 32638 Lemma for ~ itgaddnc ; cf....
itgaddnclem2 32639 Lemma for ~ itgaddnc ; cf....
itgaddnc 32640 Choice-free analogue of ~ ...
iblsubnc 32641 Choice-free analogue of ~ ...
itgsubnc 32642 Choice-free analogue of ~ ...
iblabsnclem 32643 Lemma for ~ iblabsnc ; cf....
iblabsnc 32644 Choice-free analogue of ~ ...
iblmulc2nc 32645 Choice-free analogue of ~ ...
itgmulc2nclem1 32646 Lemma for ~ itgmulc2nc ; c...
itgmulc2nclem2 32647 Lemma for ~ itgmulc2nc ; c...
itgmulc2nc 32648 Choice-free analogue of ~ ...
itgabsnc 32649 Choice-free analogue of ~ ...
bddiblnc 32650 Choice-free proof of ~ bdd...
cnicciblnc 32651 Choice-free proof of ~ cni...
itggt0cn 32652 ~ itggt0 holds for continu...
ftc1cnnclem 32653 Lemma for ~ ftc1cnnc ; cf....
ftc1cnnc 32654 Choice-free proof of ~ ftc...
ftc1anclem1 32655 Lemma for ~ ftc1anc - the ...
ftc1anclem2 32656 Lemma for ~ ftc1anc - rest...
ftc1anclem3 32657 Lemma for ~ ftc1anc - the ...
ftc1anclem4 32658 Lemma for ~ ftc1anc . (Co...
ftc1anclem5 32659 Lemma for ~ ftc1anc , the ...
ftc1anclem6 32660 Lemma for ~ ftc1anc - cons...
ftc1anclem7 32661 Lemma for ~ ftc1anc . (Co...
ftc1anclem8 32662 Lemma for ~ ftc1anc . (Co...
ftc1anc 32663 ~ ftc1a holds for function...
ftc2nc 32664 Choice-free proof of ~ ftc...
asindmre 32665 Real part of domain of dif...
dvasin 32666 Derivative of arcsine. (C...
dvacos 32667 Derivative of arccosine. ...
dvreasin 32668 Real derivative of arcsine...
dvreacos 32669 Real derivative of arccosi...
areacirclem1 32670 Antiderivative of cross-se...
areacirclem2 32671 Endpoint-inclusive continu...
areacirclem3 32672 Integrability of cross-sec...
areacirclem4 32673 Endpoint-inclusive continu...
areacirclem5 32674 Finding the cross-section ...
areacirc 32675 The area of a circle of ra...
anim12da 32676 Conjoin antecedents and co...
unirep 32677 Define a quantity whose de...
cover2 32678 Two ways of expressing the...
cover2g 32679 Two ways of expressing the...
brabg2 32680 Relation by a binary relat...
opelopab3 32681 Ordered pair membership in...
cocanfo 32682 Cancellation of a surjecti...
brresi 32683 Restriction of a binary re...
fnopabeqd 32684 Equality deduction for fun...
fvopabf4g 32685 Function value of an opera...
eqfnun 32686 Two functions on ` A u. B ...
fnopabco 32687 Composition of a function ...
opropabco 32688 Composition of an operator...
f1opr 32689 Condition for an operation...
cocnv 32690 Composition with a functio...
f1ocan1fv 32691 Cancel a composition by a ...
f1ocan2fv 32692 Cancel a composition by th...
inixp 32693 Intersection of Cartesian ...
upixp 32694 Universal property of the ...
abrexdom 32695 An indexed set is dominate...
abrexdom2 32696 An indexed set is dominate...
ac6gf 32697 Axiom of Choice. (Contrib...
indexa 32698 If for every element of an...
indexdom 32699 If for every element of an...
frinfm 32700 A subset of a well-founded...
welb 32701 A nonempty subset of a wel...
supex2g 32702 Existence of supremum. (C...
supclt 32703 Closure of supremum. (Con...
supubt 32704 Upper bound property of su...
filbcmb 32705 Combine a finite set of lo...
rdr 32706 Two ways of expressing the...
fzmul 32707 Membership of a product in...
sdclem2 32708 Lemma for ~ sdc . (Contri...
sdclem1 32709 Lemma for ~ sdc . (Contri...
sdc 32710 Strong dependent choice. ...
fdc 32711 Finite version of dependen...
fdc1 32712 Variant of ~ fdc with no s...
seqpo 32713 Two ways to say that a seq...
incsequz 32714 An increasing sequence of ...
incsequz2 32715 An increasing sequence of ...
nnubfi 32716 A bounded above set of pos...
nninfnub 32717 An infinite set of positiv...
subspopn 32718 An open set is open in the...
neificl 32719 Neighborhoods are closed u...
lpss2 32720 Limit points of a subset a...
metf1o 32721 Use a bijection with a met...
blssp 32722 A ball in the subspace met...
mettrifi 32723 Generalized triangle inequ...
lmclim2 32724 A sequence in a metric spa...
geomcau 32725 If the distance between co...
caures 32726 The restriction of a Cauch...
caushft 32727 A shifted Cauchy sequence ...
constcncf 32728 A constant function is a c...
idcncf 32729 The identity function is a...
sub1cncf 32730 Subtracting a constant is ...
sub2cncf 32731 Subtraction from a constan...
cnres2 32732 The restriction of a conti...
cnresima 32733 A continuous function is c...
cncfres 32734 A continuous function on c...
istotbnd 32738 The predicate "is a totall...
istotbnd2 32739 The predicate "is a totall...
istotbnd3 32740 A metric space is totally ...
totbndmet 32741 The predicate "totally bou...
0totbnd 32742 The metric (there is only ...
sstotbnd2 32743 Condition for a subset of ...
sstotbnd 32744 Condition for a subset of ...
sstotbnd3 32745 Use a net that is not nece...
totbndss 32746 A subset of a totally boun...
equivtotbnd 32747 If the metric ` M ` is "st...
isbnd 32749 The predicate "is a bounde...
bndmet 32750 A bounded metric space is ...
isbndx 32751 A "bounded extended metric...
isbnd2 32752 The predicate "is a bounde...
isbnd3 32753 A metric space is bounded ...
isbnd3b 32754 A metric space is bounded ...
bndss 32755 A subset of a bounded metr...
blbnd 32756 A ball is bounded. (Contr...
ssbnd 32757 A subset of a metric space...
totbndbnd 32758 A totally bounded metric s...
equivbnd 32759 If the metric ` M ` is "st...
bnd2lem 32760 Lemma for ~ equivbnd2 and ...
equivbnd2 32761 If balls are totally bound...
prdsbnd 32762 The product metric over fi...
prdstotbnd 32763 The product metric over fi...
prdsbnd2 32764 If balls are totally bound...
cntotbnd 32765 A subset of the complex nu...
cnpwstotbnd 32766 A subset of ` A ^ I ` , wh...
ismtyval 32769 The set of isometries betw...
isismty 32770 The condition "is an isome...
ismtycnv 32771 The inverse of an isometry...
ismtyima 32772 The image of a ball under ...
ismtyhmeolem 32773 Lemma for ~ ismtyhmeo . (...
ismtyhmeo 32774 An isometry is a homeomorp...
ismtybndlem 32775 Lemma for ~ ismtybnd . (C...
ismtybnd 32776 Isometries preserve bounde...
ismtyres 32777 A restriction of an isomet...
heibor1lem 32778 Lemma for ~ heibor1 . A c...
heibor1 32779 One half of ~ heibor , tha...
heiborlem1 32780 Lemma for ~ heibor . We w...
heiborlem2 32781 Lemma for ~ heibor . Subs...
heiborlem3 32782 Lemma for ~ heibor . Usin...
heiborlem4 32783 Lemma for ~ heibor . Usin...
heiborlem5 32784 Lemma for ~ heibor . The ...
heiborlem6 32785 Lemma for ~ heibor . Sinc...
heiborlem7 32786 Lemma for ~ heibor . Sinc...
heiborlem8 32787 Lemma for ~ heibor . The ...
heiborlem9 32788 Lemma for ~ heibor . Disc...
heiborlem10 32789 Lemma for ~ heibor . The ...
heibor 32790 Generalized Heine-Borel Th...
bfplem1 32791 Lemma for ~ bfp . The seq...
bfplem2 32792 Lemma for ~ bfp . Using t...
bfp 32793 Banach fixed point theorem...
rrnval 32796 The n-dimensional Euclidea...
rrnmval 32797 The value of the Euclidean...
rrnmet 32798 Euclidean space is a metri...
rrndstprj1 32799 The distance between two p...
rrndstprj2 32800 Bound on the distance betw...
rrncmslem 32801 Lemma for ~ rrncms . (Con...
rrncms 32802 Euclidean space is complet...
repwsmet 32803 The supremum metric on ` R...
rrnequiv 32804 The supremum metric on ` R...
rrntotbnd 32805 A set in Euclidean space i...
rrnheibor 32806 Heine-Borel theorem for Eu...
ismrer1 32807 An isometry between ` RR `...
reheibor 32808 Heine-Borel theorem for re...
iccbnd 32809 A closed interval in ` RR ...
icccmpALT 32810 A closed interval in ` RR ...
isass 32815 The predicate "is an assoc...
isexid 32816 The predicate ` G ` has a ...
ismgmOLD 32819 Obsolete version of ~ ismg...
clmgmOLD 32820 Obsolete version of ~ mgmc...
opidonOLD 32821 Obsolete version of ~ mndp...
rngopidOLD 32822 Obsolete version of ~ mndp...
opidon2OLD 32823 Obsolete version of ~ mndp...
isexid2 32824 If ` G e. ( Magma i^i ExId...
exidu1 32825 Unicity of the left and ri...
idrval 32826 The value of the identity ...
iorlid 32827 A magma right and left ide...
cmpidelt 32828 A magma right and left ide...
smgrpismgmOLD 32831 Obsolete version of ~ sgrp...
issmgrpOLD 32832 Obsolete version of ~ issg...
smgrpmgm 32833 A semi-group is a magma. ...
smgrpassOLD 32834 Obsolete version of ~ sgrp...
mndoissmgrpOLD 32837 Obsolete version of ~ mnds...
mndoisexid 32838 A monoid has an identity e...
mndoismgmOLD 32839 Obsolete version of ~ mndm...
mndomgmid 32840 A monoid is a magma with a...
ismndo 32841 The predicate "is a monoid...
ismndo1 32842 The predicate "is a monoid...
ismndo2 32843 The predicate "is a monoid...
grpomndo 32844 A group is a monoid. (Con...
exidcl 32845 Closure of the binary oper...
exidreslem 32846 Lemma for ~ exidres and ~ ...
exidres 32847 The restriction of a binar...
exidresid 32848 The restriction of a binar...
ablo4pnp 32849 A commutative/associative ...
grpoeqdivid 32850 Two group elements are equ...
grposnOLD 32851 The group operation for th...
elghomlem1OLD 32854 Obsolete as of 15-Mar-2020...
elghomlem2OLD 32855 Obsolete as of 15-Mar-2020...
elghomOLD 32856 Obsolete version of ~ isgh...
ghomlinOLD 32857 Obsolete version of ~ ghml...
ghomidOLD 32858 Obsolete version of ~ ghmi...
ghomf 32859 Mapping property of a grou...
ghomco 32860 The composition of two gro...
ghomdiv 32861 Group homomorphisms preser...
grpokerinj 32862 A group homomorphism is in...
relrngo 32865 The class of all unital ri...
isrngo 32866 The predicate "is a (unita...
isrngod 32867 Conditions that determine ...
rngoi 32868 The properties of a unital...
rngosm 32869 Functionality of the multi...
rngocl 32870 Closure of the multiplicat...
rngoid 32871 The multiplication operati...
rngoideu 32872 The unit element of a ring...
rngodi 32873 Distributive law for the m...
rngodir 32874 Distributive law for the m...
rngoass 32875 Associative law for the mu...
rngo2 32876 A ring element plus itself...
rngoablo 32877 A ring's addition operatio...
rngoablo2 32878 In a unital ring the addit...
rngogrpo 32879 A ring's addition operatio...
rngone0 32880 The base set of a ring is ...
rngogcl 32881 Closure law for the additi...
rngocom 32882 The addition operation of ...
rngoaass 32883 The addition operation of ...
rngoa32 32884 The addition operation of ...
rngoa4 32885 Rearrangement of 4 terms i...
rngorcan 32886 Right cancellation law for...
rngolcan 32887 Left cancellation law for ...
rngo0cl 32888 A ring has an additive ide...
rngo0rid 32889 The additive identity of a...
rngo0lid 32890 The additive identity of a...
rngolz 32891 The zero of a unital ring ...
rngorz 32892 The zero of a unital ring ...
rngosn3 32893 Obsolete as of 25-Jan-2020...
rngosn4 32894 Obsolete as of 25-Jan-2020...
rngosn6 32895 Obsolete as of 25-Jan-2020...
rngonegcl 32896 A ring is closed under neg...
rngoaddneg1 32897 Adding the negative in a r...
rngoaddneg2 32898 Adding the negative in a r...
rngosub 32899 Subtraction in a ring, in ...
rngmgmbs4 32900 The range of an internal o...
rngodm1dm2 32901 In a unital ring the domai...
rngorn1 32902 In a unital ring the range...
rngorn1eq 32903 In a unital ring the range...
rngomndo 32904 In a unital ring the multi...
rngoidmlem 32905 The unit of a ring is an i...
rngolidm 32906 The unit of a ring is an i...
rngoridm 32907 The unit of a ring is an i...
rngo1cl 32908 The unit of a ring belongs...
rngoueqz 32909 Obsolete as of 23-Jan-2020...
rngonegmn1l 32910 Negation in a ring is the ...
rngonegmn1r 32911 Negation in a ring is the ...
rngoneglmul 32912 Negation of a product in a...
rngonegrmul 32913 Negation of a product in a...
rngosubdi 32914 Ring multiplication distri...
rngosubdir 32915 Ring multiplication distri...
zerdivemp1x 32916 In a unitary ring a left i...
isdivrngo 32919 The predicate "is a divisi...
drngoi 32920 The properties of a divisi...
gidsn 32921 Obsolete as of 23-Jan-2020...
zrdivrng 32922 The zero ring is not a div...
dvrunz 32923 In a division ring the uni...
isgrpda 32924 Properties that determine ...
isdrngo1 32925 The predicate "is a divisi...
divrngcl 32926 The product of two nonzero...
isdrngo2 32927 A division ring is a ring ...
isdrngo3 32928 A division ring is a ring ...
rngohomval 32933 The set of ring homomorphi...
isrngohom 32934 The predicate "is a ring h...
rngohomf 32935 A ring homomorphism is a f...
rngohomcl 32936 Closure law for a ring hom...
rngohom1 32937 A ring homomorphism preser...
rngohomadd 32938 Ring homomorphisms preserv...
rngohommul 32939 Ring homomorphisms preserv...
rngogrphom 32940 A ring homomorphism is a g...
rngohom0 32941 A ring homomorphism preser...
rngohomsub 32942 Ring homomorphisms preserv...
rngohomco 32943 The composition of two rin...
rngokerinj 32944 A ring homomorphism is inj...
rngoisoval 32946 The set of ring isomorphis...
isrngoiso 32947 The predicate "is a ring i...
rngoiso1o 32948 A ring isomorphism is a bi...
rngoisohom 32949 A ring isomorphism is a ri...
rngoisocnv 32950 The inverse of a ring isom...
rngoisoco 32951 The composition of two rin...
isriscg 32953 The ring isomorphism relat...
isrisc 32954 The ring isomorphism relat...
risc 32955 The ring isomorphism relat...
risci 32956 Determine that two rings a...
riscer 32957 Ring isomorphism is an equ...
iscom2 32964 A device to add commutativ...
iscrngo 32965 The predicate "is a commut...
iscrngo2 32966 The predicate "is a commut...
iscringd 32967 Conditions that determine ...
flddivrng 32968 A field is a division ring...
crngorngo 32969 A commutative ring is a ri...
crngocom 32970 The multiplication operati...
crngm23 32971 Commutative/associative la...
crngm4 32972 Commutative/associative la...
fldcrng 32973 A field is a commutative r...
isfld2 32974 The predicate "is a field"...
crngohomfo 32975 The image of a homomorphis...
idlval 32982 The class of ideals of a r...
isidl 32983 The predicate "is an ideal...
isidlc 32984 The predicate "is an ideal...
idlss 32985 An ideal of ` R ` is a sub...
idlcl 32986 An element of an ideal is ...
idl0cl 32987 An ideal contains ` 0 ` . ...
idladdcl 32988 An ideal is closed under a...
idllmulcl 32989 An ideal is closed under m...
idlrmulcl 32990 An ideal is closed under m...
idlnegcl 32991 An ideal is closed under n...
idlsubcl 32992 An ideal is closed under s...
rngoidl 32993 A ring ` R ` is an ` R ` i...
0idl 32994 The set containing only ` ...
1idl 32995 Two ways of expressing the...
0rngo 32996 In a ring, ` 0 = 1 ` iff t...
divrngidl 32997 The only ideals in a divis...
intidl 32998 The intersection of a none...
inidl 32999 The intersection of two id...
unichnidl 33000 The union of a nonempty ch...
keridl 33001 The kernel of a ring homom...
pridlval 33002 The class of prime ideals ...
ispridl 33003 The predicate "is a prime ...
pridlidl 33004 A prime ideal is an ideal....
pridlnr 33005 A prime ideal is a proper ...
pridl 33006 The main property of a pri...
ispridl2 33007 A condition that shows an ...
maxidlval 33008 The set of maximal ideals ...
ismaxidl 33009 The predicate "is a maxima...
maxidlidl 33010 A maximal ideal is an idea...
maxidlnr 33011 A maximal ideal is proper....
maxidlmax 33012 A maximal ideal is a maxim...
maxidln1 33013 One is not contained in an...
maxidln0 33014 A ring with a maximal idea...
isprrngo 33019 The predicate "is a prime ...
prrngorngo 33020 A prime ring is a ring. (...
smprngopr 33021 A simple ring (one whose o...
divrngpr 33022 A division ring is a prime...
isdmn 33023 The predicate "is a domain...
isdmn2 33024 The predicate "is a domain...
dmncrng 33025 A domain is a commutative ...
dmnrngo 33026 A domain is a ring. (Cont...
flddmn 33027 A field is a domain. (Con...
igenval 33030 The ideal generated by a s...
igenss 33031 A set is a subset of the i...
igenidl 33032 The ideal generated by a s...
igenmin 33033 The ideal generated by a s...
igenidl2 33034 The ideal generated by an ...
igenval2 33035 The ideal generated by a s...
prnc 33036 A principal ideal (an idea...
isfldidl 33037 Determine if a ring is a f...
isfldidl2 33038 Determine if a ring is a f...
ispridlc 33039 The predicate "is a prime ...
pridlc 33040 Property of a prime ideal ...
pridlc2 33041 Property of a prime ideal ...
pridlc3 33042 Property of a prime ideal ...
isdmn3 33043 The predicate "is a domain...
dmnnzd 33044 A domain has no zero-divis...
dmncan1 33045 Cancellation law for domai...
dmncan2 33046 Cancellation law for domai...
efald2 33047 A proof by contradiction. ...
notbinot1 33048 Simplification rule of neg...
bicontr 33049 Biimplication of its own n...
impor 33050 An equivalent formula for ...
orfa 33051 The falsum ` F. ` can be r...
notbinot2 33052 Commutation rule between n...
biimpor 33053 A rewriting rule for biimp...
unitresl 33054 A lemma for Conjunctive No...
unitresr 33055 A lemma for Conjunctive No...
orfa1 33056 Add a contradicting disjun...
orfa2 33057 Remove a contradicting dis...
bifald 33058 Infer the equivalence to a...
orsild 33059 A lemma for not-or-not eli...
orsird 33060 A lemma for not-or-not eli...
orcomdd 33061 Commutativity of logic dis...
cnf1dd 33062 A lemma for Conjunctive No...
cnf2dd 33063 A lemma for Conjunctive No...
cnfn1dd 33064 A lemma for Conjunctive No...
cnfn2dd 33065 A lemma for Conjunctive No...
or32dd 33066 A rearrangement of disjunc...
notornotel1 33067 A lemma for not-or-not eli...
notornotel2 33068 A lemma for not-or-not eli...
contrd 33069 A proof by contradiction, ...
an12i 33070 An inference from commutin...
exmid2 33071 An excluded middle law. (...
selconj 33072 An inference for selecting...
truconj 33073 Add true as a conjunct. (...
orel 33074 An inference for disjuncti...
negel 33075 An inference for negation ...
botel 33076 An inference for bottom el...
tradd 33077 Add top ad a conjunct. (C...
sbtru 33078 Substitution does not chan...
sbfal 33079 Substitution does not chan...
sbcani 33080 Distribution of class subs...
sbcori 33081 Distribution of class subs...
sbcimi 33082 Distribution of class subs...
sbceqi 33083 Distribution of class subs...
sbcni 33084 Move class substitution in...
sbali 33085 Discard class substitution...
sbexi 33086 Discard class substitution...
sbcalf 33087 Move universal quantifier ...
sbcexf 33088 Move existential quantifie...
sbcalfi 33089 Move universal quantifier ...
sbcexfi 33090 Move existential quantifie...
csbvargi 33091 The proper substitution of...
csbconstgi 33092 The proper substitution of...
spsbcdi 33093 A lemma for eliminating a ...
alrimii 33094 A lemma for introducing a ...
spesbcdi 33095 A lemma for introducing an...
exlimddvf 33096 A lemma for eliminating an...
exlimddvfi 33097 A lemma for eliminating an...
sbceq1ddi 33098 A lemma for eliminating in...
sbccom2lem 33099 Lemma for ~ sbccom2 . (Co...
sbccom2 33100 Commutative law for double...
sbccom2f 33101 Commutative law for double...
sbccom2fi 33102 Commutative law for double...
sbcgfi 33103 Substitution for a variabl...
csbcom2fi 33104 Commutative law for double...
csbgfi 33105 Substitution for a variabl...
fald 33106 Refutation of falsity, in ...
tsim1 33107 A Tseitin axiom for logica...
tsim2 33108 A Tseitin axiom for logica...
tsim3 33109 A Tseitin axiom for logica...
tsbi1 33110 A Tseitin axiom for logica...
tsbi2 33111 A Tseitin axiom for logica...
tsbi3 33112 A Tseitin axiom for logica...
tsbi4 33113 A Tseitin axiom for logica...
tsxo1 33114 A Tseitin axiom for logica...
tsxo2 33115 A Tseitin axiom for logica...
tsxo3 33116 A Tseitin axiom for logica...
tsxo4 33117 A Tseitin axiom for logica...
tsan1 33118 A Tseitin axiom for logica...
tsan2 33119 A Tseitin axiom for logica...
tsan3 33120 A Tseitin axiom for logica...
tsna1 33121 A Tseitin axiom for logica...
tsna2 33122 A Tseitin axiom for logica...
tsna3 33123 A Tseitin axiom for logica...
tsor1 33124 A Tseitin axiom for logica...
tsor2 33125 A Tseitin axiom for logica...
tsor3 33126 A Tseitin axiom for logica...
ts3an1 33127 A Tseitin axiom for triple...
ts3an2 33128 A Tseitin axiom for triple...
ts3an3 33129 A Tseitin axiom for triple...
ts3or1 33130 A Tseitin axiom for triple...
ts3or2 33131 A Tseitin axiom for triple...
ts3or3 33132 A Tseitin axiom for triple...
iuneq2f 33133 Equality deduction for ind...
abeq12 33134 Equality deduction for cla...
rabeq12f 33135 Equality deduction for res...
csbeq12 33136 Equality deduction for sub...
nfbii2 33137 Equality deduction for not...
sbeqi 33138 Equality deduction for sub...
ralbi12f 33139 Equality deduction for res...
oprabbi 33140 Equality deduction for cla...
mpt2bi123f 33141 Equality deduction for map...
iuneq12f 33142 Equality deduction for ind...
iineq12f 33143 Equality deduction for ind...
opabbi 33144 Equality deduction for cla...
mptbi12f 33145 Equality deduction for map...
scottexf 33146 A version of ~ scottex wit...
scott0f 33147 A version of ~ scott0 with...
scottn0f 33148 A version of ~ scott0f wit...
ac6s3f 33149 Generalization of the Axio...
ac6s6 33150 Generalization of the Axio...
ac6s6f 33151 Generalization of the Axio...
prtlem60 33152 Lemma for ~ prter3 . (Con...
bicomdd 33153 Commute two sides of a bic...
jca2 33154 Inference conjoining the c...
jca2r 33155 Inference conjoining the c...
jca3 33156 Inference conjoining the c...
prtlem70 33157 Lemma for ~ prter3 : a rea...
ibdr 33158 Reverse of ~ ibd . (Contr...
pm5.31r 33159 Variant of ~ pm5.31 . (Co...
2r19.29 33160 Double the quantifiers of ...
prtlem100 33161 Lemma for ~ prter3 . (Con...
prtlem5 33162 Lemma for ~ prter1 , ~ prt...
prtlem80 33163 Lemma for ~ prter2 . (Con...
n0el 33164 Negated membership of the ...
brabsb2 33165 A closed form of ~ brabsb ...
eqbrrdv2 33166 Other version of ~ eqbrrdi...
prtlem9 33167 Lemma for ~ prter3 . (Con...
prtlem10 33168 Lemma for ~ prter3 . (Con...
prtlem11 33169 Lemma for ~ prter2 . (Con...
prtlem12 33170 Lemma for ~ prtex and ~ pr...
prtlem13 33171 Lemma for ~ prter1 , ~ prt...
prtlem16 33172 Lemma for ~ prtex , ~ prte...
prtlem400 33173 Lemma for ~ prter2 and als...
erprt 33176 The quotient set of an equ...
prtlem14 33177 Lemma for ~ prter1 , ~ prt...
prtlem15 33178 Lemma for ~ prter1 and ~ p...
prtlem17 33179 Lemma for ~ prter2 . (Con...
prtlem18 33180 Lemma for ~ prter2 . (Con...
prtlem19 33181 Lemma for ~ prter2 . (Con...
prter1 33182 Every partition generates ...
prtex 33183 The equivalence relation g...
prter2 33184 The quotient set of the eq...
prter3 33185 For every partition there ...
axc5 33196 This theorem repeats ~ sp ...
ax4fromc4 33197 Rederivation of axiom ~ ax...
ax10fromc7 33198 Rederivation of axiom ~ ax...
ax6fromc10 33199 Rederivation of axiom ~ ax...
hba1-o 33200 The setvar ` x ` is not fr...
axc4i-o 33201 Inference version of ~ ax-...
equid1 33202 Proof of ~ equid from our ...
equcomi1 33203 Proof of ~ equcomi from ~ ...
aecom-o 33204 Commutation law for identi...
aecoms-o 33205 A commutation rule for ide...
hbae-o 33206 All variables are effectiv...
dral1-o 33207 Formula-building lemma for...
ax12fromc15 33208 Rederivation of axiom ~ ax...
ax13fromc9 33209 Derive ~ ax-13 from ~ ax-c...
ax5ALT 33210 Axiom to quantify a variab...
sps-o 33211 Generalization of antecede...
hbequid 33212 Bound-variable hypothesis ...
nfequid-o 33213 Bound-variable hypothesis ...
axc5c7 33214 Proof of a single axiom th...
axc5c7toc5 33215 Rederivation of ~ ax-c5 fr...
axc5c7toc7 33216 Rederivation of ~ ax-c7 fr...
axc711 33217 Proof of a single axiom th...
nfa1-o 33218 ` x ` is not free in ` A. ...
axc711toc7 33219 Rederivation of ~ ax-c7 fr...
axc711to11 33220 Rederivation of ~ ax-11 fr...
axc5c711 33221 Proof of a single axiom th...
axc5c711toc5 33222 Rederivation of ~ ax-c5 fr...
axc5c711toc7 33223 Rederivation of ~ ax-c7 fr...
axc5c711to11 33224 Rederivation of ~ ax-11 fr...
equidqe 33225 ~ equid with existential q...
axc5sp1 33226 A special case of ~ ax-c5 ...
equidq 33227 ~ equid with universal qua...
equid1ALT 33228 Alternate proof of ~ equid...
axc11nfromc11 33229 Rederivation of ~ ax-c11n ...
naecoms-o 33230 A commutation rule for dis...
hbnae-o 33231 All variables are effectiv...
dvelimf-o 33232 Proof of ~ dvelimh that us...
dral2-o 33233 Formula-building lemma for...
aev-o 33234 A "distinctor elimination"...
ax5eq 33235 Theorem to add distinct qu...
dveeq2-o 33236 Quantifier introduction wh...
axc16g-o 33237 A generalization of axiom ...
dveeq1-o 33238 Quantifier introduction wh...
dveeq1-o16 33239 Version of ~ dveeq1 using ...
ax5el 33240 Theorem to add distinct qu...
axc11n-16 33241 This theorem shows that, g...
dveel2ALT 33242 Alternate proof of ~ dveel...
ax12f 33243 Basis step for constructin...
ax12eq 33244 Basis step for constructin...
ax12el 33245 Basis step for constructin...
ax12indn 33246 Induction step for constru...
ax12indi 33247 Induction step for constru...
ax12indalem 33248 Lemma for ~ ax12inda2 and ...
ax12inda2ALT 33249 Alternate proof of ~ ax12i...
ax12inda2 33250 Induction step for constru...
ax12inda 33251 Induction step for constru...
ax12v2-o 33252 Rederivation of ~ ax-c15 f...
ax12a2-o 33253 Derive ~ ax-c15 from a hyp...
axc11-o 33254 Show that ~ ax-c11 can be ...
fsumshftd 33255 Index shift of a finite su...
fsumshftdOLD 33256 Obsolete version of ~ fsum...
riotaclbgBAD 33258 Closure of restricted iota...
riotaclbBAD 33259 Closure of restricted iota...
riotasvd 33260 Deduction version of ~ rio...
riotasv2d 33261 Value of description binde...
riotasv2s 33262 The value of description b...
riotasv 33263 Value of description binde...
riotasv3d 33264 A property ` ch ` holding ...
elimhyps 33265 A version of ~ elimhyp usi...
dedths 33266 A version of weak deductio...
renegclALT 33267 Closure law for negative o...
elimhyps2 33268 Generalization of ~ elimhy...
dedths2 33269 Generalization of ~ dedths...
19.9alt 33270 Version of ~ 19.9t for uni...
nfcxfrdf 33271 A utility lemma to transfe...
nfded 33272 A deduction theorem that c...
nfded2 33273 A deduction theorem that c...
nfunidALT2 33274 Deduction version of ~ nfu...
nfunidALT 33275 Deduction version of ~ nfu...
nfopdALT 33276 Deduction version of bound...
cnaddcom 33277 Recover the commutative la...
toycom 33278 Show the commutative law f...
lshpset 33283 The set of all hyperplanes...
islshp 33284 The predicate "is a hyperp...
islshpsm 33285 Hyperplane properties expr...
lshplss 33286 A hyperplane is a subspace...
lshpne 33287 A hyperplane is not equal ...
lshpnel 33288 A hyperplane's generating ...
lshpnelb 33289 The subspace sum of a hype...
lshpnel2N 33290 Condition that determines ...
lshpne0 33291 The member of the span in ...
lshpdisj 33292 A hyperplane and the span ...
lshpcmp 33293 If two hyperplanes are com...
lshpinN 33294 The intersection of two di...
lsatset 33295 The set of all 1-dim subsp...
islsat 33296 The predicate "is a 1-dim ...
lsatlspsn2 33297 The span of a nonzero sing...
lsatlspsn 33298 The span of a nonzero sing...
islsati 33299 A 1-dim subspace (atom) (o...
lsateln0 33300 A 1-dim subspace (atom) (o...
lsatlss 33301 The set of 1-dim subspaces...
lsatlssel 33302 An atom is a subspace. (C...
lsatssv 33303 An atom is a set of vector...
lsatn0 33304 A 1-dim subspace (atom) of...
lsatspn0 33305 The span of a vector is an...
lsator0sp 33306 The span of a vector is ei...
lsatssn0 33307 A subspace (or any class) ...
lsatcmp 33308 If two atoms are comparabl...
lsatcmp2 33309 If an atom is included in ...
lsatel 33310 A nonzero vector in an ato...
lsatelbN 33311 A nonzero vector in an ato...
lsat2el 33312 Two atoms sharing a nonzer...
lsmsat 33313 Convert comparison of atom...
lsatfixedN 33314 Show equality with the spa...
lsmsatcv 33315 Subspace sum has the cover...
lssatomic 33316 The lattice of subspaces i...
lssats 33317 The lattice of subspaces i...
lpssat 33318 Two subspaces in a proper ...
lrelat 33319 Subspaces are relatively a...
lssatle 33320 The ordering of two subspa...
lssat 33321 Two subspaces in a proper ...
islshpat 33322 Hyperplane properties expr...
lcvfbr 33325 The covers relation for a ...
lcvbr 33326 The covers relation for a ...
lcvbr2 33327 The covers relation for a ...
lcvbr3 33328 The covers relation for a ...
lcvpss 33329 The covers relation implie...
lcvnbtwn 33330 The covers relation implie...
lcvntr 33331 The covers relation is not...
lcvnbtwn2 33332 The covers relation implie...
lcvnbtwn3 33333 The covers relation implie...
lsmcv2 33334 Subspace sum has the cover...
lcvat 33335 If a subspace covers anoth...
lsatcv0 33336 An atom covers the zero su...
lsatcveq0 33337 A subspace covered by an a...
lsat0cv 33338 A subspace is an atom iff ...
lcvexchlem1 33339 Lemma for ~ lcvexch . (Co...
lcvexchlem2 33340 Lemma for ~ lcvexch . (Co...
lcvexchlem3 33341 Lemma for ~ lcvexch . (Co...
lcvexchlem4 33342 Lemma for ~ lcvexch . (Co...
lcvexchlem5 33343 Lemma for ~ lcvexch . (Co...
lcvexch 33344 Subspaces satisfy the exch...
lcvp 33345 Covering property of Defin...
lcv1 33346 Covering property of a sub...
lcv2 33347 Covering property of a sub...
lsatexch 33348 The atom exchange property...
lsatnle 33349 The meet of a subspace and...
lsatnem0 33350 The meet of distinct atoms...
lsatexch1 33351 The atom exch1ange propert...
lsatcv0eq 33352 If the sum of two atoms co...
lsatcv1 33353 Two atoms covering the zer...
lsatcvatlem 33354 Lemma for ~ lsatcvat . (C...
lsatcvat 33355 A nonzero subspace less th...
lsatcvat2 33356 A subspace covered by the ...
lsatcvat3 33357 A condition implying that ...
islshpcv 33358 Hyperplane properties expr...
l1cvpat 33359 A subspace covered by the ...
l1cvat 33360 Create an atom under an el...
lshpat 33361 Create an atom under a hyp...
lflset 33364 The set of linear function...
islfl 33365 The predicate "is a linear...
lfli 33366 Property of a linear funct...
islfld 33367 Properties that determine ...
lflf 33368 A linear functional is a f...
lflcl 33369 A linear functional value ...
lfl0 33370 A linear functional is zer...
lfladd 33371 Property of a linear funct...
lflsub 33372 Property of a linear funct...
lflmul 33373 Property of a linear funct...
lfl0f 33374 The zero function is a fun...
lfl1 33375 A nonzero functional has a...
lfladdcl 33376 Closure of addition of two...
lfladdcom 33377 Commutativity of functiona...
lfladdass 33378 Associativity of functiona...
lfladd0l 33379 Functional addition with t...
lflnegcl 33380 Closure of the negative of...
lflnegl 33381 A functional plus its nega...
lflvscl 33382 Closure of a scalar produc...
lflvsdi1 33383 Distributive law for (righ...
lflvsdi2 33384 Reverse distributive law f...
lflvsdi2a 33385 Reverse distributive law f...
lflvsass 33386 Associative law for (right...
lfl0sc 33387 The (right vector space) s...
lflsc0N 33388 The scalar product with th...
lfl1sc 33389 The (right vector space) s...
lkrfval 33392 The kernel of a functional...
lkrval 33393 Value of the kernel of a f...
ellkr 33394 Membership in the kernel o...
lkrval2 33395 Value of the kernel of a f...
ellkr2 33396 Membership in the kernel o...
lkrcl 33397 A member of the kernel of ...
lkrf0 33398 The value of a functional ...
lkr0f 33399 The kernel of the zero fun...
lkrlss 33400 The kernel of a linear fun...
lkrssv 33401 The kernel of a linear fun...
lkrsc 33402 The kernel of a nonzero sc...
lkrscss 33403 The kernel of a scalar pro...
eqlkr 33404 Two functionals with the s...
eqlkr2 33405 Two functionals with the s...
eqlkr3 33406 Two functionals with the s...
lkrlsp 33407 The subspace sum of a kern...
lkrlsp2 33408 The subspace sum of a kern...
lkrlsp3 33409 The subspace sum of a kern...
lkrshp 33410 The kernel of a nonzero fu...
lkrshp3 33411 The kernels of nonzero fun...
lkrshpor 33412 The kernel of a functional...
lkrshp4 33413 A kernel is a hyperplane i...
lshpsmreu 33414 Lemma for ~ lshpkrex . Sh...
lshpkrlem1 33415 Lemma for ~ lshpkrex . Th...
lshpkrlem2 33416 Lemma for ~ lshpkrex . Th...
lshpkrlem3 33417 Lemma for ~ lshpkrex . De...
lshpkrlem4 33418 Lemma for ~ lshpkrex . Pa...
lshpkrlem5 33419 Lemma for ~ lshpkrex . Pa...
lshpkrlem6 33420 Lemma for ~ lshpkrex . Sh...
lshpkrcl 33421 The set ` G ` defined by h...
lshpkr 33422 The kernel of functional `...
lshpkrex 33423 There exists a functional ...
lshpset2N 33424 The set of all hyperplanes...
islshpkrN 33425 The predicate "is a hyperp...
lfl1dim 33426 Equivalent expressions for...
lfl1dim2N 33427 Equivalent expressions for...
ldualset 33430 Define the (left) dual of ...
ldualvbase 33431 The vectors of a dual spac...
ldualelvbase 33432 Utility theorem for conver...
ldualfvadd 33433 Vector addition in the dua...
ldualvadd 33434 Vector addition in the dua...
ldualvaddcl 33435 The value of vector additi...
ldualvaddval 33436 The value of the value of ...
ldualsca 33437 The ring of scalars of the...
ldualsbase 33438 Base set of scalar ring fo...
ldualsaddN 33439 Scalar addition for the du...
ldualsmul 33440 Scalar multiplication for ...
ldualfvs 33441 Scalar product operation f...
ldualvs 33442 Scalar product operation v...
ldualvsval 33443 Value of scalar product op...
ldualvscl 33444 The scalar product operati...
ldualvaddcom 33445 Commutative law for vector...
ldualvsass 33446 Associative law for scalar...
ldualvsass2 33447 Associative law for scalar...
ldualvsdi1 33448 Distributive law for scala...
ldualvsdi2 33449 Reverse distributive law f...
ldualgrplem 33450 Lemma for ~ ldualgrp . (C...
ldualgrp 33451 The dual of a vector space...
ldual0 33452 The zero scalar of the dua...
ldual1 33453 The unit scalar of the dua...
ldualneg 33454 The negative of a scalar o...
ldual0v 33455 The zero vector of the dua...
ldual0vcl 33456 The dual zero vector is a ...
lduallmodlem 33457 Lemma for ~ lduallmod . (...
lduallmod 33458 The dual of a left module ...
lduallvec 33459 The dual of a left vector ...
ldualvsub 33460 The value of vector subtra...
ldualvsubcl 33461 Closure of vector subtract...
ldualvsubval 33462 The value of the value of ...
ldualssvscl 33463 Closure of scalar product ...
ldualssvsubcl 33464 Closure of vector subtract...
ldual0vs 33465 Scalar zero times a functi...
lkr0f2 33466 The kernel of the zero fun...
lduallkr3 33467 The kernels of nonzero fun...
lkrpssN 33468 Proper subset relation bet...
lkrin 33469 Intersection of the kernel...
eqlkr4 33470 Two functionals with the s...
ldual1dim 33471 Equivalent expressions for...
ldualkrsc 33472 The kernel of a nonzero sc...
lkrss 33473 The kernel of a scalar pro...
lkrss2N 33474 Two functionals with kerne...
lkreqN 33475 Proportional functionals h...
lkrlspeqN 33476 Condition for colinear fun...
isopos 33485 The predicate "is an ortho...
opposet 33486 Every orthoposet is a pose...
oposlem 33487 Lemma for orthoposet prope...
op01dm 33488 Conditions necessary for z...
op0cl 33489 An orthoposet has a zero e...
op1cl 33490 An orthoposet has a unit e...
op0le 33491 Orthoposet zero is less th...
ople0 33492 An element less than or eq...
opnlen0 33493 An element not less than a...
lub0N 33494 The least upper bound of t...
opltn0 33495 A lattice element greater ...
ople1 33496 Any element is less than t...
op1le 33497 If the orthoposet unit is ...
glb0N 33498 The greatest lower bound o...
opoccl 33499 Closure of orthocomplement...
opococ 33500 Double negative law for or...
opcon3b 33501 Contraposition law for ort...
opcon2b 33502 Orthocomplement contraposi...
opcon1b 33503 Orthocomplement contraposi...
oplecon3 33504 Contraposition law for ort...
oplecon3b 33505 Contraposition law for ort...
oplecon1b 33506 Contraposition law for str...
opoc1 33507 Orthocomplement of orthopo...
opoc0 33508 Orthocomplement of orthopo...
opltcon3b 33509 Contraposition law for str...
opltcon1b 33510 Contraposition law for str...
opltcon2b 33511 Contraposition law for str...
opexmid 33512 Law of excluded middle for...
opnoncon 33513 Law of contradiction for o...
riotaocN 33514 The orthocomplement of the...
cmtfvalN 33515 Value of commutes relation...
cmtvalN 33516 Equivalence for commutes r...
isolat 33517 The predicate "is an ortho...
ollat 33518 An ortholattice is a latti...
olop 33519 An ortholattice is an orth...
olposN 33520 An ortholattice is a poset...
isolatiN 33521 Properties that determine ...
oldmm1 33522 De Morgan's law for meet i...
oldmm2 33523 De Morgan's law for meet i...
oldmm3N 33524 De Morgan's law for meet i...
oldmm4 33525 De Morgan's law for meet i...
oldmj1 33526 De Morgan's law for join i...
oldmj2 33527 De Morgan's law for join i...
oldmj3 33528 De Morgan's law for join i...
oldmj4 33529 De Morgan's law for join i...
olj01 33530 An ortholattice element jo...
olj02 33531 An ortholattice element jo...
olm11 33532 The meet of an ortholattic...
olm12 33533 The meet of an ortholattic...
latmassOLD 33534 Ortholattice meet is assoc...
latm12 33535 A rearrangement of lattice...
latm32 33536 A rearrangement of lattice...
latmrot 33537 Rotate lattice meet of 3 c...
latm4 33538 Rearrangement of lattice m...
latmmdiN 33539 Lattice meet distributes o...
latmmdir 33540 Lattice meet distributes o...
olm01 33541 Meet with lattice zero is ...
olm02 33542 Meet with lattice zero is ...
isoml 33543 The predicate "is an ortho...
isomliN 33544 Properties that determine ...
omlol 33545 An orthomodular lattice is...
omlop 33546 An orthomodular lattice is...
omllat 33547 An orthomodular lattice is...
omllaw 33548 The orthomodular law. (Co...
omllaw2N 33549 Variation of orthomodular ...
omllaw3 33550 Orthomodular law equivalen...
omllaw4 33551 Orthomodular law equivalen...
omllaw5N 33552 The orthomodular law. Rem...
cmtcomlemN 33553 Lemma for ~ cmtcomN . ( ~...
cmtcomN 33554 Commutation is symmetric. ...
cmt2N 33555 Commutation with orthocomp...
cmt3N 33556 Commutation with orthocomp...
cmt4N 33557 Commutation with orthocomp...
cmtbr2N 33558 Alternate definition of th...
cmtbr3N 33559 Alternate definition for t...
cmtbr4N 33560 Alternate definition for t...
lecmtN 33561 Ordered elements commute. ...
cmtidN 33562 Any element commutes with ...
omlfh1N 33563 Foulis-Holland Theorem, pa...
omlfh3N 33564 Foulis-Holland Theorem, pa...
omlmod1i2N 33565 Analogue of modular law ~ ...
omlspjN 33566 Contraction of a Sasaki pr...
cvrfval 33573 Value of covers relation "...
cvrval 33574 Binary relation expressing...
cvrlt 33575 The covers relation implie...
cvrnbtwn 33576 There is no element betwee...
ncvr1 33577 No element covers the latt...
cvrletrN 33578 Property of an element abo...
cvrval2 33579 Binary relation expressing...
cvrnbtwn2 33580 The covers relation implie...
cvrnbtwn3 33581 The covers relation implie...
cvrcon3b 33582 Contraposition law for the...
cvrle 33583 The covers relation implie...
cvrnbtwn4 33584 The covers relation implie...
cvrnle 33585 The covers relation implie...
cvrne 33586 The covers relation implie...
cvrnrefN 33587 The covers relation is not...
cvrcmp 33588 If two lattice elements th...
cvrcmp2 33589 If two lattice elements co...
pats 33590 The set of atoms in a pose...
isat 33591 The predicate "is an atom"...
isat2 33592 The predicate "is an atom"...
atcvr0 33593 An atom covers zero. ( ~ ...
atbase 33594 An atom is a member of the...
atssbase 33595 The set of atoms is a subs...
0ltat 33596 An atom is greater than ze...
leatb 33597 A poset element less than ...
leat 33598 A poset element less than ...
leat2 33599 A nonzero poset element le...
leat3 33600 A poset element less than ...
meetat 33601 The meet of any element wi...
meetat2 33602 The meet of any element wi...
isatl 33604 The predicate "is an atomi...
atllat 33605 An atomic lattice is a lat...
atlpos 33606 An atomic lattice is a pos...
atl0dm 33607 Condition necessary for ze...
atl0cl 33608 An atomic lattice has a ze...
atl0le 33609 Orthoposet zero is less th...
atlle0 33610 An element less than or eq...
atlltn0 33611 A lattice element greater ...
isat3 33612 The predicate "is an atom"...
atn0 33613 An atom is not zero. ( ~ ...
atnle0 33614 An atom is not less than o...
atlen0 33615 A lattice element is nonze...
atcmp 33616 If two atoms are comparabl...
atncmp 33617 Frequently-used variation ...
atnlt 33618 Two atoms cannot satisfy t...
atcvreq0 33619 An element covered by an a...
atncvrN 33620 Two atoms cannot satisfy t...
atlex 33621 Every nonzero element of a...
atnle 33622 Two ways of expressing "an...
atnem0 33623 The meet of distinct atoms...
atlatmstc 33624 An atomic, complete, ortho...
atlatle 33625 The ordering of two Hilber...
atlrelat1 33626 An atomistic lattice with ...
iscvlat 33628 The predicate "is an atomi...
iscvlat2N 33629 The predicate "is an atomi...
cvlatl 33630 An atomic lattice with the...
cvllat 33631 An atomic lattice with the...
cvlposN 33632 An atomic lattice with the...
cvlexch1 33633 An atomic covering lattice...
cvlexch2 33634 An atomic covering lattice...
cvlexchb1 33635 An atomic covering lattice...
cvlexchb2 33636 An atomic covering lattice...
cvlexch3 33637 An atomic covering lattice...
cvlexch4N 33638 An atomic covering lattice...
cvlatexchb1 33639 A version of ~ cvlexchb1 f...
cvlatexchb2 33640 A version of ~ cvlexchb2 f...
cvlatexch1 33641 Atom exchange property. (...
cvlatexch2 33642 Atom exchange property. (...
cvlatexch3 33643 Atom exchange property. (...
cvlcvr1 33644 The covering property. Pr...
cvlcvrp 33645 A Hilbert lattice satisfie...
cvlatcvr1 33646 An atom is covered by its ...
cvlatcvr2 33647 An atom is covered by its ...
cvlsupr2 33648 Two equivalent ways of exp...
cvlsupr3 33649 Two equivalent ways of exp...
cvlsupr4 33650 Consequence of superpositi...
cvlsupr5 33651 Consequence of superpositi...
cvlsupr6 33652 Consequence of superpositi...
cvlsupr7 33653 Consequence of superpositi...
cvlsupr8 33654 Consequence of superpositi...
ishlat1 33657 The predicate "is a Hilber...
ishlat2 33658 The predicate "is a Hilber...
ishlat3N 33659 The predicate "is a Hilber...
ishlatiN 33660 Properties that determine ...
hlomcmcv 33661 A Hilbert lattice is ortho...
hloml 33662 A Hilbert lattice is ortho...
hlclat 33663 A Hilbert lattice is compl...
hlcvl 33664 A Hilbert lattice is an at...
hlatl 33665 A Hilbert lattice is atomi...
hlol 33666 A Hilbert lattice is an or...
hlop 33667 A Hilbert lattice is an or...
hllat 33668 A Hilbert lattice is a lat...
hlomcmat 33669 A Hilbert lattice is ortho...
hlpos 33670 A Hilbert lattice is a pos...
hlatjcl 33671 Closure of join operation....
hlatjcom 33672 Commutatitivity of join op...
hlatjidm 33673 Idempotence of join operat...
hlatjass 33674 Lattice join is associativ...
hlatj12 33675 Swap 1st and 2nd members o...
hlatj32 33676 Swap 2nd and 3rd members o...
hlatjrot 33677 Rotate lattice join of 3 c...
hlatj4 33678 Rearrangement of lattice j...
hlatlej1 33679 A join's first argument is...
hlatlej2 33680 A join's second argument i...
glbconN 33681 De Morgan's law for GLB an...
glbconxN 33682 De Morgan's law for GLB an...
atnlej1 33683 If an atom is not less tha...
atnlej2 33684 If an atom is not less tha...
hlsuprexch 33685 A Hilbert lattice has the ...
hlexch1 33686 A Hilbert lattice has the ...
hlexch2 33687 A Hilbert lattice has the ...
hlexchb1 33688 A Hilbert lattice has the ...
hlexchb2 33689 A Hilbert lattice has the ...
hlsupr 33690 A Hilbert lattice has the ...
hlsupr2 33691 A Hilbert lattice has the ...
hlhgt4 33692 A Hilbert lattice has a he...
hlhgt2 33693 A Hilbert lattice has a he...
hl0lt1N 33694 Lattice 0 is less than lat...
hlexch3 33695 A Hilbert lattice has the ...
hlexch4N 33696 A Hilbert lattice has the ...
hlatexchb1 33697 A version of ~ hlexchb1 fo...
hlatexchb2 33698 A version of ~ hlexchb2 fo...
hlatexch1 33699 Atom exchange property. (...
hlatexch2 33700 Atom exchange property. (...
hlatmstcOLDN 33701 An atomic, complete, ortho...
hlatle 33702 The ordering of two Hilber...
hlateq 33703 The equality of two Hilber...
hlrelat1 33704 An atomistic lattice with ...
hlrelat5N 33705 An atomistic lattice with ...
hlrelat 33706 A Hilbert lattice is relat...
hlrelat2 33707 A consequence of relative ...
exatleN 33708 A condition for an atom to...
hl2at 33709 A Hilbert lattice has at l...
atex 33710 At least one atom exists. ...
intnatN 33711 If the intersection with a...
2llnne2N 33712 Condition implying that tw...
2llnneN 33713 Condition implying that tw...
cvr1 33714 A Hilbert lattice has the ...
cvr2N 33715 Less-than and covers equiv...
hlrelat3 33716 The Hilbert lattice is rel...
cvrval3 33717 Binary relation expressing...
cvrval4N 33718 Binary relation expressing...
cvrval5 33719 Binary relation expressing...
cvrp 33720 A Hilbert lattice satisfie...
atcvr1 33721 An atom is covered by its ...
atcvr2 33722 An atom is covered by its ...
cvrexchlem 33723 Lemma for ~ cvrexch . ( ~...
cvrexch 33724 A Hilbert lattice satisfie...
cvratlem 33725 Lemma for ~ cvrat . ( ~ a...
cvrat 33726 A nonzero Hilbert lattice ...
ltltncvr 33727 A chained strong ordering ...
ltcvrntr 33728 Non-transitive condition f...
cvrntr 33729 The covers relation is not...
atcvr0eq 33730 The covers relation is not...
lnnat 33731 A line (the join of two di...
atcvrj0 33732 Two atoms covering the zer...
cvrat2 33733 A Hilbert lattice element ...
atcvrneN 33734 Inequality derived from at...
atcvrj1 33735 Condition for an atom to b...
atcvrj2b 33736 Condition for an atom to b...
atcvrj2 33737 Condition for an atom to b...
atleneN 33738 Inequality derived from at...
atltcvr 33739 An equivalence of less-tha...
atle 33740 Any nonzero element has an...
atlt 33741 Two atoms are unequal iff ...
atlelt 33742 Transfer less-than relatio...
2atlt 33743 Given an atom less than an...
atexchcvrN 33744 Atom exchange property. V...
atexchltN 33745 Atom exchange property. V...
cvrat3 33746 A condition implying that ...
cvrat4 33747 A condition implying exist...
cvrat42 33748 Commuted version of ~ cvra...
2atjm 33749 The meet of a line (expres...
atbtwn 33750 Property of a 3rd atom ` R...
atbtwnexOLDN 33751 There exists a 3rd atom ` ...
atbtwnex 33752 Given atoms ` P ` in ` X `...
3noncolr2 33753 Two ways to express 3 non-...
3noncolr1N 33754 Two ways to express 3 non-...
hlatcon3 33755 Atom exchange combined wit...
hlatcon2 33756 Atom exchange combined wit...
4noncolr3 33757 A way to express 4 non-col...
4noncolr2 33758 A way to express 4 non-col...
4noncolr1 33759 A way to express 4 non-col...
athgt 33760 A Hilbert lattice, whose h...
3dim0 33761 There exists a 3-dimension...
3dimlem1 33762 Lemma for ~ 3dim1 . (Cont...
3dimlem2 33763 Lemma for ~ 3dim1 . (Cont...
3dimlem3a 33764 Lemma for ~ 3dim3 . (Cont...
3dimlem3 33765 Lemma for ~ 3dim1 . (Cont...
3dimlem3OLDN 33766 Lemma for ~ 3dim1 . (Cont...
3dimlem4a 33767 Lemma for ~ 3dim3 . (Cont...
3dimlem4 33768 Lemma for ~ 3dim1 . (Cont...
3dimlem4OLDN 33769 Lemma for ~ 3dim1 . (Cont...
3dim1lem5 33770 Lemma for ~ 3dim1 . (Cont...
3dim1 33771 Construct a 3-dimensional ...
3dim2 33772 Construct 2 new layers on ...
3dim3 33773 Construct a new layer on t...
2dim 33774 Generate a height-3 elemen...
1dimN 33775 An atom is covered by a he...
1cvrco 33776 The orthocomplement of an ...
1cvratex 33777 There exists an atom less ...
1cvratlt 33778 An atom less than or equal...
1cvrjat 33779 An element covered by the ...
1cvrat 33780 Create an atom under an el...
ps-1 33781 The join of two atoms ` R ...
ps-2 33782 Lattice analogue for the p...
2atjlej 33783 Two atoms are different if...
hlatexch3N 33784 Rearrange join of atoms in...
hlatexch4 33785 Exchange 2 atoms. (Contri...
ps-2b 33786 Variation of projective ge...
3atlem1 33787 Lemma for ~ 3at . (Contri...
3atlem2 33788 Lemma for ~ 3at . (Contri...
3atlem3 33789 Lemma for ~ 3at . (Contri...
3atlem4 33790 Lemma for ~ 3at . (Contri...
3atlem5 33791 Lemma for ~ 3at . (Contri...
3atlem6 33792 Lemma for ~ 3at . (Contri...
3atlem7 33793 Lemma for ~ 3at . (Contri...
3at 33794 Any three non-colinear ato...
llnset 33809 The set of lattice lines i...
islln 33810 The predicate "is a lattic...
islln4 33811 The predicate "is a lattic...
llni 33812 Condition implying a latti...
llnbase 33813 A lattice line is a lattic...
islln3 33814 The predicate "is a lattic...
islln2 33815 The predicate "is a lattic...
llni2 33816 The join of two different ...
llnnleat 33817 An atom cannot majorize a ...
llnneat 33818 A lattice line is not an a...
2atneat 33819 The join of two distinct a...
llnn0 33820 A lattice line is nonzero....
islln2a 33821 The predicate "is a lattic...
llnle 33822 Any element greater than 0...
atcvrlln2 33823 An atom under a line is co...
atcvrlln 33824 An element covering an ato...
llnexatN 33825 Given an atom on a line, t...
llncmp 33826 If two lattice lines are c...
llnnlt 33827 Two lattice lines cannot s...
2llnmat 33828 Two intersecting lines int...
2at0mat0 33829 Special case of ~ 2atmat0 ...
2atmat0 33830 The meet of two unequal li...
2atm 33831 An atom majorized by two d...
ps-2c 33832 Variation of projective ge...
lplnset 33833 The set of lattice planes ...
islpln 33834 The predicate "is a lattic...
islpln4 33835 The predicate "is a lattic...
lplni 33836 Condition implying a latti...
islpln3 33837 The predicate "is a lattic...
lplnbase 33838 A lattice plane is a latti...
islpln5 33839 The predicate "is a lattic...
islpln2 33840 The predicate "is a lattic...
lplni2 33841 The join of 3 different at...
lvolex3N 33842 There is an atom outside o...
llnmlplnN 33843 The intersection of a line...
lplnle 33844 Any element greater than 0...
lplnnle2at 33845 A lattice line (or atom) c...
lplnnleat 33846 A lattice plane cannot maj...
lplnnlelln 33847 A lattice plane is not les...
2atnelpln 33848 The join of two atoms is n...
lplnneat 33849 No lattice plane is an ato...
lplnnelln 33850 No lattice plane is a latt...
lplnn0N 33851 A lattice plane is nonzero...
islpln2a 33852 The predicate "is a lattic...
islpln2ah 33853 The predicate "is a lattic...
lplnriaN 33854 Property of a lattice plan...
lplnribN 33855 Property of a lattice plan...
lplnric 33856 Property of a lattice plan...
lplnri1 33857 Property of a lattice plan...
lplnri2N 33858 Property of a lattice plan...
lplnri3N 33859 Property of a lattice plan...
lplnllnneN 33860 Two lattice lines defined ...
llncvrlpln2 33861 A lattice line under a lat...
llncvrlpln 33862 An element covering a latt...
2lplnmN 33863 If the join of two lattice...
2llnmj 33864 The meet of two lattice li...
2atmat 33865 The meet of two intersecti...
lplncmp 33866 If two lattice planes are ...
lplnexatN 33867 Given a lattice line on a ...
lplnexllnN 33868 Given an atom on a lattice...
lplnnlt 33869 Two lattice planes cannot ...
2llnjaN 33870 The join of two different ...
2llnjN 33871 The join of two different ...
2llnm2N 33872 The meet of two different ...
2llnm3N 33873 Two lattice lines in a lat...
2llnm4 33874 Two lattice lines that maj...
2llnmeqat 33875 An atom equals the interse...
lvolset 33876 The set of 3-dim lattice v...
islvol 33877 The predicate "is a 3-dim ...
islvol4 33878 The predicate "is a 3-dim ...
lvoli 33879 Condition implying a 3-dim...
islvol3 33880 The predicate "is a 3-dim ...
lvoli3 33881 Condition implying a 3-dim...
lvolbase 33882 A 3-dim lattice volume is ...
islvol5 33883 The predicate "is a 3-dim ...
islvol2 33884 The predicate "is a 3-dim ...
lvoli2 33885 The join of 4 different at...
lvolnle3at 33886 A lattice plane (or lattic...
lvolnleat 33887 An atom cannot majorize a ...
lvolnlelln 33888 A lattice line cannot majo...
lvolnlelpln 33889 A lattice plane cannot maj...
3atnelvolN 33890 The join of 3 atoms is not...
2atnelvolN 33891 The join of two atoms is n...
lvolneatN 33892 No lattice volume is an at...
lvolnelln 33893 No lattice volume is a lat...
lvolnelpln 33894 No lattice volume is a lat...
lvoln0N 33895 A lattice volume is nonzer...
islvol2aN 33896 The predicate "is a lattic...
4atlem0a 33897 Lemma for ~ 4at . (Contri...
4atlem0ae 33898 Lemma for ~ 4at . (Contri...
4atlem0be 33899 Lemma for ~ 4at . (Contri...
4atlem3 33900 Lemma for ~ 4at . Break i...
4atlem3a 33901 Lemma for ~ 4at . Break i...
4atlem3b 33902 Lemma for ~ 4at . Break i...
4atlem4a 33903 Lemma for ~ 4at . Frequen...
4atlem4b 33904 Lemma for ~ 4at . Frequen...
4atlem4c 33905 Lemma for ~ 4at . Frequen...
4atlem4d 33906 Lemma for ~ 4at . Frequen...
4atlem9 33907 Lemma for ~ 4at . Substit...
4atlem10a 33908 Lemma for ~ 4at . Substit...
4atlem10b 33909 Lemma for ~ 4at . Substit...
4atlem10 33910 Lemma for ~ 4at . Combine...
4atlem11a 33911 Lemma for ~ 4at . Substit...
4atlem11b 33912 Lemma for ~ 4at . Substit...
4atlem11 33913 Lemma for ~ 4at . Combine...
4atlem12a 33914 Lemma for ~ 4at . Substit...
4atlem12b 33915 Lemma for ~ 4at . Substit...
4atlem12 33916 Lemma for ~ 4at . Combine...
4at 33917 Four atoms determine a lat...
4at2 33918 Four atoms determine a lat...
lplncvrlvol2 33919 A lattice line under a lat...
lplncvrlvol 33920 An element covering a latt...
lvolcmp 33921 If two lattice planes are ...
lvolnltN 33922 Two lattice volumes cannot...
2lplnja 33923 The join of two different ...
2lplnj 33924 The join of two different ...
2lplnm2N 33925 The meet of two different ...
2lplnmj 33926 The meet of two lattice pl...
dalemkehl 33927 Lemma for ~ dath . Freque...
dalemkelat 33928 Lemma for ~ dath . Freque...
dalemkeop 33929 Lemma for ~ dath . Freque...
dalempea 33930 Lemma for ~ dath . Freque...
dalemqea 33931 Lemma for ~ dath . Freque...
dalemrea 33932 Lemma for ~ dath . Freque...
dalemsea 33933 Lemma for ~ dath . Freque...
dalemtea 33934 Lemma for ~ dath . Freque...
dalemuea 33935 Lemma for ~ dath . Freque...
dalemyeo 33936 Lemma for ~ dath . Freque...
dalemzeo 33937 Lemma for ~ dath . Freque...
dalemclpjs 33938 Lemma for ~ dath . Freque...
dalemclqjt 33939 Lemma for ~ dath . Freque...
dalemclrju 33940 Lemma for ~ dath . Freque...
dalem-clpjq 33941 Lemma for ~ dath . Freque...
dalemceb 33942 Lemma for ~ dath . Freque...
dalempeb 33943 Lemma for ~ dath . Freque...
dalemqeb 33944 Lemma for ~ dath . Freque...
dalemreb 33945 Lemma for ~ dath . Freque...
dalemseb 33946 Lemma for ~ dath . Freque...
dalemteb 33947 Lemma for ~ dath . Freque...
dalemueb 33948 Lemma for ~ dath . Freque...
dalempjqeb 33949 Lemma for ~ dath . Freque...
dalemsjteb 33950 Lemma for ~ dath . Freque...
dalemtjueb 33951 Lemma for ~ dath . Freque...
dalemqrprot 33952 Lemma for ~ dath . Freque...
dalemyeb 33953 Lemma for ~ dath . Freque...
dalemcnes 33954 Lemma for ~ dath . Freque...
dalempnes 33955 Lemma for ~ dath . Freque...
dalemqnet 33956 Lemma for ~ dath . Freque...
dalempjsen 33957 Lemma for ~ dath . Freque...
dalemply 33958 Lemma for ~ dath . Freque...
dalemsly 33959 Lemma for ~ dath . Freque...
dalemswapyz 33960 Lemma for ~ dath . Swap t...
dalemrot 33961 Lemma for ~ dath . Rotate...
dalemrotyz 33962 Lemma for ~ dath . Rotate...
dalem1 33963 Lemma for ~ dath . Show t...
dalemcea 33964 Lemma for ~ dath . Freque...
dalem2 33965 Lemma for ~ dath . Show t...
dalemdea 33966 Lemma for ~ dath . Freque...
dalemeea 33967 Lemma for ~ dath . Freque...
dalem3 33968 Lemma for ~ dalemdnee . (...
dalem4 33969 Lemma for ~ dalemdnee . (...
dalemdnee 33970 Lemma for ~ dath . Axis o...
dalem5 33971 Lemma for ~ dath . Atom `...
dalem6 33972 Lemma for ~ dath . Analog...
dalem7 33973 Lemma for ~ dath . Analog...
dalem8 33974 Lemma for ~ dath . Plane ...
dalem-cly 33975 Lemma for ~ dalem9 . Cent...
dalem9 33976 Lemma for ~ dath . Since ...
dalem10 33977 Lemma for ~ dath . Atom `...
dalem11 33978 Lemma for ~ dath . Analog...
dalem12 33979 Lemma for ~ dath . Analog...
dalem13 33980 Lemma for ~ dalem14 . (Co...
dalem14 33981 Lemma for ~ dath . Planes...
dalem15 33982 Lemma for ~ dath . The ax...
dalem16 33983 Lemma for ~ dath . The at...
dalem17 33984 Lemma for ~ dath . When p...
dalem18 33985 Lemma for ~ dath . Show t...
dalem19 33986 Lemma for ~ dath . Show t...
dalemccea 33987 Lemma for ~ dath . Freque...
dalemddea 33988 Lemma for ~ dath . Freque...
dalem-ccly 33989 Lemma for ~ dath . Freque...
dalem-ddly 33990 Lemma for ~ dath . Freque...
dalemccnedd 33991 Lemma for ~ dath . Freque...
dalemclccjdd 33992 Lemma for ~ dath . Freque...
dalemcceb 33993 Lemma for ~ dath . Freque...
dalemswapyzps 33994 Lemma for ~ dath . Swap t...
dalemrotps 33995 Lemma for ~ dath . Rotate...
dalemcjden 33996 Lemma for ~ dath . Show t...
dalem20 33997 Lemma for ~ dath . Show t...
dalem21 33998 Lemma for ~ dath . Show t...
dalem22 33999 Lemma for ~ dath . Show t...
dalem23 34000 Lemma for ~ dath . Show t...
dalem24 34001 Lemma for ~ dath . Show t...
dalem25 34002 Lemma for ~ dath . Show t...
dalem27 34003 Lemma for ~ dath . Show t...
dalem28 34004 Lemma for ~ dath . Lemma ...
dalem29 34005 Lemma for ~ dath . Analog...
dalem30 34006 Lemma for ~ dath . Analog...
dalem31N 34007 Lemma for ~ dath . Analog...
dalem32 34008 Lemma for ~ dath . Analog...
dalem33 34009 Lemma for ~ dath . Analog...
dalem34 34010 Lemma for ~ dath . Analog...
dalem35 34011 Lemma for ~ dath . Analog...
dalem36 34012 Lemma for ~ dath . Analog...
dalem37 34013 Lemma for ~ dath . Analog...
dalem38 34014 Lemma for ~ dath . Plane ...
dalem39 34015 Lemma for ~ dath . Auxili...
dalem40 34016 Lemma for ~ dath . Analog...
dalem41 34017 Lemma for ~ dath . (Contr...
dalem42 34018 Lemma for ~ dath . Auxili...
dalem43 34019 Lemma for ~ dath . Planes...
dalem44 34020 Lemma for ~ dath . Dummy ...
dalem45 34021 Lemma for ~ dath . Dummy ...
dalem46 34022 Lemma for ~ dath . Analog...
dalem47 34023 Lemma for ~ dath . Analog...
dalem48 34024 Lemma for ~ dath . Analog...
dalem49 34025 Lemma for ~ dath . Analog...
dalem50 34026 Lemma for ~ dath . Analog...
dalem51 34027 Lemma for ~ dath . Constr...
dalem52 34028 Lemma for ~ dath . Lines ...
dalem53 34029 Lemma for ~ dath . The au...
dalem54 34030 Lemma for ~ dath . Line `...
dalem55 34031 Lemma for ~ dath . Lines ...
dalem56 34032 Lemma for ~ dath . Analog...
dalem57 34033 Lemma for ~ dath . Axis o...
dalem58 34034 Lemma for ~ dath . Analog...
dalem59 34035 Lemma for ~ dath . Analog...
dalem60 34036 Lemma for ~ dath . ` B ` i...
dalem61 34037 Lemma for ~ dath . Show t...
dalem62 34038 Lemma for ~ dath . Elimin...
dalem63 34039 Lemma for ~ dath . Combin...
dath 34040 Desargues' Theorem of proj...
dath2 34041 Version of Desargues' Theo...
lineset 34042 The set of lines in a Hilb...
isline 34043 The predicate "is a line"....
islinei 34044 Condition implying "is a l...
pointsetN 34045 The set of points in a Hil...
ispointN 34046 The predicate "is a point"...
atpointN 34047 The singleton of an atom i...
psubspset 34048 The set of projective subs...
ispsubsp 34049 The predicate "is a projec...
ispsubsp2 34050 The predicate "is a projec...
psubspi 34051 Property of a projective s...
psubspi2N 34052 Property of a projective s...
0psubN 34053 The empty set is a project...
snatpsubN 34054 The singleton of an atom i...
pointpsubN 34055 A point (singleton of an a...
linepsubN 34056 A line is a projective sub...
atpsubN 34057 The set of all atoms is a ...
psubssat 34058 A projective subspace cons...
psubatN 34059 A member of a projective s...
pmapfval 34060 The projective map of a Hi...
pmapval 34061 Value of the projective ma...
elpmap 34062 Member of a projective map...
pmapssat 34063 The projective map of a Hi...
pmapssbaN 34064 A weakening of ~ pmapssat ...
pmaple 34065 The projective map of a Hi...
pmap11 34066 The projective map of a Hi...
pmapat 34067 The projective map of an a...
elpmapat 34068 Member of the projective m...
pmap0 34069 Value of the projective ma...
pmapeq0 34070 A projective map value is ...
pmap1N 34071 Value of the projective ma...
pmapsub 34072 The projective map of a Hi...
pmapglbx 34073 The projective map of the ...
pmapglb 34074 The projective map of the ...
pmapglb2N 34075 The projective map of the ...
pmapglb2xN 34076 The projective map of the ...
pmapmeet 34077 The projective map of a me...
isline2 34078 Definition of line in term...
linepmap 34079 A line described with a pr...
isline3 34080 Definition of line in term...
isline4N 34081 Definition of line in term...
lneq2at 34082 A line equals the join of ...
lnatexN 34083 There is an atom in a line...
lnjatN 34084 Given an atom in a line, t...
lncvrelatN 34085 A lattice element covered ...
lncvrat 34086 A line covers the atoms it...
lncmp 34087 If two lines are comparabl...
2lnat 34088 Two intersecting lines int...
2atm2atN 34089 Two joins with a common at...
2llnma1b 34090 Generalization of ~ 2llnma...
2llnma1 34091 Two different intersecting...
2llnma3r 34092 Two different intersecting...
2llnma2 34093 Two different intersecting...
2llnma2rN 34094 Two different intersecting...
cdlema1N 34095 A condition for required f...
cdlema2N 34096 A condition for required f...
cdlemblem 34097 Lemma for ~ cdlemb . (Con...
cdlemb 34098 Given two atoms not less t...
paddfval 34101 Projective subspace sum op...
paddval 34102 Projective subspace sum op...
elpadd 34103 Member of a projective sub...
elpaddn0 34104 Member of projective subsp...
paddvaln0N 34105 Projective subspace sum op...
elpaddri 34106 Condition implying members...
elpaddatriN 34107 Condition implying members...
elpaddat 34108 Membership in a projective...
elpaddatiN 34109 Consequence of membership ...
elpadd2at 34110 Membership in a projective...
elpadd2at2 34111 Membership in a projective...
paddunssN 34112 Projective subspace sum in...
elpadd0 34113 Member of projective subsp...
paddval0 34114 Projective subspace sum wi...
padd01 34115 Projective subspace sum wi...
padd02 34116 Projective subspace sum wi...
paddcom 34117 Projective subspace sum co...
paddssat 34118 A projective subspace sum ...
sspadd1 34119 A projective subspace sum ...
sspadd2 34120 A projective subspace sum ...
paddss1 34121 Subset law for projective ...
paddss2 34122 Subset law for projective ...
paddss12 34123 Subset law for projective ...
paddasslem1 34124 Lemma for ~ paddass . (Co...
paddasslem2 34125 Lemma for ~ paddass . (Co...
paddasslem3 34126 Lemma for ~ paddass . Res...
paddasslem4 34127 Lemma for ~ paddass . Com...
paddasslem5 34128 Lemma for ~ paddass . Sho...
paddasslem6 34129 Lemma for ~ paddass . (Co...
paddasslem7 34130 Lemma for ~ paddass . Com...
paddasslem8 34131 Lemma for ~ paddass . (Co...
paddasslem9 34132 Lemma for ~ paddass . Com...
paddasslem10 34133 Lemma for ~ paddass . Use...
paddasslem11 34134 Lemma for ~ paddass . The...
paddasslem12 34135 Lemma for ~ paddass . The...
paddasslem13 34136 Lemma for ~ paddass . The...
paddasslem14 34137 Lemma for ~ paddass . Rem...
paddasslem15 34138 Lemma for ~ paddass . Use...
paddasslem16 34139 Lemma for ~ paddass . Use...
paddasslem17 34140 Lemma for ~ paddass . The...
paddasslem18 34141 Lemma for ~ paddass . Com...
paddass 34142 Projective subspace sum is...
padd12N 34143 Commutative/associative la...
padd4N 34144 Rearrangement of 4 terms i...
paddidm 34145 Projective subspace sum is...
paddclN 34146 The projective sum of two ...
paddssw1 34147 Subset law for projective ...
paddssw2 34148 Subset law for projective ...
paddss 34149 Subset law for projective ...
pmodlem1 34150 Lemma for ~ pmod1i . (Con...
pmodlem2 34151 Lemma for ~ pmod1i . (Con...
pmod1i 34152 The modular law holds in a...
pmod2iN 34153 Dual of the modular law. ...
pmodN 34154 The modular law for projec...
pmodl42N 34155 Lemma derived from modular...
pmapjoin 34156 The projective map of the ...
pmapjat1 34157 The projective map of the ...
pmapjat2 34158 The projective map of the ...
pmapjlln1 34159 The projective map of the ...
hlmod1i 34160 A version of the modular l...
atmod1i1 34161 Version of modular law ~ p...
atmod1i1m 34162 Version of modular law ~ p...
atmod1i2 34163 Version of modular law ~ p...
llnmod1i2 34164 Version of modular law ~ p...
atmod2i1 34165 Version of modular law ~ p...
atmod2i2 34166 Version of modular law ~ p...
llnmod2i2 34167 Version of modular law ~ p...
atmod3i1 34168 Version of modular law tha...
atmod3i2 34169 Version of modular law tha...
atmod4i1 34170 Version of modular law tha...
atmod4i2 34171 Version of modular law tha...
llnexchb2lem 34172 Lemma for ~ llnexchb2 . (...
llnexchb2 34173 Line exchange property (co...
llnexch2N 34174 Line exchange property (co...
dalawlem1 34175 Lemma for ~ dalaw . Speci...
dalawlem2 34176 Lemma for ~ dalaw . Utili...
dalawlem3 34177 Lemma for ~ dalaw . First...
dalawlem4 34178 Lemma for ~ dalaw . Secon...
dalawlem5 34179 Lemma for ~ dalaw . Speci...
dalawlem6 34180 Lemma for ~ dalaw . First...
dalawlem7 34181 Lemma for ~ dalaw . Secon...
dalawlem8 34182 Lemma for ~ dalaw . Speci...
dalawlem9 34183 Lemma for ~ dalaw . Speci...
dalawlem10 34184 Lemma for ~ dalaw . Combi...
dalawlem11 34185 Lemma for ~ dalaw . First...
dalawlem12 34186 Lemma for ~ dalaw . Secon...
dalawlem13 34187 Lemma for ~ dalaw . Speci...
dalawlem14 34188 Lemma for ~ dalaw . Combi...
dalawlem15 34189 Lemma for ~ dalaw . Swap ...
dalaw 34190 Desargues' law, derived fr...
pclfvalN 34193 The projective subspace cl...
pclvalN 34194 Value of the projective su...
pclclN 34195 Closure of the projective ...
elpclN 34196 Membership in the projecti...
elpcliN 34197 Implication of membership ...
pclssN 34198 Ordering is preserved by s...
pclssidN 34199 A set of atoms is included...
pclidN 34200 The projective subspace cl...
pclbtwnN 34201 A projective subspace sand...
pclunN 34202 The projective subspace cl...
pclun2N 34203 The projective subspace cl...
pclfinN 34204 The projective subspace cl...
pclcmpatN 34205 The set of projective subs...
polfvalN 34208 The projective subspace po...
polvalN 34209 Value of the projective su...
polval2N 34210 Alternate expression for v...
polsubN 34211 The polarity of a set of a...
polssatN 34212 The polarity of a set of a...
pol0N 34213 The polarity of the empty ...
pol1N 34214 The polarity of the whole ...
2pol0N 34215 The closed subspace closur...
polpmapN 34216 The polarity of a projecti...
2polpmapN 34217 Double polarity of a proje...
2polvalN 34218 Value of double polarity. ...
2polssN 34219 A set of atoms is a subset...
3polN 34220 Triple polarity cancels to...
polcon3N 34221 Contraposition law for pol...
2polcon4bN 34222 Contraposition law for pol...
polcon2N 34223 Contraposition law for pol...
polcon2bN 34224 Contraposition law for pol...
pclss2polN 34225 The projective subspace cl...
pcl0N 34226 The projective subspace cl...
pcl0bN 34227 The projective subspace cl...
pmaplubN 34228 The LUB of a projective ma...
sspmaplubN 34229 A set of atoms is a subset...
2pmaplubN 34230 Double projective map of a...
paddunN 34231 The closure of the project...
poldmj1N 34232 De Morgan's law for polari...
pmapj2N 34233 The projective map of the ...
pmapocjN 34234 The projective map of the ...
polatN 34235 The polarity of the single...
2polatN 34236 Double polarity of the sin...
pnonsingN 34237 The intersection of a set ...
psubclsetN 34240 The set of closed projecti...
ispsubclN 34241 The predicate "is a closed...
psubcliN 34242 Property of a closed proje...
psubcli2N 34243 Property of a closed proje...
psubclsubN 34244 A closed projective subspa...
psubclssatN 34245 A closed projective subspa...
pmapidclN 34246 Projective map of the LUB ...
0psubclN 34247 The empty set is a closed ...
1psubclN 34248 The set of all atoms is a ...
atpsubclN 34249 A point (singleton of an a...
pmapsubclN 34250 A projective map value is ...
ispsubcl2N 34251 Alternate predicate for "i...
psubclinN 34252 The intersection of two cl...
paddatclN 34253 The projective sum of a cl...
pclfinclN 34254 The projective subspace cl...
linepsubclN 34255 A line is a closed project...
polsubclN 34256 A polarity is a closed pro...
poml4N 34257 Orthomodular law for proje...
poml5N 34258 Orthomodular law for proje...
poml6N 34259 Orthomodular law for proje...
osumcllem1N 34260 Lemma for ~ osumclN . (Co...
osumcllem2N 34261 Lemma for ~ osumclN . (Co...
osumcllem3N 34262 Lemma for ~ osumclN . (Co...
osumcllem4N 34263 Lemma for ~ osumclN . (Co...
osumcllem5N 34264 Lemma for ~ osumclN . (Co...
osumcllem6N 34265 Lemma for ~ osumclN . Use...
osumcllem7N 34266 Lemma for ~ osumclN . (Co...
osumcllem8N 34267 Lemma for ~ osumclN . (Co...
osumcllem9N 34268 Lemma for ~ osumclN . (Co...
osumcllem10N 34269 Lemma for ~ osumclN . Con...
osumcllem11N 34270 Lemma for ~ osumclN . (Co...
osumclN 34271 Closure of orthogonal sum....
pmapojoinN 34272 For orthogonal elements, p...
pexmidN 34273 Excluded middle law for cl...
pexmidlem1N 34274 Lemma for ~ pexmidN . Hol...
pexmidlem2N 34275 Lemma for ~ pexmidN . (Co...
pexmidlem3N 34276 Lemma for ~ pexmidN . Use...
pexmidlem4N 34277 Lemma for ~ pexmidN . (Co...
pexmidlem5N 34278 Lemma for ~ pexmidN . (Co...
pexmidlem6N 34279 Lemma for ~ pexmidN . (Co...
pexmidlem7N 34280 Lemma for ~ pexmidN . Con...
pexmidlem8N 34281 Lemma for ~ pexmidN . The...
pexmidALTN 34282 Excluded middle law for cl...
pl42lem1N 34283 Lemma for ~ pl42N . (Cont...
pl42lem2N 34284 Lemma for ~ pl42N . (Cont...
pl42lem3N 34285 Lemma for ~ pl42N . (Cont...
pl42lem4N 34286 Lemma for ~ pl42N . (Cont...
pl42N 34287 Law holding in a Hilbert l...
watfvalN 34296 The W atoms function. (Co...
watvalN 34297 Value of the W atoms funct...
iswatN 34298 The predicate "is a W atom...
lhpset 34299 The set of co-atoms (latti...
islhp 34300 The predicate "is a co-ato...
islhp2 34301 The predicate "is a co-ato...
lhpbase 34302 A co-atom is a member of t...
lhp1cvr 34303 The lattice unit covers a ...
lhplt 34304 An atom under a co-atom is...
lhp2lt 34305 The join of two atoms unde...
lhpexlt 34306 There exists an atom less ...
lhp0lt 34307 A co-atom is greater than ...
lhpn0 34308 A co-atom is nonzero. TOD...
lhpexle 34309 There exists an atom under...
lhpexnle 34310 There exists an atom not u...
lhpexle1lem 34311 Lemma for ~ lhpexle1 and o...
lhpexle1 34312 There exists an atom under...
lhpexle2lem 34313 Lemma for ~ lhpexle2 . (C...
lhpexle2 34314 There exists atom under a ...
lhpexle3lem 34315 There exists atom under a ...
lhpexle3 34316 There exists atom under a ...
lhpex2leN 34317 There exist at least two d...
lhpoc 34318 The orthocomplement of a c...
lhpoc2N 34319 The orthocomplement of an ...
lhpocnle 34320 The orthocomplement of a c...
lhpocat 34321 The orthocomplement of a c...
lhpocnel 34322 The orthocomplement of a c...
lhpocnel2 34323 The orthocomplement of a c...
lhpjat1 34324 The join of a co-atom (hyp...
lhpjat2 34325 The join of a co-atom (hyp...
lhpj1 34326 The join of a co-atom (hyp...
lhpmcvr 34327 The meet of a lattice hype...
lhpmcvr2 34328 Alternate way to express t...
lhpmcvr3 34329 Specialization of ~ lhpmcv...
lhpmcvr4N 34330 Specialization of ~ lhpmcv...
lhpmcvr5N 34331 Specialization of ~ lhpmcv...
lhpmcvr6N 34332 Specialization of ~ lhpmcv...
lhpm0atN 34333 If the meet of a lattice h...
lhpmat 34334 An element covered by the ...
lhpmatb 34335 An element covered by the ...
lhp2at0 34336 Join and meet with differe...
lhp2atnle 34337 Inequality for 2 different...
lhp2atne 34338 Inequality for joins with ...
lhp2at0nle 34339 Inequality for 2 different...
lhp2at0ne 34340 Inequality for joins with ...
lhpelim 34341 Eliminate an atom not unde...
lhpmod2i2 34342 Modular law for hyperplane...
lhpmod6i1 34343 Modular law for hyperplane...
lhprelat3N 34344 The Hilbert lattice is rel...
cdlemb2 34345 Given two atoms not under ...
lhple 34346 Property of a lattice elem...
lhpat 34347 Create an atom under a co-...
lhpat4N 34348 Property of an atom under ...
lhpat2 34349 Create an atom under a co-...
lhpat3 34350 There is only one atom und...
4atexlemk 34351 Lemma for ~ 4atexlem7 . (...
4atexlemw 34352 Lemma for ~ 4atexlem7 . (...
4atexlempw 34353 Lemma for ~ 4atexlem7 . (...
4atexlemp 34354 Lemma for ~ 4atexlem7 . (...
4atexlemq 34355 Lemma for ~ 4atexlem7 . (...
4atexlems 34356 Lemma for ~ 4atexlem7 . (...
4atexlemt 34357 Lemma for ~ 4atexlem7 . (...
4atexlemutvt 34358 Lemma for ~ 4atexlem7 . (...
4atexlempnq 34359 Lemma for ~ 4atexlem7 . (...
4atexlemnslpq 34360 Lemma for ~ 4atexlem7 . (...
4atexlemkl 34361 Lemma for ~ 4atexlem7 . (...
4atexlemkc 34362 Lemma for ~ 4atexlem7 . (...
4atexlemwb 34363 Lemma for ~ 4atexlem7 . (...
4atexlempsb 34364 Lemma for ~ 4atexlem7 . (...
4atexlemqtb 34365 Lemma for ~ 4atexlem7 . (...
4atexlempns 34366 Lemma for ~ 4atexlem7 . (...
4atexlemswapqr 34367 Lemma for ~ 4atexlem7 . S...
4atexlemu 34368 Lemma for ~ 4atexlem7 . (...
4atexlemv 34369 Lemma for ~ 4atexlem7 . (...
4atexlemunv 34370 Lemma for ~ 4atexlem7 . (...
4atexlemtlw 34371 Lemma for ~ 4atexlem7 . (...
4atexlemntlpq 34372 Lemma for ~ 4atexlem7 . (...
4atexlemc 34373 Lemma for ~ 4atexlem7 . (...
4atexlemnclw 34374 Lemma for ~ 4atexlem7 . (...
4atexlemex2 34375 Lemma for ~ 4atexlem7 . S...
4atexlemcnd 34376 Lemma for ~ 4atexlem7 . (...
4atexlemex4 34377 Lemma for ~ 4atexlem7 . S...
4atexlemex6 34378 Lemma for ~ 4atexlem7 . (...
4atexlem7 34379 Whenever there are at leas...
4atex 34380 Whenever there are at leas...
4atex2 34381 More general version of ~ ...
4atex2-0aOLDN 34382 Same as ~ 4atex2 except th...
4atex2-0bOLDN 34383 Same as ~ 4atex2 except th...
4atex2-0cOLDN 34384 Same as ~ 4atex2 except th...
4atex3 34385 More general version of ~ ...
lautset 34386 The set of lattice automor...
islaut 34387 The predictate "is a latti...
lautle 34388 Less-than or equal propert...
laut1o 34389 A lattice automorphism is ...
laut11 34390 One-to-one property of a l...
lautcl 34391 A lattice automorphism val...
lautcnvclN 34392 Reverse closure of a latti...
lautcnvle 34393 Less-than or equal propert...
lautcnv 34394 The converse of a lattice ...
lautlt 34395 Less-than property of a la...
lautcvr 34396 Covering property of a lat...
lautj 34397 Meet property of a lattice...
lautm 34398 Meet property of a lattice...
lauteq 34399 A lattice automorphism arg...
idlaut 34400 The identity function is a...
lautco 34401 The composition of two lat...
pautsetN 34402 The set of projective auto...
ispautN 34403 The predictate "is a proje...
ldilfset 34412 The mapping from fiducial ...
ldilset 34413 The set of lattice dilatio...
isldil 34414 The predicate "is a lattic...
ldillaut 34415 A lattice dilation is an a...
ldil1o 34416 A lattice dilation is a on...
ldilval 34417 Value of a lattice dilatio...
idldil 34418 The identity function is a...
ldilcnv 34419 The converse of a lattice ...
ldilco 34420 The composition of two lat...
ltrnfset 34421 The set of all lattice tra...
ltrnset 34422 The set of lattice transla...
isltrn 34423 The predicate "is a lattic...
isltrn2N 34424 The predicate "is a lattic...
ltrnu 34425 Uniqueness property of a l...
ltrnldil 34426 A lattice translation is a...
ltrnlaut 34427 A lattice translation is a...
ltrn1o 34428 A lattice translation is a...
ltrncl 34429 Closure of a lattice trans...
ltrn11 34430 One-to-one property of a l...
ltrncnvnid 34431 If a translation is differ...
ltrncoidN 34432 Two translations are equal...
ltrnle 34433 Less-than or equal propert...
ltrncnvleN 34434 Less-than or equal propert...
ltrnm 34435 Lattice translation of a m...
ltrnj 34436 Lattice translation of a m...
ltrncvr 34437 Covering property of a lat...
ltrnval1 34438 Value of a lattice transla...
ltrnid 34439 A lattice translation is t...
ltrnnid 34440 If a lattice translation i...
ltrnatb 34441 The lattice translation of...
ltrncnvatb 34442 The converse of the lattic...
ltrnel 34443 The lattice translation of...
ltrnat 34444 The lattice translation of...
ltrncnvat 34445 The converse of the lattic...
ltrncnvel 34446 The converse of the lattic...
ltrncoelN 34447 Composition of lattice tra...
ltrncoat 34448 Composition of lattice tra...
ltrncoval 34449 Two ways to express value ...
ltrncnv 34450 The converse of a lattice ...
ltrn11at 34451 Frequently used one-to-one...
ltrneq2 34452 The equality of two transl...
ltrneq 34453 The equality of two transl...
idltrn 34454 The identity function is a...
ltrnmw 34455 Property of lattice transl...
ltrnmwOLD 34456 Property of lattice transl...
dilfsetN 34457 The mapping from fiducial ...
dilsetN 34458 The set of dilations for a...
isdilN 34459 The predicate "is a dilati...
trnfsetN 34460 The mapping from fiducial ...
trnsetN 34461 The set of translations fo...
istrnN 34462 The predicate "is a transl...
trlfset 34465 The set of all traces of l...
trlset 34466 The set of traces of latti...
trlval 34467 The value of the trace of ...
trlval2 34468 The value of the trace of ...
trlcl 34469 Closure of the trace of a ...
trlcnv 34470 The trace of the converse ...
trljat1 34471 The value of a translation...
trljat2 34472 The value of a translation...
trljat3 34473 The value of a translation...
trlat 34474 If an atom differs from it...
trl0 34475 If an atom not under the f...
trlator0 34476 The trace of a lattice tra...
trlatn0 34477 The trace of a lattice tra...
trlnidat 34478 The trace of a lattice tra...
ltrnnidn 34479 If a lattice translation i...
ltrnideq 34480 Property of the identity l...
trlid0 34481 The trace of the identity ...
trlnidatb 34482 A lattice translation is n...
trlid0b 34483 A lattice translation is t...
trlnid 34484 Different translations wit...
ltrn2ateq 34485 Property of the equality o...
ltrnateq 34486 If any atom (under ` W ` )...
ltrnatneq 34487 If any atom (under ` W ` )...
ltrnatlw 34488 If the value of an atom eq...
trlle 34489 The trace of a lattice tra...
trlne 34490 The trace of a lattice tra...
trlnle 34491 The atom not under the fid...
trlval3 34492 The value of the trace of ...
trlval4 34493 The value of the trace of ...
trlval5 34494 The value of the trace of ...
arglem1N 34495 Lemma for Desargues' law. ...
cdlemc1 34496 Part of proof of Lemma C i...
cdlemc2 34497 Part of proof of Lemma C i...
cdlemc3 34498 Part of proof of Lemma C i...
cdlemc4 34499 Part of proof of Lemma C i...
cdlemc5 34500 Lemma for ~ cdlemc . (Con...
cdlemc6 34501 Lemma for ~ cdlemc . (Con...
cdlemc 34502 Lemma C in [Crawley] p. 11...
cdlemd1 34503 Part of proof of Lemma D i...
cdlemd2 34504 Part of proof of Lemma D i...
cdlemd3 34505 Part of proof of Lemma D i...
cdlemd4 34506 Part of proof of Lemma D i...
cdlemd5 34507 Part of proof of Lemma D i...
cdlemd6 34508 Part of proof of Lemma D i...
cdlemd7 34509 Part of proof of Lemma D i...
cdlemd8 34510 Part of proof of Lemma D i...
cdlemd9 34511 Part of proof of Lemma D i...
cdlemd 34512 If two translations agree ...
ltrneq3 34513 Two translations agree at ...
cdleme00a 34514 Part of proof of Lemma E i...
cdleme0aa 34515 Part of proof of Lemma E i...
cdleme0a 34516 Part of proof of Lemma E i...
cdleme0b 34517 Part of proof of Lemma E i...
cdleme0c 34518 Part of proof of Lemma E i...
cdleme0cp 34519 Part of proof of Lemma E i...
cdleme0cq 34520 Part of proof of Lemma E i...
cdleme0dN 34521 Part of proof of Lemma E i...
cdleme0e 34522 Part of proof of Lemma E i...
cdleme0fN 34523 Part of proof of Lemma E i...
cdleme0gN 34524 Part of proof of Lemma E i...
cdlemeulpq 34525 Part of proof of Lemma E i...
cdleme01N 34526 Part of proof of Lemma E i...
cdleme02N 34527 Part of proof of Lemma E i...
cdleme0ex1N 34528 Part of proof of Lemma E i...
cdleme0ex2N 34529 Part of proof of Lemma E i...
cdleme0moN 34530 Part of proof of Lemma E i...
cdleme1b 34531 Part of proof of Lemma E i...
cdleme1 34532 Part of proof of Lemma E i...
cdleme2 34533 Part of proof of Lemma E i...
cdleme3b 34534 Part of proof of Lemma E i...
cdleme3c 34535 Part of proof of Lemma E i...
cdleme3d 34536 Part of proof of Lemma E i...
cdleme3e 34537 Part of proof of Lemma E i...
cdleme3fN 34538 Part of proof of Lemma E i...
cdleme3g 34539 Part of proof of Lemma E i...
cdleme3h 34540 Part of proof of Lemma E i...
cdleme3fa 34541 Part of proof of Lemma E i...
cdleme3 34542 Part of proof of Lemma E i...
cdleme4 34543 Part of proof of Lemma E i...
cdleme4a 34544 Part of proof of Lemma E i...
cdleme5 34545 Part of proof of Lemma E i...
cdleme6 34546 Part of proof of Lemma E i...
cdleme7aa 34547 Part of proof of Lemma E i...
cdleme7a 34548 Part of proof of Lemma E i...
cdleme7b 34549 Part of proof of Lemma E i...
cdleme7c 34550 Part of proof of Lemma E i...
cdleme7d 34551 Part of proof of Lemma E i...
cdleme7e 34552 Part of proof of Lemma E i...
cdleme7ga 34553 Part of proof of Lemma E i...
cdleme7 34554 Part of proof of Lemma E i...
cdleme8 34555 Part of proof of Lemma E i...
cdleme9a 34556 Part of proof of Lemma E i...
cdleme9b 34557 Utility lemma for Lemma E ...
cdleme9 34558 Part of proof of Lemma E i...
cdleme10 34559 Part of proof of Lemma E i...
cdleme8tN 34560 Part of proof of Lemma E i...
cdleme9taN 34561 Part of proof of Lemma E i...
cdleme9tN 34562 Part of proof of Lemma E i...
cdleme10tN 34563 Part of proof of Lemma E i...
cdleme16aN 34564 Part of proof of Lemma E i...
cdleme11a 34565 Part of proof of Lemma E i...
cdleme11c 34566 Part of proof of Lemma E i...
cdleme11dN 34567 Part of proof of Lemma E i...
cdleme11e 34568 Part of proof of Lemma E i...
cdleme11fN 34569 Part of proof of Lemma E i...
cdleme11g 34570 Part of proof of Lemma E i...
cdleme11h 34571 Part of proof of Lemma E i...
cdleme11j 34572 Part of proof of Lemma E i...
cdleme11k 34573 Part of proof of Lemma E i...
cdleme11l 34574 Part of proof of Lemma E i...
cdleme11 34575 Part of proof of Lemma E i...
cdleme12 34576 Part of proof of Lemma E i...
cdleme13 34577 Part of proof of Lemma E i...
cdleme14 34578 Part of proof of Lemma E i...
cdleme15a 34579 Part of proof of Lemma E i...
cdleme15b 34580 Part of proof of Lemma E i...
cdleme15c 34581 Part of proof of Lemma E i...
cdleme15d 34582 Part of proof of Lemma E i...
cdleme15 34583 Part of proof of Lemma E i...
cdleme16b 34584 Part of proof of Lemma E i...
cdleme16c 34585 Part of proof of Lemma E i...
cdleme16d 34586 Part of proof of Lemma E i...
cdleme16e 34587 Part of proof of Lemma E i...
cdleme16f 34588 Part of proof of Lemma E i...
cdleme16g 34589 Part of proof of Lemma E i...
cdleme16 34590 Part of proof of Lemma E i...
cdleme17a 34591 Part of proof of Lemma E i...
cdleme17b 34592 Lemma leading to ~ cdleme1...
cdleme17c 34593 Part of proof of Lemma E i...
cdleme17d1 34594 Part of proof of Lemma E i...
cdleme0nex 34595 Part of proof of Lemma E i...
cdleme18a 34596 Part of proof of Lemma E i...
cdleme18b 34597 Part of proof of Lemma E i...
cdleme18c 34598 Part of proof of Lemma E i...
cdleme22gb 34599 Utility lemma for Lemma E ...
cdleme18d 34600 Part of proof of Lemma E i...
cdlemesner 34601 Part of proof of Lemma E i...
cdlemedb 34602 Part of proof of Lemma E i...
cdlemeda 34603 Part of proof of Lemma E i...
cdlemednpq 34604 Part of proof of Lemma E i...
cdlemednuN 34605 Part of proof of Lemma E i...
cdleme20zN 34606 Part of proof of Lemma E i...
cdleme20y 34607 Part of proof of Lemma E i...
cdleme20yOLD 34608 Part of proof of Lemma E i...
cdleme19a 34609 Part of proof of Lemma E i...
cdleme19b 34610 Part of proof of Lemma E i...
cdleme19c 34611 Part of proof of Lemma E i...
cdleme19d 34612 Part of proof of Lemma E i...
cdleme19e 34613 Part of proof of Lemma E i...
cdleme19f 34614 Part of proof of Lemma E i...
cdleme20aN 34615 Part of proof of Lemma E i...
cdleme20bN 34616 Part of proof of Lemma E i...
cdleme20c 34617 Part of proof of Lemma E i...
cdleme20d 34618 Part of proof of Lemma E i...
cdleme20e 34619 Part of proof of Lemma E i...
cdleme20f 34620 Part of proof of Lemma E i...
cdleme20g 34621 Part of proof of Lemma E i...
cdleme20h 34622 Part of proof of Lemma E i...
cdleme20i 34623 Part of proof of Lemma E i...
cdleme20j 34624 Part of proof of Lemma E i...
cdleme20k 34625 Part of proof of Lemma E i...
cdleme20l1 34626 Part of proof of Lemma E i...
cdleme20l2 34627 Part of proof of Lemma E i...
cdleme20l 34628 Part of proof of Lemma E i...
cdleme20m 34629 Part of proof of Lemma E i...
cdleme20 34630 Combine ~ cdleme19f and ~ ...
cdleme21a 34631 Part of proof of Lemma E i...
cdleme21b 34632 Part of proof of Lemma E i...
cdleme21c 34633 Part of proof of Lemma E i...
cdleme21at 34634 Part of proof of Lemma E i...
cdleme21ct 34635 Part of proof of Lemma E i...
cdleme21d 34636 Part of proof of Lemma E i...
cdleme21e 34637 Part of proof of Lemma E i...
cdleme21f 34638 Part of proof of Lemma E i...
cdleme21g 34639 Part of proof of Lemma E i...
cdleme21h 34640 Part of proof of Lemma E i...
cdleme21i 34641 Part of proof of Lemma E i...
cdleme21j 34642 Combine ~ cdleme20 and ~ c...
cdleme21 34643 Part of proof of Lemma E i...
cdleme21k 34644 Eliminate ` S =/= T ` cond...
cdleme22aa 34645 Part of proof of Lemma E i...
cdleme22a 34646 Part of proof of Lemma E i...
cdleme22b 34647 Part of proof of Lemma E i...
cdleme22cN 34648 Part of proof of Lemma E i...
cdleme22d 34649 Part of proof of Lemma E i...
cdleme22e 34650 Part of proof of Lemma E i...
cdleme22eALTN 34651 Part of proof of Lemma E i...
cdleme22f 34652 Part of proof of Lemma E i...
cdleme22f2 34653 Part of proof of Lemma E i...
cdleme22g 34654 Part of proof of Lemma E i...
cdleme23a 34655 Part of proof of Lemma E i...
cdleme23b 34656 Part of proof of Lemma E i...
cdleme23c 34657 Part of proof of Lemma E i...
cdleme24 34658 Quantified version of ~ cd...
cdleme25a 34659 Lemma for ~ cdleme25b . (...
cdleme25b 34660 Transform ~ cdleme24 . TO...
cdleme25c 34661 Transform ~ cdleme25b . (...
cdleme25dN 34662 Transform ~ cdleme25c . (...
cdleme25cl 34663 Show closure of the unique...
cdleme25cv 34664 Change bound variables in ...
cdleme26e 34665 Part of proof of Lemma E i...
cdleme26ee 34666 Part of proof of Lemma E i...
cdleme26eALTN 34667 Part of proof of Lemma E i...
cdleme26fALTN 34668 Part of proof of Lemma E i...
cdleme26f 34669 Part of proof of Lemma E i...
cdleme26f2ALTN 34670 Part of proof of Lemma E i...
cdleme26f2 34671 Part of proof of Lemma E i...
cdleme27cl 34672 Part of proof of Lemma E i...
cdleme27a 34673 Part of proof of Lemma E i...
cdleme27b 34674 Lemma for ~ cdleme27N . (...
cdleme27N 34675 Part of proof of Lemma E i...
cdleme28a 34676 Lemma for ~ cdleme25b . T...
cdleme28b 34677 Lemma for ~ cdleme25b . T...
cdleme28c 34678 Part of proof of Lemma E i...
cdleme28 34679 Quantified version of ~ cd...
cdleme29ex 34680 Lemma for ~ cdleme29b . (...
cdleme29b 34681 Transform ~ cdleme28 . (C...
cdleme29c 34682 Transform ~ cdleme28b . (...
cdleme29cl 34683 Show closure of the unique...
cdleme30a 34684 Part of proof of Lemma E i...
cdleme31so 34685 Part of proof of Lemma E i...
cdleme31sn 34686 Part of proof of Lemma E i...
cdleme31sn1 34687 Part of proof of Lemma E i...
cdleme31se 34688 Part of proof of Lemma D i...
cdleme31se2 34689 Part of proof of Lemma D i...
cdleme31sc 34690 Part of proof of Lemma E i...
cdleme31sde 34691 Part of proof of Lemma D i...
cdleme31snd 34692 Part of proof of Lemma D i...
cdleme31sdnN 34693 Part of proof of Lemma E i...
cdleme31sn1c 34694 Part of proof of Lemma E i...
cdleme31sn2 34695 Part of proof of Lemma E i...
cdleme31fv 34696 Part of proof of Lemma E i...
cdleme31fv1 34697 Part of proof of Lemma E i...
cdleme31fv1s 34698 Part of proof of Lemma E i...
cdleme31fv2 34699 Part of proof of Lemma E i...
cdleme31id 34700 Part of proof of Lemma E i...
cdlemefrs29pre00 34701 ***START OF VALUE AT ATOM ...
cdlemefrs29bpre0 34702 TODO fix comment. (Contri...
cdlemefrs29bpre1 34703 TODO: FIX COMMENT. (Contr...
cdlemefrs29cpre1 34704 TODO: FIX COMMENT. (Contr...
cdlemefrs29clN 34705 TODO: NOT USED? Show clo...
cdlemefrs32fva 34706 Part of proof of Lemma E i...
cdlemefrs32fva1 34707 Part of proof of Lemma E i...
cdlemefr29exN 34708 Lemma for ~ cdlemefs29bpre...
cdlemefr27cl 34709 Part of proof of Lemma E i...
cdlemefr32sn2aw 34710 Show that ` [_ R / s ]_ N ...
cdlemefr32snb 34711 Show closure of ` [_ R / s...
cdlemefr29bpre0N 34712 TODO fix comment. (Contri...
cdlemefr29clN 34713 Show closure of the unique...
cdleme43frv1snN 34714 Value of ` [_ R / s ]_ N `...
cdlemefr32fvaN 34715 Part of proof of Lemma E i...
cdlemefr32fva1 34716 Part of proof of Lemma E i...
cdlemefr31fv1 34717 Value of ` ( F `` R ) ` wh...
cdlemefs29pre00N 34718 FIX COMMENT. TODO: see if ...
cdlemefs27cl 34719 Part of proof of Lemma E i...
cdlemefs32sn1aw 34720 Show that ` [_ R / s ]_ N ...
cdlemefs32snb 34721 Show closure of ` [_ R / s...
cdlemefs29bpre0N 34722 TODO: FIX COMMENT. (Contr...
cdlemefs29bpre1N 34723 TODO: FIX COMMENT. (Contr...
cdlemefs29cpre1N 34724 TODO: FIX COMMENT. (Contr...
cdlemefs29clN 34725 Show closure of the unique...
cdleme43fsv1snlem 34726 Value of ` [_ R / s ]_ N `...
cdleme43fsv1sn 34727 Value of ` [_ R / s ]_ N `...
cdlemefs32fvaN 34728 Part of proof of Lemma E i...
cdlemefs32fva1 34729 Part of proof of Lemma E i...
cdlemefs31fv1 34730 Value of ` ( F `` R ) ` wh...
cdlemefr44 34731 Value of f(r) when r is an...
cdlemefs44 34732 Value of f_s(r) when r is ...
cdlemefr45 34733 Value of f(r) when r is an...
cdlemefr45e 34734 Explicit expansion of ~ cd...
cdlemefs45 34735 Value of f_s(r) when r is ...
cdlemefs45ee 34736 Explicit expansion of ~ cd...
cdlemefs45eN 34737 Explicit expansion of ~ cd...
cdleme32sn1awN 34738 Show that ` [_ R / s ]_ N ...
cdleme41sn3a 34739 Show that ` [_ R / s ]_ N ...
cdleme32sn2awN 34740 Show that ` [_ R / s ]_ N ...
cdleme32snaw 34741 Show that ` [_ R / s ]_ N ...
cdleme32snb 34742 Show closure of ` [_ R / s...
cdleme32fva 34743 Part of proof of Lemma D i...
cdleme32fva1 34744 Part of proof of Lemma D i...
cdleme32fvaw 34745 Show that ` ( F `` R ) ` i...
cdleme32fvcl 34746 Part of proof of Lemma D i...
cdleme32a 34747 Part of proof of Lemma D i...
cdleme32b 34748 Part of proof of Lemma D i...
cdleme32c 34749 Part of proof of Lemma D i...
cdleme32d 34750 Part of proof of Lemma D i...
cdleme32e 34751 Part of proof of Lemma D i...
cdleme32f 34752 Part of proof of Lemma D i...
cdleme32le 34753 Part of proof of Lemma D i...
cdleme35a 34754 Part of proof of Lemma E i...
cdleme35fnpq 34755 Part of proof of Lemma E i...
cdleme35b 34756 Part of proof of Lemma E i...
cdleme35c 34757 Part of proof of Lemma E i...
cdleme35d 34758 Part of proof of Lemma E i...
cdleme35e 34759 Part of proof of Lemma E i...
cdleme35f 34760 Part of proof of Lemma E i...
cdleme35g 34761 Part of proof of Lemma E i...
cdleme35h 34762 Part of proof of Lemma E i...
cdleme35h2 34763 Part of proof of Lemma E i...
cdleme35sn2aw 34764 Part of proof of Lemma E i...
cdleme35sn3a 34765 Part of proof of Lemma E i...
cdleme36a 34766 Part of proof of Lemma E i...
cdleme36m 34767 Part of proof of Lemma E i...
cdleme37m 34768 Part of proof of Lemma E i...
cdleme38m 34769 Part of proof of Lemma E i...
cdleme38n 34770 Part of proof of Lemma E i...
cdleme39a 34771 Part of proof of Lemma E i...
cdleme39n 34772 Part of proof of Lemma E i...
cdleme40m 34773 Part of proof of Lemma E i...
cdleme40n 34774 Part of proof of Lemma E i...
cdleme40v 34775 Part of proof of Lemma E i...
cdleme40w 34776 Part of proof of Lemma E i...
cdleme42a 34777 Part of proof of Lemma E i...
cdleme42c 34778 Part of proof of Lemma E i...
cdleme42d 34779 Part of proof of Lemma E i...
cdleme41sn3aw 34780 Part of proof of Lemma E i...
cdleme41sn4aw 34781 Part of proof of Lemma E i...
cdleme41snaw 34782 Part of proof of Lemma E i...
cdleme41fva11 34783 Part of proof of Lemma E i...
cdleme42b 34784 Part of proof of Lemma E i...
cdleme42e 34785 Part of proof of Lemma E i...
cdleme42f 34786 Part of proof of Lemma E i...
cdleme42g 34787 Part of proof of Lemma E i...
cdleme42h 34788 Part of proof of Lemma E i...
cdleme42i 34789 Part of proof of Lemma E i...
cdleme42k 34790 Part of proof of Lemma E i...
cdleme42ke 34791 Part of proof of Lemma E i...
cdleme42keg 34792 Part of proof of Lemma E i...
cdleme42mN 34793 Part of proof of Lemma E i...
cdleme42mgN 34794 Part of proof of Lemma E i...
cdleme43aN 34795 Part of proof of Lemma E i...
cdleme43bN 34796 Lemma for Lemma E in [Craw...
cdleme43cN 34797 Part of proof of Lemma E i...
cdleme43dN 34798 Part of proof of Lemma E i...
cdleme46f2g2 34799 Conversion for ` G ` to re...
cdleme46f2g1 34800 Conversion for ` G ` to re...
cdleme17d2 34801 Part of proof of Lemma E i...
cdleme17d3 34802 TODO: FIX COMMENT. (Contr...
cdleme17d4 34803 TODO: FIX COMMENT. (Contr...
cdleme17d 34804 Part of proof of Lemma E i...
cdleme48fv 34805 Part of proof of Lemma D i...
cdleme48fvg 34806 Remove ` P =/= Q ` conditi...
cdleme46fvaw 34807 Show that ` ( F `` R ) ` i...
cdleme48bw 34808 TODO: fix comment. TODO: ...
cdleme48b 34809 TODO: fix comment. (Contr...
cdleme46frvlpq 34810 Show that ` ( F `` S ) ` i...
cdleme46fsvlpq 34811 Show that ` ( F `` R ) ` i...
cdlemeg46fvcl 34812 TODO: fix comment. (Contr...
cdleme4gfv 34813 Part of proof of Lemma D i...
cdlemeg47b 34814 TODO: FIX COMMENT. (Contr...
cdlemeg47rv 34815 Value of g_s(r) when r is ...
cdlemeg47rv2 34816 Value of g_s(r) when r is ...
cdlemeg49le 34817 Part of proof of Lemma D i...
cdlemeg46bOLDN 34818 TODO FIX COMMENT. (Contrib...
cdlemeg46c 34819 TODO FIX COMMENT. (Contrib...
cdlemeg46rvOLDN 34820 Value of g_s(r) when r is ...
cdlemeg46rv2OLDN 34821 Value of g_s(r) when r is ...
cdlemeg46fvaw 34822 Show that ` ( F `` R ) ` i...
cdlemeg46nlpq 34823 Show that ` ( G `` S ) ` i...
cdlemeg46ngfr 34824 TODO FIX COMMENT g(f(s))=s...
cdlemeg46nfgr 34825 TODO FIX COMMENT f(g(s))=s...
cdlemeg46sfg 34826 TODO FIX COMMENT f(r) ` \/...
cdlemeg46fjgN 34827 NOT NEEDED? TODO FIX COMM...
cdlemeg46rjgN 34828 NOT NEEDED? TODO FIX COMM...
cdlemeg46fjv 34829 TODO FIX COMMENT f(r) ` \/...
cdlemeg46fsfv 34830 TODO FIX COMMENT f(r) ` \/...
cdlemeg46frv 34831 TODO FIX COMMENT. (f(r) ` ...
cdlemeg46v1v2 34832 TODO FIX COMMENT v_1 = v_2...
cdlemeg46vrg 34833 TODO FIX COMMENT v_1 ` <_ ...
cdlemeg46rgv 34834 TODO FIX COMMENT r ` <_ ` ...
cdlemeg46req 34835 TODO FIX COMMENT r = (v_1 ...
cdlemeg46gfv 34836 TODO FIX COMMENT p. 115 pe...
cdlemeg46gfr 34837 TODO FIX COMMENT p. 116 pe...
cdlemeg46gfre 34838 TODO FIX COMMENT p. 116 pe...
cdlemeg46gf 34839 TODO FIX COMMENT Eliminate...
cdlemeg46fgN 34840 TODO FIX COMMENT p. 116 pe...
cdleme48d 34841 TODO: fix comment. (Contr...
cdleme48gfv1 34842 TODO: fix comment. (Contr...
cdleme48gfv 34843 TODO: fix comment. (Contr...
cdleme48fgv 34844 TODO: fix comment. (Contr...
cdlemeg49lebilem 34845 Part of proof of Lemma D i...
cdleme50lebi 34846 Part of proof of Lemma D i...
cdleme50eq 34847 Part of proof of Lemma D i...
cdleme50f 34848 Part of proof of Lemma D i...
cdleme50f1 34849 Part of proof of Lemma D i...
cdleme50rnlem 34850 Part of proof of Lemma D i...
cdleme50rn 34851 Part of proof of Lemma D i...
cdleme50f1o 34852 Part of proof of Lemma D i...
cdleme50laut 34853 Part of proof of Lemma D i...
cdleme50ldil 34854 Part of proof of Lemma D i...
cdleme50trn1 34855 Part of proof that ` F ` i...
cdleme50trn2a 34856 Part of proof that ` F ` i...
cdleme50trn2 34857 Part of proof that ` F ` i...
cdleme50trn12 34858 Part of proof that ` F ` i...
cdleme50trn3 34859 Part of proof that ` F ` i...
cdleme50trn123 34860 Part of proof that ` F ` i...
cdleme51finvfvN 34861 Part of proof of Lemma E i...
cdleme51finvN 34862 Part of proof of Lemma E i...
cdleme50ltrn 34863 Part of proof of Lemma E i...
cdleme51finvtrN 34864 Part of proof of Lemma E i...
cdleme50ex 34865 Part of Lemma E in [Crawle...
cdleme 34866 Lemma E in [Crawley] p. 11...
cdlemf1 34867 Part of Lemma F in [Crawle...
cdlemf2 34868 Part of Lemma F in [Crawle...
cdlemf 34869 Lemma F in [Crawley] p. 11...
cdlemfnid 34870 ~ cdlemf with additional c...
cdlemftr3 34871 Special case of ~ cdlemf s...
cdlemftr2 34872 Special case of ~ cdlemf s...
cdlemftr1 34873 Part of proof of Lemma G o...
cdlemftr0 34874 Special case of ~ cdlemf s...
trlord 34875 The ordering of two Hilber...
cdlemg1a 34876 Shorter expression for ` G...
cdlemg1b2 34877 This theorem can be used t...
cdlemg1idlemN 34878 Lemma for ~ cdlemg1idN . ...
cdlemg1fvawlemN 34879 Lemma for ~ ltrniotafvawN ...
cdlemg1ltrnlem 34880 Lemma for ~ ltrniotacl . ...
cdlemg1finvtrlemN 34881 Lemma for ~ ltrniotacnvN ....
cdlemg1bOLDN 34882 This theorem can be used t...
cdlemg1idN 34883 Version of ~ cdleme31id wi...
ltrniotafvawN 34884 Version of ~ cdleme46fvaw ...
ltrniotacl 34885 Version of ~ cdleme50ltrn ...
ltrniotacnvN 34886 Version of ~ cdleme51finvt...
ltrniotaval 34887 Value of the unique transl...
ltrniotacnvval 34888 Converse value of the uniq...
ltrniotaidvalN 34889 Value of the unique transl...
ltrniotavalbN 34890 Value of the unique transl...
cdlemeiota 34891 A translation is uniquely ...
cdlemg1ci2 34892 Any function of the form o...
cdlemg1cN 34893 Any translation belongs to...
cdlemg1cex 34894 Any translation is one of ...
cdlemg2cN 34895 Any translation belongs to...
cdlemg2dN 34896 This theorem can be used t...
cdlemg2cex 34897 Any translation is one of ...
cdlemg2ce 34898 Utility theorem to elimina...
cdlemg2jlemOLDN 34899 Part of proof of Lemma E i...
cdlemg2fvlem 34900 Lemma for ~ cdlemg2fv . (...
cdlemg2klem 34901 ~ cdleme42keg with simpler...
cdlemg2idN 34902 Version of ~ cdleme31id wi...
cdlemg3a 34903 Part of proof of Lemma G i...
cdlemg2jOLDN 34904 TODO: Replace this with ~...
cdlemg2fv 34905 Value of a translation in ...
cdlemg2fv2 34906 Value of a translation in ...
cdlemg2k 34907 ~ cdleme42keg with simpler...
cdlemg2kq 34908 ~ cdlemg2k with ` P ` and ...
cdlemg2l 34909 TODO: FIX COMMENT. (Contr...
cdlemg2m 34910 TODO: FIX COMMENT. (Contr...
cdlemg5 34911 TODO: Is there a simpler ...
cdlemb3 34912 Given two atoms not under ...
cdlemg7fvbwN 34913 Properties of a translatio...
cdlemg4a 34914 TODO: FIX COMMENT If fg(p...
cdlemg4b1 34915 TODO: FIX COMMENT. (Contr...
cdlemg4b2 34916 TODO: FIX COMMENT. (Contr...
cdlemg4b12 34917 TODO: FIX COMMENT. (Contr...
cdlemg4c 34918 TODO: FIX COMMENT. (Contr...
cdlemg4d 34919 TODO: FIX COMMENT. (Contr...
cdlemg4e 34920 TODO: FIX COMMENT. (Contr...
cdlemg4f 34921 TODO: FIX COMMENT. (Contr...
cdlemg4g 34922 TODO: FIX COMMENT. (Contr...
cdlemg4 34923 TODO: FIX COMMENT. (Contr...
cdlemg6a 34924 TODO: FIX COMMENT. TODO: ...
cdlemg6b 34925 TODO: FIX COMMENT. TODO: ...
cdlemg6c 34926 TODO: FIX COMMENT. (Contr...
cdlemg6d 34927 TODO: FIX COMMENT. (Contr...
cdlemg6e 34928 TODO: FIX COMMENT. (Contr...
cdlemg6 34929 TODO: FIX COMMENT. (Contr...
cdlemg7fvN 34930 Value of a translation com...
cdlemg7aN 34931 TODO: FIX COMMENT. (Contr...
cdlemg7N 34932 TODO: FIX COMMENT. (Contr...
cdlemg8a 34933 TODO: FIX COMMENT. (Contr...
cdlemg8b 34934 TODO: FIX COMMENT. (Contr...
cdlemg8c 34935 TODO: FIX COMMENT. (Contr...
cdlemg8d 34936 TODO: FIX COMMENT. (Contr...
cdlemg8 34937 TODO: FIX COMMENT. (Contr...
cdlemg9a 34938 TODO: FIX COMMENT. (Contr...
cdlemg9b 34939 The triples ` <. P , ( F `...
cdlemg9 34940 The triples ` <. P , ( F `...
cdlemg10b 34941 TODO: FIX COMMENT. TODO: ...
cdlemg10bALTN 34942 TODO: FIX COMMENT. TODO: ...
cdlemg11a 34943 TODO: FIX COMMENT. (Contr...
cdlemg11aq 34944 TODO: FIX COMMENT. TODO: ...
cdlemg10c 34945 TODO: FIX COMMENT. TODO: ...
cdlemg10a 34946 TODO: FIX COMMENT. (Contr...
cdlemg10 34947 TODO: FIX COMMENT. (Contr...
cdlemg11b 34948 TODO: FIX COMMENT. (Contr...
cdlemg12a 34949 TODO: FIX COMMENT. (Contr...
cdlemg12b 34950 The triples ` <. P , ( F `...
cdlemg12c 34951 The triples ` <. P , ( F `...
cdlemg12d 34952 TODO: FIX COMMENT. (Contr...
cdlemg12e 34953 TODO: FIX COMMENT. (Contr...
cdlemg12f 34954 TODO: FIX COMMENT. (Contr...
cdlemg12g 34955 TODO: FIX COMMENT. TODO: ...
cdlemg12 34956 TODO: FIX COMMENT. (Contr...
cdlemg13a 34957 TODO: FIX COMMENT. (Contr...
cdlemg13 34958 TODO: FIX COMMENT. (Contr...
cdlemg14f 34959 TODO: FIX COMMENT. (Contr...
cdlemg14g 34960 TODO: FIX COMMENT. (Contr...
cdlemg15a 34961 Eliminate the ` ( F `` P )...
cdlemg15 34962 Eliminate the ` ( (...
cdlemg16 34963 Part of proof of Lemma G o...
cdlemg16ALTN 34964 This version of ~ cdlemg16...
cdlemg16z 34965 Eliminate ` ( ( F `...
cdlemg16zz 34966 Eliminate ` P =/= Q ` from...
cdlemg17a 34967 TODO: FIX COMMENT. (Contr...
cdlemg17b 34968 Part of proof of Lemma G i...
cdlemg17dN 34969 TODO: fix comment. (Contr...
cdlemg17dALTN 34970 Same as ~ cdlemg17dN with ...
cdlemg17e 34971 TODO: fix comment. (Contr...
cdlemg17f 34972 TODO: fix comment. (Contr...
cdlemg17g 34973 TODO: fix comment. (Contr...
cdlemg17h 34974 TODO: fix comment. (Contr...
cdlemg17i 34975 TODO: fix comment. (Contr...
cdlemg17ir 34976 TODO: fix comment. (Contr...
cdlemg17j 34977 TODO: fix comment. (Contr...
cdlemg17pq 34978 Utility theorem for swappi...
cdlemg17bq 34979 ~ cdlemg17b with ` P ` and...
cdlemg17iqN 34980 ~ cdlemg17i with ` P ` and...
cdlemg17irq 34981 ~ cdlemg17ir with ` P ` an...
cdlemg17jq 34982 ~ cdlemg17j with ` P ` and...
cdlemg17 34983 Part of Lemma G of [Crawle...
cdlemg18a 34984 Show two lines are differe...
cdlemg18b 34985 Lemma for ~ cdlemg18c . T...
cdlemg18c 34986 Show two lines intersect a...
cdlemg18d 34987 Show two lines intersect a...
cdlemg18 34988 Show two lines intersect a...
cdlemg19a 34989 Show two lines intersect a...
cdlemg19 34990 Show two lines intersect a...
cdlemg20 34991 Show two lines intersect a...
cdlemg21 34992 Version of cdlemg19 with `...
cdlemg22 34993 ~ cdlemg21 with ` ( F `` P...
cdlemg24 34994 Combine ~ cdlemg16z and ~ ...
cdlemg37 34995 Use ~ cdlemg8 to eliminate...
cdlemg25zz 34996 ~ cdlemg16zz restated for ...
cdlemg26zz 34997 ~ cdlemg16zz restated for ...
cdlemg27a 34998 For use with case when ` (...
cdlemg28a 34999 Part of proof of Lemma G o...
cdlemg31b0N 35000 TODO: Fix comment. (Cont...
cdlemg31b0a 35001 TODO: Fix comment. (Cont...
cdlemg27b 35002 TODO: Fix comment. (Cont...
cdlemg31a 35003 TODO: fix comment. (Contr...
cdlemg31b 35004 TODO: fix comment. (Contr...
cdlemg31c 35005 Show that when ` N ` is an...
cdlemg31d 35006 Eliminate ` ( F `` P ) =/=...
cdlemg33b0 35007 TODO: Fix comment. (Cont...
cdlemg33c0 35008 TODO: Fix comment. (Cont...
cdlemg28b 35009 Part of proof of Lemma G o...
cdlemg28 35010 Part of proof of Lemma G o...
cdlemg29 35011 Eliminate ` ( F `` P ) =/=...
cdlemg33a 35012 TODO: Fix comment. (Cont...
cdlemg33b 35013 TODO: Fix comment. (Cont...
cdlemg33c 35014 TODO: Fix comment. (Cont...
cdlemg33d 35015 TODO: Fix comment. (Cont...
cdlemg33e 35016 TODO: Fix comment. (Cont...
cdlemg33 35017 Combine ~ cdlemg33b , ~ cd...
cdlemg34 35018 Use cdlemg33 to eliminate ...
cdlemg35 35019 TODO: Fix comment. TODO:...
cdlemg36 35020 Use cdlemg35 to eliminate ...
cdlemg38 35021 Use ~ cdlemg37 to eliminat...
cdlemg39 35022 Eliminate ` =/= ` conditio...
cdlemg40 35023 Eliminate ` P =/= Q ` cond...
cdlemg41 35024 Convert ~ cdlemg40 to func...
ltrnco 35025 The composition of two tra...
trlcocnv 35026 Swap the arguments of the ...
trlcoabs 35027 Absorption into a composit...
trlcoabs2N 35028 Absorption of the trace of...
trlcoat 35029 The trace of a composition...
trlcocnvat 35030 Commonly used special case...
trlconid 35031 The composition of two dif...
trlcolem 35032 Lemma for ~ trlco . (Cont...
trlco 35033 The trace of a composition...
trlcone 35034 If two translations have d...
cdlemg42 35035 Part of proof of Lemma G o...
cdlemg43 35036 Part of proof of Lemma G o...
cdlemg44a 35037 Part of proof of Lemma G o...
cdlemg44b 35038 Eliminate ` ( F `` P ) =/=...
cdlemg44 35039 Part of proof of Lemma G o...
cdlemg47a 35040 TODO: fix comment. TODO: ...
cdlemg46 35041 Part of proof of Lemma G o...
cdlemg47 35042 Part of proof of Lemma G o...
cdlemg48 35043 Elmininate ` h ` from ~ cd...
ltrncom 35044 Composition is commutative...
ltrnco4 35045 Rearrange a composition of...
trljco 35046 Trace joined with trace of...
trljco2 35047 Trace joined with trace of...
tgrpfset 35050 The translation group maps...
tgrpset 35051 The translation group for ...
tgrpbase 35052 The base set of the transl...
tgrpopr 35053 The group operation of the...
tgrpov 35054 The group operation value ...
tgrpgrplem 35055 Lemma for ~ tgrpgrp . (Co...
tgrpgrp 35056 The translation group is a...
tgrpabl 35057 The translation group is a...
tendofset 35064 The set of all trace-prese...
tendoset 35065 The set of trace-preservin...
istendo 35066 The predicate "is a trace-...
tendotp 35067 Trace-preserving property ...
istendod 35068 Deduce the predicate "is a...
tendof 35069 Functionality of a trace-p...
tendoeq1 35070 Condition determining equa...
tendovalco 35071 Value of composition of tr...
tendocoval 35072 Value of composition of en...
tendocl 35073 Closure of a trace-preserv...
tendoco2 35074 Distribution of compositio...
tendoidcl 35075 The identity is a trace-pr...
tendo1mul 35076 Multiplicative identity mu...
tendo1mulr 35077 Multiplicative identity mu...
tendococl 35078 The composition of two tra...
tendoid 35079 The identity value of a tr...
tendoeq2 35080 Condition determining equa...
tendoplcbv 35081 Define sum operation for t...
tendopl 35082 Value of endomorphism sum ...
tendopl2 35083 Value of result of endomor...
tendoplcl2 35084 Value of result of endomor...
tendoplco2 35085 Value of result of endomor...
tendopltp 35086 Trace-preserving property ...
tendoplcl 35087 Endomorphism sum is a trac...
tendoplcom 35088 The endomorphism sum opera...
tendoplass 35089 The endomorphism sum opera...
tendodi1 35090 Endomorphism composition d...
tendodi2 35091 Endomorphism composition d...
tendo0cbv 35092 Define additive identity f...
tendo02 35093 Value of additive identity...
tendo0co2 35094 The additive identity trac...
tendo0tp 35095 Trace-preserving property ...
tendo0cl 35096 The additive identity is a...
tendo0pl 35097 Property of the additive i...
tendo0plr 35098 Property of the additive i...
tendoicbv 35099 Define inverse function fo...
tendoi 35100 Value of inverse endomorph...
tendoi2 35101 Value of additive inverse ...
tendoicl 35102 Closure of the additive in...
tendoipl 35103 Property of the additive i...
tendoipl2 35104 Property of the additive i...
erngfset 35105 The division rings on trac...
erngset 35106 The division ring on trace...
erngbase 35107 The base set of the divisi...
erngfplus 35108 Ring addition operation. ...
erngplus 35109 Ring addition operation. ...
erngplus2 35110 Ring addition operation. ...
erngfmul 35111 Ring multiplication operat...
erngmul 35112 Ring addition operation. ...
erngfset-rN 35113 The division rings on trac...
erngset-rN 35114 The division ring on trace...
erngbase-rN 35115 The base set of the divisi...
erngfplus-rN 35116 Ring addition operation. ...
erngplus-rN 35117 Ring addition operation. ...
erngplus2-rN 35118 Ring addition operation. ...
erngfmul-rN 35119 Ring multiplication operat...
erngmul-rN 35120 Ring addition operation. ...
cdlemh1 35121 Part of proof of Lemma H o...
cdlemh2 35122 Part of proof of Lemma H o...
cdlemh 35123 Lemma H of [Crawley] p. 11...
cdlemi1 35124 Part of proof of Lemma I o...
cdlemi2 35125 Part of proof of Lemma I o...
cdlemi 35126 Lemma I of [Crawley] p. 11...
cdlemj1 35127 Part of proof of Lemma J o...
cdlemj2 35128 Part of proof of Lemma J o...
cdlemj3 35129 Part of proof of Lemma J o...
tendocan 35130 Cancellation law: if the v...
tendoid0 35131 A trace-preserving endomor...
tendo0mul 35132 Additive identity multipli...
tendo0mulr 35133 Additive identity multipli...
tendo1ne0 35134 The identity (unity) is no...
tendoconid 35135 The composition (product) ...
tendotr 35136 The trace of the value of ...
cdlemk1 35137 Part of proof of Lemma K o...
cdlemk2 35138 Part of proof of Lemma K o...
cdlemk3 35139 Part of proof of Lemma K o...
cdlemk4 35140 Part of proof of Lemma K o...
cdlemk5a 35141 Part of proof of Lemma K o...
cdlemk5 35142 Part of proof of Lemma K o...
cdlemk6 35143 Part of proof of Lemma K o...
cdlemk8 35144 Part of proof of Lemma K o...
cdlemk9 35145 Part of proof of Lemma K o...
cdlemk9bN 35146 Part of proof of Lemma K o...
cdlemki 35147 Part of proof of Lemma K o...
cdlemkvcl 35148 Part of proof of Lemma K o...
cdlemk10 35149 Part of proof of Lemma K o...
cdlemksv 35150 Part of proof of Lemma K o...
cdlemksel 35151 Part of proof of Lemma K o...
cdlemksat 35152 Part of proof of Lemma K o...
cdlemksv2 35153 Part of proof of Lemma K o...
cdlemk7 35154 Part of proof of Lemma K o...
cdlemk11 35155 Part of proof of Lemma K o...
cdlemk12 35156 Part of proof of Lemma K o...
cdlemkoatnle 35157 Utility lemma. (Contribut...
cdlemk13 35158 Part of proof of Lemma K o...
cdlemkole 35159 Utility lemma. (Contribut...
cdlemk14 35160 Part of proof of Lemma K o...
cdlemk15 35161 Part of proof of Lemma K o...
cdlemk16a 35162 Part of proof of Lemma K o...
cdlemk16 35163 Part of proof of Lemma K o...
cdlemk17 35164 Part of proof of Lemma K o...
cdlemk1u 35165 Part of proof of Lemma K o...
cdlemk5auN 35166 Part of proof of Lemma K o...
cdlemk5u 35167 Part of proof of Lemma K o...
cdlemk6u 35168 Part of proof of Lemma K o...
cdlemkj 35169 Part of proof of Lemma K o...
cdlemkuvN 35170 Part of proof of Lemma K o...
cdlemkuel 35171 Part of proof of Lemma K o...
cdlemkuat 35172 Part of proof of Lemma K o...
cdlemkuv2 35173 Part of proof of Lemma K o...
cdlemk18 35174 Part of proof of Lemma K o...
cdlemk19 35175 Part of proof of Lemma K o...
cdlemk7u 35176 Part of proof of Lemma K o...
cdlemk11u 35177 Part of proof of Lemma K o...
cdlemk12u 35178 Part of proof of Lemma K o...
cdlemk21N 35179 Part of proof of Lemma K o...
cdlemk20 35180 Part of proof of Lemma K o...
cdlemkoatnle-2N 35181 Utility lemma. (Contribut...
cdlemk13-2N 35182 Part of proof of Lemma K o...
cdlemkole-2N 35183 Utility lemma. (Contribut...
cdlemk14-2N 35184 Part of proof of Lemma K o...
cdlemk15-2N 35185 Part of proof of Lemma K o...
cdlemk16-2N 35186 Part of proof of Lemma K o...
cdlemk17-2N 35187 Part of proof of Lemma K o...
cdlemkj-2N 35188 Part of proof of Lemma K o...
cdlemkuv-2N 35189 Part of proof of Lemma K o...
cdlemkuel-2N 35190 Part of proof of Lemma K o...
cdlemkuv2-2 35191 Part of proof of Lemma K o...
cdlemk18-2N 35192 Part of proof of Lemma K o...
cdlemk19-2N 35193 Part of proof of Lemma K o...
cdlemk7u-2N 35194 Part of proof of Lemma K o...
cdlemk11u-2N 35195 Part of proof of Lemma K o...
cdlemk12u-2N 35196 Part of proof of Lemma K o...
cdlemk21-2N 35197 Part of proof of Lemma K o...
cdlemk20-2N 35198 Part of proof of Lemma K o...
cdlemk22 35199 Part of proof of Lemma K o...
cdlemk30 35200 Part of proof of Lemma K o...
cdlemkuu 35201 Convert between function a...
cdlemk31 35202 Part of proof of Lemma K o...
cdlemk32 35203 Part of proof of Lemma K o...
cdlemkuel-3 35204 Part of proof of Lemma K o...
cdlemkuv2-3N 35205 Part of proof of Lemma K o...
cdlemk18-3N 35206 Part of proof of Lemma K o...
cdlemk22-3 35207 Part of proof of Lemma K o...
cdlemk23-3 35208 Part of proof of Lemma K o...
cdlemk24-3 35209 Part of proof of Lemma K o...
cdlemk25-3 35210 Part of proof of Lemma K o...
cdlemk26b-3 35211 Part of proof of Lemma K o...
cdlemk26-3 35212 Part of proof of Lemma K o...
cdlemk27-3 35213 Part of proof of Lemma K o...
cdlemk28-3 35214 Part of proof of Lemma K o...
cdlemk33N 35215 Part of proof of Lemma K o...
cdlemk34 35216 Part of proof of Lemma K o...
cdlemk29-3 35217 Part of proof of Lemma K o...
cdlemk35 35218 Part of proof of Lemma K o...
cdlemk36 35219 Part of proof of Lemma K o...
cdlemk37 35220 Part of proof of Lemma K o...
cdlemk38 35221 Part of proof of Lemma K o...
cdlemk39 35222 Part of proof of Lemma K o...
cdlemk40 35223 TODO: fix comment. (Contr...
cdlemk40t 35224 TODO: fix comment. (Contr...
cdlemk40f 35225 TODO: fix comment. (Contr...
cdlemk41 35226 Part of proof of Lemma K o...
cdlemkfid1N 35227 Lemma for ~ cdlemkfid3N . ...
cdlemkid1 35228 Lemma for ~ cdlemkid . (C...
cdlemkfid2N 35229 Lemma for ~ cdlemkfid3N . ...
cdlemkid2 35230 Lemma for ~ cdlemkid . (C...
cdlemkfid3N 35231 TODO: is this useful or sh...
cdlemky 35232 Part of proof of Lemma K o...
cdlemkyu 35233 Convert between function a...
cdlemkyuu 35234 ~ cdlemkyu with some hypot...
cdlemk11ta 35235 Part of proof of Lemma K o...
cdlemk19ylem 35236 Lemma for ~ cdlemk19y . (...
cdlemk11tb 35237 Part of proof of Lemma K o...
cdlemk19y 35238 ~ cdlemk19 with simpler hy...
cdlemkid3N 35239 Lemma for ~ cdlemkid . (C...
cdlemkid4 35240 Lemma for ~ cdlemkid . (C...
cdlemkid5 35241 Lemma for ~ cdlemkid . (C...
cdlemkid 35242 The value of the tau funct...
cdlemk35s 35243 Substitution version of ~ ...
cdlemk35s-id 35244 Substitution version of ~ ...
cdlemk39s 35245 Substitution version of ~ ...
cdlemk39s-id 35246 Substitution version of ~ ...
cdlemk42 35247 Part of proof of Lemma K o...
cdlemk19xlem 35248 Lemma for ~ cdlemk19x . (...
cdlemk19x 35249 ~ cdlemk19 with simpler hy...
cdlemk42yN 35250 Part of proof of Lemma K o...
cdlemk11tc 35251 Part of proof of Lemma K o...
cdlemk11t 35252 Part of proof of Lemma K o...
cdlemk45 35253 Part of proof of Lemma K o...
cdlemk46 35254 Part of proof of Lemma K o...
cdlemk47 35255 Part of proof of Lemma K o...
cdlemk48 35256 Part of proof of Lemma K o...
cdlemk49 35257 Part of proof of Lemma K o...
cdlemk50 35258 Part of proof of Lemma K o...
cdlemk51 35259 Part of proof of Lemma K o...
cdlemk52 35260 Part of proof of Lemma K o...
cdlemk53a 35261 Lemma for ~ cdlemk53 . (C...
cdlemk53b 35262 Lemma for ~ cdlemk53 . (C...
cdlemk53 35263 Part of proof of Lemma K o...
cdlemk54 35264 Part of proof of Lemma K o...
cdlemk55a 35265 Lemma for ~ cdlemk55 . (C...
cdlemk55b 35266 Lemma for ~ cdlemk55 . (C...
cdlemk55 35267 Part of proof of Lemma K o...
cdlemkyyN 35268 Part of proof of Lemma K o...
cdlemk43N 35269 Part of proof of Lemma K o...
cdlemk35u 35270 Substitution version of ~ ...
cdlemk55u1 35271 Lemma for ~ cdlemk55u . (...
cdlemk55u 35272 Part of proof of Lemma K o...
cdlemk39u1 35273 Lemma for ~ cdlemk39u . (...
cdlemk39u 35274 Part of proof of Lemma K o...
cdlemk19u1 35275 ~ cdlemk19 with simpler hy...
cdlemk19u 35276 Part of Lemma K of [Crawle...
cdlemk56 35277 Part of Lemma K of [Crawle...
cdlemk19w 35278 Use a fixed element to eli...
cdlemk56w 35279 Use a fixed element to eli...
cdlemk 35280 Lemma K of [Crawley] p. 11...
tendoex 35281 Generalization of Lemma K ...
cdleml1N 35282 Part of proof of Lemma L o...
cdleml2N 35283 Part of proof of Lemma L o...
cdleml3N 35284 Part of proof of Lemma L o...
cdleml4N 35285 Part of proof of Lemma L o...
cdleml5N 35286 Part of proof of Lemma L o...
cdleml6 35287 Part of proof of Lemma L o...
cdleml7 35288 Part of proof of Lemma L o...
cdleml8 35289 Part of proof of Lemma L o...
cdleml9 35290 Part of proof of Lemma L o...
dva1dim 35291 Two expressions for the 1-...
dvhb1dimN 35292 Two expressions for the 1-...
erng1lem 35293 Value of the endomorphism ...
erngdvlem1 35294 Lemma for ~ eringring . (...
erngdvlem2N 35295 Lemma for ~ eringring . (...
erngdvlem3 35296 Lemma for ~ eringring . (...
erngdvlem4 35297 Lemma for ~ erngdv . (Con...
eringring 35298 An endomorphism ring is a ...
erngdv 35299 An endomorphism ring is a ...
erng0g 35300 The division ring zero of ...
erng1r 35301 The division ring unit of ...
erngdvlem1-rN 35302 Lemma for ~ eringring . (...
erngdvlem2-rN 35303 Lemma for ~ eringring . (...
erngdvlem3-rN 35304 Lemma for ~ eringring . (...
erngdvlem4-rN 35305 Lemma for ~ erngdv . (Con...
erngring-rN 35306 An endomorphism ring is a ...
erngdv-rN 35307 An endomorphism ring is a ...
dvafset 35310 The constructed partial ve...
dvaset 35311 The constructed partial ve...
dvasca 35312 The ring base set of the c...
dvabase 35313 The ring base set of the c...
dvafplusg 35314 Ring addition operation fo...
dvaplusg 35315 Ring addition operation fo...
dvaplusgv 35316 Ring addition operation fo...
dvafmulr 35317 Ring multiplication operat...
dvamulr 35318 Ring multiplication operat...
dvavbase 35319 The vectors (vector base s...
dvafvadd 35320 The vector sum operation f...
dvavadd 35321 Ring addition operation fo...
dvafvsca 35322 Ring addition operation fo...
dvavsca 35323 Ring addition operation fo...
tendospid 35324 Identity property of endom...
tendospcl 35325 Closure of endomorphism sc...
tendospass 35326 Associative law for endomo...
tendospdi1 35327 Forward distributive law f...
tendocnv 35328 Converse of a trace-preser...
tendospdi2 35329 Reverse distributive law f...
tendospcanN 35330 Cancellation law for trace...
dvaabl 35331 The constructed partial ve...
dvalveclem 35332 Lemma for ~ dvalvec . (Co...
dvalvec 35333 The constructed partial ve...
dva0g 35334 The zero vector of partial...
diaffval 35337 The partial isomorphism A ...
diafval 35338 The partial isomorphism A ...
diaval 35339 The partial isomorphism A ...
diaelval 35340 Member of the partial isom...
diafn 35341 Functionality and domain o...
diadm 35342 Domain of the partial isom...
diaeldm 35343 Member of domain of the pa...
diadmclN 35344 A member of domain of the ...
diadmleN 35345 A member of domain of the ...
dian0 35346 The value of the partial i...
dia0eldmN 35347 The lattice zero belongs t...
dia1eldmN 35348 The fiducial hyperplane (t...
diass 35349 The value of the partial i...
diael 35350 A member of the value of t...
diatrl 35351 Trace of a member of the p...
diaelrnN 35352 Any value of the partial i...
dialss 35353 The value of partial isomo...
diaord 35354 The partial isomorphism A ...
dia11N 35355 The partial isomorphism A ...
diaf11N 35356 The partial isomorphism A ...
diaclN 35357 Closure of partial isomorp...
diacnvclN 35358 Closure of partial isomorp...
dia0 35359 The value of the partial i...
dia1N 35360 The value of the partial i...
dia1elN 35361 The largest subspace in th...
diaglbN 35362 Partial isomorphism A of a...
diameetN 35363 Partial isomorphism A of a...
diainN 35364 Inverse partial isomorphis...
diaintclN 35365 The intersection of partia...
diasslssN 35366 The partial isomorphism A ...
diassdvaN 35367 The partial isomorphism A ...
dia1dim 35368 Two expressions for the 1-...
dia1dim2 35369 Two expressions for a 1-di...
dia1dimid 35370 A vector (translation) bel...
dia2dimlem1 35371 Lemma for ~ dia2dim . Sho...
dia2dimlem2 35372 Lemma for ~ dia2dim . Def...
dia2dimlem3 35373 Lemma for ~ dia2dim . Def...
dia2dimlem4 35374 Lemma for ~ dia2dim . Sho...
dia2dimlem5 35375 Lemma for ~ dia2dim . The...
dia2dimlem6 35376 Lemma for ~ dia2dim . Eli...
dia2dimlem7 35377 Lemma for ~ dia2dim . Eli...
dia2dimlem8 35378 Lemma for ~ dia2dim . Eli...
dia2dimlem9 35379 Lemma for ~ dia2dim . Eli...
dia2dimlem10 35380 Lemma for ~ dia2dim . Con...
dia2dimlem11 35381 Lemma for ~ dia2dim . Con...
dia2dimlem12 35382 Lemma for ~ dia2dim . Obt...
dia2dimlem13 35383 Lemma for ~ dia2dim . Eli...
dia2dim 35384 A two-dimensional subspace...
dvhfset 35387 The constructed full vecto...
dvhset 35388 The constructed full vecto...
dvhsca 35389 The ring of scalars of the...
dvhbase 35390 The ring base set of the c...
dvhfplusr 35391 Ring addition operation fo...
dvhfmulr 35392 Ring multiplication operat...
dvhmulr 35393 Ring multiplication operat...
dvhvbase 35394 The vectors (vector base s...
dvhelvbasei 35395 Vector membership in the c...
dvhvaddcbv 35396 Change bound variables to ...
dvhvaddval 35397 The vector sum operation f...
dvhfvadd 35398 The vector sum operation f...
dvhvadd 35399 The vector sum operation f...
dvhopvadd 35400 The vector sum operation f...
dvhopvadd2 35401 The vector sum operation f...
dvhvaddcl 35402 Closure of the vector sum ...
dvhvaddcomN 35403 Commutativity of vector su...
dvhvaddass 35404 Associativity of vector su...
dvhvscacbv 35405 Change bound variables to ...
dvhvscaval 35406 The scalar product operati...
dvhfvsca 35407 Scalar product operation f...
dvhvsca 35408 Scalar product operation f...
dvhopvsca 35409 Scalar product operation f...
dvhvscacl 35410 Closure of the scalar prod...
tendoinvcl 35411 Closure of multiplicative ...
tendolinv 35412 Left multiplicative invers...
tendorinv 35413 Right multiplicative inver...
dvhgrp 35414 The full vector space ` U ...
dvhlveclem 35415 Lemma for ~ dvhlvec . TOD...
dvhlvec 35416 The full vector space ` U ...
dvhlmod 35417 The full vector space ` U ...
dvh0g 35418 The zero vector of vector ...
dvheveccl 35419 Properties of a unit vecto...
dvhopclN 35420 Closure of a ` DVecH ` vec...
dvhopaddN 35421 Sum of ` DVecH ` vectors e...
dvhopspN 35422 Scalar product of ` DVecH ...
dvhopN 35423 Decompose a ` DVecH ` vect...
dvhopellsm 35424 Ordered pair membership in...
cdlemm10N 35425 The image of the map ` G `...
docaffvalN 35428 Subspace orthocomplement f...
docafvalN 35429 Subspace orthocomplement f...
docavalN 35430 Subspace orthocomplement f...
docaclN 35431 Closure of subspace orthoc...
diaocN 35432 Value of partial isomorphi...
doca2N 35433 Double orthocomplement of ...
doca3N 35434 Double orthocomplement of ...
dvadiaN 35435 Any closed subspace is a m...
diarnN 35436 Partial isomorphism A maps...
diaf1oN 35437 The partial isomorphism A ...
djaffvalN 35440 Subspace join for ` DVecA ...
djafvalN 35441 Subspace join for ` DVecA ...
djavalN 35442 Subspace join for ` DVecA ...
djaclN 35443 Closure of subspace join f...
djajN 35444 Transfer lattice join to `...
dibffval 35447 The partial isomorphism B ...
dibfval 35448 The partial isomorphism B ...
dibval 35449 The partial isomorphism B ...
dibopelvalN 35450 Member of the partial isom...
dibval2 35451 Value of the partial isomo...
dibopelval2 35452 Member of the partial isom...
dibval3N 35453 Value of the partial isomo...
dibelval3 35454 Member of the partial isom...
dibopelval3 35455 Member of the partial isom...
dibelval1st 35456 Membership in value of the...
dibelval1st1 35457 Membership in value of the...
dibelval1st2N 35458 Membership in value of the...
dibelval2nd 35459 Membership in value of the...
dibn0 35460 The value of the partial i...
dibfna 35461 Functionality and domain o...
dibdiadm 35462 Domain of the partial isom...
dibfnN 35463 Functionality and domain o...
dibdmN 35464 Domain of the partial isom...
dibeldmN 35465 Member of domain of the pa...
dibord 35466 The isomorphism B for a la...
dib11N 35467 The isomorphism B for a la...
dibf11N 35468 The partial isomorphism A ...
dibclN 35469 Closure of partial isomorp...
dibvalrel 35470 The value of partial isomo...
dib0 35471 The value of partial isomo...
dib1dim 35472 Two expressions for the 1-...
dibglbN 35473 Partial isomorphism B of a...
dibintclN 35474 The intersection of partia...
dib1dim2 35475 Two expressions for a 1-di...
dibss 35476 The partial isomorphism B ...
diblss 35477 The value of partial isomo...
diblsmopel 35478 Membership in subspace sum...
dicffval 35481 The partial isomorphism C ...
dicfval 35482 The partial isomorphism C ...
dicval 35483 The partial isomorphism C ...
dicopelval 35484 Membership in value of the...
dicelvalN 35485 Membership in value of the...
dicval2 35486 The partial isomorphism C ...
dicelval3 35487 Member of the partial isom...
dicopelval2 35488 Membership in value of the...
dicelval2N 35489 Membership in value of the...
dicfnN 35490 Functionality and domain o...
dicdmN 35491 Domain of the partial isom...
dicvalrelN 35492 The value of partial isomo...
dicssdvh 35493 The partial isomorphism C ...
dicelval1sta 35494 Membership in value of the...
dicelval1stN 35495 Membership in value of the...
dicelval2nd 35496 Membership in value of the...
dicvaddcl 35497 Membership in value of the...
dicvscacl 35498 Membership in value of the...
dicn0 35499 The value of the partial i...
diclss 35500 The value of partial isomo...
diclspsn 35501 The value of isomorphism C...
cdlemn2 35502 Part of proof of Lemma N o...
cdlemn2a 35503 Part of proof of Lemma N o...
cdlemn3 35504 Part of proof of Lemma N o...
cdlemn4 35505 Part of proof of Lemma N o...
cdlemn4a 35506 Part of proof of Lemma N o...
cdlemn5pre 35507 Part of proof of Lemma N o...
cdlemn5 35508 Part of proof of Lemma N o...
cdlemn6 35509 Part of proof of Lemma N o...
cdlemn7 35510 Part of proof of Lemma N o...
cdlemn8 35511 Part of proof of Lemma N o...
cdlemn9 35512 Part of proof of Lemma N o...
cdlemn10 35513 Part of proof of Lemma N o...
cdlemn11a 35514 Part of proof of Lemma N o...
cdlemn11b 35515 Part of proof of Lemma N o...
cdlemn11c 35516 Part of proof of Lemma N o...
cdlemn11pre 35517 Part of proof of Lemma N o...
cdlemn11 35518 Part of proof of Lemma N o...
cdlemn 35519 Lemma N of [Crawley] p. 12...
dihordlem6 35520 Part of proof of Lemma N o...
dihordlem7 35521 Part of proof of Lemma N o...
dihordlem7b 35522 Part of proof of Lemma N o...
dihjustlem 35523 Part of proof after Lemma ...
dihjust 35524 Part of proof after Lemma ...
dihord1 35525 Part of proof after Lemma ...
dihord2a 35526 Part of proof after Lemma ...
dihord2b 35527 Part of proof after Lemma ...
dihord2cN 35528 Part of proof after Lemma ...
dihord11b 35529 Part of proof after Lemma ...
dihord10 35530 Part of proof after Lemma ...
dihord11c 35531 Part of proof after Lemma ...
dihord2pre 35532 Part of proof after Lemma ...
dihord2pre2 35533 Part of proof after Lemma ...
dihord2 35534 Part of proof after Lemma ...
dihffval 35537 The isomorphism H for a la...
dihfval 35538 Isomorphism H for a lattic...
dihval 35539 Value of isomorphism H for...
dihvalc 35540 Value of isomorphism H for...
dihlsscpre 35541 Closure of isomorphism H f...
dihvalcqpre 35542 Value of isomorphism H for...
dihvalcq 35543 Value of isomorphism H for...
dihvalb 35544 Value of isomorphism H for...
dihopelvalbN 35545 Ordered pair member of the...
dihvalcqat 35546 Value of isomorphism H for...
dih1dimb 35547 Two expressions for a 1-di...
dih1dimb2 35548 Isomorphism H at an atom u...
dih1dimc 35549 Isomorphism H at an atom n...
dib2dim 35550 Extend ~ dia2dim to partia...
dih2dimb 35551 Extend ~ dib2dim to isomor...
dih2dimbALTN 35552 Extend ~ dia2dim to isomor...
dihopelvalcqat 35553 Ordered pair member of the...
dihvalcq2 35554 Value of isomorphism H for...
dihopelvalcpre 35555 Member of value of isomorp...
dihopelvalc 35556 Member of value of isomorp...
dihlss 35557 The value of isomorphism H...
dihss 35558 The value of isomorphism H...
dihssxp 35559 An isomorphism H value is ...
dihopcl 35560 Closure of an ordered pair...
xihopellsmN 35561 Ordered pair membership in...
dihopellsm 35562 Ordered pair membership in...
dihord6apre 35563 Part of proof that isomorp...
dihord3 35564 The isomorphism H for a la...
dihord4 35565 The isomorphism H for a la...
dihord5b 35566 Part of proof that isomorp...
dihord6b 35567 Part of proof that isomorp...
dihord6a 35568 Part of proof that isomorp...
dihord5apre 35569 Part of proof that isomorp...
dihord5a 35570 Part of proof that isomorp...
dihord 35571 The isomorphism H is order...
dih11 35572 The isomorphism H is one-t...
dihf11lem 35573 Functionality of the isomo...
dihf11 35574 The isomorphism H for a la...
dihfn 35575 Functionality and domain o...
dihdm 35576 Domain of isomorphism H. (...
dihcl 35577 Closure of isomorphism H. ...
dihcnvcl 35578 Closure of isomorphism H c...
dihcnvid1 35579 The converse isomorphism o...
dihcnvid2 35580 The isomorphism of a conve...
dihcnvord 35581 Ordering property for conv...
dihcnv11 35582 The converse of isomorphis...
dihsslss 35583 The isomorphism H maps to ...
dihrnlss 35584 The isomorphism H maps to ...
dihrnss 35585 The isomorphism H maps to ...
dihvalrel 35586 The value of isomorphism H...
dih0 35587 The value of isomorphism H...
dih0bN 35588 A lattice element is zero ...
dih0vbN 35589 A vector is zero iff its s...
dih0cnv 35590 The isomorphism H converse...
dih0rn 35591 The zero subspace belongs ...
dih0sb 35592 A subspace is zero iff the...
dih1 35593 The value of isomorphism H...
dih1rn 35594 The full vector space belo...
dih1cnv 35595 The isomorphism H converse...
dihwN 35596 Value of isomorphism H at ...
dihmeetlem1N 35597 Isomorphism H of a conjunc...
dihglblem5apreN 35598 A conjunction property of ...
dihglblem5aN 35599 A conjunction property of ...
dihglblem2aN 35600 Lemma for isomorphism H of...
dihglblem2N 35601 The GLB of a set of lattic...
dihglblem3N 35602 Isomorphism H of a lattice...
dihglblem3aN 35603 Isomorphism H of a lattice...
dihglblem4 35604 Isomorphism H of a lattice...
dihglblem5 35605 Isomorphism H of a lattice...
dihmeetlem2N 35606 Isomorphism H of a conjunc...
dihglbcpreN 35607 Isomorphism H of a lattice...
dihglbcN 35608 Isomorphism H of a lattice...
dihmeetcN 35609 Isomorphism H of a lattice...
dihmeetbN 35610 Isomorphism H of a lattice...
dihmeetbclemN 35611 Lemma for isomorphism H of...
dihmeetlem3N 35612 Lemma for isomorphism H of...
dihmeetlem4preN 35613 Lemma for isomorphism H of...
dihmeetlem4N 35614 Lemma for isomorphism H of...
dihmeetlem5 35615 Part of proof that isomorp...
dihmeetlem6 35616 Lemma for isomorphism H of...
dihmeetlem7N 35617 Lemma for isomorphism H of...
dihjatc1 35618 Lemma for isomorphism H of...
dihjatc2N 35619 Isomorphism H of join with...
dihjatc3 35620 Isomorphism H of join with...
dihmeetlem8N 35621 Lemma for isomorphism H of...
dihmeetlem9N 35622 Lemma for isomorphism H of...
dihmeetlem10N 35623 Lemma for isomorphism H of...
dihmeetlem11N 35624 Lemma for isomorphism H of...
dihmeetlem12N 35625 Lemma for isomorphism H of...
dihmeetlem13N 35626 Lemma for isomorphism H of...
dihmeetlem14N 35627 Lemma for isomorphism H of...
dihmeetlem15N 35628 Lemma for isomorphism H of...
dihmeetlem16N 35629 Lemma for isomorphism H of...
dihmeetlem17N 35630 Lemma for isomorphism H of...
dihmeetlem18N 35631 Lemma for isomorphism H of...
dihmeetlem19N 35632 Lemma for isomorphism H of...
dihmeetlem20N 35633 Lemma for isomorphism H of...
dihmeetALTN 35634 Isomorphism H of a lattice...
dih1dimatlem0 35635 Lemma for ~ dih1dimat . (...
dih1dimatlem 35636 Lemma for ~ dih1dimat . (...
dih1dimat 35637 Any 1-dimensional subspace...
dihlsprn 35638 The span of a vector belon...
dihlspsnssN 35639 A subspace included in a 1...
dihlspsnat 35640 The inverse isomorphism H ...
dihatlat 35641 The isomorphism H of an at...
dihat 35642 There exists at least one ...
dihpN 35643 The value of isomorphism H...
dihlatat 35644 The reverse isomorphism H ...
dihatexv 35645 There is a nonzero vector ...
dihatexv2 35646 There is a nonzero vector ...
dihglblem6 35647 Isomorphism H of a lattice...
dihglb 35648 Isomorphism H of a lattice...
dihglb2 35649 Isomorphism H of a lattice...
dihmeet 35650 Isomorphism H of a lattice...
dihintcl 35651 The intersection of closed...
dihmeetcl 35652 Closure of closed subspace...
dihmeet2 35653 Reverse isomorphism H of a...
dochffval 35656 Subspace orthocomplement f...
dochfval 35657 Subspace orthocomplement f...
dochval 35658 Subspace orthocomplement f...
dochval2 35659 Subspace orthocomplement f...
dochcl 35660 Closure of subspace orthoc...
dochlss 35661 A subspace orthocomplement...
dochssv 35662 A subspace orthocomplement...
dochfN 35663 Domain and codomain of the...
dochvalr 35664 Orthocomplement of a close...
doch0 35665 Orthocomplement of the zer...
doch1 35666 Orthocomplement of the uni...
dochoc0 35667 The zero subspace is close...
dochoc1 35668 The unit subspace (all vec...
dochvalr2 35669 Orthocomplement of a close...
dochvalr3 35670 Orthocomplement of a close...
doch2val2 35671 Double orthocomplement for...
dochss 35672 Subset law for orthocomple...
dochocss 35673 Double negative law for or...
dochoc 35674 Double negative law for or...
dochsscl 35675 If a set of vectors is inc...
dochoccl 35676 A set of vectors is closed...
dochord 35677 Ordering law for orthocomp...
dochord2N 35678 Ordering law for orthocomp...
dochord3 35679 Ordering law for orthocomp...
doch11 35680 Orthocomplement is one-to-...
dochsordN 35681 Strict ordering law for or...
dochn0nv 35682 An orthocomplement is nonz...
dihoml4c 35683 Version of ~ dihoml4 with ...
dihoml4 35684 Orthomodular law for const...
dochspss 35685 The span of a set of vecto...
dochocsp 35686 The span of an orthocomple...
dochspocN 35687 The span of an orthocomple...
dochocsn 35688 The double orthocomplement...
dochsncom 35689 Swap vectors in an orthoco...
dochsat 35690 The double orthocomplement...
dochshpncl 35691 If a hyperplane is not clo...
dochlkr 35692 Equivalent conditions for ...
dochkrshp 35693 The closure of a kernel is...
dochkrshp2 35694 Properties of the closure ...
dochkrshp3 35695 Properties of the closure ...
dochkrshp4 35696 Properties of the closure ...
dochdmj1 35697 De Morgan-like law for sub...
dochnoncon 35698 Law of noncontradiction. ...
dochnel2 35699 A nonzero member of a subs...
dochnel 35700 A nonzero vector doesn't b...
djhffval 35703 Subspace join for ` DVecH ...
djhfval 35704 Subspace join for ` DVecH ...
djhval 35705 Subspace join for ` DVecH ...
djhval2 35706 Value of subspace join for...
djhcl 35707 Closure of subspace join f...
djhlj 35708 Transfer lattice join to `...
djhljjN 35709 Lattice join in terms of `...
djhjlj 35710 ` DVecH ` vector space clo...
djhj 35711 ` DVecH ` vector space clo...
djhcom 35712 Subspace join commutes. (...
djhspss 35713 Subspace span of union is ...
djhsumss 35714 Subspace sum is a subset o...
dihsumssj 35715 The subspace sum of two is...
djhunssN 35716 Subspace union is a subset...
dochdmm1 35717 De Morgan-like law for clo...
djhexmid 35718 Excluded middle property o...
djh01 35719 Closed subspace join with ...
djh02 35720 Closed subspace join with ...
djhlsmcl 35721 A closed subspace sum equa...
djhcvat42 35722 A covering property. ( ~ ...
dihjatb 35723 Isomorphism H of lattice j...
dihjatc 35724 Isomorphism H of lattice j...
dihjatcclem1 35725 Lemma for isomorphism H of...
dihjatcclem2 35726 Lemma for isomorphism H of...
dihjatcclem3 35727 Lemma for ~ dihjatcc . (C...
dihjatcclem4 35728 Lemma for isomorphism H of...
dihjatcc 35729 Isomorphism H of lattice j...
dihjat 35730 Isomorphism H of lattice j...
dihprrnlem1N 35731 Lemma for ~ dihprrn , show...
dihprrnlem2 35732 Lemma for ~ dihprrn . (Co...
dihprrn 35733 The span of a vector pair ...
djhlsmat 35734 The sum of two subspace at...
dihjat1lem 35735 Subspace sum of a closed s...
dihjat1 35736 Subspace sum of a closed s...
dihsmsprn 35737 Subspace sum of a closed s...
dihjat2 35738 The subspace sum of a clos...
dihjat3 35739 Isomorphism H of lattice j...
dihjat4 35740 Transfer the subspace sum ...
dihjat6 35741 Transfer the subspace sum ...
dihsmsnrn 35742 The subspace sum of two si...
dihsmatrn 35743 The subspace sum of a clos...
dihjat5N 35744 Transfer lattice join with...
dvh4dimat 35745 There is an atom that is o...
dvh3dimatN 35746 There is an atom that is o...
dvh2dimatN 35747 Given an atom, there exist...
dvh1dimat 35748 There exists an atom. (Co...
dvh1dim 35749 There exists a nonzero vec...
dvh4dimlem 35750 Lemma for ~ dvh4dimN . (C...
dvhdimlem 35751 Lemma for ~ dvh2dim and ~ ...
dvh2dim 35752 There is a vector that is ...
dvh3dim 35753 There is a vector that is ...
dvh4dimN 35754 There is a vector that is ...
dvh3dim2 35755 There is a vector that is ...
dvh3dim3N 35756 There is a vector that is ...
dochsnnz 35757 The orthocomplement of a s...
dochsatshp 35758 The orthocomplement of a s...
dochsatshpb 35759 The orthocomplement of a s...
dochsnshp 35760 The orthocomplement of a n...
dochshpsat 35761 A hyperplane is closed iff...
dochkrsat 35762 The orthocomplement of a k...
dochkrsat2 35763 The orthocomplement of a k...
dochsat0 35764 The orthocomplement of a k...
dochkrsm 35765 The subspace sum of a clos...
dochexmidat 35766 Special case of excluded m...
dochexmidlem1 35767 Lemma for ~ dochexmid . H...
dochexmidlem2 35768 Lemma for ~ dochexmid . (...
dochexmidlem3 35769 Lemma for ~ dochexmid . U...
dochexmidlem4 35770 Lemma for ~ dochexmid . (...
dochexmidlem5 35771 Lemma for ~ dochexmid . (...
dochexmidlem6 35772 Lemma for ~ dochexmid . (...
dochexmidlem7 35773 Lemma for ~ dochexmid . C...
dochexmidlem8 35774 Lemma for ~ dochexmid . T...
dochexmid 35775 Excluded middle law for cl...
dochsnkrlem1 35776 Lemma for ~ dochsnkr . (C...
dochsnkrlem2 35777 Lemma for ~ dochsnkr . (C...
dochsnkrlem3 35778 Lemma for ~ dochsnkr . (C...
dochsnkr 35779 A (closed) kernel expresse...
dochsnkr2 35780 Kernel of the explicit fun...
dochsnkr2cl 35781 The ` X ` determining func...
dochflcl 35782 Closure of the explicit fu...
dochfl1 35783 The value of the explicit ...
dochfln0 35784 The value of a functional ...
dochkr1 35785 A nonzero functional has a...
dochkr1OLDN 35786 A nonzero functional has a...
lpolsetN 35789 The set of polarities of a...
islpolN 35790 The predicate "is a polari...
islpoldN 35791 Properties that determine ...
lpolfN 35792 Functionality of a polarit...
lpolvN 35793 The polarity of the whole ...
lpolconN 35794 Contraposition property of...
lpolsatN 35795 The polarity of an atomic ...
lpolpolsatN 35796 Property of a polarity. (...
dochpolN 35797 The subspace orthocompleme...
lcfl1lem 35798 Property of a functional w...
lcfl1 35799 Property of a functional w...
lcfl2 35800 Property of a functional w...
lcfl3 35801 Property of a functional w...
lcfl4N 35802 Property of a functional w...
lcfl5 35803 Property of a functional w...
lcfl5a 35804 Property of a functional w...
lcfl6lem 35805 Lemma for ~ lcfl6 . A fun...
lcfl7lem 35806 Lemma for ~ lcfl7N . If t...
lcfl6 35807 Property of a functional w...
lcfl7N 35808 Property of a functional w...
lcfl8 35809 Property of a functional w...
lcfl8a 35810 Property of a functional w...
lcfl8b 35811 Property of a nonzero func...
lcfl9a 35812 Property implying that a f...
lclkrlem1 35813 The set of functionals hav...
lclkrlem2a 35814 Lemma for ~ lclkr . Use ~...
lclkrlem2b 35815 Lemma for ~ lclkr . (Cont...
lclkrlem2c 35816 Lemma for ~ lclkr . (Cont...
lclkrlem2d 35817 Lemma for ~ lclkr . (Cont...
lclkrlem2e 35818 Lemma for ~ lclkr . The k...
lclkrlem2f 35819 Lemma for ~ lclkr . Const...
lclkrlem2g 35820 Lemma for ~ lclkr . Compa...
lclkrlem2h 35821 Lemma for ~ lclkr . Elimi...
lclkrlem2i 35822 Lemma for ~ lclkr . Elimi...
lclkrlem2j 35823 Lemma for ~ lclkr . Kerne...
lclkrlem2k 35824 Lemma for ~ lclkr . Kerne...
lclkrlem2l 35825 Lemma for ~ lclkr . Elimi...
lclkrlem2m 35826 Lemma for ~ lclkr . Const...
lclkrlem2n 35827 Lemma for ~ lclkr . (Cont...
lclkrlem2o 35828 Lemma for ~ lclkr . When ...
lclkrlem2p 35829 Lemma for ~ lclkr . When ...
lclkrlem2q 35830 Lemma for ~ lclkr . The s...
lclkrlem2r 35831 Lemma for ~ lclkr . When ...
lclkrlem2s 35832 Lemma for ~ lclkr . Thus,...
lclkrlem2t 35833 Lemma for ~ lclkr . We el...
lclkrlem2u 35834 Lemma for ~ lclkr . ~ lclk...
lclkrlem2v 35835 Lemma for ~ lclkr . When ...
lclkrlem2w 35836 Lemma for ~ lclkr . This ...
lclkrlem2x 35837 Lemma for ~ lclkr . Elimi...
lclkrlem2y 35838 Lemma for ~ lclkr . Resta...
lclkrlem2 35839 The set of functionals hav...
lclkr 35840 The set of functionals wit...
lcfls1lem 35841 Property of a functional w...
lcfls1N 35842 Property of a functional w...
lcfls1c 35843 Property of a functional w...
lclkrslem1 35844 The set of functionals hav...
lclkrslem2 35845 The set of functionals hav...
lclkrs 35846 The set of functionals hav...
lclkrs2 35847 The set of functionals wit...
lcfrvalsnN 35848 Reconstruction from the du...
lcfrlem1 35849 Lemma for ~ lcfr . Note t...
lcfrlem2 35850 Lemma for ~ lcfr . (Contr...
lcfrlem3 35851 Lemma for ~ lcfr . (Contr...
lcfrlem4 35852 Lemma for ~ lcfr . (Contr...
lcfrlem5 35853 Lemma for ~ lcfr . The se...
lcfrlem6 35854 Lemma for ~ lcfr . Closur...
lcfrlem7 35855 Lemma for ~ lcfr . Closur...
lcfrlem8 35856 Lemma for ~ lcf1o and ~ lc...
lcfrlem9 35857 Lemma for ~ lcf1o . (This...
lcf1o 35858 Define a function ` J ` th...
lcfrlem10 35859 Lemma for ~ lcfr . (Contr...
lcfrlem11 35860 Lemma for ~ lcfr . (Contr...
lcfrlem12N 35861 Lemma for ~ lcfr . (Contr...
lcfrlem13 35862 Lemma for ~ lcfr . (Contr...
lcfrlem14 35863 Lemma for ~ lcfr . (Contr...
lcfrlem15 35864 Lemma for ~ lcfr . (Contr...
lcfrlem16 35865 Lemma for ~ lcfr . (Contr...
lcfrlem17 35866 Lemma for ~ lcfr . Condit...
lcfrlem18 35867 Lemma for ~ lcfr . (Contr...
lcfrlem19 35868 Lemma for ~ lcfr . (Contr...
lcfrlem20 35869 Lemma for ~ lcfr . (Contr...
lcfrlem21 35870 Lemma for ~ lcfr . (Contr...
lcfrlem22 35871 Lemma for ~ lcfr . (Contr...
lcfrlem23 35872 Lemma for ~ lcfr . TODO: ...
lcfrlem24 35873 Lemma for ~ lcfr . (Contr...
lcfrlem25 35874 Lemma for ~ lcfr . Specia...
lcfrlem26 35875 Lemma for ~ lcfr . Specia...
lcfrlem27 35876 Lemma for ~ lcfr . Specia...
lcfrlem28 35877 Lemma for ~ lcfr . TODO: ...
lcfrlem29 35878 Lemma for ~ lcfr . (Contr...
lcfrlem30 35879 Lemma for ~ lcfr . (Contr...
lcfrlem31 35880 Lemma for ~ lcfr . (Contr...
lcfrlem32 35881 Lemma for ~ lcfr . (Contr...
lcfrlem33 35882 Lemma for ~ lcfr . (Contr...
lcfrlem34 35883 Lemma for ~ lcfr . (Contr...
lcfrlem35 35884 Lemma for ~ lcfr . (Contr...
lcfrlem36 35885 Lemma for ~ lcfr . (Contr...
lcfrlem37 35886 Lemma for ~ lcfr . (Contr...
lcfrlem38 35887 Lemma for ~ lcfr . Combin...
lcfrlem39 35888 Lemma for ~ lcfr . Elimin...
lcfrlem40 35889 Lemma for ~ lcfr . Elimin...
lcfrlem41 35890 Lemma for ~ lcfr . Elimin...
lcfrlem42 35891 Lemma for ~ lcfr . Elimin...
lcfr 35892 Reconstruction of a subspa...
lcdfval 35895 Dual vector space of funct...
lcdval 35896 Dual vector space of funct...
lcdval2 35897 Dual vector space of funct...
lcdlvec 35898 The dual vector space of f...
lcdlmod 35899 The dual vector space of f...
lcdvbase 35900 Vector base set of a dual ...
lcdvbasess 35901 The vector base set of the...
lcdvbaselfl 35902 A vector in the base set o...
lcdvbasecl 35903 Closure of the value of a ...
lcdvadd 35904 Vector addition for the cl...
lcdvaddval 35905 The value of the value of ...
lcdsca 35906 The ring of scalars of the...
lcdsbase 35907 Base set of scalar ring fo...
lcdsadd 35908 Scalar addition for the cl...
lcdsmul 35909 Scalar multiplication for ...
lcdvs 35910 Scalar product for the clo...
lcdvsval 35911 Value of scalar product op...
lcdvscl 35912 The scalar product operati...
lcdlssvscl 35913 Closure of scalar product ...
lcdvsass 35914 Associative law for scalar...
lcd0 35915 The zero scalar of the clo...
lcd1 35916 The unit scalar of the clo...
lcdneg 35917 The unit scalar of the clo...
lcd0v 35918 The zero functional in the...
lcd0v2 35919 The zero functional in the...
lcd0vvalN 35920 Value of the zero function...
lcd0vcl 35921 Closure of the zero functi...
lcd0vs 35922 A scalar zero times a func...
lcdvs0N 35923 A scalar times the zero fu...
lcdvsub 35924 The value of vector subtra...
lcdvsubval 35925 The value of the value of ...
lcdlss 35926 Subspaces of a dual vector...
lcdlss2N 35927 Subspaces of a dual vector...
lcdlsp 35928 Span in the set of functio...
lcdlkreqN 35929 Colinear functionals have ...
lcdlkreq2N 35930 Colinear functionals have ...
mapdffval 35933 Projectivity from vector s...
mapdfval 35934 Projectivity from vector s...
mapdval 35935 Value of projectivity from...
mapdvalc 35936 Value of projectivity from...
mapdval2N 35937 Value of projectivity from...
mapdval3N 35938 Value of projectivity from...
mapdval4N 35939 Value of projectivity from...
mapdval5N 35940 Value of projectivity from...
mapdordlem1a 35941 Lemma for ~ mapdord . (Co...
mapdordlem1bN 35942 Lemma for ~ mapdord . (Co...
mapdordlem1 35943 Lemma for ~ mapdord . (Co...
mapdordlem2 35944 Lemma for ~ mapdord . Ord...
mapdord 35945 Ordering property of the m...
mapd11 35946 The map defined by ~ df-ma...
mapddlssN 35947 The mapping of a subspace ...
mapdsn 35948 Value of the map defined b...
mapdsn2 35949 Value of the map defined b...
mapdsn3 35950 Value of the map defined b...
mapd1dim2lem1N 35951 Value of the map defined b...
mapdrvallem2 35952 Lemma for ~ mapdrval . TO...
mapdrvallem3 35953 Lemma for ~ mapdrval . (C...
mapdrval 35954 Given a dual subspace ` R ...
mapd1o 35955 The map defined by ~ df-ma...
mapdrn 35956 Range of the map defined b...
mapdunirnN 35957 Union of the range of the ...
mapdrn2 35958 Range of the map defined b...
mapdcnvcl 35959 Closure of the converse of...
mapdcl 35960 Closure the value of the m...
mapdcnvid1N 35961 Converse of the value of t...
mapdsord 35962 Strong ordering property o...
mapdcl2 35963 The mapping of a subspace ...
mapdcnvid2 35964 Value of the converse of t...
mapdcnvordN 35965 Ordering property of the c...
mapdcnv11N 35966 The converse of the map de...
mapdcv 35967 Covering property of the c...
mapdincl 35968 Closure of dual subspace i...
mapdin 35969 Subspace intersection is p...
mapdlsmcl 35970 Closure of dual subspace s...
mapdlsm 35971 Subspace sum is preserved ...
mapd0 35972 Projectivity map of the ze...
mapdcnvatN 35973 Atoms are preserved by the...
mapdat 35974 Atoms are preserved by the...
mapdspex 35975 The map of a span equals t...
mapdn0 35976 Transfer nonzero property ...
mapdncol 35977 Transfer non-colinearity f...
mapdindp 35978 Transfer (part of) vector ...
mapdpglem1 35979 Lemma for ~ mapdpg . Baer...
mapdpglem2 35980 Lemma for ~ mapdpg . Baer...
mapdpglem2a 35981 Lemma for ~ mapdpg . (Con...
mapdpglem3 35982 Lemma for ~ mapdpg . Baer...
mapdpglem4N 35983 Lemma for ~ mapdpg . (Con...
mapdpglem5N 35984 Lemma for ~ mapdpg . (Con...
mapdpglem6 35985 Lemma for ~ mapdpg . Baer...
mapdpglem8 35986 Lemma for ~ mapdpg . Baer...
mapdpglem9 35987 Lemma for ~ mapdpg . Baer...
mapdpglem10 35988 Lemma for ~ mapdpg . Baer...
mapdpglem11 35989 Lemma for ~ mapdpg . (Con...
mapdpglem12 35990 Lemma for ~ mapdpg . TODO...
mapdpglem13 35991 Lemma for ~ mapdpg . (Con...
mapdpglem14 35992 Lemma for ~ mapdpg . (Con...
mapdpglem15 35993 Lemma for ~ mapdpg . (Con...
mapdpglem16 35994 Lemma for ~ mapdpg . Baer...
mapdpglem17N 35995 Lemma for ~ mapdpg . Baer...
mapdpglem18 35996 Lemma for ~ mapdpg . Baer...
mapdpglem19 35997 Lemma for ~ mapdpg . Baer...
mapdpglem20 35998 Lemma for ~ mapdpg . Baer...
mapdpglem21 35999 Lemma for ~ mapdpg . (Con...
mapdpglem22 36000 Lemma for ~ mapdpg . Baer...
mapdpglem23 36001 Lemma for ~ mapdpg . Baer...
mapdpglem30a 36002 Lemma for ~ mapdpg . (Con...
mapdpglem30b 36003 Lemma for ~ mapdpg . (Con...
mapdpglem25 36004 Lemma for ~ mapdpg . Baer...
mapdpglem26 36005 Lemma for ~ mapdpg . Baer...
mapdpglem27 36006 Lemma for ~ mapdpg . Baer...
mapdpglem29 36007 Lemma for ~ mapdpg . Baer...
mapdpglem28 36008 Lemma for ~ mapdpg . Baer...
mapdpglem30 36009 Lemma for ~ mapdpg . Baer...
mapdpglem31 36010 Lemma for ~ mapdpg . Baer...
mapdpglem24 36011 Lemma for ~ mapdpg . Exis...
mapdpglem32 36012 Lemma for ~ mapdpg . Uniq...
mapdpg 36013 Part 1 of proof of the fir...
baerlem3lem1 36014 Lemma for ~ baerlem3 . (C...
baerlem5alem1 36015 Lemma for ~ baerlem5a . (...
baerlem5blem1 36016 Lemma for ~ baerlem5b . (...
baerlem3lem2 36017 Lemma for ~ baerlem3 . (C...
baerlem5alem2 36018 Lemma for ~ baerlem5a . (...
baerlem5blem2 36019 Lemma for ~ baerlem5b . (...
baerlem3 36020 An equality that holds whe...
baerlem5a 36021 An equality that holds whe...
baerlem5b 36022 An equality that holds whe...
baerlem5amN 36023 An equality that holds whe...
baerlem5bmN 36024 An equality that holds whe...
baerlem5abmN 36025 An equality that holds whe...
mapdindp0 36026 Vector independence lemma....
mapdindp1 36027 Vector independence lemma....
mapdindp2 36028 Vector independence lemma....
mapdindp3 36029 Vector independence lemma....
mapdindp4 36030 Vector independence lemma....
mapdhval 36031 Lemmma for ~~? mapdh . (C...
mapdhval0 36032 Lemmma for ~~? mapdh . (C...
mapdhval2 36033 Lemmma for ~~? mapdh . (C...
mapdhcl 36034 Lemmma for ~~? mapdh . (C...
mapdheq 36035 Lemmma for ~~? mapdh . Th...
mapdheq2 36036 Lemmma for ~~? mapdh . On...
mapdheq2biN 36037 Lemmma for ~~? mapdh . Pa...
mapdheq4lem 36038 Lemma for ~ mapdheq4 . Pa...
mapdheq4 36039 Lemma for ~~? mapdh . Par...
mapdh6lem1N 36040 Lemma for ~ mapdh6N . Par...
mapdh6lem2N 36041 Lemma for ~ mapdh6N . Par...
mapdh6aN 36042 Lemma for ~ mapdh6N . Par...
mapdh6b0N 36043 Lemmma for ~ mapdh6N . (C...
mapdh6bN 36044 Lemmma for ~ mapdh6N . (C...
mapdh6cN 36045 Lemmma for ~ mapdh6N . (C...
mapdh6dN 36046 Lemmma for ~ mapdh6N . (C...
mapdh6eN 36047 Lemmma for ~ mapdh6N . Pa...
mapdh6fN 36048 Lemmma for ~ mapdh6N . Pa...
mapdh6gN 36049 Lemmma for ~ mapdh6N . Pa...
mapdh6hN 36050 Lemmma for ~ mapdh6N . Pa...
mapdh6iN 36051 Lemmma for ~ mapdh6N . El...
mapdh6jN 36052 Lemmma for ~ mapdh6N . El...
mapdh6kN 36053 Lemmma for ~ mapdh6N . El...
mapdh6N 36054 Part (6) of [Baer] p. 47 l...
mapdh7eN 36055 Part (7) of [Baer] p. 48 l...
mapdh7cN 36056 Part (7) of [Baer] p. 48 l...
mapdh7dN 36057 Part (7) of [Baer] p. 48 l...
mapdh7fN 36058 Part (7) of [Baer] p. 48 l...
mapdh75e 36059 Part (7) of [Baer] p. 48 l...
mapdh75cN 36060 Part (7) of [Baer] p. 48 l...
mapdh75d 36061 Part (7) of [Baer] p. 48 l...
mapdh75fN 36062 Part (7) of [Baer] p. 48 l...
hvmapffval 36065 Map from nonzero vectors t...
hvmapfval 36066 Map from nonzero vectors t...
hvmapval 36067 Value of map from nonzero ...
hvmapvalvalN 36068 Value of value of map (i.e...
hvmapidN 36069 The value of the vector to...
hvmap1o 36070 The vector to functional m...
hvmapclN 36071 Closure of the vector to f...
hvmap1o2 36072 The vector to functional m...
hvmapcl2 36073 Closure of the vector to f...
hvmaplfl 36074 The vector to functional m...
hvmaplkr 36075 Kernel of the vector to fu...
mapdhvmap 36076 Relationship between ` map...
lspindp5 36077 Obtain an independent vect...
hdmaplem1 36078 Lemma to convert a frequen...
hdmaplem2N 36079 Lemma to convert a frequen...
hdmaplem3 36080 Lemma to convert a frequen...
hdmaplem4 36081 Lemma to convert a frequen...
mapdh8a 36082 Part of Part (8) in [Baer]...
mapdh8aa 36083 Part of Part (8) in [Baer]...
mapdh8ab 36084 Part of Part (8) in [Baer]...
mapdh8ac 36085 Part of Part (8) in [Baer]...
mapdh8ad 36086 Part of Part (8) in [Baer]...
mapdh8b 36087 Part of Part (8) in [Baer]...
mapdh8c 36088 Part of Part (8) in [Baer]...
mapdh8d0N 36089 Part of Part (8) in [Baer]...
mapdh8d 36090 Part of Part (8) in [Baer]...
mapdh8e 36091 Part of Part (8) in [Baer]...
mapdh8fN 36092 Part of Part (8) in [Baer]...
mapdh8g 36093 Part of Part (8) in [Baer]...
mapdh8i 36094 Part of Part (8) in [Baer]...
mapdh8j 36095 Part of Part (8) in [Baer]...
mapdh8 36096 Part (8) in [Baer] p. 48. ...
mapdh9a 36097 Lemma for part (9) in [Bae...
mapdh9aOLDN 36098 Lemma for part (9) in [Bae...
hdmap1ffval 36103 Preliminary map from vecto...
hdmap1fval 36104 Preliminary map from vecto...
hdmap1vallem 36105 Value of preliminary map f...
hdmap1val 36106 Value of preliminary map f...
hdmap1val0 36107 Value of preliminary map f...
hdmap1val2 36108 Value of preliminary map f...
hdmap1eq 36109 The defining equation for ...
hdmap1cbv 36110 Frequently used lemma to c...
hdmap1valc 36111 Connect the value of the p...
hdmap1cl 36112 Convert closure theorem ~ ...
hdmap1eq2 36113 Convert ~ mapdheq2 to use ...
hdmap1eq4N 36114 Convert ~ mapdheq4 to use ...
hdmap1l6lem1 36115 Lemma for ~ hdmap1l6 . Pa...
hdmap1l6lem2 36116 Lemma for ~ hdmap1l6 . Pa...
hdmap1l6a 36117 Lemma for ~ hdmap1l6 . Pa...
hdmap1l6b0N 36118 Lemmma for ~ hdmap1l6 . (...
hdmap1l6b 36119 Lemmma for ~ hdmap1l6 . (...
hdmap1l6c 36120 Lemmma for ~ hdmap1l6 . (...
hdmap1l6d 36121 Lemmma for ~ hdmap1l6 . (...
hdmap1l6e 36122 Lemmma for ~ hdmap1l6 . P...
hdmap1l6f 36123 Lemmma for ~ hdmap1l6 . P...
hdmap1l6g 36124 Lemmma for ~ hdmap1l6 . P...
hdmap1l6h 36125 Lemmma for ~ hdmap1l6 . P...
hdmap1l6i 36126 Lemmma for ~ hdmap1l6 . E...
hdmap1l6j 36127 Lemmma for ~ hdmap1l6 . E...
hdmap1l6k 36128 Lemmma for ~ hdmap1l6 . E...
hdmap1l6 36129 Part (6) of [Baer] p. 47 l...
hdmap1p6N 36130 (Convert ~ mapdh6N to use ...
hdmap1eulem 36131 Lemma for ~ hdmap1eu . TO...
hdmap1eulemOLDN 36132 Lemma for ~ hdmap1euOLDN ....
hdmap1eu 36133 Convert ~ mapdh9a to use t...
hdmap1euOLDN 36134 Convert ~ mapdh9aOLDN to u...
hdmap1neglem1N 36135 Lemma for ~ hdmapneg . TO...
hdmapffval 36136 Map from vectors to functi...
hdmapfval 36137 Map from vectors to functi...
hdmapval 36138 Value of map from vectors ...
hdmapfnN 36139 Functionality of map from ...
hdmapcl 36140 Closure of map from vector...
hdmapval2lem 36141 Lemma for ~ hdmapval2 . (...
hdmapval2 36142 Value of map from vectors ...
hdmapval0 36143 Value of map from vectors ...
hdmapeveclem 36144 Lemma for ~ hdmapevec . T...
hdmapevec 36145 Value of map from vectors ...
hdmapevec2 36146 The inner product of the r...
hdmapval3lemN 36147 Value of map from vectors ...
hdmapval3N 36148 Value of map from vectors ...
hdmap10lem 36149 Lemma for ~ hdmap10 . (Co...
hdmap10 36150 Part 10 in [Baer] p. 48 li...
hdmap11lem1 36151 Lemma for ~ hdmapadd . (C...
hdmap11lem2 36152 Lemma for ~ hdmapadd . (C...
hdmapadd 36153 Part 11 in [Baer] p. 48 li...
hdmapeq0 36154 Part of proof of part 12 i...
hdmapnzcl 36155 Nonzero vector closure of ...
hdmapneg 36156 Part of proof of part 12 i...
hdmapsub 36157 Part of proof of part 12 i...
hdmap11 36158 Part of proof of part 12 i...
hdmaprnlem1N 36159 Part of proof of part 12 i...
hdmaprnlem3N 36160 Part of proof of part 12 i...
hdmaprnlem3uN 36161 Part of proof of part 12 i...
hdmaprnlem4tN 36162 Lemma for ~ hdmaprnN . TO...
hdmaprnlem4N 36163 Part of proof of part 12 i...
hdmaprnlem6N 36164 Part of proof of part 12 i...
hdmaprnlem7N 36165 Part of proof of part 12 i...
hdmaprnlem8N 36166 Part of proof of part 12 i...
hdmaprnlem9N 36167 Part of proof of part 12 i...
hdmaprnlem3eN 36168 Lemma for ~ hdmaprnN . (C...
hdmaprnlem10N 36169 Lemma for ~ hdmaprnN . Sh...
hdmaprnlem11N 36170 Lemma for ~ hdmaprnN . Sh...
hdmaprnlem15N 36171 Lemma for ~ hdmaprnN . El...
hdmaprnlem16N 36172 Lemma for ~ hdmaprnN . El...
hdmaprnlem17N 36173 Lemma for ~ hdmaprnN . In...
hdmaprnN 36174 Part of proof of part 12 i...
hdmapf1oN 36175 Part 12 in [Baer] p. 49. ...
hdmap14lem1a 36176 Prior to part 14 in [Baer]...
hdmap14lem2a 36177 Prior to part 14 in [Baer]...
hdmap14lem1 36178 Prior to part 14 in [Baer]...
hdmap14lem2N 36179 Prior to part 14 in [Baer]...
hdmap14lem3 36180 Prior to part 14 in [Baer]...
hdmap14lem4a 36181 Simplify ` ( A \ { Q } ) `...
hdmap14lem4 36182 Simplify ` ( A \ { Q } ) `...
hdmap14lem6 36183 Case where ` F ` is zero. ...
hdmap14lem7 36184 Combine cases of ` F ` . ...
hdmap14lem8 36185 Part of proof of part 14 i...
hdmap14lem9 36186 Part of proof of part 14 i...
hdmap14lem10 36187 Part of proof of part 14 i...
hdmap14lem11 36188 Part of proof of part 14 i...
hdmap14lem12 36189 Lemma for proof of part 14...
hdmap14lem13 36190 Lemma for proof of part 14...
hdmap14lem14 36191 Part of proof of part 14 i...
hdmap14lem15 36192 Part of proof of part 14 i...
hgmapffval 36195 Map from the scalar divisi...
hgmapfval 36196 Map from the scalar divisi...
hgmapval 36197 Value of map from the scal...
hgmapfnN 36198 Functionality of scalar si...
hgmapcl 36199 Closure of scalar sigma ma...
hgmapdcl 36200 Closure of the vector spac...
hgmapvs 36201 Part 15 of [Baer] p. 50 li...
hgmapval0 36202 Value of the scalar sigma ...
hgmapval1 36203 Value of the scalar sigma ...
hgmapadd 36204 Part 15 of [Baer] p. 50 li...
hgmapmul 36205 Part 15 of [Baer] p. 50 li...
hgmaprnlem1N 36206 Lemma for ~ hgmaprnN . (C...
hgmaprnlem2N 36207 Lemma for ~ hgmaprnN . Pa...
hgmaprnlem3N 36208 Lemma for ~ hgmaprnN . El...
hgmaprnlem4N 36209 Lemma for ~ hgmaprnN . El...
hgmaprnlem5N 36210 Lemma for ~ hgmaprnN . El...
hgmaprnN 36211 Part of proof of part 16 i...
hgmap11 36212 The scalar sigma map is on...
hgmapf1oN 36213 The scalar sigma map is a ...
hgmapeq0 36214 The scalar sigma map is ze...
hdmapipcl 36215 The inner product (Hermiti...
hdmapln1 36216 Linearity property that wi...
hdmaplna1 36217 Additive property of first...
hdmaplns1 36218 Subtraction property of fi...
hdmaplnm1 36219 Multiplicative property of...
hdmaplna2 36220 Additive property of secon...
hdmapglnm2 36221 g-linear property of secon...
hdmapgln2 36222 g-linear property that wil...
hdmaplkr 36223 Kernel of the vector to du...
hdmapellkr 36224 Membership in the kernel (...
hdmapip0 36225 Zero property that will be...
hdmapip1 36226 Construct a proportional v...
hdmapip0com 36227 Commutation property of Ba...
hdmapinvlem1 36228 Line 27 in [Baer] p. 110. ...
hdmapinvlem2 36229 Line 28 in [Baer] p. 110, ...
hdmapinvlem3 36230 Line 30 in [Baer] p. 110, ...
hdmapinvlem4 36231 Part 1.1 of Proposition 1 ...
hdmapglem5 36232 Part 1.2 in [Baer] p. 110 ...
hgmapvvlem1 36233 Involution property of sca...
hgmapvvlem2 36234 Lemma for ~ hgmapvv . Eli...
hgmapvvlem3 36235 Lemma for ~ hgmapvv . Eli...
hgmapvv 36236 Value of a double involuti...
hdmapglem7a 36237 Lemma for ~ hdmapg . (Con...
hdmapglem7b 36238 Lemma for ~ hdmapg . (Con...
hdmapglem7 36239 Lemma for ~ hdmapg . Line...
hdmapg 36240 Apply the scalar sigma fun...
hdmapoc 36241 Express our constructed or...
hlhilset 36244 The final Hilbert space co...
hlhilsca 36245 The scalar of the final co...
hlhilbase 36246 The base set of the final ...
hlhilplus 36247 The vector addition for th...
hlhilslem 36248 Lemma for ~ hlhilsbase2 . ...
hlhilsbase 36249 The scalar base set of the...
hlhilsplus 36250 Scalar addition for the fi...
hlhilsmul 36251 Scalar multiplication for ...
hlhilsbase2 36252 The scalar base set of the...
hlhilsplus2 36253 Scalar addition for the fi...
hlhilsmul2 36254 Scalar multiplication for ...
hlhils0 36255 The scalar ring zero for t...
hlhils1N 36256 The scalar ring unity for ...
hlhilvsca 36257 The scalar product for the...
hlhilip 36258 Inner product operation fo...
hlhilipval 36259 Value of inner product ope...
hlhilnvl 36260 The involution operation o...
hlhillvec 36261 The final constructed Hilb...
hlhildrng 36262 The star division ring for...
hlhilsrnglem 36263 Lemma for ~ hlhilsrng . (...
hlhilsrng 36264 The star division ring for...
hlhil0 36265 The zero vector for the fi...
hlhillsm 36266 The vector sum operation f...
hlhilocv 36267 The orthocomplement for th...
hlhillcs 36268 The closed subspaces of th...
hlhilphllem 36269 Lemma for ~ hlhil . (Cont...
hlhilhillem 36270 Lemma for ~ hlhil . (Cont...
hlathil 36271 Construction of a Hilbert ...
rntrclfvOAI 36272 The range of the transitiv...
moxfr 36273 Transfer at-most-one betwe...
imaiinfv 36274 Indexed intersection of an...
elrfi 36275 Elementhood in a set of re...
elrfirn 36276 Elementhood in a set of re...
elrfirn2 36277 Elementhood in a set of re...
cmpfiiin 36278 In a compact topology, a s...
ismrcd1 36279 Any function from the subs...
ismrcd2 36280 Second half of ~ ismrcd1 ....
istopclsd 36281 A closure function which s...
ismrc 36282 A function is a Moore clos...
isnacs 36285 Expand definition of Noeth...
nacsfg 36286 In a Noetherian-type closu...
isnacs2 36287 Express Noetherian-type cl...
mrefg2 36288 Slight variation on finite...
mrefg3 36289 Slight variation on finite...
nacsacs 36290 A closure system of Noethe...
isnacs3 36291 A choice-free order equiva...
incssnn0 36292 Transitivity induction of ...
nacsfix 36293 An increasing sequence of ...
constmap 36294 A constant (represented wi...
mapco2g 36295 Renaming indexes in a tupl...
mapco2 36296 Post-composition (renaming...
mapfzcons 36297 Extending a one-based mapp...
mapfzcons1 36298 Recover prefix mapping fro...
mapfzcons1cl 36299 A nonempty mapping has a p...
mapfzcons2 36300 Recover added element from...
mptfcl 36301 Interpret range of a maps-...
mzpclval 36306 Substitution lemma for ` m...
elmzpcl 36307 Double substitution lemma ...
mzpclall 36308 The set of all functions w...
mzpcln0 36309 Corrolary of ~ mzpclall : ...
mzpcl1 36310 Defining property 1 of a p...
mzpcl2 36311 Defining property 2 of a p...
mzpcl34 36312 Defining properties 3 and ...
mzpval 36313 Value of the ` mzPoly ` fu...
dmmzp 36314 ` mzPoly ` is defined for ...
mzpincl 36315 Polynomial closedness is a...
mzpconst 36316 Constant functions are pol...
mzpf 36317 A polynomial function is a...
mzpproj 36318 A projection function is p...
mzpadd 36319 The pointwise sum of two p...
mzpmul 36320 The pointwise product of t...
mzpconstmpt 36321 A constant function expres...
mzpaddmpt 36322 Sum of polynomial function...
mzpmulmpt 36323 Product of polynomial func...
mzpsubmpt 36324 The difference of two poly...
mzpnegmpt 36325 Negation of a polynomial f...
mzpexpmpt 36326 Raise a polynomial functio...
mzpindd 36327 "Structural" induction to ...
mzpmfp 36328 Relationship between multi...
mzpsubst 36329 Substituting polynomials f...
mzprename 36330 Simplified version of ~ mz...
mzpresrename 36331 A polynomial is a polynomi...
mzpcompact2lem 36332 Lemma for ~ mzpcompact2 . ...
mzpcompact2 36333 Polynomials are finitary o...
coeq0i 36334 ~ coeq0 but without explic...
fzsplit1nn0 36335 Split a finite 1-based set...
eldiophb 36338 Initial expression of Diop...
eldioph 36339 Condition for a set to be ...
diophrw 36340 Renaming and adding unused...
eldioph2lem1 36341 Lemma for ~ eldioph2 . Co...
eldioph2lem2 36342 Lemma for ~ eldioph2 . Co...
eldioph2 36343 Construct a Diophantine se...
eldioph2b 36344 While Diophantine sets wer...
eldiophelnn0 36345 Remove antecedent on ` B `...
eldioph3b 36346 Define Diophantine sets in...
eldioph3 36347 Inference version of ~ eld...
ellz1 36348 Membership in a lower set ...
lzunuz 36349 The union of a lower set o...
fz1eqin 36350 Express a one-based finite...
lzenom 36351 Lower integers are countab...
elmapresaun 36352 ~ fresaun transposed to ma...
elmapresaunres2 36353 ~ fresaunres2 transposed t...
diophin 36354 If two sets are Diophantin...
diophun 36355 If two sets are Diophantin...
eldiophss 36356 Diophantine sets are sets ...
diophrex 36357 Projecting a Diophantine s...
eq0rabdioph 36358 This is the first of a num...
eqrabdioph 36359 Diophantine set builder fo...
0dioph 36360 The null set is Diophantin...
vdioph 36361 The "universal" set (as la...
anrabdioph 36362 Diophantine set builder fo...
orrabdioph 36363 Diophantine set builder fo...
3anrabdioph 36364 Diophantine set builder fo...
3orrabdioph 36365 Diophantine set builder fo...
2sbcrex 36366 Exchange an existential qu...
sbcrexgOLD 36367 Interchange class substitu...
2sbcrexOLD 36368 Exchange an existential qu...
sbc2rex 36369 Exchange a substitution wi...
sbc2rexgOLD 36370 Exchange a substitution wi...
sbc4rex 36371 Exchange a substitution wi...
sbc4rexgOLD 36372 Exchange a substitution wi...
sbcrot3 36373 Rotate a sequence of three...
sbcrot5 36374 Rotate a sequence of five ...
sbccomieg 36375 Commute two explicit subst...
rexrabdioph 36376 Diophantine set builder fo...
rexfrabdioph 36377 Diophantine set builder fo...
2rexfrabdioph 36378 Diophantine set builder fo...
3rexfrabdioph 36379 Diophantine set builder fo...
4rexfrabdioph 36380 Diophantine set builder fo...
6rexfrabdioph 36381 Diophantine set builder fo...
7rexfrabdioph 36382 Diophantine set builder fo...
rabdiophlem1 36383 Lemma for arithmetic dioph...
rabdiophlem2 36384 Lemma for arithmetic dioph...
elnn0rabdioph 36385 Diophantine set builder fo...
rexzrexnn0 36386 Rewrite a quantification o...
lerabdioph 36387 Diophantine set builder fo...
eluzrabdioph 36388 Diophantine set builder fo...
elnnrabdioph 36389 Diophantine set builder fo...
ltrabdioph 36390 Diophantine set builder fo...
nerabdioph 36391 Diophantine set builder fo...
dvdsrabdioph 36392 Divisibility is a Diophant...
eldioph4b 36393 Membership in ` Dioph ` ex...
eldioph4i 36394 Forward-only version of ~ ...
diophren 36395 Change variables in a Diop...
rabrenfdioph 36396 Change variable numbers in...
rabren3dioph 36397 Change variable numbers in...
fphpd 36398 Pigeonhole principle expre...
fphpdo 36399 Pigeonhole principle for s...
ctbnfien 36400 An infinite subset of a co...
fiphp3d 36401 Infinite pigeonhole princi...
rencldnfilem 36402 Lemma for ~ rencldnfi . (...
rencldnfi 36403 A set of real numbers whic...
irrapxlem1 36404 Lemma for ~ irrapx1 . Div...
irrapxlem2 36405 Lemma for ~ irrapx1 . Two...
irrapxlem3 36406 Lemma for ~ irrapx1 . By ...
irrapxlem4 36407 Lemma for ~ irrapx1 . Eli...
irrapxlem5 36408 Lemma for ~ irrapx1 . Swi...
irrapxlem6 36409 Lemma for ~ irrapx1 . Exp...
irrapx1 36410 Dirichlet's approximation ...
pellexlem1 36411 Lemma for ~ pellex . Arit...
pellexlem2 36412 Lemma for ~ pellex . Arit...
pellexlem3 36413 Lemma for ~ pellex . To e...
pellexlem4 36414 Lemma for ~ pellex . Invo...
pellexlem5 36415 Lemma for ~ pellex . Invo...
pellexlem6 36416 Lemma for ~ pellex . Doin...
pellex 36417 Every Pell equation has a ...
pell1qrval 36428 Value of the set of first-...
elpell1qr 36429 Membership in a first-quad...
pell14qrval 36430 Value of the set of positi...
elpell14qr 36431 Membership in the set of p...
pell1234qrval 36432 Value of the set of genera...
elpell1234qr 36433 Membership in the set of g...
pell1234qrre 36434 General Pell solutions are...
pell1234qrne0 36435 No solution to a Pell equa...
pell1234qrreccl 36436 General solutions of the P...
pell1234qrmulcl 36437 General solutions of the P...
pell14qrss1234 36438 A positive Pell solution i...
pell14qrre 36439 A positive Pell solution i...
pell14qrne0 36440 A positive Pell solution i...
pell14qrgt0 36441 A positive Pell solution i...
pell14qrrp 36442 A positive Pell solution i...
pell1234qrdich 36443 A general Pell solution is...
elpell14qr2 36444 A number is a positive Pel...
pell14qrmulcl 36445 Positive Pell solutions ar...
pell14qrreccl 36446 Positive Pell solutions ar...
pell14qrdivcl 36447 Positive Pell solutions ar...
pell14qrexpclnn0 36448 Lemma for ~ pell14qrexpcl ...
pell14qrexpcl 36449 Positive Pell solutions ar...
pell1qrss14 36450 First-quadrant Pell soluti...
pell14qrdich 36451 A positive Pell solution i...
pell1qrge1 36452 A Pell solution in the fir...
pell1qr1 36453 1 is a Pell solution and i...
elpell1qr2 36454 The first quadrant solutio...
pell1qrgaplem 36455 Lemma for ~ pell1qrgap . ...
pell1qrgap 36456 First-quadrant Pell soluti...
pell14qrgap 36457 Positive Pell solutions ar...
pell14qrgapw 36458 Positive Pell solutions ar...
pellqrexplicit 36459 Condition for a calculated...
infmrgelbi 36460 Any lower bound of a nonem...
pellqrex 36461 There is a nontrivial solu...
pellfundval 36462 Value of the fundamental s...
pellfundre 36463 The fundamental solution o...
pellfundge 36464 Lower bound on the fundame...
pellfundgt1 36465 Weak lower bound on the Pe...
pellfundlb 36466 A nontrivial first quadran...
pellfundglb 36467 If a real is larger than t...
pellfundex 36468 The fundamental solution a...
pellfund14gap 36469 There are no solutions bet...
pellfundrp 36470 The fundamental Pell solut...
pellfundne1 36471 The fundamental Pell solut...
reglogcl 36472 General logarithm is a rea...
reglogltb 36473 General logarithm preserve...
reglogleb 36474 General logarithm preserve...
reglogmul 36475 Multiplication law for gen...
reglogexp 36476 Power law for general log....
reglogbas 36477 General log of the base is...
reglog1 36478 General log of 1 is 0. (C...
reglogexpbas 36479 General log of a power of ...
pellfund14 36480 Every positive Pell soluti...
pellfund14b 36481 The positive Pell solution...
rmxfval 36486 Value of the X sequence. ...
rmyfval 36487 Value of the Y sequence. ...
rmspecsqrtnq 36488 The discriminant used to d...
rmspecsqrtnqOLD 36489 Obsolete version of ~ rmsp...
rmspecnonsq 36490 The discriminant used to d...
qirropth 36491 This lemma implements the ...
rmspecfund 36492 The base of exponent used ...
rmxyelqirr 36493 The solutions used to cons...
rmxypairf1o 36494 The function used to extra...
rmxyelxp 36495 Lemma for ~ frmx and ~ frm...
frmx 36496 The X sequence is a nonneg...
frmy 36497 The Y sequence is an integ...
rmxyval 36498 Main definition of the X a...
rmspecpos 36499 The discriminant used to d...
rmxycomplete 36500 The X and Y sequences take...
rmxynorm 36501 The X and Y sequences defi...
rmbaserp 36502 The base of exponentiation...
rmxyneg 36503 Negation law for X and Y s...
rmxyadd 36504 Addition formula for X and...
rmxy1 36505 Value of the X and Y seque...
rmxy0 36506 Value of the X and Y seque...
rmxneg 36507 Negation law (even functio...
rmx0 36508 Value of X sequence at 0. ...
rmx1 36509 Value of X sequence at 1. ...
rmxadd 36510 Addition formula for X seq...
rmyneg 36511 Negation formula for Y seq...
rmy0 36512 Value of Y sequence at 0. ...
rmy1 36513 Value of Y sequence at 1. ...
rmyadd 36514 Addition formula for Y seq...
rmxp1 36515 Special addition-of-1 form...
rmyp1 36516 Special addition of 1 form...
rmxm1 36517 Subtraction of 1 formula f...
rmym1 36518 Subtraction of 1 formula f...
rmxluc 36519 The X sequence is a Lucas ...
rmyluc 36520 The Y sequence is a Lucas ...
rmyluc2 36521 Lucas sequence property of...
rmxdbl 36522 "Double-angle formula" for...
rmydbl 36523 "Double-angle formula" for...
monotuz 36524 A function defined on an u...
monotoddzzfi 36525 A function which is odd an...
monotoddzz 36526 A function (given implicit...
oddcomabszz 36527 An odd function which take...
2nn0ind 36528 Induction on nonnegative i...
zindbi 36529 Inductively transfer a pro...
expmordi 36530 Mantissa ordering relation...
rpexpmord 36531 Mantissa ordering relation...
rmxypos 36532 For all nonnegative indice...
ltrmynn0 36533 The Y-sequence is strictly...
ltrmxnn0 36534 The X-sequence is strictly...
lermxnn0 36535 The X-sequence is monotoni...
rmxnn 36536 The X-sequence is defined ...
ltrmy 36537 The Y-sequence is strictly...
rmyeq0 36538 Y is zero only at zero. (...
rmyeq 36539 Y is one-to-one. (Contrib...
lermy 36540 Y is monotonic (non-strict...
rmynn 36541 ` rmY ` is positive for po...
rmynn0 36542 ` rmY ` is nonnegative for...
rmyabs 36543 ` rmY ` commutes with ` ab...
jm2.24nn 36544 X(n) is strictly greater t...
jm2.17a 36545 First half of lemma 2.17 o...
jm2.17b 36546 Weak form of the second ha...
jm2.17c 36547 Second half of lemma 2.17 ...
jm2.24 36548 Lemma 2.24 of [JonesMatija...
rmygeid 36549 Y(n) increases faster than...
congtr 36550 A wff of the form ` A || (...
congadd 36551 If two pairs of numbers ar...
congmul 36552 If two pairs of numbers ar...
congsym 36553 Congruence mod ` A ` is a ...
congneg 36554 If two integers are congru...
congsub 36555 If two pairs of numbers ar...
congid 36556 Every integer is congruent...
mzpcong 36557 Polynomials commute with c...
congrep 36558 Every integer is congruent...
congabseq 36559 If two integers are congru...
acongid 36560 A wff like that in this th...
acongsym 36561 Symmetry of alternating co...
acongneg2 36562 Negate right side of alter...
acongtr 36563 Transitivity of alternatin...
acongeq12d 36564 Substitution deduction for...
acongrep 36565 Every integer is alternati...
fzmaxdif 36566 Bound on the difference be...
fzneg 36567 Reflection of a finite ran...
acongeq 36568 Two numbers in the fundame...
dvdsacongtr 36569 Alternating congruence pas...
coprmdvdsb 36570 Multiplication by a coprim...
modabsdifz 36571 Divisibility in terms of m...
dvdsabsmod0 36572 Divisibility in terms of m...
jm2.18 36573 Theorem 2.18 of [JonesMati...
jm2.19lem1 36574 Lemma for ~ jm2.19 . X an...
jm2.19lem2 36575 Lemma for ~ jm2.19 . (Con...
jm2.19lem3 36576 Lemma for ~ jm2.19 . (Con...
jm2.19lem4 36577 Lemma for ~ jm2.19 . Exte...
jm2.19 36578 Lemma 2.19 of [JonesMatija...
jm2.21 36579 Lemma for ~ jm2.20nn . Ex...
jm2.22 36580 Lemma for ~ jm2.20nn . Ap...
jm2.23 36581 Lemma for ~ jm2.20nn . Tr...
jm2.20nn 36582 Lemma 2.20 of [JonesMatija...
jm2.25lem1 36583 Lemma for ~ jm2.26 . (Con...
jm2.25 36584 Lemma for ~ jm2.26 . Rema...
jm2.26a 36585 Lemma for ~ jm2.26 . Reve...
jm2.26lem3 36586 Lemma for ~ jm2.26 . Use ...
jm2.26 36587 Lemma 2.26 of [JonesMatija...
jm2.15nn0 36588 Lemma 2.15 of [JonesMatija...
jm2.16nn0 36589 Lemma 2.16 of [JonesMatija...
jm2.27a 36590 Lemma for ~ jm2.27 . Reve...
jm2.27b 36591 Lemma for ~ jm2.27 . Expa...
jm2.27c 36592 Lemma for ~ jm2.27 . Forw...
jm2.27 36593 Lemma 2.27 of [JonesMatija...
jm2.27dlem1 36594 Lemma for ~ rmydioph . Su...
jm2.27dlem2 36595 Lemma for ~ rmydioph . Th...
jm2.27dlem3 36596 Lemma for ~ rmydioph . In...
jm2.27dlem4 36597 Lemma for ~ rmydioph . In...
jm2.27dlem5 36598 Lemma for ~ rmydioph . Us...
rmydioph 36599 ~ jm2.27 restated in terms...
rmxdiophlem 36600 X can be expressed in term...
rmxdioph 36601 X is a Diophantine functio...
jm3.1lem1 36602 Lemma for ~ jm3.1 . (Cont...
jm3.1lem2 36603 Lemma for ~ jm3.1 . (Cont...
jm3.1lem3 36604 Lemma for ~ jm3.1 . (Cont...
jm3.1 36605 Diophantine expression for...
expdiophlem1 36606 Lemma for ~ expdioph . Fu...
expdiophlem2 36607 Lemma for ~ expdioph . Ex...
expdioph 36608 The exponential function i...
setindtr 36609 Epsilon induction for sets...
setindtrs 36610 Epsilon induction scheme w...
dford3lem1 36611 Lemma for ~ dford3 . (Con...
dford3lem2 36612 Lemma for ~ dford3 . (Con...
dford3 36613 Ordinals are precisely the...
dford4 36614 ~ dford3 expressed in prim...
wopprc 36615 Unrelated: Wiener pairs t...
rpnnen3lem 36616 Lemma for ~ rpnnen3 . (Co...
rpnnen3 36617 Dedekind cut injection of ...
axac10 36618 Characterization of choice...
harinf 36619 The Hartogs number of an i...
wdom2d2 36620 Deduction for weak dominan...
ttac 36621 Tarski's theorem about cho...
pw2f1ocnv 36622 Define a bijection between...
pw2f1o2 36623 Define a bijection between...
pw2f1o2val 36624 Function value of the ~ pw...
pw2f1o2val2 36625 Membership in a mapped set...
soeq12d 36626 Equality deduction for tot...
freq12d 36627 Equality deduction for fou...
weeq12d 36628 Equality deduction for wel...
limsuc2 36629 Limit ordinals in the sens...
wepwsolem 36630 Transfer an ordering on ch...
wepwso 36631 A well-ordering induces a ...
dnnumch1 36632 Define an enumeration of a...
dnnumch2 36633 Define an enumeration (wea...
dnnumch3lem 36634 Value of the ordinal injec...
dnnumch3 36635 Define an injection from a...
dnwech 36636 Define a well-ordering fro...
fnwe2val 36637 Lemma for ~ fnwe2 . Subst...
fnwe2lem1 36638 Lemma for ~ fnwe2 . Subst...
fnwe2lem2 36639 Lemma for ~ fnwe2 . An el...
fnwe2lem3 36640 Lemma for ~ fnwe2 . Trich...
fnwe2 36641 A well-ordering can be con...
aomclem1 36642 Lemma for ~ dfac11 . This...
aomclem2 36643 Lemma for ~ dfac11 . Succ...
aomclem3 36644 Lemma for ~ dfac11 . Succ...
aomclem4 36645 Lemma for ~ dfac11 . Limi...
aomclem5 36646 Lemma for ~ dfac11 . Comb...
aomclem6 36647 Lemma for ~ dfac11 . Tran...
aomclem7 36648 Lemma for ~ dfac11 . ` ( R...
aomclem8 36649 Lemma for ~ dfac11 . Perf...
dfac11 36650 The right-hand side of thi...
kelac1 36651 Kelley's choice, basic for...
kelac2lem 36652 Lemma for ~ kelac2 and ~ d...
kelac2 36653 Kelley's choice, most comm...
dfac21 36654 Tychonoff's theorem is a c...
islmodfg 36657 Property of a finitely gen...
islssfg 36658 Property of a finitely gen...
islssfg2 36659 Property of a finitely gen...
islssfgi 36660 Finitely spanned subspaces...
fglmod 36661 Finitely generated left mo...
lsmfgcl 36662 The sum of two finitely ge...
islnm 36665 Property of being a Noethe...
islnm2 36666 Property of being a Noethe...
lnmlmod 36667 A Noetherian left module i...
lnmlssfg 36668 A submodule of Noetherian ...
lnmlsslnm 36669 All submodules of a Noethe...
lnmfg 36670 A Noetherian left module i...
kercvrlsm 36671 The domain of a linear fun...
lmhmfgima 36672 A homomorphism maps finite...
lnmepi 36673 Epimorphic images of Noeth...
lmhmfgsplit 36674 If the kernel and range of...
lmhmlnmsplit 36675 If the kernel and range of...
lnmlmic 36676 Noetherian is an invariant...
pwssplit4 36677 Splitting for structure po...
filnm 36678 Finite left modules are No...
pwslnmlem0 36679 Zeroeth powers are Noether...
pwslnmlem1 36680 First powers are Noetheria...
pwslnmlem2 36681 A sum of powers is Noether...
pwslnm 36682 Finite powers of Noetheria...
unxpwdom3 36683 Weaker version of ~ unxpwd...
pwfi2f1o 36684 The ~ pw2f1o bijection rel...
pwfi2en 36685 Finitely supported indicat...
frlmpwfi 36686 Formal linear combinations...
gicabl 36687 Being Abelian is a group i...
imasgim 36688 A relabeling of the elemen...
basfn 36689 Functionality of the base ...
isnumbasgrplem1 36690 A set which is equipollent...
harn0 36691 The Hartogs number of a se...
numinfctb 36692 A numerable infinite set c...
isnumbasgrplem2 36693 If the (to be thought of a...
isnumbasgrplem3 36694 Every nonempty numerable s...
isnumbasabl 36695 A set is numerable iff it ...
isnumbasgrp 36696 A set is numerable iff it ...
dfacbasgrp 36697 A choice equivalent in abs...
islnr 36700 Property of a left-Noether...
lnrring 36701 Left-Noetherian rings are ...
lnrlnm 36702 Left-Noetherian rings have...
islnr2 36703 Property of being a left-N...
islnr3 36704 Relate left-Noetherian rin...
lnr2i 36705 Given an ideal in a left-N...
lpirlnr 36706 Left principal ideal rings...
lnrfrlm 36707 Finite-dimensional free mo...
lnrfg 36708 Finitely-generated modules...
lnrfgtr 36709 A submodule of a finitely ...
hbtlem1 36712 Value of the leading coeff...
hbtlem2 36713 Leading coefficient ideals...
hbtlem7 36714 Functionality of leading c...
hbtlem4 36715 The leading ideal function...
hbtlem3 36716 The leading ideal function...
hbtlem5 36717 The leading ideal function...
hbtlem6 36718 There is a finite set of p...
hbt 36719 The Hilbert Basis Theorem ...
dgrsub2 36724 Subtracting two polynomial...
elmnc 36725 Property of a monic polyno...
mncply 36726 A monic polynomial is a po...
mnccoe 36727 A monic polynomial has lea...
mncn0 36728 A monic polynomial is not ...
dgraaval 36733 Value of the degree functi...
dgraalem 36734 Properties of the degree o...
dgraacl 36735 Closure of the degree func...
dgraaf 36736 Degree function on algebra...
dgraaub 36737 Upper bound on degree of a...
dgraa0p 36738 A rational polynomial of d...
mpaaeu 36739 An algebraic number has ex...
mpaaval 36740 Value of the minimal polyn...
mpaalem 36741 Properties of the minimal ...
mpaacl 36742 Minimal polynomial is a po...
mpaadgr 36743 Minimal polynomial has deg...
mpaaroot 36744 The minimal polynomial of ...
mpaamn 36745 Minimal polynomial is moni...
itgoval 36750 Value of the integral-over...
aaitgo 36751 The standard algebraic num...
itgoss 36752 An integral element is int...
itgocn 36753 All integral elements are ...
cnsrexpcl 36754 Exponentiation is closed i...
fsumcnsrcl 36755 Finite sums are closed in ...
cnsrplycl 36756 Polynomials are closed in ...
rgspnval 36757 Value of the ring-span of ...
rgspncl 36758 The ring-span of a set is ...
rgspnssid 36759 The ring-span of a set con...
rgspnmin 36760 The ring-span is contained...
rgspnid 36761 The span of a subring is i...
rngunsnply 36762 Adjoining one element to a...
flcidc 36763 Finite linear combinations...
algstr 36766 Lemma to shorten proofs of...
algbase 36767 The base set of a construc...
algaddg 36768 The additive operation of ...
algmulr 36769 The multiplicative operati...
algsca 36770 The set of scalars of a co...
algvsca 36771 The scalar product operati...
mendval 36772 Value of the module endomo...
mendbas 36773 Base set of the module end...
mendplusgfval 36774 Addition in the module end...
mendplusg 36775 A specific addition in the...
mendmulrfval 36776 Multiplication in the modu...
mendmulr 36777 A specific multiplication ...
mendsca 36778 The module endomorphism al...
mendvscafval 36779 Scalar multiplication in t...
mendvsca 36780 A specific scalar multipli...
mendring 36781 The module endomorphism al...
mendlmod 36782 The module endomorphism al...
mendassa 36783 The module endomorphism al...
issdrg 36786 Property of a division sub...
issdrg2 36787 Property of a division sub...
acsfn1p 36788 Construction of a closure ...
subrgacs 36789 Closure property of subrin...
sdrgacs 36790 Closure property of divisi...
cntzsdrg 36791 Centralizers in division r...
idomrootle 36792 No element of an integral ...
idomodle 36793 Limit on the number of ` N...
fiuneneq 36794 Two finite sets of equal s...
idomsubgmo 36795 The units of an integral d...
proot1mul 36796 Any primitive ` N ` -th ro...
proot1hash 36797 If an integral domain has ...
proot1ex 36798 The complex field has prim...
isdomn3 36801 Nonzero elements form a mu...
mon1pid 36802 Monicity and degree of the...
mon1psubm 36803 Monic polynomials are a mu...
deg1mhm 36804 Homomorphic property of th...
cytpfn 36805 Functionality of the cyclo...
cytpval 36806 Substitutions for the Nth ...
fgraphopab 36807 Express a function as a su...
fgraphxp 36808 Express a function as a su...
hausgraph 36809 The graph of a continuous ...
ioounsn 36814 The closure of the upper e...
iocunico 36815 Split an open interval int...
iocinico 36816 The intersection of two se...
iocmbl 36817 An open-below, closed-abov...
cnioobibld 36818 A bounded, continuous func...
itgpowd 36819 The integral of a monomial...
arearect 36820 The area of a rectangle wh...
areaquad 36821 The area of a quadrilatera...
ifpan123g 36822 Conjunction of conditional...
ifpan23 36823 Conjunction of conditional...
ifpdfor2 36824 Define or in terms of cond...
ifporcor 36825 Corollary of commutation o...
ifpdfan2 36826 Define and with conditiona...
ifpancor 36827 Corollary of commutation o...
ifpdfor 36828 Define or in terms of cond...
ifpdfan 36829 Define and with conditiona...
ifpbi2 36830 Equivalence theorem for co...
ifpbi3 36831 Equivalence theorem for co...
ifpim1 36832 Restate implication as con...
ifpnot 36833 Restate negated wff as con...
ifpid2 36834 Restate wff as conditional...
ifpim2 36835 Restate implication as con...
ifpbi23 36836 Equivalence theorem for co...
ifpdfbi 36837 Define biimplication as co...
ifpbiidcor 36838 Restatement of ~ biid . (...
ifpbicor 36839 Corollary of commutation o...
ifpxorcor 36840 Corollary of commutation o...
ifpbi1 36841 Equivalence theorem for co...
ifpnot23 36842 Negation of conditional lo...
ifpnotnotb 36843 Factor conditional logic o...
ifpnorcor 36844 Corollary of commutation o...
ifpnancor 36845 Corollary of commutation o...
ifpnot23b 36846 Negation of conditional lo...
ifpbiidcor2 36847 Restatement of ~ biid . (...
ifpnot23c 36848 Negation of conditional lo...
ifpnot23d 36849 Negation of conditional lo...
ifpdfnan 36850 Define nand as conditional...
ifpdfxor 36851 Define xor as conditional ...
ifpbi12 36852 Equivalence theorem for co...
ifpbi13 36853 Equivalence theorem for co...
ifpbi123 36854 Equivalence theorem for co...
ifpidg 36855 Restate wff as conditional...
ifpid3g 36856 Restate wff as conditional...
ifpid2g 36857 Restate wff as conditional...
ifpid1g 36858 Restate wff as conditional...
ifpim23g 36859 Restate implication as con...
ifpim3 36860 Restate implication as con...
ifpnim1 36861 Restate negated implicatio...
ifpim4 36862 Restate implication as con...
ifpnim2 36863 Restate negated implicatio...
ifpim123g 36864 Implication of conditional...
ifpim1g 36865 Implication of conditional...
ifp1bi 36866 Substitute the first eleme...
ifpbi1b 36867 When the first variable is...
ifpimimb 36868 Factor conditional logic o...
ifpororb 36869 Factor conditional logic o...
ifpananb 36870 Factor conditional logic o...
ifpnannanb 36871 Factor conditional logic o...
ifpor123g 36872 Disjunction of conditional...
ifpimim 36873 Consequnce of implication....
ifpbibib 36874 Factor conditional logic o...
ifpxorxorb 36875 Factor conditional logic o...
rp-fakeimass 36876 A special case where impli...
rp-fakeanorass 36877 A special case where a mix...
rp-fakeoranass 36878 A special case where a mix...
rp-fakenanass 36879 A special case where nand ...
rp-fakeinunass 36880 A special case where a mix...
rp-fakeuninass 36881 A special case where a mix...
rp-isfinite5 36882 A set is said to be finite...
rp-isfinite6 36883 A set is said to be finite...
pwelg 36884 The powerclass is an eleme...
pwinfig 36885 The powerclass of an infin...
pwinfi2 36886 The powerclass of an infin...
pwinfi3 36887 The powerclass of an infin...
pwinfi 36888 The powerclass of an infin...
fipjust 36889 A definition of the finite...
cllem0 36890 The class of all sets with...
superficl 36891 The class of all supersets...
superuncl 36892 The class of all supersets...
ssficl 36893 The class of all subsets o...
ssuncl 36894 The class of all subsets o...
ssdifcl 36895 The class of all subsets o...
sssymdifcl 36896 The class of all subsets o...
fiinfi 36897 If two classes have the fi...
rababg 36898 Condition when restricted ...
elintabg 36899 Two ways of saying a set i...
elinintab 36900 Two ways of saying a set i...
elmapintrab 36901 Two ways to say a set is a...
elinintrab 36902 Two ways of saying a set i...
inintabss 36903 Upper bound on intersectio...
inintabd 36904 Value of the intersection ...
xpinintabd 36905 Value of the intersection ...
relintabex 36906 If the intersection of a c...
elcnvcnvintab 36907 Two ways of saying a set i...
relintab 36908 Value of the intersection ...
nonrel 36909 A non-relation is equal to...
elnonrel 36910 Only an ordered pair where...
cnvssb 36911 Subclass theorem for conve...
relnonrel 36912 The non-relation part of a...
cnvnonrel 36913 The converse of the non-re...
brnonrel 36914 A non-relation cannot rela...
dmnonrel 36915 The domain of the non-rela...
rnnonrel 36916 The range of the non-relat...
resnonrel 36917 A restriction of the non-r...
imanonrel 36918 An image under the non-rel...
cononrel1 36919 Composition with the non-r...
cononrel2 36920 Composition with the non-r...
elmapintab 36921 Two ways to say a set is a...
fvnonrel 36922 The function value of any ...
elinlem 36923 Two ways to say a set is a...
elcnvcnvlem 36924 Two ways to say a set is a...
cnvcnvintabd 36925 Value of the relationship ...
elcnvlem 36926 Two ways to say a set is a...
elcnvintab 36927 Two ways of saying a set i...
cnvintabd 36928 Value of the converse of t...
undmrnresiss 36929 Two ways of saying the ide...
reflexg 36930 Two ways of saying a relat...
cnvssco 36931 A condition weaker than re...
refimssco 36932 Reflexive relations are su...
cleq2lem 36933 Equality implies bijection...
cbvcllem 36934 Change of bound variable i...
intabssd 36935 When for each element ` y ...
clublem 36936 If a superset ` Y ` of ` X...
clss2lem 36937 The closure of a property ...
dfid7 36938 Definition of identity rel...
mptrcllem 36939 Show two versions of a clo...
cotrintab 36940 The intersection of a clas...
rclexi 36941 The reflexive closure of a...
rtrclexlem 36942 Existence of relation impl...
rtrclex 36943 The reflexive-transitive c...
trclubgNEW 36944 If a relation exists then ...
trclubNEW 36945 If a relation exists then ...
trclexi 36946 The transitive closure of ...
rtrclexi 36947 The reflexive-transitive c...
clrellem 36948 When the property ` ps ` h...
clcnvlem 36949 When ` A ` , an upper boun...
cnvtrucl0 36950 The converse of the trivia...
cnvrcl0 36951 The converse of the reflex...
cnvtrcl0 36952 The converse of the transi...
dmtrcl 36953 The domain of the transiti...
rntrcl 36954 The range of the transitiv...
dfrtrcl5 36955 Definition of reflexive-tr...
trcleq2lemRP 36956 Equality implies bijection...
al3im 36957 Version of ~ ax-4 for a ne...
intima0 36958 Two ways of expressing the...
elimaint 36959 Element of image of inters...
csbcog 36960 Distribute proper substitu...
cnviun 36961 Converse of indexed union....
imaiun1 36962 The image of an indexed un...
coiun1 36963 Composition with an indexe...
elintima 36964 Element of intersection of...
intimass 36965 The image under the inters...
intimass2 36966 The image under the inters...
intimag 36967 Requirement for the image ...
intimasn 36968 Two ways to express the im...
intimasn2 36969 Two ways to express the im...
ss2iundf 36970 Subclass theorem for index...
ss2iundv 36971 Subclass theorem for index...
cbviuneq12df 36972 Rule used to change the bo...
cbviuneq12dv 36973 Rule used to change the bo...
conrel1d 36974 Deduction about compositio...
conrel2d 36975 Deduction about compositio...
trrelind 36976 The intersection of transi...
xpintrreld 36977 The intersection of a tran...
restrreld 36978 The restriction of a trans...
trrelsuperreldg 36979 Concrete construction of a...
trficl 36980 The class of all transitiv...
cnvtrrel 36981 The converse of a transiti...
trrelsuperrel2dg 36982 Concrete construction of a...
dfrcl2 36985 Reflexive closure of a rel...
dfrcl3 36986 Reflexive closure of a rel...
dfrcl4 36987 Reflexive closure of a rel...
relexp2 36988 A set operated on by the r...
relexpnul 36989 If the domain and range of...
eliunov2 36990 Membership in the indexed ...
eltrclrec 36991 Membership in the indexed ...
elrtrclrec 36992 Membership in the indexed ...
briunov2 36993 Two classes related by the...
brmptiunrelexpd 36994 If two elements are connec...
fvmptiunrelexplb0d 36995 If the indexed union range...
fvmptiunrelexplb0da 36996 If the indexed union range...
fvmptiunrelexplb1d 36997 If the indexed union range...
brfvid 36998 If two elements are connec...
brfvidRP 36999 If two elements are connec...
fvilbd 37000 A set is a subset of its i...
fvilbdRP 37001 A set is a subset of its i...
brfvrcld 37002 If two elements are connec...
brfvrcld2 37003 If two elements are connec...
fvrcllb0d 37004 A restriction of the ident...
fvrcllb0da 37005 A restriction of the ident...
fvrcllb1d 37006 A set is a subset of its i...
brtrclrec 37007 Two classes related by the...
brrtrclrec 37008 Two classes related by the...
briunov2uz 37009 Two classes related by the...
eliunov2uz 37010 Membership in the indexed ...
ov2ssiunov2 37011 Any particular operator va...
relexp0eq 37012 The zeroth power of relati...
iunrelexp0 37013 Simplification of zeroth p...
relexpxpnnidm 37014 Any positive power of a cr...
relexpiidm 37015 Any power of any restricti...
relexpss1d 37016 The relational power of a ...
comptiunov2i 37017 The composition two indexe...
corclrcl 37018 The reflexive closure is i...
iunrelexpmin1 37019 The indexed union of relat...
relexpmulnn 37020 With exponents limited to ...
relexpmulg 37021 With ordered exponents, th...
trclrelexplem 37022 The union of relational po...
iunrelexpmin2 37023 The indexed union of relat...
relexp01min 37024 With exponents limited to ...
relexp1idm 37025 Repeated raising a relatio...
relexp0idm 37026 Repeated raising a relatio...
relexp0a 37027 Absorbtion law for zeroth ...
relexpxpmin 37028 The composition of powers ...
relexpaddss 37029 The composition of two pow...
iunrelexpuztr 37030 The indexed union of relat...
dftrcl3 37031 Transitive closure of a re...
brfvtrcld 37032 If two elements are connec...
fvtrcllb1d 37033 A set is a subset of its i...
trclfvcom 37034 The transitive closure of ...
cnvtrclfv 37035 The converse of the transi...
cotrcltrcl 37036 The transitive closure is ...
trclimalb2 37037 Lower bound for image unde...
brtrclfv2 37038 Two ways to indicate two e...
trclfvdecomr 37039 The transitive closure of ...
trclfvdecoml 37040 The transitive closure of ...
dmtrclfvRP 37041 The domain of the transiti...
rntrclfvRP 37042 The range of the transitiv...
rntrclfv 37043 The range of the transitiv...
dfrtrcl3 37044 Reflexive-transitive closu...
brfvrtrcld 37045 If two elements are connec...
fvrtrcllb0d 37046 A restriction of the ident...
fvrtrcllb0da 37047 A restriction of the ident...
fvrtrcllb1d 37048 A set is a subset of its i...
dfrtrcl4 37049 Reflexive-transitive closu...
corcltrcl 37050 The composition of the ref...
cortrcltrcl 37051 Composition with the refle...
corclrtrcl 37052 Composition with the refle...
cotrclrcl 37053 The composition of the ref...
cortrclrcl 37054 Composition with the refle...
cotrclrtrcl 37055 Composition with the refle...
cortrclrtrcl 37056 The reflexive-transitive c...
frege77d 37057 If the images of both ` { ...
frege81d 37058 If the image of ` U ` is a...
frege83d 37059 If the image of the union ...
frege96d 37060 If ` C ` follows ` A ` in ...
frege87d 37061 If the images of both ` { ...
frege91d 37062 If ` B ` follows ` A ` in ...
frege97d 37063 If ` A ` contains all elem...
frege98d 37064 If ` C ` follows ` A ` and...
frege102d 37065 If either ` A ` and ` C ` ...
frege106d 37066 If ` B ` follows ` A ` in ...
frege108d 37067 If either ` A ` and ` C ` ...
frege109d 37068 If ` A ` contains all elem...
frege114d 37069 If either ` R ` relates ` ...
frege111d 37070 If either ` A ` and ` C ` ...
frege122d 37071 If ` F ` is a function, ` ...
frege124d 37072 If ` F ` is a function, ` ...
frege126d 37073 If ` F ` is a function, ` ...
frege129d 37074 If ` F ` is a function and...
frege131d 37075 If ` F ` is a function and...
frege133d 37076 If ` F ` is a function and...
dfxor4 37077 Express exclusive-or in te...
dfxor5 37078 Express exclusive-or in te...
df3or2 37079 Express triple-or in terms...
df3an2 37080 Express triple-and in term...
nev 37081 Express that not every set...
dfss6 37082 Another definition of subc...
ndisj 37083 Express that an intersecti...
0pssin 37084 Express that an intersecti...
rp-imass 37085 If the ` R ` -image of a c...
dfhe2 37088 The property of relation `...
dfhe3 37089 The property of relation `...
heeq12 37090 Equality law for relations...
heeq1 37091 Equality law for relations...
heeq2 37092 Equality law for relations...
sbcheg 37093 Distribute proper substitu...
hess 37094 Subclass law for relations...
xphe 37095 Any Cartesian product is h...
0he 37096 The empty relation is here...
0heALT 37097 The empty relation is here...
he0 37098 Any relation is hereditary...
unhe1 37099 The union of two relations...
snhesn 37100 Any singleton is hereditar...
idhe 37101 The identity relation is h...
psshepw 37102 The relation between sets ...
sshepw 37103 The relation between sets ...
rp-simp2-frege 37106 Simplification of triple c...
rp-simp2 37107 Simplification of triple c...
rp-frege3g 37108 Add antecedent to ~ ax-fre...
frege3 37109 Add antecedent to ~ ax-fre...
rp-misc1-frege 37110 Double-use of ~ ax-frege2 ...
rp-frege24 37111 Introducing an embedded an...
rp-frege4g 37112 Deduction related to distr...
frege4 37113 Special case of closed for...
frege5 37114 A closed form of ~ syl . ...
rp-7frege 37115 Distribute antecedent and ...
rp-4frege 37116 Elimination of a nested an...
rp-6frege 37117 Elimination of a nested an...
rp-8frege 37118 Eliminate antecedent when ...
rp-frege25 37119 Closed form for ~ a1dd . ...
frege6 37120 A closed form of ~ imim2d ...
axfrege8 37121 Swap antecedents. Identic...
frege7 37122 A closed form of ~ syl6 . ...
frege26 37124 Identical to ~ idd . Prop...
frege27 37125 We cannot (at the same tim...
frege9 37126 Closed form of ~ syl with ...
frege12 37127 A closed form of ~ com23 ....
frege11 37128 Elimination of a nested an...
frege24 37129 Closed form for ~ a1d . D...
frege16 37130 A closed form of ~ com34 ....
frege25 37131 Closed form for ~ a1dd . ...
frege18 37132 Closed form of a syllogism...
frege22 37133 A closed form of ~ com45 ....
frege10 37134 Result commuting anteceden...
frege17 37135 A closed form of ~ com3l ....
frege13 37136 A closed form of ~ com3r ....
frege14 37137 Closed form of a deduction...
frege19 37138 A closed form of ~ syl6 . ...
frege23 37139 Syllogism followed by rota...
frege15 37140 A closed form of ~ com4r ....
frege21 37141 Replace antecedent in ante...
frege20 37142 A closed form of ~ syl8 . ...
axfrege28 37143 Contraposition. Identical...
frege29 37145 Closed form of ~ con3d . ...
frege30 37146 Commuted, closed form of ~...
axfrege31 37147 Identical to ~ notnotr . ...
frege32 37149 Deduce ~ con1 from ~ con3 ...
frege33 37150 If ` ph ` or ` ps ` takes ...
frege34 37151 If as a conseqence of the ...
frege35 37152 Commuted, closed form of ~...
frege36 37153 The case in which ` ps ` i...
frege37 37154 If ` ch ` is a necessary c...
frege38 37155 Identical to ~ pm2.21 . P...
frege39 37156 Syllogism between ~ pm2.18...
frege40 37157 Anything implies ~ pm2.18 ...
axfrege41 37158 Identical to ~ notnot . A...
frege42 37160 Not not ~ id . Propositio...
frege43 37161 If there is a choice only ...
frege44 37162 Similar to a commuted ~ pm...
frege45 37163 Deduce ~ pm2.6 from ~ con1...
frege46 37164 If ` ps ` holds when ` ph ...
frege47 37165 Deduce consequence follows...
frege48 37166 Closed form of syllogism w...
frege49 37167 Closed form of deduction w...
frege50 37168 Closed form of ~ jaoi . P...
frege51 37169 Compare with ~ jaod . Pro...
axfrege52a 37170 Justification for ~ ax-fre...
frege52aid 37172 The case when the content ...
frege53aid 37173 Specialization of ~ frege5...
frege53a 37174 Lemma for ~ frege55a . Pr...
axfrege54a 37175 Justification for ~ ax-fre...
frege54cor0a 37177 Synonym for logical equiva...
frege54cor1a 37178 Reflexive equality. (Cont...
frege55aid 37179 Lemma for ~ frege57aid . ...
frege55lem1a 37180 Necessary deduction regard...
frege55lem2a 37181 Core proof of Proposition ...
frege55a 37182 Proposition 55 of [Frege18...
frege55cor1a 37183 Proposition 55 of [Frege18...
frege56aid 37184 Lemma for ~ frege57aid . ...
frege56a 37185 Proposition 56 of [Frege18...
frege57aid 37186 This is the all imporant f...
frege57a 37187 Analogue of ~ frege57aid ....
axfrege58a 37188 Identical to ~ anifp . Ju...
frege58acor 37190 Lemma for ~ frege59a . (C...
frege59a 37191 A kind of Aristotelian inf...
frege60a 37192 Swap antecedents of ~ ax-f...
frege61a 37193 Lemma for ~ frege65a . Pr...
frege62a 37194 A kind of Aristotelian inf...
frege63a 37195 Proposition 63 of [Frege18...
frege64a 37196 Lemma for ~ frege65a . Pr...
frege65a 37197 A kind of Aristotelian inf...
frege66a 37198 Swap antecedents of ~ freg...
frege67a 37199 Lemma for ~ frege68a . Pr...
frege68a 37200 Combination of applying a ...
axfrege52c 37201 Justification for ~ ax-fre...
frege52b 37203 The case when the content ...
frege53b 37204 Lemma for frege102 (via ~ ...
axfrege54c 37205 Reflexive equality of clas...
frege54b 37207 Reflexive equality of sets...
frege54cor1b 37208 Reflexive equality. (Cont...
frege55lem1b 37209 Necessary deduction regard...
frege55lem2b 37210 Lemma for ~ frege55b . Co...
frege55b 37211 Lemma for ~ frege57b . Pr...
frege56b 37212 Lemma for ~ frege57b . Pr...
frege57b 37213 Analogue of ~ frege57aid ....
axfrege58b 37214 If ` A. x ph ` is affirmed...
frege58bid 37216 If ` A. x ph ` is affirmed...
frege58bcor 37217 Lemma for ~ frege59b . (C...
frege59b 37218 A kind of Aristotelian inf...
frege60b 37219 Swap antecedents of ~ ax-f...
frege61b 37220 Lemma for ~ frege65b . Pr...
frege62b 37221 A kind of Aristotelian inf...
frege63b 37222 Lemma for ~ frege91 . Pro...
frege64b 37223 Lemma for ~ frege65b . Pr...
frege65b 37224 A kind of Aristotelian inf...
frege66b 37225 Swap antecedents of ~ freg...
frege67b 37226 Lemma for ~ frege68b . Pr...
frege68b 37227 Combination of applying a ...
frege53c 37228 Proposition 53 of [Frege18...
frege54cor1c 37229 Reflexive equality. (Cont...
frege55lem1c 37230 Necessary deduction regard...
frege55lem2c 37231 Core proof of Proposition ...
frege55c 37232 Proposition 55 of [Frege18...
frege56c 37233 Lemma for ~ frege57c . Pr...
frege57c 37234 Swap order of implication ...
frege58c 37235 Principle related to ~ sp ...
frege59c 37236 A kind of Aristotelian inf...
frege60c 37237 Swap antecedents of ~ freg...
frege61c 37238 Lemma for ~ frege65c . Pr...
frege62c 37239 A kind of Aristotelian inf...
frege63c 37240 Analogue of ~ frege63b . ...
frege64c 37241 Lemma for ~ frege65c . Pr...
frege65c 37242 A kind of Aristotelian inf...
frege66c 37243 Swap antecedents of ~ freg...
frege67c 37244 Lemma for ~ frege68c . Pr...
frege68c 37245 Combination of applying a ...
dffrege69 37246 If from the proposition th...
frege70 37247 Lemma for ~ frege72 . Pro...
frege71 37248 Lemma for ~ frege72 . Pro...
frege72 37249 If property ` A ` is hered...
frege73 37250 Lemma for ~ frege87 . Pro...
frege74 37251 If ` X ` has a property ` ...
frege75 37252 If from the proposition th...
dffrege76 37253 If from the two propositio...
frege77 37254 If ` Y ` follows ` X ` in ...
frege78 37255 Commuted form of of ~ freg...
frege79 37256 Distributed form of ~ freg...
frege80 37257 Add additional condition t...
frege81 37258 If ` X ` has a property ` ...
frege82 37259 Closed-form deduction base...
frege83 37260 Apply commuted form of ~ f...
frege84 37261 Commuted form of ~ frege81...
frege85 37262 Commuted form of ~ frege77...
frege86 37263 Conclusion about element o...
frege87 37264 If ` Z ` is a result of an...
frege88 37265 Commuted form of ~ frege87...
frege89 37266 One direction of ~ dffrege...
frege90 37267 Add antecedent to ~ frege8...
frege91 37268 Every result of an applica...
frege92 37269 Inference from ~ frege91 ....
frege93 37270 Necessary condition for tw...
frege94 37271 Looking one past a pair re...
frege95 37272 Looking one past a pair re...
frege96 37273 Every result of an applica...
frege97 37274 The property of following ...
frege98 37275 If ` Y ` follows ` X ` and...
dffrege99 37276 If ` Z ` is identical with...
frege100 37277 One direction of ~ dffrege...
frege101 37278 Lemma for ~ frege102 . Pr...
frege102 37279 If ` Z ` belongs to the ` ...
frege103 37280 Proposition 103 of [Frege1...
frege104 37281 Proposition 104 of [Frege1...
frege105 37282 Proposition 105 of [Frege1...
frege106 37283 Whatever follows ` X ` in ...
frege107 37284 Proposition 107 of [Frege1...
frege108 37285 If ` Y ` belongs to the ` ...
frege109 37286 The property of belonging ...
frege110 37287 Proposition 110 of [Frege1...
frege111 37288 If ` Y ` belongs to the ` ...
frege112 37289 Identity implies belonging...
frege113 37290 Proposition 113 of [Frege1...
frege114 37291 If ` X ` belongs to the ` ...
dffrege115 37292 If from the the circumstan...
frege116 37293 One direction of ~ dffrege...
frege117 37294 Lemma for ~ frege118 . Pr...
frege118 37295 Simplified application of ...
frege119 37296 Lemma for ~ frege120 . Pr...
frege120 37297 Simplified application of ...
frege121 37298 Lemma for ~ frege122 . Pr...
frege122 37299 If ` X ` is a result of an...
frege123 37300 Lemma for ~ frege124 . Pr...
frege124 37301 If ` X ` is a result of an...
frege125 37302 Lemma for ~ frege126 . Pr...
frege126 37303 If ` M ` follows ` Y ` in ...
frege127 37304 Communte antecedents of ~ ...
frege128 37305 Lemma for ~ frege129 . Pr...
frege129 37306 If the procedure ` R ` is ...
frege130 37307 Lemma for ~ frege131 . Pr...
frege131 37308 If the procedure ` R ` is ...
frege132 37309 Lemma for ~ frege133 . Pr...
frege133 37310 If the procedure ` R ` is ...
enrelmap 37311 The set of all possible re...
enrelmapr 37312 The set of all possible re...
enmappw 37313 The set of all mappings fr...
enmappwid 37314 The set of all mappings fr...
rfovd 37315 Value of the operator, ` (...
rfovfvd 37316 Value of the operator, ` (...
rfovfvfvd 37317 Value of the operator, ` (...
rfovcnvf1od 37318 Properties of the operator...
rfovcnvd 37319 Value of the converse of t...
rfovf1od 37320 The value of the operator,...
rfovcnvfvd 37321 Value of the converse of t...
fsovd 37322 Value of the operator, ` (...
fsovrfovd 37323 The operator which gives a...
fsovfvd 37324 Value of the operator, ` (...
fsovfvfvd 37325 Value of the operator, ` (...
fsovfd 37326 The operator, ` ( A O B ) ...
fsovcnvlem 37327 The ` O ` operator, which ...
fsovcnvd 37328 The value of the converse ...
fsovcnvfvd 37329 The value of the converse ...
fsovf1od 37330 The value of ` ( A O B ) `...
dssmapfvd 37331 Value of the duality opera...
dssmapfv2d 37332 Value of the duality opera...
dssmapfv3d 37333 Value of the duality opera...
dssmapnvod 37334 For any base set ` B ` the...
dssmapf1od 37335 For any base set ` B ` the...
dssmap2d 37336 For any base set ` B ` the...
sscon34b 37337 Relative complementation r...
rcompleq 37338 Two subclasses are equal i...
or3or 37339 Decompose disjunction into...
andi3or 37340 Distribute over triple dis...
uneqsn 37341 If a union of classes is e...
df3o2 37342 Ordinal 3 is the triplet c...
df3o3 37343 Ordinal 3 , fully expanded...
brfvimex 37344 If a binary relation holds...
brovmptimex 37345 If a binary relation holds...
brovmptimex1 37346 If a binary relation holds...
brovmptimex2 37347 If a binary relation holds...
brcoffn 37348 Conditions allowing the de...
brcofffn 37349 Conditions allowing the de...
brco2f1o 37350 Conditions allowing the de...
brco3f1o 37351 Conditions allowing the de...
ntrclsbex 37352 If (pseudo-)interior and (...
ntrclsrcomplex 37353 The relative complement of...
neik0imk0p 37354 Kuratowski's K0 axiom impl...
ntrk2imkb 37355 If an interior function is...
ntrkbimka 37356 If the interiors of disjoi...
ntrk0kbimka 37357 If the interiors of disjoi...
clsk3nimkb 37358 If the base set is not emp...
clsk1indlem0 37359 The ansatz closure functio...
clsk1indlem2 37360 The ansatz closure functio...
clsk1indlem3 37361 The ansatz closure functio...
clsk1indlem4 37362 The ansatz closure functio...
clsk1indlem1 37363 The ansatz closure functio...
clsk1independent 37364 For generalized closure fu...
neik0pk1imk0 37365 Kuratowski's K0' and K1 ax...
isotone1 37366 Two different ways to say ...
isotone2 37367 Two different ways to say ...
ntrk1k3eqk13 37368 An interior function is bo...
ntrclsf1o 37369 If (pseudo-)interior and (...
ntrclsnvobr 37370 If (pseudo-)interior and (...
ntrclsiex 37371 If (pseudo-)interior and (...
ntrclskex 37372 If (pseudo-)interior and (...
ntrclsfv1 37373 If (pseudo-)interior and (...
ntrclsfv2 37374 If (pseudo-)interior and (...
ntrclselnel1 37375 If (pseudo-)interior and (...
ntrclselnel2 37376 If (pseudo-)interior and (...
ntrclsfv 37377 The value of the interior ...
ntrclsfveq1 37378 If interior and closure fu...
ntrclsfveq2 37379 If interior and closure fu...
ntrclsfveq 37380 If interior and closure fu...
ntrclsss 37381 If interior and closure fu...
ntrclsneine0lem 37382 If (pseudo-)interior and (...
ntrclsneine0 37383 If (pseudo-)interior and (...
ntrclscls00 37384 If (pseudo-)interior and (...
ntrclsiso 37385 If (pseudo-)interior and (...
ntrclsk2 37386 An interior function is co...
ntrclskb 37387 The interiors of disjoint ...
ntrclsk3 37388 The intersection of interi...
ntrclsk13 37389 The interior of the inters...
ntrclsk4 37390 Idempotence of the interio...
ntrneibex 37391 If (pseudo-)interior and (...
ntrneircomplex 37392 The relative complement of...
ntrneif1o 37393 If (pseudo-)interior and (...
ntrneiiex 37394 If (pseudo-)interior and (...
ntrneinex 37395 If (pseudo-)interior and (...
ntrneicnv 37396 If (pseudo-)interior and (...
ntrneifv1 37397 If (pseudo-)interior and (...
ntrneifv2 37398 If (pseudo-)interior and (...
ntrneiel 37399 If (pseudo-)interior and (...
ntrneifv3 37400 The value of the neighbors...
ntrneineine0lem 37401 If (pseudo-)interior and (...
ntrneineine1lem 37402 If (pseudo-)interior and (...
ntrneifv4 37403 The value of the interior ...
ntrneiel2 37404 Membership in iterated int...
ntrneineine0 37405 If (pseudo-)interior and (...
ntrneineine1 37406 If (pseudo-)interior and (...
ntrneicls00 37407 If (pseudo-)interior and (...
ntrneicls11 37408 If (pseudo-)interior and (...
ntrneiiso 37409 If (pseudo-)interior and (...
ntrneik2 37410 An interior function is co...
ntrneix2 37411 An interior (closure) func...
ntrneikb 37412 The interiors of disjoint ...
ntrneixb 37413 The interiors (closures) o...
ntrneik3 37414 The intersection of interi...
ntrneix3 37415 The closure of the union o...
ntrneik13 37416 The interior of the inters...
ntrneix13 37417 The closure of the union o...
ntrneik4w 37418 Idempotence of the interio...
ntrneik4 37419 Idempotence of the interio...
clsneibex 37420 If (pseudo-)closure and (p...
clsneircomplex 37421 The relative complement of...
clsneif1o 37422 If a (pseudo-)closure func...
clsneicnv 37423 If a (pseudo-)closure func...
clsneikex 37424 If closure and neighborhoo...
clsneinex 37425 If closure and neighborhoo...
clsneiel1 37426 If a (pseudo-)closure func...
clsneiel2 37427 If a (pseudo-)closure func...
clsneifv3 37428 Value of the neighborhoods...
clsneifv4 37429 Value of the the closure (...
neicvgbex 37430 If (pseudo-)neighborhood a...
neicvgrcomplex 37431 The relative complement of...
neicvgf1o 37432 If neighborhood and conver...
neicvgnvo 37433 If neighborhood and conver...
neicvgnvor 37434 If neighborhood and conver...
neicvgmex 37435 If the neighborhoods and c...
neicvgnex 37436 If the neighborhoods and c...
neicvgel1 37437 A subset being an element ...
neicvgel2 37438 The complement of a subset...
neicvgfv 37439 The value of the neighborh...
ntrrn 37440 The range of the interior ...
ntrf 37441 The interior function of a...
ntrf2 37442 The interior function is a...
ntrelmap 37443 The interior function is a...
clsf2 37444 The closure function is a ...
clselmap 37445 The closure function is a ...
dssmapntrcls 37446 The interior and closure o...
dssmapclsntr 37447 The closure and interior o...
gneispa 37448 Each point ` p ` of the ne...
gneispb 37449 Given a neighborhood ` N `...
gneispace2 37450 The predicate that ` F ` i...
gneispace3 37451 The predicate that ` F ` i...
gneispace 37452 The predicate that ` F ` i...
gneispacef 37453 A generic neighborhood spa...
gneispacef2 37454 A generic neighborhood spa...
gneispacefun 37455 A generic neighborhood spa...
gneispacern 37456 A generic neighborhood spa...
gneispacern2 37457 A generic neighborhood spa...
gneispace0nelrn 37458 A generic neighborhood spa...
gneispace0nelrn2 37459 A generic neighborhood spa...
gneispace0nelrn3 37460 A generic neighborhood spa...
gneispaceel 37461 Every neighborhood of a po...
gneispaceel2 37462 Every neighborhood of a po...
gneispacess 37463 All supersets of a neighbo...
gneispacess2 37464 All supersets of a neighbo...
k0004lem1 37465 Application of ~ ssin to r...
k0004lem2 37466 A mapping with a particula...
k0004lem3 37467 When the value of a mappin...
k0004val 37468 The topological simplex of...
k0004ss1 37469 The topological simplex of...
k0004ss2 37470 The topological simplex of...
k0004ss3 37471 The topological simplex of...
k0004val0 37472 The topological simplex of...
inductionexd 37473 Simple induction example. ...
wwlemuld 37474 Natural deduction form of ...
leeq1d 37475 Specialization of ~ breq1d...
leeq2d 37476 Specialization of ~ breq2d...
absmulrposd 37477 Specialization of absmuld ...
imadisjld 37478 Natural dduction form of o...
imadisjlnd 37479 Natural deduction form of ...
wnefimgd 37480 The image of a mapping fro...
fco2d 37481 Natural deduction form of ...
suprubd 37482 Natural deduction form of ...
suprcld 37483 Natural deduction form of ...
fvco3d 37484 Natural deduction form of ...
wfximgfd 37485 The value of a function on...
extoimad 37486 If |f(x)| <= C for all x t...
imo72b2lem0 37487 Lemma for ~ imo72b2 . (Co...
suprleubrd 37488 Natural deduction form of ...
imo72b2lem2 37489 Lemma for ~ imo72b2 . (Co...
syldbl2 37490 Stacked hypotheseis implie...
funfvima2d 37491 A function's value in a pr...
suprlubrd 37492 Natural deduction form of ...
imo72b2lem1 37493 Lemma for ~ imo72b2 . (Co...
lemuldiv3d 37494 'Less than or equal to' re...
lemuldiv4d 37495 'Less than or equal to' re...
rspcdvinvd 37496 If something is true for a...
imo72b2 37497 IMO 1972 B2. (14th Intern...
int-addcomd 37498 AdditionCommutativity gene...
int-addassocd 37499 AdditionAssociativity gene...
int-addsimpd 37500 AdditionSimplification gen...
int-mulcomd 37501 MultiplicationCommutativit...
int-mulassocd 37502 MultiplicationAssociativit...
int-mulsimpd 37503 MultiplicationSimplificati...
int-leftdistd 37504 AdditionMultiplicationLeft...
int-rightdistd 37505 AdditionMultiplicationRigh...
int-sqdefd 37506 SquareDefinition generator...
int-mul11d 37507 First MultiplicationOne ge...
int-mul12d 37508 Second MultiplicationOne g...
int-add01d 37509 First AdditionZero generat...
int-add02d 37510 Second AdditionZero genera...
int-sqgeq0d 37511 SquareGEQZero generator ru...
int-eqprincd 37512 PrincipleOfEquality genera...
int-eqtransd 37513 EqualityTransitivity gener...
int-eqmvtd 37514 EquMoveTerm generator rule...
int-eqineqd 37515 EquivalenceImpliesDoubleIn...
int-ineqmvtd 37516 IneqMoveTerm generator rul...
int-ineq1stprincd 37517 FirstPrincipleOfInequality...
int-ineq2ndprincd 37518 SecondPrincipleOfInequalit...
int-ineqtransd 37519 InequalityTransitivity gen...
unitadd 37520 Theorem used in conjunctio...
gsumws3 37521 Valuation of a length 3 wo...
gsumws4 37522 Valuation of a length 4 wo...
amgm2d 37523 Arithmetic-geometric mean ...
amgm3d 37524 Arithmetic-geometric mean ...
amgm4d 37525 Arithmetic-geometric mean ...
nanorxor 37526 'nand' is equivalent to th...
undisjrab 37527 Union of two disjoint rest...
iso0 37528 The empty set is an ` R , ...
ssrecnpr 37529 ` RR ` is a subset of both...
seff 37530 Let set ` S ` be the real ...
sblpnf 37531 The infinity ball in the a...
prmunb2 37532 The primes are unbounded. ...
dvgrat 37533 Ratio test for divergence ...
cvgdvgrat 37534 Ratio test for convergence...
radcnvrat 37535 Let ` L ` be the limit, if...
reldvds 37536 The divides relation is in...
nznngen 37537 All positive integers in t...
nzss 37538 The set of multiples of _m...
nzin 37539 The intersection of the se...
nzprmdif 37540 Subtract one prime's multi...
hashnzfz 37541 Special case of ~ hashdvds...
hashnzfz2 37542 Special case of ~ hashnzfz...
hashnzfzclim 37543 As the upper bound ` K ` o...
caofcan 37544 Transfer a cancellation la...
ofsubid 37545 Function analogue of ~ sub...
ofmul12 37546 Function analogue of ~ mul...
ofdivrec 37547 Function analogue of ~ div...
ofdivcan4 37548 Function analogue of ~ div...
ofdivdiv2 37549 Function analogue of ~ div...
lhe4.4ex1a 37550 Example of the Fundamental...
dvsconst 37551 Derivative of a constant f...
dvsid 37552 Derivative of the identity...
dvsef 37553 Derivative of the exponent...
expgrowthi 37554 Exponential growth and dec...
dvconstbi 37555 The derivative of a functi...
expgrowth 37556 Exponential growth and dec...
bccval 37559 Value of the generalized b...
bcccl 37560 Closure of the generalized...
bcc0 37561 The generalized binomial c...
bccp1k 37562 Generalized binomial coeff...
bccm1k 37563 Generalized binomial coeff...
bccn0 37564 Generalized binomial coeff...
bccn1 37565 Generalized binomial coeff...
bccbc 37566 The binomial coefficient a...
uzmptshftfval 37567 When ` F ` is a maps-to fu...
dvradcnv2 37568 The radius of convergence ...
binomcxplemwb 37569 Lemma for ~ binomcxp . Th...
binomcxplemnn0 37570 Lemma for ~ binomcxp . Wh...
binomcxplemrat 37571 Lemma for ~ binomcxp . As...
binomcxplemfrat 37572 Lemma for ~ binomcxp . ~ b...
binomcxplemradcnv 37573 Lemma for ~ binomcxp . By...
binomcxplemdvbinom 37574 Lemma for ~ binomcxp . By...
binomcxplemcvg 37575 Lemma for ~ binomcxp . Th...
binomcxplemdvsum 37576 Lemma for ~ binomcxp . Th...
binomcxplemnotnn0 37577 Lemma for ~ binomcxp . Wh...
binomcxp 37578 Generalize the binomial th...
pm10.12 37579 Theorem *10.12 in [Whitehe...
pm10.14 37580 Theorem *10.14 in [Whitehe...
pm10.251 37581 Theorem *10.251 in [Whiteh...
pm10.252 37582 Theorem *10.252 in [Whiteh...
pm10.253 37583 Theorem *10.253 in [Whiteh...
albitr 37584 Theorem *10.301 in [Whiteh...
pm10.42 37585 Theorem *10.42 in [Whitehe...
pm10.52 37586 Theorem *10.52 in [Whitehe...
pm10.53 37587 Theorem *10.53 in [Whitehe...
pm10.541 37588 Theorem *10.541 in [Whiteh...
pm10.542 37589 Theorem *10.542 in [Whiteh...
pm10.55 37590 Theorem *10.55 in [Whitehe...
pm10.56 37591 Theorem *10.56 in [Whitehe...
pm10.57 37592 Theorem *10.57 in [Whitehe...
2alanimi 37593 Removes two universal quan...
2al2imi 37594 Removes two universal quan...
pm11.11 37595 Theorem *11.11 in [Whitehe...
pm11.12 37596 Theorem *11.12 in [Whitehe...
19.21vv 37597 Compare Theorem *11.3 in [...
2alim 37598 Theorem *11.32 in [Whitehe...
2albi 37599 Theorem *11.33 in [Whitehe...
2exim 37600 Theorem *11.34 in [Whitehe...
2exbi 37601 Theorem *11.341 in [Whiteh...
spsbce-2 37602 Theorem *11.36 in [Whitehe...
19.33-2 37603 Theorem *11.421 in [Whiteh...
19.36vv 37604 Theorem *11.43 in [Whitehe...
19.31vv 37605 Theorem *11.44 in [Whitehe...
19.37vv 37606 Theorem *11.46 in [Whitehe...
19.28vv 37607 Theorem *11.47 in [Whitehe...
pm11.52 37608 Theorem *11.52 in [Whitehe...
2exanali 37609 Theorem *11.521 in [Whiteh...
aaanv 37610 Theorem *11.56 in [Whitehe...
pm11.57 37611 Theorem *11.57 in [Whitehe...
pm11.58 37612 Theorem *11.58 in [Whitehe...
pm11.59 37613 Theorem *11.59 in [Whitehe...
pm11.6 37614 Theorem *11.6 in [Whitehea...
pm11.61 37615 Theorem *11.61 in [Whitehe...
pm11.62 37616 Theorem *11.62 in [Whitehe...
pm11.63 37617 Theorem *11.63 in [Whitehe...
pm11.7 37618 Theorem *11.7 in [Whitehea...
pm11.71 37619 Theorem *11.71 in [Whitehe...
sbeqal1 37620 If ` x = y ` always implie...
sbeqal1i 37621 Suppose you know ` x = y `...
sbeqal2i 37622 If ` x = y ` implies ` x =...
sbeqalbi 37623 When both ` x ` and ` z ` ...
axc5c4c711 37624 Proof of a theorem that ca...
axc5c4c711toc5 37625 Rederivation of ~ sp from ...
axc5c4c711toc4 37626 Rederivation of ~ axc4 fro...
axc5c4c711toc7 37627 Rederivation of ~ axc7 fro...
axc5c4c711to11 37628 Rederivation of ~ ax-11 fr...
axc11next 37629 This theorem shows that, g...
pm13.13a 37630 One result of theorem *13....
pm13.13b 37631 Theorem *13.13 in [Whitehe...
pm13.14 37632 Theorem *13.14 in [Whitehe...
pm13.192 37633 Theorem *13.192 in [Whiteh...
pm13.193 37634 Theorem *13.193 in [Whiteh...
pm13.194 37635 Theorem *13.194 in [Whiteh...
pm13.195 37636 Theorem *13.195 in [Whiteh...
pm13.196a 37637 Theorem *13.196 in [Whiteh...
2sbc6g 37638 Theorem *13.21 in [Whitehe...
2sbc5g 37639 Theorem *13.22 in [Whitehe...
iotain 37640 Equivalence between two di...
iotaexeu 37641 The iota class exists. Th...
iotasbc 37642 Definition *14.01 in [Whit...
iotasbc2 37643 Theorem *14.111 in [Whiteh...
pm14.12 37644 Theorem *14.12 in [Whitehe...
pm14.122a 37645 Theorem *14.122 in [Whiteh...
pm14.122b 37646 Theorem *14.122 in [Whiteh...
pm14.122c 37647 Theorem *14.122 in [Whiteh...
pm14.123a 37648 Theorem *14.123 in [Whiteh...
pm14.123b 37649 Theorem *14.123 in [Whiteh...
pm14.123c 37650 Theorem *14.123 in [Whiteh...
pm14.18 37651 Theorem *14.18 in [Whitehe...
iotaequ 37652 Theorem *14.2 in [Whitehea...
iotavalb 37653 Theorem *14.202 in [Whiteh...
iotasbc5 37654 Theorem *14.205 in [Whiteh...
pm14.24 37655 Theorem *14.24 in [Whitehe...
iotavalsb 37656 Theorem *14.242 in [Whiteh...
sbiota1 37657 Theorem *14.25 in [Whitehe...
sbaniota 37658 Theorem *14.26 in [Whitehe...
eubi 37659 Theorem *14.271 in [Whiteh...
iotasbcq 37660 Theorem *14.272 in [Whiteh...
elnev 37661 Any set that contains one ...
rusbcALT 37662 A version of Russell's par...
compel 37663 Equivalence between two wa...
compeq 37664 Equality between two ways ...
compne 37665 The complement of ` A ` is...
compab 37666 Two ways of saying "the co...
conss34OLD 37667 Obsolete proof of ~ compls...
conss2 37668 Contrapositive law for sub...
conss1 37669 Contrapositive law for sub...
ralbidar 37670 More general form of ~ ral...
rexbidar 37671 More general form of ~ rex...
dropab1 37672 Theorem to aid use of the ...
dropab2 37673 Theorem to aid use of the ...
ipo0 37674 If the identity relation p...
ifr0 37675 A class that is founded by...
ordpss 37676 ~ ordelpss with an anteced...
fvsb 37677 Explicit substitution of a...
fveqsb 37678 Implicit substitution of a...
xpexb 37679 A Cartesian product exists...
trelpss 37680 An element of a transitive...
addcomgi 37681 Generalization of commutat...
addrval 37691 Value of the operation of ...
subrval 37692 Value of the operation of ...
mulvval 37693 Value of the operation of ...
addrfv 37694 Vector addition at a value...
subrfv 37695 Vector subtraction at a va...
mulvfv 37696 Scalar multiplication at a...
addrfn 37697 Vector addition produces a...
subrfn 37698 Vector subtraction produce...
mulvfn 37699 Scalar multiplication prod...
addrcom 37700 Vector addition is commuta...
idiALT 37704 Placeholder for ~ idi . T...
exbir 37705 Exportation implication al...
3impexpbicom 37706 Version of ~ 3impexp where...
3impexpbicomi 37707 Inference associated with ...
bi1imp 37708 Importation inference simi...
bi2imp 37709 Importation inference simi...
bi3impb 37710 Similar to ~ 3impb with im...
bi3impa 37711 Similar to ~ 3impa with im...
bi23impib 37712 ~ 3impib with the inner im...
bi13impib 37713 ~ 3impib with the outer im...
bi123impib 37714 ~ 3impib with the implicat...
bi13impia 37715 ~ 3impia with the outer im...
bi123impia 37716 ~ 3impia with the implicat...
bi33imp12 37717 ~ 3imp with innermost impl...
bi23imp13 37718 ~ 3imp with middle implica...
bi13imp23 37719 ~ 3imp with outermost impl...
bi13imp2 37720 Similar to ~ 3imp except t...
bi12imp3 37721 Similar to ~ 3imp except a...
bi23imp1 37722 Similar to ~ 3imp except a...
bi123imp0 37723 Similar to ~ 3imp except a...
4animp1 37724 A single hypothesis unific...
4an31 37725 A rearrangement of conjunc...
4an4132 37726 A rearrangement of conjunc...
expcomdg 37727 Biconditional form of ~ ex...
iidn3 37728 ~ idn3 without virtual ded...
ee222 37729 ~ e222 without virtual ded...
ee3bir 37730 Right-biconditional form o...
ee13 37731 ~ e13 without virtual dedu...
ee121 37732 ~ e121 without virtual ded...
ee122 37733 ~ e122 without virtual ded...
ee333 37734 ~ e333 without virtual ded...
ee323 37735 ~ e323 without virtual ded...
3ornot23 37736 If the second and third di...
orbi1r 37737 ~ orbi1 with order of disj...
3orbi123 37738 ~ pm4.39 with a 3-conjunct...
syl5imp 37739 Closed form of ~ syl5 . D...
impexpd 37740 The following User's Proof...
com3rgbi 37741 The following User's Proof...
impexpdcom 37742 The following User's Proof...
ee1111 37743 Non-virtual deduction form...
pm2.43bgbi 37744 Logical equivalence of a 2...
pm2.43cbi 37745 Logical equivalence of a 3...
ee233 37746 Non-virtual deduction form...
imbi13 37747 Join three logical equival...
ee33 37748 Non-virtual deduction form...
con5 37749 Biconditional contrapositi...
con5i 37750 Inference form of ~ con5 ....
exlimexi 37751 Inference similar to Theor...
sb5ALT 37752 Equivalence for substituti...
eexinst01 37753 ~ exinst01 without virtual...
eexinst11 37754 ~ exinst11 without virtual...
vk15.4j 37755 Excercise 4j of Unit 15 of...
notnotrALT 37756 Converse of double negatio...
con3ALT2 37757 Contraposition. Alternate...
ssralv2 37758 Quantification restricted ...
sbc3or 37759 ~ sbcor with a 3-disjuncts...
sbcangOLD 37760 Distribution of class subs...
sbcorgOLD 37761 Distribution of class subs...
sbcbiiOLD 37762 Formula-building inference...
sbc3orgOLD 37763 ~ sbcorgOLD with a 3-disju...
alrim3con13v 37764 Closed form of ~ alrimi wi...
rspsbc2 37765 ~ rspsbc with two quantify...
sbcoreleleq 37766 Substitution of a setvar v...
tratrb 37767 If a class is transitive a...
ordelordALT 37768 An element of an ordinal c...
sbcim2g 37769 Distribution of class subs...
sbcbi 37770 Implication form of ~ sbcb...
trsbc 37771 Formula-building inference...
truniALT 37772 The union of a class of tr...
sbcalgOLD 37773 Move universal quantifier ...
sbcexgOLD 37774 Move existential quantifie...
sbcel12gOLD 37775 Distribute proper substitu...
sbcel2gOLD 37776 Move proper substitution i...
sbcssOLD 37777 Distribute proper substitu...
onfrALTlem5 37778 Lemma for ~ onfrALT . (Co...
onfrALTlem4 37779 Lemma for ~ onfrALT . (Co...
onfrALTlem3 37780 Lemma for ~ onfrALT . (Co...
ggen31 37781 ~ gen31 without virtual de...
onfrALTlem2 37782 Lemma for ~ onfrALT . (Co...
cbvexsv 37783 A theorem pertaining to th...
onfrALTlem1 37784 Lemma for ~ onfrALT . (Co...
onfrALT 37785 The epsilon relation is fo...
csbeq2gOLD 37786 Formula-building implicati...
19.41rg 37787 Closed form of right-to-le...
opelopab4 37788 Ordered pair membership in...
2pm13.193 37789 ~ pm13.193 for two variabl...
hbntal 37790 A closed form of ~ hbn . ~...
hbimpg 37791 A closed form of ~ hbim . ...
hbalg 37792 Closed form of ~ hbal . D...
hbexg 37793 Closed form of ~ nfex . D...
ax6e2eq 37794 Alternate form of ~ ax6e f...
ax6e2nd 37795 If at least two sets exist...
ax6e2ndeq 37796 "At least two sets exist" ...
2sb5nd 37797 Equivalence for double sub...
2uasbanh 37798 Distribute the unabbreviat...
2uasban 37799 Distribute the unabbreviat...
e2ebind 37800 Absorption of an existenti...
elpwgded 37801 ~ elpwgdedVD in convention...
trelded 37802 Deduction form of ~ trel ....
jaoded 37803 Deduction form of ~ jao . ...
sbtT 37804 A substitution into a theo...
not12an2impnot1 37805 If a double conjunction is...
in1 37808 Inference form of ~ df-vd1...
iin1 37809 ~ in1 without virtual dedu...
dfvd1ir 37810 Inference form of ~ df-vd1...
idn1 37811 Virtual deduction identity...
dfvd1imp 37812 Left-to-right part of defi...
dfvd1impr 37813 Right-to-left part of defi...
dfvd2 37816 Definition of a 2-hypothes...
dfvd2an 37819 Definition of a 2-hypothes...
dfvd2ani 37820 Inference form of ~ dfvd2a...
dfvd2anir 37821 Right-to-left inference fo...
dfvd2i 37822 Inference form of ~ dfvd2 ...
dfvd2ir 37823 Right-to-left inference fo...
dfvd3 37828 Definition of a 3-hypothes...
dfvd3i 37829 Inference form of ~ dfvd3 ...
dfvd3ir 37830 Right-to-left inference fo...
dfvd3an 37831 Definition of a 3-hypothes...
dfvd3ani 37832 Inference form of ~ dfvd3a...
dfvd3anir 37833 Right-to-left inference fo...
vd01 37843 A virtual hypothesis virtu...
vd02 37844 Two virtual hypotheses vir...
vd03 37845 A theorem is virtually inf...
vd12 37846 A virtual deduction with 1...
vd13 37847 A virtual deduction with 1...
vd23 37848 A virtual deduction with 2...
dfvd2imp 37849 The virtual deduction form...
dfvd2impr 37850 A 2-antecedent nested impl...
in2 37851 The virtual deduction intr...
int2 37852 The virtual deduction intr...
iin2 37853 ~ in2 without virtual dedu...
in2an 37854 The virtual deduction intr...
in3 37855 The virtual deduction intr...
iin3 37856 ~ in3 without virtual dedu...
in3an 37857 The virtual deduction intr...
int3 37858 The virtual deduction intr...
idn2 37859 Virtual deduction identity...
iden2 37860 Virtual deduction identity...
idn3 37861 Virtual deduction identity...
gen11 37862 Virtual deduction generali...
gen11nv 37863 Virtual deduction generali...
gen12 37864 Virtual deduction generali...
gen21 37865 Virtual deduction generali...
gen21nv 37866 Virtual deduction form of ...
gen31 37867 Virtual deduction generali...
gen22 37868 Virtual deduction generali...
ggen22 37869 ~ gen22 without virtual de...
exinst 37870 Existential Instantiation....
exinst01 37871 Existential Instantiation....
exinst11 37872 Existential Instantiation....
e1a 37873 A Virtual deduction elimin...
el1 37874 A Virtual deduction elimin...
e1bi 37875 Biconditional form of ~ e1...
e1bir 37876 Right biconditional form o...
e2 37877 A virtual deduction elimin...
e2bi 37878 Biconditional form of ~ e2...
e2bir 37879 Right biconditional form o...
ee223 37880 ~ e223 without virtual ded...
e223 37881 A virtual deduction elimin...
e222 37882 A virtual deduction elimin...
e220 37883 A virtual deduction elimin...
ee220 37884 ~ e220 without virtual ded...
e202 37885 A virtual deduction elimin...
ee202 37886 ~ e202 without virtual ded...
e022 37887 A virtual deduction elimin...
ee022 37888 ~ e022 without virtual ded...
e002 37889 A virtual deduction elimin...
ee002 37890 ~ e002 without virtual ded...
e020 37891 A virtual deduction elimin...
ee020 37892 ~ e020 without virtual ded...
e200 37893 A virtual deduction elimin...
ee200 37894 ~ e200 without virtual ded...
e221 37895 A virtual deduction elimin...
ee221 37896 ~ e221 without virtual ded...
e212 37897 A virtual deduction elimin...
ee212 37898 ~ e212 without virtual ded...
e122 37899 A virtual deduction elimin...
e112 37900 A virtual deduction elimin...
ee112 37901 ~ e112 without virtual ded...
e121 37902 A virtual deduction elimin...
e211 37903 A virtual deduction elimin...
ee211 37904 ~ e211 without virtual ded...
e210 37905 A virtual deduction elimin...
ee210 37906 ~ e210 without virtual ded...
e201 37907 A virtual deduction elimin...
ee201 37908 ~ e201 without virtual ded...
e120 37909 A virtual deduction elimin...
ee120 37910 Virtual deduction rule ~ e...
e021 37911 A virtual deduction elimin...
ee021 37912 ~ e021 without virtual ded...
e012 37913 A virtual deduction elimin...
ee012 37914 ~ e012 without virtual ded...
e102 37915 A virtual deduction elimin...
ee102 37916 ~ e102 without virtual ded...
e22 37917 A virtual deduction elimin...
e22an 37918 Conjunction form of ~ e22 ...
ee22an 37919 ~ e22an without virtual de...
e111 37920 A virtual deduction elimin...
e1111 37921 A virtual deduction elimin...
e110 37922 A virtual deduction elimin...
ee110 37923 ~ e110 without virtual ded...
e101 37924 A virtual deduction elimin...
ee101 37925 ~ e101 without virtual ded...
e011 37926 A virtual deduction elimin...
ee011 37927 ~ e011 without virtual ded...
e100 37928 A virtual deduction elimin...
ee100 37929 ~ e100 without virtual ded...
e010 37930 A virtual deduction elimin...
ee010 37931 ~ e010 without virtual ded...
e001 37932 A virtual deduction elimin...
ee001 37933 ~ e001 without virtual ded...
e11 37934 A virtual deduction elimin...
e11an 37935 Conjunction form of ~ e11 ...
ee11an 37936 ~ e11an without virtual de...
e01 37937 A virtual deduction elimin...
e01an 37938 Conjunction form of ~ e01 ...
ee01an 37939 ~ e01an without virtual de...
e10 37940 A virtual deduction elimin...
e10an 37941 Conjunction form of ~ e10 ...
ee10an 37942 ~ e10an without virtual de...
e02 37943 A virtual deduction elimin...
e02an 37944 Conjunction form of ~ e02 ...
ee02an 37945 ~ e02an without virtual de...
eel021old 37946 ~ el021old without virtual...
el021old 37947 A virtual deduction elimin...
eel132 37948 ~ syl2an with antecedents ...
eel000cT 37949 An elimination deduction. ...
eel0TT 37950 An elimination deduction. ...
eelT00 37951 An elimination deduction. ...
eelTTT 37952 An elimination deduction. ...
eelT11 37953 An elimination deduction. ...
eelT1 37954 Syllogism inference combin...
eelT12 37955 An elimination deduction. ...
eelTT1 37956 An elimination deduction. ...
eelT01 37957 An elimination deduction. ...
eel0T1 37958 An elimination deduction. ...
eel12131 37959 An elimination deduction. ...
eel2131 37960 ~ syl2an with antecedents ...
eel3132 37961 ~ syl2an with antecedents ...
eel0321old 37962 ~ el0321old without virtua...
el0321old 37963 A virtual deduction elimin...
eel2122old 37964 ~ el2122old without virtua...
el2122old 37965 A virtual deduction elimin...
eel0001 37966 An elimination deduction. ...
eel0000 37967 Elimination rule similar t...
eel1111 37968 Four-hypothesis eliminatio...
eel00001 37969 An elimination deduction. ...
eel00000 37970 Elimination rule similar ~...
eel11111 37971 Five-hypothesis eliminatio...
e12 37972 A virtual deduction elimin...
e12an 37973 Conjunction form of ~ e12 ...
el12 37974 Virtual deduction form of ...
e20 37975 A virtual deduction elimin...
e20an 37976 Conjunction form of ~ e20 ...
ee20an 37977 ~ e20an without virtual de...
e21 37978 A virtual deduction elimin...
e21an 37979 Conjunction form of ~ e21 ...
ee21an 37980 ~ e21an without virtual de...
e333 37981 A virtual deduction elimin...
e33 37982 A virtual deduction elimin...
e33an 37983 Conjunction form of ~ e33 ...
ee33an 37984 ~ e33an without virtual de...
e3 37985 Meta-connective form of ~ ...
e3bi 37986 Biconditional form of ~ e3...
e3bir 37987 Right biconditional form o...
e03 37988 A virtual deduction elimin...
ee03 37989 ~ e03 without virtual dedu...
e03an 37990 Conjunction form of ~ e03 ...
ee03an 37991 Conjunction form of ~ ee03...
e30 37992 A virtual deduction elimin...
ee30 37993 ~ e30 without virtual dedu...
e30an 37994 A virtual deduction elimin...
ee30an 37995 Conjunction form of ~ ee30...
e13 37996 A virtual deduction elimin...
e13an 37997 A virtual deduction elimin...
ee13an 37998 ~ e13an without virtual de...
e31 37999 A virtual deduction elimin...
ee31 38000 ~ e31 without virtual dedu...
e31an 38001 A virtual deduction elimin...
ee31an 38002 ~ e31an without virtual de...
e23 38003 A virtual deduction elimin...
e23an 38004 A virtual deduction elimin...
ee23an 38005 ~ e23an without virtual de...
e32 38006 A virtual deduction elimin...
ee32 38007 ~ e32 without virtual dedu...
e32an 38008 A virtual deduction elimin...
ee32an 38009 ~ e33an without virtual de...
e123 38010 A virtual deduction elimin...
ee123 38011 ~ e123 without virtual ded...
el123 38012 A virtual deduction elimin...
e233 38013 A virtual deduction elimin...
e323 38014 A virtual deduction elimin...
e000 38015 A virtual deduction elimin...
e00 38016 Elimination rule identical...
e00an 38017 Elimination rule identical...
eel00cT 38018 An elimination deduction. ...
eelTT 38019 An elimination deduction. ...
e0a 38020 Elimination rule identical...
eelT 38021 An elimination deduction. ...
eel0cT 38022 An elimination deduction. ...
eelT0 38023 An elimination deduction. ...
e0bi 38024 Elimination rule identical...
e0bir 38025 Elimination rule identical...
uun0.1 38026 Convention notation form o...
un0.1 38027 ` T. ` is the constant tru...
uunT1 38028 A deduction unionizing a n...
uunT1p1 38029 A deduction unionizing a n...
uunT21 38030 A deduction unionizing a n...
uun121 38031 A deduction unionizing a n...
uun121p1 38032 A deduction unionizing a n...
uun132 38033 A deduction unionizing a n...
uun132p1 38034 A deduction unionizing a n...
anabss7p1 38035 A deduction unionizing a n...
un10 38036 A unionizing deduction. (...
un01 38037 A unionizing deduction. (...
un2122 38038 A deduction unionizing a n...
uun2131 38039 A deduction unionizing a n...
uun2131p1 38040 A deduction unionizing a n...
uunTT1 38041 A deduction unionizing a n...
uunTT1p1 38042 A deduction unionizing a n...
uunTT1p2 38043 A deduction unionizing a n...
uunT11 38044 A deduction unionizing a n...
uunT11p1 38045 A deduction unionizing a n...
uunT11p2 38046 A deduction unionizing a n...
uunT12 38047 A deduction unionizing a n...
uunT12p1 38048 A deduction unionizing a n...
uunT12p2 38049 A deduction unionizing a n...
uunT12p3 38050 A deduction unionizing a n...
uunT12p4 38051 A deduction unionizing a n...
uunT12p5 38052 A deduction unionizing a n...
uun111 38053 A deduction unionizing a n...
3anidm12p1 38054 A deduction unionizing a n...
3anidm12p2 38055 A deduction unionizing a n...
uun123 38056 A deduction unionizing a n...
uun123p1 38057 A deduction unionizing a n...
uun123p2 38058 A deduction unionizing a n...
uun123p3 38059 A deduction unionizing a n...
uun123p4 38060 A deduction unionizing a n...
uun2221 38061 A deduction unionizing a n...
uun2221p1 38062 A deduction unionizing a n...
uun2221p2 38063 A deduction unionizing a n...
3impdirp1 38064 A deduction unionizing a n...
3impcombi 38065 A 1-hypothesis proposition...
3imp231 38066 Importation inference. (C...
trsspwALT 38067 Virtual deduction proof of...
trsspwALT2 38068 Virtual deduction proof of...
trsspwALT3 38069 Short predicate calculus p...
sspwtr 38070 Virtual deduction proof of...
sspwtrALT 38071 Virtual deduction proof of...
csbabgOLD 38072 Move substitution into a c...
csbunigOLD 38073 Distribute proper substitu...
csbfv12gALTOLD 38074 Move class substitution in...
csbxpgOLD 38075 Distribute proper substitu...
csbingOLD 38076 Distribute proper substitu...
csbresgOLD 38077 Distribute proper substitu...
csbrngOLD 38078 Distribute proper substitu...
csbima12gALTOLD 38079 Move class substitution in...
sspwtrALT2 38080 Short predicate calculus p...
pwtrVD 38081 Virtual deduction proof of...
pwtrrVD 38082 Virtual deduction proof of...
suctrALT 38083 The successor of a transit...
snssiALTVD 38084 Virtual deduction proof of...
snssiALT 38085 If a class is an element o...
snsslVD 38086 Virtual deduction proof of...
snssl 38087 If a singleton is a subcla...
snelpwrVD 38088 Virtual deduction proof of...
unipwrVD 38089 Virtual deduction proof of...
unipwr 38090 A class is a subclass of t...
sstrALT2VD 38091 Virtual deduction proof of...
sstrALT2 38092 Virtual deduction proof of...
suctrALT2VD 38093 Virtual deduction proof of...
suctrALT2 38094 Virtual deduction proof of...
elex2VD 38095 Virtual deduction proof of...
elex22VD 38096 Virtual deduction proof of...
eqsbc3rVD 38097 Virtual deduction proof of...
zfregs2VD 38098 Virtual deduction proof of...
tpid3gVD 38099 Virtual deduction proof of...
en3lplem1VD 38100 Virtual deduction proof of...
en3lplem2VD 38101 Virtual deduction proof of...
en3lpVD 38102 Virtual deduction proof of...
simplbi2VD 38103 Virtual deduction proof of...
3ornot23VD 38104 Virtual deduction proof of...
orbi1rVD 38105 Virtual deduction proof of...
bitr3VD 38106 Virtual deduction proof of...
3orbi123VD 38107 Virtual deduction proof of...
sbc3orgVD 38108 Virtual deduction proof of...
19.21a3con13vVD 38109 Virtual deduction proof of...
exbirVD 38110 Virtual deduction proof of...
exbiriVD 38111 Virtual deduction proof of...
rspsbc2VD 38112 Virtual deduction proof of...
3impexpVD 38113 Virtual deduction proof of...
3impexpbicomVD 38114 Virtual deduction proof of...
3impexpbicomiVD 38115 Virtual deduction proof of...
sbcel1gvOLD 38116 Class substitution into a ...
sbcoreleleqVD 38117 Virtual deduction proof of...
hbra2VD 38118 Virtual deduction proof of...
tratrbVD 38119 Virtual deduction proof of...
al2imVD 38120 Virtual deduction proof of...
syl5impVD 38121 Virtual deduction proof of...
idiVD 38122 Virtual deduction proof of...
ancomstVD 38123 Closed form of ~ ancoms . ...
ssralv2VD 38124 Quantification restricted ...
ordelordALTVD 38125 An element of an ordinal c...
equncomVD 38126 If a class equals the unio...
equncomiVD 38127 Inference form of ~ equnco...
sucidALTVD 38128 A set belongs to its succe...
sucidALT 38129 A set belongs to its succe...
sucidVD 38130 A set belongs to its succe...
imbi12VD 38131 Implication form of ~ imbi...
imbi13VD 38132 Join three logical equival...
sbcim2gVD 38133 Distribution of class subs...
sbcbiVD 38134 Implication form of ~ sbcb...
trsbcVD 38135 Formula-building inference...
truniALTVD 38136 The union of a class of tr...
ee33VD 38137 Non-virtual deduction form...
trintALTVD 38138 The intersection of a clas...
trintALT 38139 The intersection of a clas...
undif3VD 38140 The first equality of Exer...
sbcssgVD 38141 Virtual deduction proof of...
csbingVD 38142 Virtual deduction proof of...
onfrALTlem5VD 38143 Virtual deduction proof of...
onfrALTlem4VD 38144 Virtual deduction proof of...
onfrALTlem3VD 38145 Virtual deduction proof of...
simplbi2comtVD 38146 Virtual deduction proof of...
onfrALTlem2VD 38147 Virtual deduction proof of...
onfrALTlem1VD 38148 Virtual deduction proof of...
onfrALTVD 38149 Virtual deduction proof of...
csbeq2gVD 38150 Virtual deduction proof of...
csbsngVD 38151 Virtual deduction proof of...
csbxpgVD 38152 Virtual deduction proof of...
csbresgVD 38153 Virtual deduction proof of...
csbrngVD 38154 Virtual deduction proof of...
csbima12gALTVD 38155 Virtual deduction proof of...
csbunigVD 38156 Virtual deduction proof of...
csbfv12gALTVD 38157 Virtual deduction proof of...
con5VD 38158 Virtual deduction proof of...
relopabVD 38159 Virtual deduction proof of...
19.41rgVD 38160 Virtual deduction proof of...
2pm13.193VD 38161 Virtual deduction proof of...
hbimpgVD 38162 Virtual deduction proof of...
hbalgVD 38163 Virtual deduction proof of...
hbexgVD 38164 Virtual deduction proof of...
ax6e2eqVD 38165 The following User's Proof...
ax6e2ndVD 38166 The following User's Proof...
ax6e2ndeqVD 38167 The following User's Proof...
2sb5ndVD 38168 The following User's Proof...
2uasbanhVD 38169 The following User's Proof...
e2ebindVD 38170 The following User's Proof...
sb5ALTVD 38171 The following User's Proof...
vk15.4jVD 38172 The following User's Proof...
notnotrALTVD 38173 The following User's Proof...
con3ALTVD 38174 The following User's Proof...
elpwgdedVD 38175 Membership in a power clas...
sspwimp 38176 If a class is a subclass o...
sspwimpVD 38177 The following User's Proof...
sspwimpcf 38178 If a class is a subclass o...
sspwimpcfVD 38179 The following User's Proof...
suctrALTcf 38180 The sucessor of a transiti...
suctrALTcfVD 38181 The following User's Proof...
suctrALT3 38182 The successor of a transit...
sspwimpALT 38183 If a class is a subclass o...
unisnALT 38184 A set equals the union of ...
notnotrALT2 38185 Converse of double negatio...
sspwimpALT2 38186 If a class is a subclass o...
e2ebindALT 38187 Absorption of an existenti...
ax6e2ndALT 38188 If at least two sets exist...
ax6e2ndeqALT 38189 "At least two sets exist" ...
2sb5ndALT 38190 Equivalence for double sub...
chordthmALT 38191 The intersecting chords th...
isosctrlem1ALT 38192 Lemma for ~ isosctr . Thi...
iunconlem2 38193 The indexed union of conne...
iunconALT 38194 The indexed union of conne...
sineq0ALT 38195 A complex number whose sin...
fnvinran 38196 the function value belongs...
evth2f 38197 A version of ~ evth2 using...
elunif 38198 A version of ~ eluni using...
rzalf 38199 A version of ~ rzal using ...
fvelrnbf 38200 A version of ~ fvelrnb usi...
rfcnpre1 38201 If F is a continuous funct...
ubelsupr 38202 If U belongs to A and U is...
fsumcnf 38203 A finite sum of functions ...
mulltgt0 38204 The product of a negative ...
rspcegf 38205 A version of ~ rspcev usin...
rabexgf 38206 A version of ~ rabexg usin...
fcnre 38207 A function continuous with...
sumsnd 38208 A sum of a singleton is th...
evthf 38209 A version of ~ evth using ...
cnfex 38210 The class of continuous fu...
fnchoice 38211 For a finite set, a choice...
refsumcn 38212 A finite sum of continuous...
rfcnpre2 38213 If ` F ` is a continuous f...
cncmpmax 38214 When the hypothesis for th...
rfcnpre3 38215 If F is a continuous funct...
rfcnpre4 38216 If F is a continuous funct...
sumpair 38217 Sum of two distinct comple...
rfcnnnub 38218 Given a real continuous fu...
refsum2cnlem1 38219 This is the core Lemma for...
refsum2cn 38220 The sum of two continuus r...
elunnel2 38221 A member of a union that i...
adantlllr 38222 Deduction adding a conjunc...
3adantlr3 38223 Deduction adding a conjunc...
nnxrd 38224 A natural number is an ext...
3adantll2 38225 Deduction adding a conjunc...
3adantll3 38226 Deduction adding a conjunc...
ssnel 38227 If not element of a set, t...
jcn 38228 Inference joining the cons...
elabrexg 38229 Elementhood in an image se...
unicntop 38230 The underlying set of the ...
ifeq123d 38231 Equality deduction for con...
sncldre 38232 A singleton is closed w.r....
cnopn 38233 The set of complex numbers...
n0p 38234 A polynomial with a nonzer...
pm2.65ni 38235 Inference rule for proof b...
pwssfi 38236 Every element of the power...
iuneq2df 38237 Equality deduction for ind...
nnfoctb 38238 There exists a mapping fro...
ssinss1d 38239 Intersection preserves sub...
0un 38240 The union of the empty set...
elpwinss 38241 An element of the powerset...
unidmex 38242 If ` F ` is a set, then ` ...
ndisj2 38243 A non disjointness conditi...
zenom 38244 The set of integer numbers...
rexsngf 38245 Restricted existential qua...
uzwo4 38246 Well-ordering principle: a...
unisn0 38247 The union of the singleton...
ssin0 38248 If two classes are disjoin...
inabs3 38249 Absorption law for interse...
pwpwuni 38250 Relationship between power...
disjiun2 38251 In a disjoint collection, ...
0pwfi 38252 The empty set is in any po...
ssinss2d 38253 Intersection preserves sub...
zct 38254 The set of integer numbers...
iunxsngf2 38255 A singleton index picks ou...
pwfin0 38256 A finite set always belong...
uzct 38257 An upper integer set is co...
iunxsnf 38258 A singleton index picks ou...
fiiuncl 38259 If a set is closed under t...
iunp1 38260 The addition of the next s...
fiunicl 38261 If a set is closed under t...
ixpeq2d 38262 Equality theorem for infin...
disjxp1 38263 The sets of a cartesian pr...
elpwd 38264 Membership in a power clas...
disjsnxp 38265 The sets in the cartesian ...
eliind 38266 Membership in indexed inte...
rspcef 38267 Restricted existential spe...
inn0f 38268 A non-empty intersection. ...
ixpssmapc 38269 An infinite Cartesian prod...
inn0 38270 A non-empty intersection. ...
elintd 38271 Membership in class inters...
eqneltri 38272 If a class is not an eleme...
ssdf 38273 A sufficient condition for...
brneqtrd 38274 Substitution of equal clas...
ssnct 38275 A set containing an uncoun...
ssuniint 38276 Sufficient condition for b...
elintdv 38277 Membership in class inters...
ssd 38278 A sufficient condition for...
ralimralim 38279 Introducing any antecedent...
snelmap 38280 Membership of the element ...
dfcleqf 38281 Equality connective betwee...
xrnmnfpnf 38282 An extended real that is n...
nelrnmpt 38283 Non-membership in the rang...
snn0d 38284 The singleton of a set is ...
rabid3 38285 Membership in a restricted...
iuneq1i 38286 Equality theorem for index...
nssrex 38287 Negation of subclass relat...
nelpr2 38288 If a class is not an eleme...
nelpr1 38289 If a class is not an eleme...
iunssf 38290 Subset theorem for an inde...
elpwi2 38291 Membership in a power clas...
ssinc 38292 Inclusion relation for a m...
ssdec 38293 Inclusion relation for a m...
elixpconstg 38294 Membership in an infinite ...
iineq1d 38295 Equality theorem for index...
metpsmet 38296 A metric is a pseudometric...
ixpssixp 38297 Subclass theorem for infin...
ballss3 38298 A sufficient condition for...
iunssd 38299 Subset theorem for an inde...
iunincfi 38300 Given a sequence of increa...
nsstr 38301 If it's not a subclass, it...
rabbida 38302 Equivalent wff's yield equ...
rexanuz3 38303 Combine two different uppe...
rabeqd 38304 Equality theorem for restr...
cbvmpt22 38305 Rule to change the second ...
cbvmpt21 38306 Rule to change the first b...
eliuniin 38307 Indexed union of indexed i...
ssabf 38308 Subclass of a class abstra...
rabbia2 38309 Equivalent wff's yield equ...
uniexd 38310 Deduction version of the Z...
pwexd 38311 Deduction version of the p...
pssnssi 38312 A proper subclass does not...
rabidim2 38313 Membership in a restricted...
xpexd 38314 The Cartesian product of t...
eluni2f 38315 Membership in class union....
eliin2f 38316 Membership in indexed inte...
nssd 38317 Negation of subclass relat...
rabidim1 38318 Membership in a restricted...
iineq12dv 38319 Equality deduction for ind...
rabeqif 38320 Equality theorem for restr...
supxrcld 38321 The supremum of an arbitra...
elrestd 38322 A sufficient condition for...
eliuniincex 38323 Counterexample to show tha...
eliincex 38324 Counterexample to show tha...
eliinid 38325 Membership in an indexed i...
abssf 38326 Class abstraction in a sub...
fexd 38327 If the domain of a mapping...
supxrubd 38328 A member of a set of exten...
ssrabf 38329 Subclass of a restricted c...
eliin2 38330 Membership in indexed inte...
ssrab2f 38331 Subclass relation for a re...
rabeqi 38332 Equality theorem for restr...
restuni3 38333 The underlying set of a su...
rabssf 38334 Restricted class abstracti...
eliuniin2 38335 Indexed union of indexed i...
restuni4 38336 The underlying set of a su...
restuni6 38337 The underlying set of a su...
restuni5 38338 The underlying set of a su...
unirestss 38339 The union of an elementwis...
unima 38340 Image of a union. (Contri...
feq1dd 38341 Equality deduction for fun...
fnresdmss 38342 A function does not change...
fmptsnxp 38343 Maps-to notation and cross...
mptex2 38344 If a class given as a map-...
fvmpt2bd 38345 Value of a function given ...
rnmptfi 38346 The range of a function wi...
fresin2 38347 Restriction of a function ...
rnmptc 38348 Range of a constant functi...
ffi 38349 A function with finite dom...
suprnmpt 38350 An explicit bound for the ...
rnffi 38351 The range of a function wi...
mptelpm 38352 A function in maps-to nota...
rnmptpr 38353 Range of a function define...
resmpti 38354 Restriction of the mapping...
founiiun 38355 Union expressed as an inde...
f1oeq2d 38356 Equality deduction for one...
rnresun 38357 Distribution law for range...
f1oeq1d 38358 Equality deduction for one...
dffo3f 38359 An onto mapping expressed ...
rnresss 38360 The range of a restriction...
elrnmptd 38361 The range of a function in...
elrnmptf 38362 The range of a function in...
rnmptssrn 38363 Inclusion relation for two...
disjf1 38364 A 1 to 1 mapping built fro...
rnsnf 38365 The range of a function wh...
wessf1ornlem 38366 Given a function ` F ` on ...
wessf1orn 38367 Given a function ` F ` on ...
foelrnf 38368 Property of a surjective f...
nelrnres 38369 If ` A ` is not in the ran...
disjrnmpt2 38370 Disjointness of the range ...
elrnmpt1sf 38371 Elementhood in an image se...
founiiun0 38372 Union expressed as an inde...
disjf1o 38373 A bijection built from dis...
fompt 38374 Express being onto for a m...
disjinfi 38375 Only a finite number of di...
fvovco 38376 Value of the composition o...
ssnnf1octb 38377 There exists a bijection b...
mapdm0 38378 The empty set is the only ...
nnf1oxpnn 38379 There is a bijection betwe...
rnmptssd 38380 The range of an operation ...
projf1o 38381 A biijection from a set to...
fvmap 38382 Function value for a membe...
mapsnd 38383 The value of set exponenti...
fvixp2 38384 Projection of a factor of ...
fidmfisupp 38385 A function with a finite d...
mapsnend 38386 Set exponentiation to a si...
choicefi 38387 For a finite set, a choice...
mpct 38388 The exponentiation of a co...
cnmetcoval 38389 Value of the distance func...
fcomptss 38390 Express composition of two...
elmapsnd 38391 Membership in a set expone...
mapss2 38392 Subset inheritance for set...
fsneq 38393 Equality condition for two...
difmap 38394 Difference of two sets exp...
unirnmap 38395 Given a subset of a set ex...
inmap 38396 Intersection of two sets e...
fcoss 38397 Composition of two mapping...
fsneqrn 38398 Equality condition for two...
difmapsn 38399 Difference of two sets exp...
mapssbi 38400 Subset inheritance for set...
unirnmapsn 38401 Equality theorem for a sub...
iunmapss 38402 The indexed union of set e...
ssmapsn 38403 A subset ` C ` of a set ex...
iunmapsn 38404 The indexed union of set e...
absfico 38405 Mapping domain and codomai...
icof 38406 The set of left-closed rig...
rnmpt0 38407 The range of a function in...
rnmptn0 38408 The range of a function in...
elpmrn 38409 The range of a partial fun...
imaexi 38410 The image of a set is a se...
axccdom 38411 Relax the constraint on ax...
dmmptdf 38412 The domain of the mapping ...
elpmi2 38413 The domain of a partial fu...
dmrelrnrel 38414 A relation preserving func...
fdmd 38415 The domain of a mapping. ...
fco3 38416 Functionality of a composi...
dmexd 38417 The domain of a set is a s...
fvcod 38418 Value of a function compos...
fcod 38419 Composition of two mapping...
freld 38420 A mapping is a relation. ...
frnd 38421 The range of a mapping. (...
elrnmpt2id 38422 Membership in the range of...
fvmptelrn 38423 A function's value belongs...
axccd 38424 An alternative version of ...
axccd2 38425 An alternative version of ...
sub2times 38426 Subtracting from a number,...
xrltled 38427 'Less than' implies 'less ...
abssubrp 38428 The distance of two distin...
elfzfzo 38429 Relationship between membe...
oddfl 38430 Odd number representation ...
abscosbd 38431 Bound for the absolute val...
mul13d 38432 Commutative/associative la...
negpilt0 38433 Negative ` _pi ` is negati...
dstregt0 38434 A complex number ` A ` tha...
subadd4b 38435 Rearrangement of 4 terms i...
xrlttri5d 38436 Not equal and not larger i...
neglt 38437 The negative of a positive...
zltlesub 38438 If an integer ` N ` is sma...
divlt0gt0d 38439 The ratio of a negative nu...
subsub23d 38440 Swap subtrahend and result...
2timesgt 38441 Double of a positive real ...
reopn 38442 The reals are open with re...
elfzop1le2 38443 A member in a half-open in...
sub31 38444 Swap the first and third t...
nnne1ge2 38445 A positive integer which i...
lefldiveq 38446 A closed enough, smaller r...
negsubdi3d 38447 Distribution of negative o...
ltdiv2dd 38448 Division of a positive num...
absnpncand 38449 Triangular inequality, com...
abssinbd 38450 Bound for the absolute val...
halffl 38451 Floor of ` ( 1 / 2 ) ` . ...
monoords 38452 Ordering relation for a st...
hashssle 38453 The size of a subset of a ...
lttri5d 38454 Not equal and not larger i...
fzisoeu 38455 A finite ordered set has a...
lt3addmuld 38456 If three real numbers are ...
absnpncan2d 38457 Triangular inequality, com...
fperiodmullem 38458 A function with period T i...
fperiodmul 38459 A function with period T i...
upbdrech 38460 Choice of an upper bound f...
lt4addmuld 38461 If four real numbers are l...
absnpncan3d 38462 Triangular inequality, com...
upbdrech2 38463 Choice of an upper bound f...
ssfiunibd 38464 A finite union of bounded ...
fz1ssfz0 38465 Subset relationship for fi...
fzdifsuc2 38466 Remove a successor from th...
fzsscn 38467 A finite sequence of integ...
divcan8d 38468 A cancellation law for div...
dmmcand 38469 Cancellation law for divis...
fzssre 38470 A finite sequence of integ...
elfzelzd 38471 A member of a finite set o...
bccld 38472 A binomial coefficient, in...
leadd12dd 38473 Addition to both sides of ...
fzssnn0 38474 A finite set of sequential...
xreqle 38475 Equality implies 'less tha...
xaddid2d 38476 ` 0 ` is a left identity f...
xadd0ge 38477 A number is less than or e...
elfzolem1 38478 A member in a half-open in...
xrgtned 38479 'Greater than' implies not...
xrleneltd 38480 'Less than or equal to' an...
xaddcomd 38481 The extended real addition...
supxrre3 38482 The supremum of a nonempty...
uzfissfz 38483 For any finite subset of t...
xleadd2d 38484 Addition of extended reals...
suprltrp 38485 The supremum of a nonempty...
xleadd1d 38486 Addition of extended reals...
xreqled 38487 Equality implies 'less tha...
xrgepnfd 38488 An extended real greater o...
xrge0nemnfd 38489 A nonnegative extended rea...
supxrgere 38490 If a real number can be ap...
iuneqfzuzlem 38491 Lemma for ~ iuneqfzuz : he...
iuneqfzuz 38492 If two unions indexed by u...
xle2addd 38493 Adding both side of two in...
supxrgelem 38494 If an extended real number...
supxrge 38495 If an extended real number...
suplesup 38496 If any element of ` A ` ca...
infxrglb 38497 The infimum of a set of ex...
xadd0ge2 38498 A number is less than or e...
nepnfltpnf 38499 An extended real that is n...
ltadd12dd 38500 Addition to both sides of ...
nemnftgtmnft 38501 An extended real that is n...
xrgtso 38502 'Greater than' is a strict...
rpex 38503 The positive reals form a ...
xrge0ge0 38504 A nonnegative extended rea...
xrssre 38505 A subset of extended reals...
ssuzfz 38506 A finite subset of the upp...
absfun 38507 The absolute value is a fu...
infrpge 38508 The infimum of a non empty...
xrlexaddrp 38509 If an extended real number...
supsubc 38510 The supremum function dist...
xralrple2 38511 Show that ` A ` is less th...
nnuzdisj 38512 The first ` N ` elements o...
ltdivgt1 38513 Divsion by a number greate...
xrltned 38514 'Less than' implies not eq...
nnsplit 38515 Express the set of positiv...
divdiv3d 38516 Division into a fraction. ...
abslt2sqd 38517 Comparison of the square o...
qenom 38518 The set of rational number...
qct 38519 The set of rational number...
xrltnled 38520 'Less than' in terms of 'l...
lenlteq 38521 'less than or equal to' bu...
xrred 38522 An extended real that is n...
rr2sscn2 38523 ` RR^ 2 ` is a subset of C...
infxr 38524 The infimum of a set of ex...
infxrunb2 38525 The infimum of an unbounde...
infxrbnd2 38526 The infimum of a bounded-b...
infleinflem1 38527 Lemma for ~ infleinf , cas...
infleinflem2 38528 Lemma for ~ infleinf , whe...
infleinf 38529 If any element of ` B ` ca...
xralrple4 38530 Show that ` A ` is less th...
xralrple3 38531 Show that ` A ` is less th...
eluzelzd 38532 A member of an upper set o...
suplesup2 38533 If any element of ` A ` is...
recnnltrp 38534 ` N ` is a natural number ...
fiminre2 38535 A nonempty finite set of r...
nnn0 38536 The set of positive intege...
fzct 38537 A finite set of sequential...
rpgtrecnn 38538 Any positive real number i...
fzossuz 38539 A half-open integer interv...
fzossz 38540 A half-open integer interv...
infrefilb 38541 The infimum of a finite se...
infxrrefi 38542 The real and extended real...
xrralrecnnle 38543 Show that ` A ` is less th...
fzoct 38544 A finite set of sequential...
frexr 38545 A function taking real val...
nnrecrp 38546 The reciprocal of a positi...
qred 38547 A rational number is a rea...
reclt0d 38548 The reciprocal of a negati...
lt0neg1dd 38549 If a number is negative, i...
mnfled 38550 Minus infinity is less tha...
xrleidd 38551 'Less than or equal to' is...
negelrpd 38552 The negation of a negative...
infxrcld 38553 The infimum of an arbitrar...
xrralrecnnge 38554 Show that ` A ` is less th...
reclt0 38555 The reciprocal of a negati...
ltmulneg 38556 Multiplying by a negative ...
allbutfi 38557 For all but finitely many....
ltdiv23neg 38558 Swap denominator with othe...
gtnelioc 38559 A real number larger than ...
ioossioc 38560 An open interval is a subs...
ioondisj2 38561 A condition for two open i...
ioondisj1 38562 A condition for two open i...
ioosscn 38563 An open interval is a set ...
ioogtlb 38564 An element of a closed int...
evthiccabs 38565 Extreme Value Theorem on y...
ltnelicc 38566 A real number smaller than...
eliood 38567 Membership in an open real...
iooabslt 38568 An upper bound for the dis...
gtnelicc 38569 A real number greater than...
iooinlbub 38570 An open interval has empty...
iocgtlb 38571 An element of a left open ...
iocleub 38572 An element of a left open ...
eliccd 38573 Membership in a closed rea...
iccssred 38574 A closed real interval is ...
eliccre 38575 A member of a closed inter...
eliooshift 38576 Element of an open interva...
eliocd 38577 Membership in a left open,...
snunioo2 38578 The closure of one end of ...
icoltub 38579 An element of a left close...
tgiooss 38580 The restriction of the com...
eliocre 38581 A member of a left open, r...
iooltub 38582 An element of an open inte...
ioontr 38583 The interior of an interva...
eliccxr 38584 A member of a closed inter...
snunioo1 38585 The closure of one end of ...
lbioc 38586 An left open right closed ...
ioomidp 38587 The midpoint is an element...
iccdifioo 38588 If the open inverval is re...
iccdifprioo 38589 An open interval is the cl...
ioossioobi 38590 Biconditional form of ~ io...
iccshift 38591 A closed interval shifted ...
iccsuble 38592 An upper bound to the dist...
iocopn 38593 A left open right closed i...
eliccelioc 38594 Membership in a closed int...
iooshift 38595 An open interval shifted b...
iccintsng 38596 Intersection of two adiace...
icoiccdif 38597 Left closed, right open in...
icoopn 38598 A left closed right open i...
icoub 38599 A left-closed, right-open ...
eliccxrd 38600 Membership in a closed rea...
pnfel0pnf 38601 ` +oo ` is a nonnegative e...
ge0nemnf2 38602 A nonnegative extended rea...
eliccnelico 38603 An element of a closed int...
eliccelicod 38604 A member of a closed inter...
ge0xrre 38605 A nonnegative extended rea...
ge0lere 38606 A nonnegative extended Rea...
elicores 38607 Membership in a left-close...
inficc 38608 The infimum of a nonempty ...
qinioo 38609 The rational numbers are d...
lenelioc 38610 A real number smaller than...
ioonct 38611 C non empty open interval ...
xrgtnelicc 38612 A real number greater than...
iccdificc 38613 The difference of two clos...
iocnct 38614 A non empty left-open, rig...
iccnct 38615 A closed interval, with mo...
iooiinicc 38616 A closed interval expresse...
iccgelbd 38617 An element of a closed int...
iooltubd 38618 An element of an open inte...
icoltubd 38619 An element of a left close...
qelioo 38620 The rational numbers are d...
tgqioo2 38621 Every open set of reals is...
iccleubd 38622 An element of a closed int...
elioored 38623 A member of an open interv...
ioogtlbd 38624 An element of a closed int...
ioofun 38625 ` (,) ` is a function. (C...
icomnfinre 38626 A left-closed, right-open,...
sqrlearg 38627 The square compared with i...
ressiocsup 38628 If the supremum belongs to...
ressioosup 38629 If the supremum does not b...
iooiinioc 38630 A left-open, right-closed ...
ressiooinf 38631 If the infimum does not be...
sumeq2ad 38632 Equality deduction for sum...
fsumclf 38633 Closure of a finite sum of...
fsumsplitf 38634 Split a sum into two parts...
fsummulc1f 38635 Closure of a finite sum of...
sumsnf 38636 A sum of a singleton is th...
fsumsplitsn 38637 Separate out a term in a f...
fsumnncl 38638 Closure of a non empty, fi...
fsumsplit1 38639 Separate out a term in a f...
fsumge0cl 38640 The finite sum of nonnegat...
fsumf1of 38641 Re-index a finite sum usin...
fsumiunss 38642 Sum over a disjoint indexe...
fsumreclf 38643 Closure of a finite sum of...
fsumlessf 38644 A shorter sum of nonnegati...
fsumsupp0 38645 Finite sum of function val...
fsumsermpt 38646 A finite sum expressed in ...
fmul01 38647 Multiplying a finite numbe...
fmulcl 38648 If ' Y ' is closed under t...
fmuldfeqlem1 38649 induction step for the pro...
fmuldfeq 38650 X and Z are two equivalent...
fmul01lt1lem1 38651 Given a finite multiplicat...
fmul01lt1lem2 38652 Given a finite multiplicat...
fmul01lt1 38653 Given a finite multiplicat...
cncfmptss 38654 A continuous complex funct...
rrpsscn 38655 The positive reals are a s...
mulc1cncfg 38656 A version of ~ mulc1cncf u...
infrglb 38657 The infimum of a nonempty ...
expcnfg 38658 If ` F ` is a complex cont...
prodeq2ad 38659 Equality deduction for pro...
fprodsplit1 38660 Separate out a term in a f...
fprodexp 38661 Positive integer exponenti...
fprodabs2 38662 The absolute value of a fi...
fprod0 38663 A finite product with a ze...
mccllem 38664 * Induction step for ~ mcc...
mccl 38665 A multinomial coefficient,...
fprodcnlem 38666 A finite product of functi...
fprodcn 38667 A finite product of functi...
clim1fr1 38668 A class of sequences of fr...
isumneg 38669 Negation of a converging s...
climrec 38670 Limit of the reciprocal of...
climmulf 38671 A version of ~ climmul usi...
climexp 38672 The limit of natural power...
climinf 38673 A bounded monotonic non in...
climsuselem1 38674 The subsequence index ` I ...
climsuse 38675 A subsequence ` G ` of a c...
climrecf 38676 A version of ~ climrec usi...
climneg 38677 Complex limit of the negat...
climinff 38678 A version of ~ climinf usi...
climdivf 38679 Limit of the ratio of two ...
climreeq 38680 If ` F ` is a real functio...
ellimciota 38681 An explicit value for the ...
climaddf 38682 A version of ~ climadd usi...
mullimc 38683 Limit of the product of tw...
ellimcabssub0 38684 An equivalent condition fo...
limcdm0 38685 If a function has empty do...
islptre 38686 An equivalence condition f...
limccog 38687 Limit of the composition o...
limciccioolb 38688 The limit of a function at...
climf 38689 Express the predicate: Th...
mullimcf 38690 Limit of the multiplicatio...
constlimc 38691 Limit of constant function...
rexlim2d 38692 Inference removing two res...
idlimc 38693 Limit of the identity func...
divcnvg 38694 The sequence of reciprocal...
limcperiod 38695 If ` F ` is a periodic fun...
limcrecl 38696 If ` F ` is a real valued ...
sumnnodd 38697 A series indexed by ` NN `...
lptioo2 38698 The upper bound of an open...
lptioo1 38699 The lower bound of an open...
elprn1 38700 A member of an unordered p...
elprn2 38701 A member of an unordered p...
limcmptdm 38702 The domain of a map-to fun...
clim2f 38703 Express the predicate: Th...
limcicciooub 38704 The limit of a function at...
ltmod 38705 A sufficient condition for...
islpcn 38706 A characterization for a l...
lptre2pt 38707 If a set in the real line ...
limsupre 38708 If a sequence is bounded, ...
limcresiooub 38709 The left limit doesn't cha...
limcresioolb 38710 The right limit doesn't ch...
limcleqr 38711 If the left and the right ...
lptioo2cn 38712 The upper bound of an open...
lptioo1cn 38713 The lower bound of an open...
neglimc 38714 Limit of the negative func...
addlimc 38715 Sum of two limits. (Contr...
0ellimcdiv 38716 If the numerator converges...
clim2cf 38717 Express the predicate ` F ...
limclner 38718 For a limit point, both fr...
sublimc 38719 Subtraction of two limits....
reclimc 38720 Limit of the reciprocal of...
clim0cf 38721 Express the predicate ` F ...
limclr 38722 For a limit point, both fr...
divlimc 38723 Limit of the quotient of t...
expfac 38724 Factorial grows faster tha...
climconstmpt 38725 A constant sequence conver...
climresmpt 38726 A function restricted to u...
climsubmpt 38727 Limit of the difference of...
climsubc2mpt 38728 Limit of the difference of...
climsubc1mpt 38729 Limit of the difference of...
fnlimfv 38730 The value of the limit fun...
climreclf 38731 The limit of a convergent ...
climeldmeq 38732 Two functions that are eve...
climf2 38733 Express the predicate: Th...
fnlimcnv 38734 The sequence of function v...
climeldmeqmpt 38735 Two functions that are eve...
climfveq 38736 Two functions that are eve...
clim2f2 38737 Express the predicate: Th...
climfveqmpt 38738 Two functions that are eve...
climd 38739 Express the predicate: Th...
clim2d 38740 The limit of complex numbe...
fnlimfvre 38741 The limit function of real...
allbutfifvre 38742 Given a sequence of real v...
climleltrp 38743 The limit of complex numbe...
fnlimfvre2 38744 The limit function of real...
fnlimf 38745 The limit function of real...
fnlimabslt 38746 A sequence of function val...
coseq0 38747 A complex number whose cos...
sinmulcos 38748 Multiplication formula for...
coskpi2 38749 The cosine of an integer m...
cosnegpi 38750 The cosine of negative ` _...
sinaover2ne0 38751 If ` A ` in ` ( 0 , 2 _pi ...
cosknegpi 38752 The cosine of an integer m...
mulcncff 38753 The multiplication of two ...
subcncf 38754 The addition of two contin...
cncfmptssg 38755 A continuous complex funct...
constcncfg 38756 A constant function is a c...
idcncfg 38757 The identity function is a...
addcncf 38758 The addition of two contin...
cncfshift 38759 A periodic continuous func...
resincncf 38760 ` sin ` restricted to real...
addccncf2 38761 Adding a constant is a con...
0cnf 38762 The empty set is a continu...
fsumcncf 38763 The finite sum of continuo...
cncfperiod 38764 A periodic continuous func...
subcncff 38765 The subtraction of two con...
negcncfg 38766 The opposite of a continuo...
cnfdmsn 38767 A function with a singleto...
cncfcompt 38768 Composition of continuous ...
divcncf 38769 The quotient of two contin...
addcncff 38770 The addition of two contin...
ioccncflimc 38771 Limit at the upper bound, ...
cncfuni 38772 A function is continuous i...
icccncfext 38773 A continuous function on a...
cncficcgt0 38774 A the absolute value of a ...
icocncflimc 38775 Limit at the lower bound, ...
cncfdmsn 38776 A complex function with a ...
divcncff 38777 The quotient of two contin...
cncfshiftioo 38778 A periodic continuous func...
cncfiooicclem1 38779 A continuous function ` F ...
cncfiooicc 38780 A continuous function ` F ...
cncfiooiccre 38781 A continuous function ` F ...
cncfioobdlem 38782 ` G ` actually extends ` F...
cncfioobd 38783 A continuous function ` F ...
jumpncnp 38784 Jump discontinuity or disc...
cncfcompt2 38785 Composition of continuous ...
cxpcncf2 38786 The complex power function...
fprodcncf 38787 The finite product of cont...
add1cncf 38788 Addition to a constant is ...
add2cncf 38789 Addition to a constant is ...
sub1cncfd 38790 Subtracting a constant is ...
sub2cncfd 38791 Subtraction from a constan...
fprodsub2cncf 38792 ` F ` is continuous. (Con...
fprodadd2cncf 38793 ` F ` is continuous. (Con...
fprodsubrecnncnvlem 38794 The sequence ` S ` of fini...
fprodsubrecnncnv 38795 The sequence ` S ` of fini...
fprodaddrecnncnvlem 38796 The sequence ` S ` of fini...
fprodaddrecnncnv 38797 The sequence ` S ` of fini...
dvsinexp 38798 The derivative of sin^N . ...
dvcosre 38799 The real derivative of the...
dvrecg 38800 Derivative of the reciproc...
dvsinax 38801 Derivative exercise: the d...
dvsubf 38802 The subtraction rule for e...
dvmptconst 38803 Function-builder for deriv...
dvcnre 38804 From compex differentiatio...
dvmptidg 38805 Function-builder for deriv...
dvresntr 38806 Function-builder for deriv...
dvmptdiv 38807 Function-builder for deriv...
fperdvper 38808 The derivative of a period...
dvmptresicc 38809 Derivative of a function r...
dvasinbx 38810 Derivative exercise: the d...
dvresioo 38811 Restriction of a derivativ...
dvdivf 38812 The quotient rule for ever...
dvdivbd 38813 A sufficient condition for...
dvsubcncf 38814 A sufficient condition for...
dvmulcncf 38815 A sufficient condition for...
dvcosax 38816 Derivative exercise: the d...
dvdivcncf 38817 A sufficient condition for...
dvbdfbdioolem1 38818 Given a function with boun...
dvbdfbdioolem2 38819 A function on an open inte...
dvbdfbdioo 38820 A function on an open inte...
ioodvbdlimc1lem1 38821 If ` F ` has bounded deriv...
ioodvbdlimc1lem2 38822 Limit at the lower bound o...
ioodvbdlimc1 38823 A real function with bound...
ioodvbdlimc2lem 38824 Limit at the upper bound o...
ioodvbdlimc2 38825 A real function with bound...
dvdmsscn 38826 ` X ` is a subset of ` CC ...
dvmptmulf 38827 Function-builder for deriv...
dvnmptdivc 38828 Function-builder for itera...
dvdsn1add 38829 If ` K ` divides ` N ` but...
dvxpaek 38830 Derivative of the polynomi...
dvnmptconst 38831 The ` N ` -th derivative o...
dvnxpaek 38832 The ` n ` -th derivative o...
dvnmul 38833 Function-builder for the `...
dvmptfprodlem 38834 Induction step for ~ dvmpt...
dvmptfprod 38835 Function-builder for deriv...
dvnprodlem1 38836 ` D ` is bijective. (Cont...
dvnprodlem2 38837 Induction step for ~ dvnpr...
dvnprodlem3 38838 The multinomial formula fo...
dvnprod 38839 The multinomial formula fo...
volioo 38840 The measure of an open int...
itgsin0pilem1 38841 Calculation of the integra...
ibliccsinexp 38842 sin^n on a closed interval...
itgsin0pi 38843 Calculation of the integra...
iblioosinexp 38844 sin^n on an open integral ...
itgsinexplem1 38845 Integration by parts is ap...
itgsinexp 38846 A recursive formula for th...
iblconstmpt 38847 A constant function is int...
itgeq1d 38848 Equality theorem for an in...
mbf0 38849 The empty set is a measura...
mbfres2cn 38850 Measurability of a piecewi...
vol0 38851 The measure of the empty s...
ditgeqiooicc 38852 A function ` F ` on an ope...
volge0 38853 The volume of a set is alw...
cnbdibl 38854 A continuous bounded funct...
snmbl 38855 A singleton is measurable....
ditgeq3d 38856 Equality theorem for the d...
iblempty 38857 The empty function is inte...
iblsplit 38858 The union of two integrabl...
volsn 38859 A singleton has 0 Lebesgue...
itgvol0 38860 If the domani is negligibl...
itgcoscmulx 38861 Exercise: the integral of ...
iblsplitf 38862 A version of ~ iblsplit us...
ibliooicc 38863 If a function is integrabl...
volioc 38864 The measure of left open, ...
iblspltprt 38865 If a function is integrabl...
itgsincmulx 38866 Exercise: the integral of ...
itgsubsticclem 38867 lemma for ~ itgsubsticc . ...
itgsubsticc 38868 Integration by u-substitut...
itgioocnicc 38869 The integral of a piecewis...
iblcncfioo 38870 A continuous function ` F ...
itgspltprt 38871 The ` S. ` integral splits...
itgiccshift 38872 The integral of a function...
itgperiod 38873 The integral of a periodic...
itgsbtaddcnst 38874 Integral substitution, add...
itgeq2d 38875 Equality theorem for an in...
volico 38876 The measure of left closed...
sublevolico 38877 The Lebesgue measure of a ...
dmvolss 38878 Lebesgue measurable sets a...
ismbl3 38879 The predicate " ` A ` is L...
volioof 38880 The function that assigns ...
ovolsplit 38881 The Lebesgue outer measure...
fvvolioof 38882 The function value of the ...
volioore 38883 The measure of an open int...
fvvolicof 38884 The function value of the ...
voliooico 38885 An open interval and a lef...
ismbl4 38886 The predicate " ` A ` is L...
volioofmpt 38887 ` ( ( vol o. (,) ) o. F ) ...
volicoff 38888 ` ( ( vol o. [,) ) o. F ) ...
voliooicof 38889 The Lebesgue measure of op...
volicofmpt 38890 ` ( ( vol o. [,) ) o. F ) ...
volicc 38891 The Lebesgue measure of a ...
voliccico 38892 A closed interval and a le...
mbfdmssre 38893 The domain of a measurable...
stoweidlem1 38894 Lemma for ~ stoweid . Thi...
stoweidlem2 38895 lemma for ~ stoweid : here...
stoweidlem3 38896 Lemma for ~ stoweid : if `...
stoweidlem4 38897 Lemma for ~ stoweid : a cl...
stoweidlem5 38898 There exists a δ as ...
stoweidlem6 38899 Lemma for ~ stoweid : two ...
stoweidlem7 38900 This lemma is used to prov...
stoweidlem8 38901 Lemma for ~ stoweid : two ...
stoweidlem9 38902 Lemma for ~ stoweid : here...
stoweidlem10 38903 Lemma for ~ stoweid . Thi...
stoweidlem11 38904 This lemma is used to prov...
stoweidlem12 38905 Lemma for ~ stoweid . Thi...
stoweidlem13 38906 Lemma for ~ stoweid . Thi...
stoweidlem14 38907 There exists a ` k ` as in...
stoweidlem15 38908 This lemma is used to prov...
stoweidlem16 38909 Lemma for ~ stoweid . The...
stoweidlem17 38910 This lemma proves that the...
stoweidlem18 38911 This theorem proves Lemma ...
stoweidlem19 38912 If a set of real functions...
stoweidlem20 38913 If a set A of real functio...
stoweidlem21 38914 Once the Stone Weierstrass...
stoweidlem22 38915 If a set of real functions...
stoweidlem23 38916 This lemma is used to prov...
stoweidlem24 38917 This lemma proves that for...
stoweidlem25 38918 This lemma proves that for...
stoweidlem26 38919 This lemma is used to prov...
stoweidlem27 38920 This lemma is used to prov...
stoweidlem28 38921 There exists a δ as ...
stoweidlem29 38922 When the hypothesis for th...
stoweidlem30 38923 This lemma is used to prov...
stoweidlem31 38924 This lemma is used to prov...
stoweidlem32 38925 If a set A of real functio...
stoweidlem33 38926 If a set of real functions...
stoweidlem34 38927 This lemma proves that for...
stoweidlem35 38928 This lemma is used to prov...
stoweidlem36 38929 This lemma is used to prov...
stoweidlem37 38930 This lemma is used to prov...
stoweidlem38 38931 This lemma is used to prov...
stoweidlem39 38932 This lemma is used to prov...
stoweidlem40 38933 This lemma proves that q_n...
stoweidlem41 38934 This lemma is used to prov...
stoweidlem42 38935 This lemma is used to prov...
stoweidlem43 38936 This lemma is used to prov...
stoweidlem44 38937 This lemma is used to prov...
stoweidlem45 38938 This lemma proves that, gi...
stoweidlem46 38939 This lemma proves that set...
stoweidlem47 38940 Subtracting a constant fro...
stoweidlem48 38941 This lemma is used to prov...
stoweidlem49 38942 There exists a function q_...
stoweidlem50 38943 This lemma proves that set...
stoweidlem51 38944 There exists a function x ...
stoweidlem52 38945 There exists a neighborood...
stoweidlem53 38946 This lemma is used to prov...
stoweidlem54 38947 There exists a function ` ...
stoweidlem55 38948 This lemma proves the exis...
stoweidlem56 38949 This theorem proves Lemma ...
stoweidlem57 38950 There exists a function x ...
stoweidlem58 38951 This theorem proves Lemma ...
stoweidlem59 38952 This lemma proves that the...
stoweidlem60 38953 This lemma proves that the...
stoweidlem61 38954 This lemma proves that the...
stoweidlem62 38955 This theorem proves the St...
stoweid 38956 This theorem proves the St...
stowei 38957 This theorem proves the St...
wallispilem1 38958 ` I ` is monotone: increas...
wallispilem2 38959 A first set of properties ...
wallispilem3 38960 I maps to real values. (C...
wallispilem4 38961 ` F ` maps to explicit exp...
wallispilem5 38962 The sequence ` H ` converg...
wallispi 38963 Wallis' formula for π :...
wallispi2lem1 38964 An intermediate step betwe...
wallispi2lem2 38965 Two expressions are proven...
wallispi2 38966 An alternative version of ...
stirlinglem1 38967 A simple limit of fraction...
stirlinglem2 38968 ` A ` maps to positive rea...
stirlinglem3 38969 Long but simple algebraic ...
stirlinglem4 38970 Algebraic manipulation of ...
stirlinglem5 38971 If ` T ` is between ` 0 ` ...
stirlinglem6 38972 A series that converges to...
stirlinglem7 38973 Algebraic manipulation of ...
stirlinglem8 38974 If ` A ` converges to ` C ...
stirlinglem9 38975 ` ( ( B `` N ) - ( B `` ( ...
stirlinglem10 38976 A bound for any B(N)-B(N +...
stirlinglem11 38977 ` B ` is decreasing. (Con...
stirlinglem12 38978 The sequence ` B ` is boun...
stirlinglem13 38979 ` B ` is decreasing and ha...
stirlinglem14 38980 The sequence ` A ` converg...
stirlinglem15 38981 The Stirling's formula is ...
stirling 38982 Stirling's approximation f...
stirlingr 38983 Stirling's approximation f...
dirkerval 38984 The N_th Dirichlet Kernel....
dirker2re 38985 The Dirchlet Kernel value ...
dirkerdenne0 38986 The Dirchlet Kernel denomi...
dirkerval2 38987 The N_th Dirichlet Kernel ...
dirkerre 38988 The Dirichlet Kernel at an...
dirkerper 38989 the Dirichlet Kernel has p...
dirkerf 38990 For any natural number ` N...
dirkertrigeqlem1 38991 Sum of an even number of a...
dirkertrigeqlem2 38992 Trigonomic equality lemma ...
dirkertrigeqlem3 38993 Trigonometric equality lem...
dirkertrigeq 38994 Trigonometric equality for...
dirkeritg 38995 The definite integral of t...
dirkercncflem1 38996 If ` Y ` is a multiple of ...
dirkercncflem2 38997 Lemma used to prove that t...
dirkercncflem3 38998 The Dirichlet Kernel is co...
dirkercncflem4 38999 The Dirichlet Kernel is co...
dirkercncf 39000 For any natural number ` N...
fourierdlem1 39001 A partition interval is a ...
fourierdlem2 39002 Membership in a partition....
fourierdlem3 39003 Membership in a partition....
fourierdlem4 39004 ` E ` is a function that m...
fourierdlem5 39005 ` S ` is a function. (Con...
fourierdlem6 39006 ` X ` is in the periodic p...
fourierdlem7 39007 The difference between a p...
fourierdlem8 39008 A partition interval is a ...
fourierdlem9 39009 ` H ` is a complex functio...
fourierdlem10 39010 Condition on the bounds of...
fourierdlem11 39011 If there is a partition, t...
fourierdlem12 39012 A point of a partition is ...
fourierdlem13 39013 Value of ` V ` in terms of...
fourierdlem14 39014 Given the partition ` V ` ...
fourierdlem15 39015 The range of the partition...
fourierdlem16 39016 The coefficients of the fo...
fourierdlem17 39017 The defined ` L ` is actua...
fourierdlem18 39018 The function ` S ` is cont...
fourierdlem19 39019 If two elements of ` D ` h...
fourierdlem20 39020 Every interval in the part...
fourierdlem21 39021 The coefficients of the fo...
fourierdlem22 39022 The coefficients of the fo...
fourierdlem23 39023 If ` F ` is continuous and...
fourierdlem24 39024 A sufficient condition for...
fourierdlem25 39025 If ` C ` is not in the ran...
fourierdlem26 39026 Periodic image of a point ...
fourierdlem27 39027 A partition open interval ...
fourierdlem28 39028 Derivative of ` ( F `` ( X...
fourierdlem29 39029 Explicit function value fo...
fourierdlem30 39030 Sum of three small pieces ...
fourierdlem31 39031 If ` A ` is finite and for...
fourierdlem32 39032 Limit of a continuous func...
fourierdlem33 39033 Limit of a continuous func...
fourierdlem34 39034 A partition is one to one....
fourierdlem35 39035 There is a single point in...
fourierdlem36 39036 ` F ` is an isomorphism. ...
fourierdlem37 39037 ` I ` is a function that m...
fourierdlem38 39038 The function ` F ` is cont...
fourierdlem39 39039 Integration by parts of ...
fourierdlem40 39040 ` H ` is a continuous func...
fourierdlem41 39041 Lemma used to prove that e...
fourierdlem42 39042 The set of points in a mov...
fourierdlem43 39043 ` K ` is a real function. ...
fourierdlem44 39044 A condition for having ` (...
fourierdlem46 39045 The function ` F ` has a l...
fourierdlem47 39046 For ` r ` large enough, th...
fourierdlem48 39047 The given periodic functio...
fourierdlem49 39048 The given periodic functio...
fourierdlem50 39049 Continuity of ` O ` and it...
fourierdlem51 39050 ` X ` is in the periodic p...
fourierdlem52 39051 d16:d17,d18:jca |- ( ph ->...
fourierdlem53 39052 The limit of ` F ( s ) ` a...
fourierdlem54 39053 Given a partition ` Q ` an...
fourierdlem55 39054 ` U ` is a real function. ...
fourierdlem56 39055 Derivative of the ` K ` fu...
fourierdlem57 39056 The derivative of ` O ` . ...
fourierdlem58 39057 The derivative of ` K ` is...
fourierdlem59 39058 The derivative of ` H ` is...
fourierdlem60 39059 Given a differentiable fun...
fourierdlem61 39060 Given a differentiable fun...
fourierdlem62 39061 The function ` K ` is cont...
fourierdlem63 39062 The upper bound of interva...
fourierdlem64 39063 The partition ` V ` is fin...
fourierdlem65 39064 The distance of two adjace...
fourierdlem66 39065 Value of the ` G ` functio...
fourierdlem67 39066 ` G ` is a function. (Con...
fourierdlem68 39067 The derivative of ` O ` is...
fourierdlem69 39068 A piecewise continuous fun...
fourierdlem70 39069 A piecewise continuous fun...
fourierdlem71 39070 A periodic piecewise conti...
fourierdlem72 39071 The derivative of ` O ` is...
fourierdlem73 39072 A version of the Riemann L...
fourierdlem74 39073 Given a piecewise smooth f...
fourierdlem75 39074 Given a piecewise smooth f...
fourierdlem76 39075 Continuity of ` O ` and it...
fourierdlem77 39076 If ` H ` is bounded, then ...
fourierdlem78 39077 ` G ` is continuous when r...
fourierdlem79 39078 ` E ` projects every inter...
fourierdlem80 39079 The derivative of ` O ` is...
fourierdlem81 39080 The integral of a piecewis...
fourierdlem82 39081 Integral by substitution, ...
fourierdlem83 39082 The fourier partial sum fo...
fourierdlem84 39083 If ` F ` is piecewise coni...
fourierdlem85 39084 Limit of the function ` G ...
fourierdlem86 39085 Continuity of ` O ` and it...
fourierdlem87 39086 The integral of ` G ` goes...
fourierdlem88 39087 Given a piecewise continuo...
fourierdlem89 39088 Given a piecewise continuo...
fourierdlem90 39089 Given a piecewise continuo...
fourierdlem91 39090 Given a piecewise continuo...
fourierdlem92 39091 The integral of a piecewis...
fourierdlem93 39092 Integral by substitution (...
fourierdlem94 39093 For a piecewise smooth fun...
fourierdlem95 39094 Algebraic manipulation of ...
fourierdlem96 39095 limit for ` F ` at the low...
fourierdlem97 39096 ` F ` is continuous on the...
fourierdlem98 39097 ` F ` is continuous on the...
fourierdlem99 39098 limit for ` F ` at the upp...
fourierdlem100 39099 A piecewise continuous fun...
fourierdlem101 39100 Integral by substitution f...
fourierdlem102 39101 For a piecewise smooth fun...
fourierdlem103 39102 The half lower part of the...
fourierdlem104 39103 The half upper part of the...
fourierdlem105 39104 A piecewise continuous fun...
fourierdlem106 39105 For a piecewise smooth fun...
fourierdlem107 39106 The integral of a piecewis...
fourierdlem108 39107 The integral of a piecewis...
fourierdlem109 39108 The integral of a piecewis...
fourierdlem110 39109 The integral of a piecewis...
fourierdlem111 39110 The fourier partial sum fo...
fourierdlem112 39111 Here abbreviations (local ...
fourierdlem113 39112 Fourier series convergence...
fourierdlem114 39113 Fourier series convergence...
fourierdlem115 39114 Fourier serier convergence...
fourierd 39115 Fourier series convergence...
fourierclimd 39116 Fourier series convergence...
fourierclim 39117 Fourier series convergence...
fourier 39118 Fourier series convergence...
fouriercnp 39119 If ` F ` is continuous at ...
fourier2 39120 Fourier series convergence...
sqwvfoura 39121 Fourier coefficients for t...
sqwvfourb 39122 Fourier series ` B ` coeff...
fourierswlem 39123 The Fourier series for the...
fouriersw 39124 Fourier series convergence...
fouriercn 39125 If the derivative of ` F `...
elaa2lem 39126 Elementhood in the set of ...
elaa2 39127 Elementhood in the set of ...
etransclem1 39128 ` H ` is a function. (Con...
etransclem2 39129 Derivative of ` G ` . (Co...
etransclem3 39130 The given ` if ` term is a...
etransclem4 39131 ` F ` expressed as a finit...
etransclem5 39132 A change of bound variable...
etransclem6 39133 A change of bound variable...
etransclem7 39134 The given product is an in...
etransclem8 39135 ` F ` is a function. (Con...
etransclem9 39136 If ` K ` divides ` N ` but...
etransclem10 39137 The given ` if ` term is a...
etransclem11 39138 A change of bound variable...
etransclem12 39139 ` C ` applied to ` N ` . ...
etransclem13 39140 ` F ` applied to ` Y ` . ...
etransclem14 39141 Value of the term ` T ` , ...
etransclem15 39142 Value of the term ` T ` , ...
etransclem16 39143 Every element in the range...
etransclem17 39144 The ` N ` -th derivative o...
etransclem18 39145 The given function is inte...
etransclem19 39146 The ` N ` -th derivative o...
etransclem20 39147 ` H ` is smooth. (Contrib...
etransclem21 39148 The ` N ` -th derivative o...
etransclem22 39149 The ` N ` -th derivative o...
etransclem23 39150 This is the claim proof in...
etransclem24 39151 ` P ` divides the I -th de...
etransclem25 39152 ` P ` factorial divides th...
etransclem26 39153 Every term in the sum of t...
etransclem27 39154 The ` N ` -th derivative o...
etransclem28 39155 ` ( P - 1 ) ` factorial di...
etransclem29 39156 The ` N ` -th derivative o...
etransclem30 39157 The ` N ` -th derivative o...
etransclem31 39158 The ` N ` -th derivative o...
etransclem32 39159 This is the proof for the ...
etransclem33 39160 ` F ` is smooth. (Contrib...
etransclem34 39161 The ` N ` -th derivative o...
etransclem35 39162 ` P ` does not divide the ...
etransclem36 39163 The ` N ` -th derivative o...
etransclem37 39164 ` ( P - 1 ) ` factorial di...
etransclem38 39165 ` P ` divides the I -th de...
etransclem39 39166 ` G ` is a function. (Con...
etransclem40 39167 The ` N ` -th derivative o...
etransclem41 39168 ` P ` does not divide the ...
etransclem42 39169 The ` N ` -th derivative o...
etransclem43 39170 ` G ` is a continuous func...
etransclem44 39171 The given finite sum is no...
etransclem45 39172 ` K ` is an integer. (Con...
etransclem46 39173 This is the proof for equa...
etransclem47 39174 ` _e ` is transcendental. ...
etransclem48 39175 ` _e ` is transcendental. ...
etransc 39176 ` _e ` is transcendental. ...
rrxtopn 39177 The topology of the genera...
rrxngp 39178 Generalized Euclidean real...
rrxbasefi 39179 The base of the generalize...
rrxtps 39180 Generalized Euclidean real...
rrxdsfi 39181 The distance over generali...
rrxtopnfi 39182 The topology of the n-dime...
rrxmetfi 39183 Euclidean space is a metri...
rrxtopon 39184 The topology on Generalize...
rrxtop 39185 The topology on Generalize...
rrndistlt 39186 Given two points in the sp...
rrxtoponfi 39187 The topology on n-dimensio...
rrxunitopnfi 39188 The base set of the standa...
rrxtopn0 39189 The topology of the zero-d...
qndenserrnbllem 39190 n-dimensional rational num...
qndenserrnbl 39191 n-dimensional rational num...
rrxtopn0b 39192 The topology of the zero-d...
qndenserrnopnlem 39193 n-dimensional rational num...
qndenserrnopn 39194 n-dimensional rational num...
qndenserrn 39195 n-dimensional rational num...
rrxsnicc 39196 A multidimensional singlet...
rrnprjdstle 39197 The distance between two p...
rrndsmet 39198 ` D ` is a metric for the ...
rrndsxmet 39199 ` D ` is an extended metri...
ioorrnopnlem 39200 The a point in an indexed ...
ioorrnopn 39201 The indexed product of ope...
ioorrnopnxrlem 39202 Given a point ` F ` that b...
ioorrnopnxr 39203 The indexed product of ope...
issal 39210 Express the predicate " ` ...
pwsal 39211 The power set of a given s...
salunicl 39212 SAlg sigma-algebra is clos...
saluncl 39213 The union of two sets in a...
prsal 39214 The pair of the empty set ...
saldifcl 39215 The complement of an eleme...
0sal 39216 The empty set belongs to e...
salgenval 39217 The sigma-algebra generate...
saliuncl 39218 SAlg sigma-algebra is clos...
salincl 39219 The intersection of two se...
saluni 39220 A set is an element of any...
saliincl 39221 SAlg sigma-algebra is clos...
saldifcl2 39222 The difference of two elem...
intsaluni 39223 The union of an arbitrary ...
intsal 39224 The arbitrary intersection...
salgenn0 39225 The set used in the defini...
salgencl 39226 ` SalGen ` actually genera...
issald 39227 Sufficient condition to pr...
salexct 39228 An example of non trivial ...
sssalgen 39229 A set is a subset of the s...
salgenss 39230 The sigma-algebra generate...
salgenuni 39231 The base set of the sigma-...
issalgend 39232 One side of ~ dfsalgen2 . ...
salexct2 39233 An example of a subset tha...
unisalgen 39234 The union of a set belongs...
dfsalgen2 39235 Alternate characterization...
salexct3 39236 An example of a sigma-alge...
salgencntex 39237 This counterexample shows ...
salgensscntex 39238 This counterexample shows ...
issalnnd 39239 Sufficient condition to pr...
dmvolsal 39240 Lebesgue measurable sets f...
saldifcld 39241 The complement of an eleme...
saluncld 39242 The union of two sets in a...
salgencld 39243 ` SalGen ` actually genera...
0sald 39244 The empty set belongs to e...
iooborel 39245 An open interval is a Bore...
salincld 39246 The intersection of two se...
salunid 39247 A set is an element of any...
unisalgen2 39248 The union of a set belongs...
bor1sal 39249 The Borel sigma-algebra on...
iocborel 39250 A left-open, right-closed ...
subsaliuncllem 39251 A subspace sigma-algebra i...
subsaliuncl 39252 A subspace sigma-algebra i...
subsalsal 39253 A subspace sigma-algebra i...
subsaluni 39254 A set belongs to the subsp...
sge0rnre 39257 When ` sum^ ` is applied t...
fge0icoicc 39258 If ` F ` maps to nonnegati...
sge0val 39259 The value of the sum of no...
fge0npnf 39260 If ` F ` maps to nonnegati...
sge0rnn0 39261 The range used in the defi...
sge0vald 39262 The value of the sum of no...
fge0iccico 39263 A range of nonnegative ext...
gsumge0cl 39264 Closure of group sum, for ...
sge0reval 39265 Value of the sum of nonneg...
sge0pnfval 39266 If a term in the sum of no...
fge0iccre 39267 A range of nonnegative ext...
sge0z 39268 Any nonnegative extended s...
sge00 39269 The sum of nonnegative ext...
fsumlesge0 39270 Every finite subsum of non...
sge0revalmpt 39271 Value of the sum of nonneg...
sge0sn 39272 A sum of a nonnegative ext...
sge0tsms 39273 ` sum^ ` applied to a nonn...
sge0cl 39274 The arbitrary sum of nonne...
sge0f1o 39275 Re-index a nonnegative ext...
sge0snmpt 39276 A sum of a nonnegative ext...
sge0ge0 39277 The sum of nonnegative ext...
sge0xrcl 39278 The arbitrary sum of nonne...
sge0repnf 39279 The of nonnegative extende...
sge0fsum 39280 The arbitrary sum of a fin...
sge0rern 39281 If the sum of nonnegative ...
sge0supre 39282 If the arbitrary sum of no...
sge0fsummpt 39283 The arbitrary sum of a fin...
sge0sup 39284 The arbitrary sum of nonne...
sge0less 39285 A shorter sum of nonnegati...
sge0rnbnd 39286 The range used in the defi...
sge0pr 39287 Sum of a pair of nonnegati...
sge0gerp 39288 The arbitrary sum of nonne...
sge0pnffigt 39289 If the sum of nonnegative ...
sge0ssre 39290 If a sum of nonnegative ex...
sge0lefi 39291 A sum of nonnegative exten...
sge0lessmpt 39292 A shorter sum of nonnegati...
sge0ltfirp 39293 If the sum of nonnegative ...
sge0prle 39294 The sum of a pair of nonne...
sge0gerpmpt 39295 The arbitrary sum of nonne...
sge0resrnlem 39296 The sum of nonnegative ext...
sge0resrn 39297 The sum of nonnegative ext...
sge0ssrempt 39298 If a sum of nonnegative ex...
sge0resplit 39299 ` sum^ ` splits into two p...
sge0le 39300 If all of the terms of sum...
sge0ltfirpmpt 39301 If the extended sum of non...
sge0split 39302 Split a sum of nonnegative...
sge0lempt 39303 If all of the terms of sum...
sge0splitmpt 39304 Split a sum of nonnegative...
sge0ss 39305 Change the index set to a ...
sge0iunmptlemfi 39306 Sum of nonnegative extende...
sge0p1 39307 The addition of the next t...
sge0iunmptlemre 39308 Sum of nonnegative extende...
sge0fodjrnlem 39309 Re-index a nonnegative ext...
sge0fodjrn 39310 Re-index a nonnegative ext...
sge0iunmpt 39311 Sum of nonnegative extende...
sge0iun 39312 Sum of nonnegative extende...
sge0nemnf 39313 The generalized sum of non...
sge0rpcpnf 39314 The sum of an infinite num...
sge0rernmpt 39315 If the sum of nonnegative ...
sge0lefimpt 39316 A sum of nonnegative exten...
nn0ssge0 39317 Nonnegative integers are n...
sge0clmpt 39318 The generalized sum of non...
sge0ltfirpmpt2 39319 If the extended sum of non...
sge0isum 39320 If a series of nonnegative...
sge0xrclmpt 39321 The generalized sum of non...
sge0xp 39322 Combine two generalized su...
sge0isummpt 39323 If a series of nonnegative...
sge0ad2en 39324 The value of the infinite ...
sge0isummpt2 39325 If a series of nonnegative...
sge0xaddlem1 39326 The extended addition of t...
sge0xaddlem2 39327 The extended addition of t...
sge0xadd 39328 The extended addition of t...
sge0fsummptf 39329 The generalized sum of a f...
sge0snmptf 39330 A sum of a nonnegative ext...
sge0ge0mpt 39331 The sum of nonnegative ext...
sge0repnfmpt 39332 The of nonnegative extende...
sge0pnffigtmpt 39333 If the generalized sum of ...
sge0splitsn 39334 Separate out a term in a g...
sge0pnffsumgt 39335 If the sum of nonnegative ...
sge0gtfsumgt 39336 If the generalized sum of ...
sge0uzfsumgt 39337 If a real number is smalle...
sge0pnfmpt 39338 If a term in the sum of no...
sge0seq 39339 A series of nonnegative re...
sge0reuz 39340 Value of the generalized s...
sge0reuzb 39341 Value of the generalized s...
ismea 39344 Express the predicate " ` ...
dmmeasal 39345 The domain of a measure is...
meaf 39346 A measure is a function th...
mea0 39347 The measure of the empty s...
nnfoctbdjlem 39348 There exists a mapping fro...
nnfoctbdj 39349 There exists a mapping fro...
meadjuni 39350 The measure of the disjoin...
meacl 39351 The measure of a set is a ...
iundjiunlem 39352 The sets in the sequence `...
iundjiun 39353 Given a sequence ` E ` of ...
meaxrcl 39354 The measure of a set is an...
meadjun 39355 The measure of the union o...
meassle 39356 The measure of a set is la...
meaunle 39357 The measure of the union o...
meadjiunlem 39358 The sum of nonnegative ext...
meadjiun 39359 The measure of the disjoin...
ismeannd 39360 Sufficient condition to pr...
meaiunlelem 39361 The measure of the union o...
meaiunle 39362 The measure of the union o...
psmeasurelem 39363 ` M ` applied to a disjoin...
psmeasure 39364 Point supported measure, R...
voliunsge0lem 39365 The Lebesgue measure funct...
voliunsge0 39366 The Lebesgue measure funct...
volmea 39367 The Lebeasgue measure on t...
meage0 39368 If the measure of a measur...
meadjunre 39369 The measure of the union o...
meassre 39370 If the measure of a measur...
meale0eq0 39371 A measure that is smaller ...
meadif 39372 The measure of the differe...
meaiuninclem 39373 Measures are continuous fr...
meaiuninc 39374 Measures are continuous fr...
meaiuninc2 39375 Measures are continuous fr...
meaiininclem 39376 Measures are continuous fr...
meaiininc 39377 Measures are continuous fr...
meaiininc2 39378 Measures are continuous fr...
caragenval 39383 The sigma-algebra generate...
isome 39384 Express the predicate " ` ...
caragenel 39385 Membership in the Caratheo...
omef 39386 An outer measure is a func...
ome0 39387 The outer measure of the e...
omessle 39388 The outer measure of a set...
omedm 39389 The domain of an outer mea...
caragensplit 39390 If ` E ` is in the set gen...
caragenelss 39391 An element of the Caratheo...
carageneld 39392 Membership in the Caratheo...
omecl 39393 The outer measure of a set...
caragenss 39394 The sigma-algebra generate...
omeunile 39395 The outer measure of the u...
caragen0 39396 The empty set belongs to a...
omexrcl 39397 The outer measure of a set...
caragenunidm 39398 The base set of an outer m...
caragensspw 39399 The sigma-algebra generate...
omessre 39400 If the outer measure of a ...
caragenuni 39401 The base set of the sigma-...
caragenuncllem 39402 The Caratheodory's constru...
caragenuncl 39403 The Caratheodory's constru...
caragendifcl 39404 The Caratheodory's constru...
caragenfiiuncl 39405 The Caratheodory's constru...
omeunle 39406 The outer measure of the u...
omeiunle 39407 The outer measure of the i...
omelesplit 39408 The outer measure of a set...
omeiunltfirp 39409 If the outer measure of a ...
omeiunlempt 39410 The outer measure of the i...
carageniuncllem1 39411 The outer measure of ` A i...
carageniuncllem2 39412 The Caratheodory's constru...
carageniuncl 39413 The Caratheodory's constru...
caragenunicl 39414 The Caratheodory's constru...
caragensal 39415 Caratheodory's method gene...
caratheodorylem1 39416 Lemma used to prove that C...
caratheodorylem2 39417 Caratheodory's constructio...
caratheodory 39418 Caratheodory's constructio...
0ome 39419 The map that assigns 0 to ...
isomenndlem 39420 ` O ` is sub-additive w.r....
isomennd 39421 Sufficient condition to pr...
caragenel2d 39422 Membership in the Caratheo...
omege0 39423 If the outer measure of a ...
omess0 39424 If the outer measure of a ...
caragencmpl 39425 A measure built with the C...
vonval 39430 Value of the Lebesgue meas...
ovnval 39431 Value of the Lebesgue oute...
elhoi 39432 Membership in a multidimen...
icoresmbl 39433 A closed-below, open-above...
hoissre 39434 The projection of a half-o...
ovnval2 39435 Value of the Lebesgue oute...
volicorecl 39436 The Lebesgue measure of a ...
hoiprodcl 39437 The pre-measure of half-op...
hoicvr 39438 ` I ` is a countable set o...
hoissrrn 39439 A half-open interval is a ...
ovn0val 39440 The Lebesgue outer measure...
ovnn0val 39441 The value of a (multidimen...
ovnval2b 39442 Value of the Lebesgue oute...
volicorescl 39443 The Lebesgue measure of a ...
ovnprodcl 39444 The product used in the de...
hoiprodcl2 39445 The pre-measure of half-op...
hoicvrrex 39446 Any subset of the multidim...
ovnsupge0 39447 The set used in the defini...
ovnlecvr 39448 Given a subset of multidim...
ovnpnfelsup 39449 ` +oo ` is an element of t...
ovnsslelem 39450 The (multidimensional, non...
ovnssle 39451 The (multidimensional) Leb...
ovnlerp 39452 The Lebesgue outer measure...
ovnf 39453 The Lebesgue outer measure...
ovncvrrp 39454 The Lebesgue outer measure...
ovn0lem 39455 For any finite dimension, ...
ovn0 39456 For any finite dimension, ...
ovncl 39457 The Lebesgue outer measure...
ovn02 39458 For the zero-dimensional s...
ovnxrcl 39459 The Lebesgue outer measure...
ovnsubaddlem1 39460 The Lebesgue outer measure...
ovnsubaddlem2 39461 ` ( voln* `` X ) ` is suba...
ovnsubadd 39462 ` ( voln* `` X ) ` is suba...
ovnome 39463 ` ( voln* `` X ) ` is an o...
vonmea 39464 ` ( voln `` X ) ` is a mea...
volicon0 39465 The measure of a nonempty ...
hsphoif 39466 ` H ` is a function (that ...
hoidmvval 39467 The dimensional volume of ...
hoissrrn2 39468 A half-open interval is a ...
hsphoival 39469 ` H ` is a function (that ...
hoiprodcl3 39470 The pre-measure of half-op...
volicore 39471 The Lebesgue measure of a ...
hoidmvcl 39472 The dimensional volume of ...
hoidmv0val 39473 The dimensional volume of ...
hoidmvn0val 39474 The dimensional volume of ...
hsphoidmvle2 39475 The dimensional volume of ...
hsphoidmvle 39476 The dimensional volume of ...
hoidmvval0 39477 The dimensional volume of ...
hoiprodp1 39478 The dimensional volume of ...
sge0hsphoire 39479 If the generalized sum of ...
hoidmvval0b 39480 The dimensional volume of ...
hoidmv1lelem1 39481 The supremum of ` U ` belo...
hoidmv1lelem2 39482 This is the contradiction ...
hoidmv1lelem3 39483 The dimensional volume of ...
hoidmv1le 39484 The dimensional volume of ...
hoidmvlelem1 39485 The supremum of ` U ` belo...
hoidmvlelem2 39486 This is the contradiction ...
hoidmvlelem3 39487 This is the contradiction ...
hoidmvlelem4 39488 The dimensional volume of ...
hoidmvlelem5 39489 The dimensional volume of ...
hoidmvle 39490 The dimensional volume of ...
ovnhoilem1 39491 The Lebesgue outer measure...
ovnhoilem2 39492 The Lebesgue outer measure...
ovnhoi 39493 The Lebesgue outer measure...
dmovn 39494 The domain of the Lebesgue...
hoicoto2 39495 The half-open interval exp...
dmvon 39496 Lebesgue measurable n-dime...
hoi2toco 39497 The half-open interval exp...
hoidifhspval 39498 ` D ` is a function that r...
hspval 39499 The value of the half-spac...
ovnlecvr2 39500 Given a subset of multidim...
ovncvr2 39501 ` B ` and ` T ` are the le...
dmovnsal 39502 The domain of the Lebesgue...
unidmovn 39503 Base set of the n-dimensio...
rrnmbl 39504 The set of n-dimensional R...
hoidifhspval2 39505 ` D ` is a function that r...
hspdifhsp 39506 A n-dimensional half-open ...
unidmvon 39507 Base set of the n-dimensio...
hoidifhspf 39508 ` D ` is a function that r...
hoidifhspval3 39509 ` D ` is a function that r...
hoidifhspdmvle 39510 The dimensional volume of ...
voncmpl 39511 The Lebesgue measure is co...
hoiqssbllem1 39512 The center of the n-dimens...
hoiqssbllem2 39513 The center of the n-dimens...
hoiqssbllem3 39514 A n-dimensional ball conta...
hoiqssbl 39515 A n-dimensional ball conta...
hspmbllem1 39516 Any half-space of the n-di...
hspmbllem2 39517 Any half-space of the n-di...
hspmbllem3 39518 Any half-space of the n-di...
hspmbl 39519 Any half-space of the n-di...
hoimbllem 39520 Any n-dimensional half-ope...
hoimbl 39521 Any n-dimensional half-ope...
opnvonmbllem1 39522 The half-open interval exp...
opnvonmbllem2 39523 An open subset of the n-di...
opnvonmbl 39524 An open subset of the n-di...
opnssborel 39525 Open sets of a generalized...
borelmbl 39526 All Borel subsets of the n...
volicorege0 39527 The Lebesgue measure of a ...
isvonmbl 39528 The predicate " ` A ` is m...
mblvon 39529 The n-dimensional Lebesgue...
vonmblss 39530 n-dimensional Lebesgue mea...
volico2 39531 The measure of left closed...
vonmblss2 39532 n-dimensional Lebesgue mea...
ovolval2lem 39533 The value of the Lebesgue ...
ovolval2 39534 The value of the Lebesgue ...
ovnsubadd2lem 39535 ` ( voln* `` X ) ` is suba...
ovnsubadd2 39536 ` ( voln* `` X ) ` is suba...
ovolval3 39537 The value of the Lebesgue ...
ovnsplit 39538 The n-dimensional Lebesgue...
ovolval4lem1 39539 |- ( ( ph /\ n e. A ) -> ...
ovolval4lem2 39540 The value of the Lebesgue ...
ovolval4 39541 The value of the Lebesgue ...
ovolval5lem1 39542 |- ( ph -> ( sum^ ` ( n e....
ovolval5lem2 39543 |- ( ( ph /\ n e. NN ) ->...
ovolval5lem3 39544 The value of the Lebesgue ...
ovolval5 39545 The value of the Lebesgue ...
ovnovollem1 39546 if ` F ` is a cover of ` B...
ovnovollem2 39547 if ` I ` is a cover of ` (...
ovnovollem3 39548 The 1-dimensional Lebesgue...
ovnovol 39549 The 1-dimensional Lebesgue...
vonvolmbllem 39550 If a subset ` B ` of real ...
vonvolmbl 39551 A subset of Real numbers i...
vonvol 39552 The 1-dimensional Lebesgue...
vonvolmbl2 39553 A subset ` X ` of the spac...
vonvol2 39554 The 1-dimensional Lebesgue...
hoimbl2 39555 Any n-dimensional half-ope...
voncl 39556 The Lebesgue measure of a ...
vonhoi 39557 The Lebesgue outer measure...
vonxrcl 39558 The Lebesgue measure of a ...
vonval2 39559 Value of the Lebesgue meas...
ioosshoi 39560 A n-dimensional open inter...
vonn0hoi 39561 The Lebesgue outer measure...
von0val 39562 The Lebesgue measure (for ...
vonhoire 39563 The Lebesgue measure of a ...
iinhoiicclem 39564 A n-dimensional closed int...
iinhoiicc 39565 A n-dimensional closed int...
iunhoiioolem 39566 A n-dimensional open inter...
iunhoiioo 39567 A n-dimensional open inter...
ioovonmbl 39568 Any n-dimensional open int...
iccvonmbllem 39569 Any n-dimensional closed i...
iccvonmbl 39570 Any n-dimensional closed i...
vonioolem1 39571 The sequence of the measur...
vonioolem2 39572 The n-dimensional Lebesgue...
vonioo 39573 The n-dimensional Lebesgue...
vonicclem1 39574 The sequence of the measur...
vonicclem2 39575 The n-dimensional Lebesgue...
vonicc 39576 The n-dimensional Lebesgue...
snvonmbl 39577 A n-dimensional singleton ...
vonn0ioo 39578 The n-dimensional Lebesgue...
vonn0icc 39579 The n-dimensional Lebesgue...
ctvonmbl 39580 Any n-dimensional countabl...
vonn0ioo2 39581 The n-dimensional Lebesgue...
vonsn 39582 The n-dimensional Lebesgue...
vonn0icc2 39583 The n-dimensional Lebesgue...
vonct 39584 The n-dimensional Lebesgue...
vitali2 39585 There are non-measurable s...
pimltmnf2 39588 Given a real valued functi...
preimagelt 39589 The preimage of a right-op...
preimalegt 39590 The preimage of a left-ope...
pimconstlt0 39591 Given a constant function,...
pimconstlt1 39592 Given a constant function,...
pimltpnf 39593 Given a real valued functi...
pimgtpnf2 39594 Given a real valued functi...
salpreimagelt 39595 If all the preimages of le...
pimrecltpos 39596 The preimage of an unbound...
salpreimalegt 39597 If all the preimages of ri...
pimiooltgt 39598 The preimage of an open in...
preimaicomnf 39599 Preimage of an open interv...
pimltpnf2 39600 Given a real valued functi...
pimgtmnf2 39601 Given a real valued functi...
pimdecfgtioc 39602 Given a non-increasing fun...
pimincfltioc 39603 Given a non decreasing fun...
pimdecfgtioo 39604 Given a non decreasing fun...
pimincfltioo 39605 Given a non decreasing fun...
preimaioomnf 39606 Preimage of an open interv...
preimageiingt 39607 A preimage of a left-close...
preimaleiinlt 39608 A preimage of a left-open,...
pimgtmnf 39609 Given a real valued functi...
pimrecltneg 39610 The preimage of an unbound...
salpreimagtge 39611 If all the preimages of le...
salpreimaltle 39612 If all the preimages of ri...
issmflem 39613 The predicate " ` F ` is a...
issmf 39614 The predicate " ` F ` is a...
salpreimalelt 39615 If all the preimages of ri...
salpreimagtlt 39616 If all the preimages of le...
smfpreimalt 39617 Given a function measurabl...
smff 39618 A function measurable w.r....
smfdmss 39619 The domain of a function m...
issmff 39620 The predicate " ` F ` is a...
issmfd 39621 A sufficient condition for...
issmfltle 39622 The definition of a measur...
smfpreimaltf 39623 Given a function measurabl...
issmfdf 39624 A sufficient condition for...
sssmf 39625 The restriction of a sigma...
mbfresmf 39626 A Real valued, measurable ...
cnfsmf 39627 A continuous function is m...
incsmflem 39628 A non decreasing function ...
incsmf 39629 A real valued, non-decreas...
smfsssmf 39630 If a function is measurabl...
issmflelem 39631 The predicate " ` F ` is a...
issmfle 39632 The predicate " ` F ` is b...
smfpimltmpt 39633 Given a function measurabl...
smfpimltxr 39634 Given a function measurabl...
issmfdmpt 39635 A sufficient condition for...
smfconst 39636 A constant function is mea...
sssmfmpt 39637 The restriction of a sigma...
cnfrrnsmf 39638 A function, continuous fro...
smfid 39639 The identity function is B...
bormflebmf 39640 A Borel measurable functio...
smfpreimale 39641 Given a function measurabl...
issmfgtlem 39642 The predicate " ` F ` is a...
issmfgt 39643 The predicate " ` F ` is b...
issmfled 39644 A sufficient condition for...
smfpimltxrmpt 39645 Given a function measurabl...
smfmbfcex 39646 A constant function, with ...
issmfgtd 39647 A sufficient condition for...
smfpreimagt 39648 Given a function measurabl...
smfaddlem1 39649 Given the sum of two funct...
smfaddlem2 39650 The sum of two sigma-measu...
smfadd 39651 The sum of two sigma-measu...
decsmflem 39652 A non-increasing function ...
decsmf 39653 A real valued, non-increas...
smfpreimagtf 39654 Given a function measurabl...
issmfgelem 39655 The predicate " ` F ` is a...
issmfge 39656 The predicate " ` F ` is b...
smflimlem1 39657 Lemma for the proof that t...
smflimlem2 39658 Lemma for the proof that t...
smflimlem3 39659 The limit of sigma-measura...
smflimlem4 39660 Lemma for the proof that t...
smflimlem5 39661 Lemma for the proof that t...
smflimlem6 39662 Lemma for the proof that t...
smflim 39663 The limit of sigma-measura...
nsssmfmbflem 39664 The sigma-measurable funct...
nsssmfmbf 39665 The sigma-measurable funct...
smfpimgtxr 39666 Given a function measurabl...
smfpimgtmpt 39667 Given a function measurabl...
smfpreimage 39668 Given a function measurabl...
mbfpsssmf 39669 Real valued, measurable fu...
smfpimgtxrmpt 39670 Given a function measurabl...
smfpimioompt 39671 Given a function measurabl...
smfpimioo 39672 Given a function measurabl...
smfresal 39673 Given a sigma-measurable f...
smfrec 39674 The reciprocal of a sigma-...
smfres 39675 The restriction of sigma-m...
smfmullem1 39676 The multiplication of two ...
smfmullem2 39677 The multiplication of two ...
smfmullem3 39678 The multiplication of two ...
smfmullem4 39679 The multiplication of two ...
smfmul 39680 The multiplication of two ...
smfmulc1 39681 A sigma-measurable functio...
smfdiv 39682 The fraction of two sigma-...
smfpimbor1lem1 39683 Every open set belongs to ...
smfpimbor1lem2 39684 Given D sigma-measurable f...
smfpimbor1 39685 Given a sigma-measurable f...
smf2id 39686 Twice the identity functio...
smfco 39687 The composition of a Borel...
sigarval 39688 Define the signed area by ...
sigarim 39689 Signed area takes value in...
sigarac 39690 Signed area is anticommuta...
sigaraf 39691 Signed area is additive by...
sigarmf 39692 Signed area is additive (w...
sigaras 39693 Signed area is additive by...
sigarms 39694 Signed area is additive (w...
sigarls 39695 Signed area is linear by t...
sigarid 39696 Signed area of a flat para...
sigarexp 39697 Expand the signed area for...
sigarperm 39698 Signed area ` ( A - C ) G ...
sigardiv 39699 If signed area between vec...
sigarimcd 39700 Signed area takes value in...
sigariz 39701 If signed area is zero, th...
sigarcol 39702 Given three points ` A ` ,...
sharhght 39703 Let ` A B C ` be a triangl...
sigaradd 39704 Subtracting (double) area ...
cevathlem1 39705 Ceva's theorem first lemma...
cevathlem2 39706 Ceva's theorem second lemm...
cevath 39707 Ceva's theorem. Let ` A B...
hirstL-ax3 39708 The third axiom of a syste...
ax3h 39709 Recovery of ~ ax-3 from ~ ...
aibandbiaiffaiffb 39710 A closed form showing (a i...
aibandbiaiaiffb 39711 A closed form showing (a i...
notatnand 39712 Do not use. Use intnanr i...
aistia 39713 Given a is equivalent to `...
aisfina 39714 Given a is equivalent to `...
bothtbothsame 39715 Given both a, b are equiva...
bothfbothsame 39716 Given both a, b are equiva...
aiffbbtat 39717 Given a is equivalent to b...
aisbbisfaisf 39718 Given a is equivalent to b...
axorbtnotaiffb 39719 Given a is exclusive to b,...
aiffnbandciffatnotciffb 39720 Given a is equivalent to (...
axorbciffatcxorb 39721 Given a is equivalent to (...
aibnbna 39722 Given a implies b, (not b)...
aibnbaif 39723 Given a implies b, not b, ...
aiffbtbat 39724 Given a is equivalent to b...
astbstanbst 39725 Given a is equivalent to T...
aistbistaandb 39726 Given a is equivalent to T...
aisbnaxb 39727 Given a is equivalent to b...
atbiffatnnb 39728 If a implies b, then a imp...
bisaiaisb 39729 Application of bicom1 with...
atbiffatnnbalt 39730 If a implies b, then a imp...
abnotbtaxb 39731 Assuming a, not b, there e...
abnotataxb 39732 Assuming not a, b, there e...
conimpf 39733 Assuming a, not b, and a i...
conimpfalt 39734 Assuming a, not b, and a i...
aistbisfiaxb 39735 Given a is equivalent to T...
aisfbistiaxb 39736 Given a is equivalent to F...
aifftbifffaibif 39737 Given a is equivalent to T...
aifftbifffaibifff 39738 Given a is equivalent to T...
atnaiana 39739 Given a, it is not the cas...
ainaiaandna 39740 Given a, a implies it is n...
abcdta 39741 Given (((a and b) and c) a...
abcdtb 39742 Given (((a and b) and c) a...
abcdtc 39743 Given (((a and b) and c) a...
abcdtd 39744 Given (((a and b) and c) a...
abciffcbatnabciffncba 39745 Operands in a biconditiona...
abciffcbatnabciffncbai 39746 Operands in a biconditiona...
nabctnabc 39747 not ( a -> ( b /\ c ) ) we...
jabtaib 39748 For when pm3.4 lacks a pm3...
onenotinotbothi 39749 From one negated implicati...
twonotinotbothi 39750 From these two negated imp...
clifte 39751 show d is the same as an i...
cliftet 39752 show d is the same as an i...
clifteta 39753 show d is the same as an i...
cliftetb 39754 show d is the same as an i...
confun 39755 Given the hypotheses there...
confun2 39756 Confun simplified to two p...
confun3 39757 Confun's more complex form...
confun4 39758 An attempt at derivative. ...
confun5 39759 An attempt at derivative. ...
plcofph 39760 Given, a,b and a "definiti...
pldofph 39761 Given, a,b c, d, "definiti...
plvcofph 39762 Given, a,b,d, and "definit...
plvcofphax 39763 Given, a,b,d, and "definit...
plvofpos 39764 rh is derivable because ON...
mdandyv0 39765 Given the equivalences set...
mdandyv1 39766 Given the equivalences set...
mdandyv2 39767 Given the equivalences set...
mdandyv3 39768 Given the equivalences set...
mdandyv4 39769 Given the equivalences set...
mdandyv5 39770 Given the equivalences set...
mdandyv6 39771 Given the equivalences set...
mdandyv7 39772 Given the equivalences set...
mdandyv8 39773 Given the equivalences set...
mdandyv9 39774 Given the equivalences set...
mdandyv10 39775 Given the equivalences set...
mdandyv11 39776 Given the equivalences set...
mdandyv12 39777 Given the equivalences set...
mdandyv13 39778 Given the equivalences set...
mdandyv14 39779 Given the equivalences set...
mdandyv15 39780 Given the equivalences set...
mdandyvr0 39781 Given the equivalences set...
mdandyvr1 39782 Given the equivalences set...
mdandyvr2 39783 Given the equivalences set...
mdandyvr3 39784 Given the equivalences set...
mdandyvr4 39785 Given the equivalences set...
mdandyvr5 39786 Given the equivalences set...
mdandyvr6 39787 Given the equivalences set...
mdandyvr7 39788 Given the equivalences set...
mdandyvr8 39789 Given the equivalences set...
mdandyvr9 39790 Given the equivalences set...
mdandyvr10 39791 Given the equivalences set...
mdandyvr11 39792 Given the equivalences set...
mdandyvr12 39793 Given the equivalences set...
mdandyvr13 39794 Given the equivalences set...
mdandyvr14 39795 Given the equivalences set...
mdandyvr15 39796 Given the equivalences set...
mdandyvrx0 39797 Given the exclusivities se...
mdandyvrx1 39798 Given the exclusivities se...
mdandyvrx2 39799 Given the exclusivities se...
mdandyvrx3 39800 Given the exclusivities se...
mdandyvrx4 39801 Given the exclusivities se...
mdandyvrx5 39802 Given the exclusivities se...
mdandyvrx6 39803 Given the exclusivities se...
mdandyvrx7 39804 Given the exclusivities se...
mdandyvrx8 39805 Given the exclusivities se...
mdandyvrx9 39806 Given the exclusivities se...
mdandyvrx10 39807 Given the exclusivities se...
mdandyvrx11 39808 Given the exclusivities se...
mdandyvrx12 39809 Given the exclusivities se...
mdandyvrx13 39810 Given the exclusivities se...
mdandyvrx14 39811 Given the exclusivities se...
mdandyvrx15 39812 Given the exclusivities se...
H15NH16TH15IH16 39813 Given 15 hypotheses and a ...
dandysum2p2e4 39814 CONTRADICTION PRO...
mdandysum2p2e4 39815 CONTRADICTION PROVED AT 1 ...
r19.32 39816 Theorem 19.32 of [Margaris...
rexsb 39817 An equivalent expression f...
rexrsb 39818 An equivalent expression f...
2rexsb 39819 An equivalent expression f...
2rexrsb 39820 An equivalent expression f...
cbvral2 39821 Change bound variables of ...
cbvrex2 39822 Change bound variables of ...
2ralbiim 39823 Split a biconditional and ...
raaan2 39824 Rearrange restricted quant...
rmoimi 39825 Restricted "at most one" i...
2reu5a 39826 Double restricted existent...
reuimrmo 39827 Restricted uniqueness impl...
rmoanim 39828 Introduction of a conjunct...
reuan 39829 Introduction of a conjunct...
2reurex 39830 Double restricted quantifi...
2reurmo 39831 Double restricted quantifi...
2reu2rex 39832 Double restricted existent...
2rmoswap 39833 A condition allowing swap ...
2rexreu 39834 Double restricted existent...
2reu1 39835 Double restricted existent...
2reu2 39836 Double restricted existent...
2reu3 39837 Double restricted existent...
2reu4a 39838 Definition of double restr...
2reu4 39839 Definition of double restr...
2reu7 39840 Two equivalent expressions...
2reu8 39841 Two equivalent expressions...
ralbinrald 39848 Elemination of a restricte...
nvelim 39849 If a class is the universa...
alneu 39850 If a statement holds for a...
eu2ndop1stv 39851 If there is a unique secon...
eldmressn 39852 Element of the domain of a...
fveqvfvv 39853 If a function's value at a...
funresfunco 39854 Composition of two functio...
fnresfnco 39855 Composition of two functio...
funcoressn 39856 A composition restricted t...
funressnfv 39857 A restriction to a singlet...
dfateq12d 39858 Equality deduction for "de...
nfdfat 39859 Bound-variable hypothesis ...
dfdfat2 39860 Alternate definition of th...
dfafv2 39861 Alternative definition of ...
afveq12d 39862 Equality deduction for fun...
afveq1 39863 Equality theorem for funct...
afveq2 39864 Equality theorem for funct...
nfafv 39865 Bound-variable hypothesis ...
csbafv12g 39866 Move class substitution in...
afvfundmfveq 39867 If a class is a function r...
afvnfundmuv 39868 If a set is not in the dom...
ndmafv 39869 The value of a class outsi...
afvvdm 39870 If the function value of a...
nfunsnafv 39871 If the restriction of a cl...
afvvfunressn 39872 If the function value of a...
afvprc 39873 A function's value at a pr...
afvvv 39874 If a function's value at a...
afvpcfv0 39875 If the value of the altern...
afvnufveq 39876 The value of the alternati...
afvvfveq 39877 The value of the alternati...
afv0fv0 39878 If the value of the altern...
afvfvn0fveq 39879 If the function's value at...
afv0nbfvbi 39880 The function's value at an...
afvfv0bi 39881 The function's value at an...
afveu 39882 The value of a function at...
fnbrafvb 39883 Equivalence of function va...
fnopafvb 39884 Equivalence of function va...
funbrafvb 39885 Equivalence of function va...
funopafvb 39886 Equivalence of function va...
funbrafv 39887 The second argument of a b...
funbrafv2b 39888 Function value in terms of...
dfafn5a 39889 Representation of a functi...
dfafn5b 39890 Representation of a functi...
fnrnafv 39891 The range of a function ex...
afvelrnb 39892 A member of a function's r...
afvelrnb0 39893 A member of a function's r...
dfaimafn 39894 Alternate definition of th...
dfaimafn2 39895 Alternate definition of th...
afvelima 39896 Function value in an image...
afvelrn 39897 A function's value belongs...
fnafvelrn 39898 A function's value belongs...
fafvelrn 39899 A function's value belongs...
ffnafv 39900 A function maps to a class...
afvres 39901 The value of a restricted ...
tz6.12-afv 39902 Function value. Theorem 6...
tz6.12-1-afv 39903 Function value (Theorem 6....
dmfcoafv 39904 Domains of a function comp...
afvco2 39905 Value of a function compos...
rlimdmafv 39906 Two ways to express that a...
aoveq123d 39907 Equality deduction for ope...
nfaov 39908 Bound-variable hypothesis ...
csbaovg 39909 Move class substitution in...
aovfundmoveq 39910 If a class is a function r...
aovnfundmuv 39911 If an ordered pair is not ...
ndmaov 39912 The value of an operation ...
ndmaovg 39913 The value of an operation ...
aovvdm 39914 If the operation value of ...
nfunsnaov 39915 If the restriction of a cl...
aovvfunressn 39916 If the operation value of ...
aovprc 39917 The value of an operation ...
aovrcl 39918 Reverse closure for an ope...
aovpcov0 39919 If the alternative value o...
aovnuoveq 39920 The alternative value of t...
aovvoveq 39921 The alternative value of t...
aov0ov0 39922 If the alternative value o...
aovovn0oveq 39923 If the operation's value a...
aov0nbovbi 39924 The operation's value on a...
aovov0bi 39925 The operation's value on a...
rspceaov 39926 A frequently used special ...
fnotaovb 39927 Equivalence of operation v...
ffnaov 39928 An operation maps to a cla...
faovcl 39929 Closure law for an operati...
aovmpt4g 39930 Value of a function given ...
aoprssdm 39931 Domain of closure of an op...
ndmaovcl 39932 The "closure" of an operat...
ndmaovrcl 39933 Reverse closure law, in co...
ndmaovcom 39934 Any operation is commutati...
ndmaovass 39935 Any operation is associati...
ndmaovdistr 39936 Any operation is distribut...
1t10e1p1e11 39937 11 is 1 times 10 to the po...
1t10e1p1e11OLD 39938 Obsolete version of ~ 1t10...
elprneb 39939 An element of a proper uno...
leltletr 39940 Transitive law, weaker for...
deccarry 39941 Add 1 to a 2 digit number ...
nltle2tri 39942 Negated extended trichotom...
zgeltp1eq 39943 If an integer is between a...
smonoord 39944 Ordering relation for a st...
fzopred 39945 Join a predecessor to the ...
fzopredsuc 39946 Join a predecessor and a s...
1fzopredsuc 39947 Join 0 and a successor to ...
el1fzopredsuc 39948 An element of an open inte...
m1mod0mod1 39949 An integer decreased by 1 ...
elmod2 39950 An integer modulo 2 is eit...
iccpval 39953 Partition consisting of a ...
iccpart 39954 A special partition. Corr...
iccpartimp 39955 Implications for a class b...
iccpartres 39956 The restriction of a parti...
iccpartxr 39957 If there is a partition, t...
iccpartgtprec 39958 If there is a partition, t...
iccpartipre 39959 If there is a partition, t...
iccpartiltu 39960 If there is a partition, t...
iccpartigtl 39961 If there is a partition, t...
iccpartlt 39962 If there is a partition, t...
iccpartltu 39963 If there is a partition, t...
iccpartgtl 39964 If there is a partition, t...
iccpartgt 39965 If there is a partition, t...
iccpartleu 39966 If there is a partition, t...
iccpartgel 39967 If there is a partition, t...
iccpartrn 39968 If there is a partition, t...
iccpartf 39969 The range of the partition...
iccpartel 39970 If there is a partition, t...
iccelpart 39971 An element of any partitio...
iccpartiun 39972 A half opened interval of ...
icceuelpartlem 39973 Lemma for ~ icceuelpart . ...
icceuelpart 39974 An element of a partitione...
iccpartdisj 39975 The segments of a partitio...
iccpartnel 39976 A point of a partition is ...
fmtno 39979 The ` N ` th Fermat number...
fmtnoge3 39980 Each Fermat number is grea...
fmtnonn 39981 Each Fermat number is a po...
fmtnom1nn 39982 A Fermat number minus one ...
fmtnoodd 39983 Each Fermat number is odd....
fmtnorn 39984 A Fermat number is a funct...
fmtnof1 39985 The enumeration of the Fer...
fmtnoinf 39986 The set of Fermat numbers ...
fmtnorec1 39987 The first recurrence relat...
sqrtpwpw2p 39988 The floor of the square ro...
fmtnosqrt 39989 The floor of the square ro...
fmtno0 39990 The ` 0 ` th Fermat number...
fmtno1 39991 The ` 1 ` st Fermat number...
fmtnorec2lem 39992 Lemma for ~ fmtnorec2 (ind...
fmtnorec2 39993 The second recurrence rela...
fmtnodvds 39994 Any Fermat number divides ...
goldbachthlem1 39995 Lemma 1 for ~ goldbachth ....
goldbachthlem2 39996 Lemma 2 for ~ goldbachth ....
goldbachth 39997 Goldbach's theorem: Two d...
fmtnorec3 39998 The third recurrence relat...
fmtnorec4 39999 The fourth recurrence rela...
fmtno2 40000 The ` 2 ` nd Fermat number...
fmtno3 40001 The ` 3 ` rd Fermat number...
fmtno4 40002 The ` 4 ` th Fermat number...
fmtno5lem1 40003 Lemma 1 for ~ fmtno5 . (C...
fmtno5lem2 40004 Lemma 2 for ~ fmtno5 . (C...
fmtno5lem3 40005 Lemma 3 for ~ fmtno5 . (C...
fmtno5lem4 40006 Lemma 4 for ~ fmtno5 . (C...
fmtno5 40007 The ` 5 ` th Fermat number...
fmtno0prm 40008 The ` 0 ` th Fermat number...
fmtno1prm 40009 The ` 1 ` st Fermat number...
fmtno2prm 40010 The ` 2 ` nd Fermat number...
257prm 40011 257 is a prime number (the...
fmtno3prm 40012 The ` 3 ` rd Fermat number...
odz2prm2pw 40013 Any power of two is coprim...
fmtnoprmfac1lem 40014 Lemma for ~ fmtnoprmfac1 :...
fmtnoprmfac1 40015 Divisor of Fermat number (...
fmtnoprmfac2lem1 40016 Lemma for ~ fmtnoprmfac2 ....
fmtnoprmfac2 40017 Divisor of Fermat number (...
fmtnofac2lem 40018 Lemma for ~ fmtnofac2 (Ind...
fmtnofac2 40019 Divisor of Fermat number (...
fmtnofac1 40020 Divisor of Fermat number (...
fmtno4sqrt 40021 The floor of the square ro...
fmtno4prmfac 40022 If P was a (prime) factor ...
fmtno4prmfac193 40023 If P was a (prime) factor ...
fmtno4nprmfac193 40024 193 is not a (prime) facto...
fmtno4prm 40025 The ` 4 `-th Fermat number...
65537prm 40026 65537 is a prime number (t...
fmtnofz04prm 40027 The first five Fermat numb...
fmtnole4prm 40028 The first five Fermat numb...
fmtno5faclem1 40029 Lemma 1 for ~ fmtno5fac . ...
fmtno5faclem2 40030 Lemma 2 for ~ fmtno5fac . ...
fmtno5faclem3 40031 Lemma 3 for ~ fmtno5fac . ...
fmtno5fac 40032 The factorisation of the `...
fmtno5nprm 40033 The ` 5 ` th Fermat number...
prmdvdsfmtnof1lem1 40034 Lemma 1 for ~ prmdvdsfmtno...
prmdvdsfmtnof1lem2 40035 Lemma 2 for ~ prmdvdsfmtno...
prmdvdsfmtnof 40036 The mapping of a Fermat nu...
prmdvdsfmtnof1 40037 The mapping of a Fermat nu...
prminf2 40038 The set of prime numbers i...
pwdif 40039 The difference of two numb...
pwm1geoserALT 40040 The n-th power of a number...
2pwp1prm 40041 For every prime number of ...
2pwp1prmfmtno 40042 Every prime number of the ...
m2prm 40043 The second Mersenne number...
m3prm 40044 The third Mersenne number ...
2exp5 40045 Two to the fifth power is ...
flsqrt 40046 A condition equivalent to ...
flsqrt5 40047 The floor of the square ro...
3ndvds4 40048 3 does not divide 4. (Con...
139prmALT 40049 139 is a prime number. In...
31prm 40050 31 is a prime number. In ...
m5prm 40051 The fifth Mersenne number ...
2exp7 40052 Two to the seventh power i...
127prm 40053 127 is a prime number. (C...
m7prm 40054 The seventh Mersenne numbe...
2exp11 40055 Two to the eleventh power ...
m11nprm 40056 The eleventh Mersenne numb...
mod42tp1mod8 40057 If a number is ` 3 ` modul...
sfprmdvdsmersenne 40058 If ` Q ` is a safe prime (...
sgprmdvdsmersenne 40059 If ` P ` is a Sophie Germa...
lighneallem1 40060 Lemma 1 for ~ lighneal . ...
lighneallem2 40061 Lemma 2 for ~ lighneal . ...
lighneallem3 40062 Lemma 3 for ~ lighneal . ...
lighneallem4a 40063 Lemma 1 for ~ lighneallem4...
lighneallem4b 40064 Lemma 2 for ~ lighneallem4...
lighneallem4 40065 Lemma 3 for ~ lighneal . ...
lighneal 40066 If a power of a prime ` P ...
modexp2m1d 40067 The square of an integer w...
proththdlem 40068 Lemma for ~ proththd . (C...
proththd 40069 Proth's theorem (1878). I...
5tcu2e40 40070 5 times the cube of 2 is 4...
3exp4mod41 40071 3 to the fourth power is -...
41prothprmlem1 40072 Lemma 1 for ~ 41prothprm ....
41prothprmlem2 40073 Lemma 2 for ~ 41prothprm ....
41prothprm 40074 41 is a _Proth prime_. (C...
iseven 40079 The predicate "is an even ...
isodd 40080 The predicate "is an odd n...
evenz 40081 An even number is an integ...
oddz 40082 An odd number is an intege...
evendiv2z 40083 The result of dividing an ...
oddp1div2z 40084 The result of dividing an ...
oddm1div2z 40085 The result of dividing an ...
isodd2 40086 The predicate "is an odd n...
dfodd2 40087 Alternate definition for o...
dfodd6 40088 Alternate definition for o...
dfeven4 40089 Alternate definition for e...
evenm1odd 40090 The predecessor of an even...
evenp1odd 40091 The successor of an even n...
oddp1eveni 40092 The successor of an odd nu...
oddm1eveni 40093 The predecessor of an odd ...
evennodd 40094 An even number is not an o...
oddneven 40095 An odd number is not an ev...
enege 40096 The negative of an even nu...
onego 40097 The negative of an odd num...
m1expevenALTV 40098 Exponentiation of -1 by an...
m1expoddALTV 40099 Exponentiation of -1 by an...
dfeven2 40100 Alternate definition for e...
dfodd3 40101 Alternate definition for o...
iseven2 40102 The predicate "is an even ...
isodd3 40103 The predicate "is an odd n...
2dvdseven 40104 2 divides an even number. ...
2ndvdsodd 40105 2 does not divide an odd n...
2dvdsoddp1 40106 2 divides an odd number in...
2dvdsoddm1 40107 2 divides an odd number de...
dfeven3 40108 Alternate definition for e...
dfodd4 40109 Alternate definition for o...
dfodd5 40110 Alternate definition for o...
zefldiv2ALTV 40111 The floor of an even numbe...
zofldiv2ALTV 40112 The floor of an odd numer ...
oddflALTV 40113 Odd number representation ...
iseven5 40114 The predicate "is an even ...
isodd7 40115 The predicate "is an odd n...
dfeven5 40116 Alternate definition for e...
dfodd7 40117 Alternate definition for o...
zneoALTV 40118 No even integer equals an ...
zeoALTV 40119 An integer is even or odd....
zeo2ALTV 40120 An integer is even or odd ...
nneoALTV 40121 A positive integer is even...
nneoiALTV 40122 A positive integer is even...
odd2np1ALTV 40123 An integer is odd iff it i...
oddm1evenALTV 40124 An integer is odd iff its ...
oddp1evenALTV 40125 An integer is odd iff its ...
oexpnegALTV 40126 The exponential of the neg...
oexpnegnz 40127 The exponential of the neg...
bits0ALTV 40128 Value of the zeroth bit. ...
bits0eALTV 40129 The zeroth bit of an even ...
bits0oALTV 40130 The zeroth bit of an odd n...
divgcdoddALTV 40131 Either ` A / ( A gcd B ) `...
opoeALTV 40132 The sum of two odds is eve...
opeoALTV 40133 The sum of an odd and an e...
omoeALTV 40134 The difference of two odds...
omeoALTV 40135 The difference of an odd a...
oddprmALTV 40136 A prime not equal to ` 2 `...
0evenALTV 40137 0 is an even number. (Con...
0noddALTV 40138 0 is not an odd number. (...
1oddALTV 40139 1 is an odd number. (Cont...
1nevenALTV 40140 1 is not an even number. ...
2evenALTV 40141 2 is an even number. (Con...
2noddALTV 40142 2 is not an odd number. (...
nn0o1gt2ALTV 40143 An odd nonnegative integer...
nnoALTV 40144 An alternate characterizat...
nn0oALTV 40145 An alternate characterizat...
nn0e 40146 An alternate characterizat...
nn0onn0exALTV 40147 For each odd nonnegative i...
nn0enn0exALTV 40148 For each even nonnegative ...
nnpw2evenALTV 40149 2 to the power of a positi...
epoo 40150 The sum of an even and an ...
emoo 40151 The difference of an even ...
epee 40152 The sum of two even number...
emee 40153 The difference of two even...
evensumeven 40154 If a summand is even, the ...
3odd 40155 3 is an odd number. (Cont...
4even 40156 4 is an even number. (Con...
5odd 40157 5 is an odd number. (Cont...
6even 40158 6 is an even number. (Con...
7odd 40159 7 is an odd number. (Cont...
8even 40160 8 is an even number. (Con...
evenprm2 40161 A prime number is even iff...
oddprmne2 40162 Every prime number not bei...
oddprmuzge3 40163 A prime number which is od...
perfectALTVlem1 40164 Lemma for ~ perfectALTV . ...
perfectALTVlem2 40165 Lemma for ~ perfectALTV . ...
perfectALTV 40166 The Euclid-Euler theorem, ...
isgbe 40173 The predicate "is an even ...
isgbo 40174 The predicate "is an odd G...
isgboa 40175 The predicate "is an odd G...
gbeeven 40176 An even Goldbach number is...
gboodd 40177 An odd Goldbach number is ...
gboagbo 40178 An odd Goldbach number (st...
gboaodd 40179 An odd Goldbach number is ...
gbepos 40180 Any even Goldbach number i...
gbopos 40181 Any odd Goldbach number is...
gboapos 40182 Any odd Goldbach number is...
gbegt5 40183 Any even Goldbach number i...
gbogt5 40184 Any odd Goldbach number is...
gboge7 40185 Any odd Goldbach number is...
gboage9 40186 Any odd Goldbach number (s...
gbege6 40187 Any even Goldbach number i...
gbpart6 40188 The Goldbach partition of ...
gbpart7 40189 The (weak) Goldbach partit...
gbpart8 40190 The Goldbach partition of ...
gbpart9 40191 The (strong) Goldbach part...
gbpart11 40192 The (strong) Goldbach part...
6gbe 40193 6 is an even Goldbach numb...
7gbo 40194 7 is an odd Goldbach numbe...
8gbe 40195 8 is an even Goldbach numb...
9gboa 40196 9 is an odd Goldbach numbe...
11gboa 40197 11 is an odd Goldbach numb...
stgoldbwt 40198 If the strong ternary Gold...
bgoldbwt 40199 If the binary Goldbach con...
bgoldbst 40200 If the binary Goldbach con...
sgoldbaltlem1 40201 Lemma 1 for ~ sgoldbalt : ...
sgoldbaltlem2 40202 Lemma 2 for ~ sgoldbalt : ...
sgoldbalt 40203 An alternate (the original...
nnsum3primes4 40204 4 is the sum of at most 3 ...
nnsum4primes4 40205 4 is the sum of at most 4 ...
nnsum3primesprm 40206 Every prime is "the sum of...
nnsum4primesprm 40207 Every prime is "the sum of...
nnsum3primesgbe 40208 Any even Goldbach number i...
nnsum4primesgbe 40209 Any even Goldbach number i...
nnsum3primesle9 40210 Every integer greater than...
nnsum4primesle9 40211 Every integer greater than...
nnsum4primesodd 40212 If the (weak) ternary Gold...
nnsum4primesoddALTV 40213 If the (strong) ternary Go...
evengpop3 40214 If the (weak) ternary Gold...
evengpoap3 40215 If the (strong) ternary Go...
nnsum4primeseven 40216 If the (weak) ternary Gold...
nnsum4primesevenALTV 40217 If the (strong) ternary Go...
wtgoldbnnsum4prm 40218 If the (weak) ternary Gold...
stgoldbnnsum4prm 40219 If the (strong) ternary Go...
bgoldbnnsum3prm 40220 If the binary Goldbach con...
bgoldbtbndlem1 40221 Lemma 1 for ~ bgoldbtbnd :...
bgoldbtbndlem2 40222 Lemma 2 for ~ bgoldbtbnd ....
bgoldbtbndlem3 40223 Lemma 3 for ~ bgoldbtbnd ....
bgoldbtbndlem4 40224 Lemma 4 for ~ bgoldbtbnd ....
bgoldbtbnd 40225 If the binary Goldbach con...
bgoldbachlt 40227 The binary Goldbach conjec...
tgblthelfgott 40229 The ternary Goldbach conje...
tgoldbachlt 40230 The ternary Goldbach conje...
tgoldbach 40232 The ternary Goldbach conje...
bgoldbachltOLD 40234 Obsolete version of ~ bgol...
tgblthelfgottOLD 40236 Obsolete version of ~ tgbl...
tgoldbachltOLD 40237 Obsolete version of ~ tgol...
tgoldbachOLD 40239 Obsolete version of ~ tgol...
wrdred1 40240 A word truncated by a symb...
wrdred1hash 40241 The length of a word trunc...
lswn0 40242 The last symbol of a not e...
ccatw2s1cl 40243 The concatenation of a wor...
pfxval 40246 Value of a prefix. (Contr...
pfx00 40247 A zero length prefix. (Co...
pfx0 40248 A prefix of an empty set i...
pfxcl 40249 Closure of the prefix extr...
pfxmpt 40250 Value of the prefix extrac...
pfxres 40251 Value of the prefix extrac...
pfxf 40252 A prefix of a word is a fu...
pfxfn 40253 Value of the prefix extrac...
pfxlen 40254 Length of a prefix. Could...
pfxid 40255 A word is a prefix of itse...
pfxrn 40256 The range of a prefix of a...
pfxn0 40257 A prefix consisting of at ...
pfxnd 40258 The value of the prefix ex...
pfxlen0 40259 Length of a prefix of a wo...
addlenrevpfx 40260 The sum of the lengths of ...
addlenpfx 40261 The sum of the lengths of ...
pfxfv 40262 A symbol in a prefix of a ...
pfxfv0 40263 The first symbol in a pref...
pfxtrcfv 40264 A symbol in a word truncat...
pfxtrcfv0 40265 The first symbol in a word...
pfxfvlsw 40266 The last symbol in a (not ...
pfxeq 40267 The prefixes of two words ...
pfxtrcfvl 40268 The last symbol in a word ...
pfxsuffeqwrdeq 40269 Two words are equal if and...
pfxsuff1eqwrdeq 40270 Two (nonempty) words are e...
disjwrdpfx 40271 Sets of words are disjoint...
ccatpfx 40272 Joining a prefix with an a...
pfxccat1 40273 Recover the left half of a...
pfx1 40274 A prefix of length 1. (Co...
pfx2 40275 A prefix of length 2. (Co...
pfxswrd 40276 A prefix of a subword. Co...
swrdpfx 40277 A subword of a prefix. Co...
pfxpfx 40278 A prefix of a prefix. Cou...
pfxpfxid 40279 A prefix of a prefix with ...
pfxcctswrd 40280 The concatenation of the p...
lenpfxcctswrd 40281 The length of the concaten...
lenrevpfxcctswrd 40282 The length of the concaten...
pfxlswccat 40283 Reconstruct a nonempty wor...
ccats1pfxeq 40284 The last symbol of a word ...
ccats1pfxeqrex 40285 There exists a symbol such...
pfxccatin12lem1 40286 Lemma 1 for ~ pfxccatin12 ...
pfxccatin12lem2 40287 Lemma 2 for ~ pfxccatin12 ...
pfxccatin12 40288 The subword of a concatena...
pfxccat3 40289 The subword of a concatena...
pfxccatpfx1 40290 A prefix of a concatenatio...
pfxccatpfx2 40291 A prefix of a concatenatio...
pfxccat3a 40292 A prefix of a concatenatio...
pfxccatid 40293 A prefix of a concatenatio...
ccats1pfxeqbi 40294 A word is a prefix of a wo...
pfxccatin12d 40295 The subword of a concatena...
reuccatpfxs1lem 40296 Lemma for ~ reuccatpfxs1 ....
reuccatpfxs1 40297 There is a unique word hav...
splvalpfx 40298 Value of the substring rep...
repswpfx 40299 A prefix of a repeated sym...
cshword2 40300 Perform a cyclical shift f...
pfxco 40301 Mapping of words commutes ...
elnelall 40302 A contradiction concerning...
clel5 40303 Alternate definition of cl...
dfss7 40304 Alternate definition of su...
sssseq 40305 If a class is a subclass o...
prcssprc 40306 The superclass of a proper...
ralnralall 40307 A contradiction concerning...
falseral0 40308 A false statement can only...
ralralimp 40309 Selecting one of two alter...
n0rex 40310 There is an element in a n...
ssn0rex 40311 There is an element in a c...
elpwdifsn 40312 A subset of a set is an el...
pr1eqbg 40313 A (proper) pair is equal t...
pr1nebg 40314 A (proper) pair is not equ...
rexdifpr 40315 Restricted existential qua...
opidg 40316 The ordered pair ` <. A , ...
otiunsndisjX 40317 The union of singletons co...
opabn1stprc 40318 An ordered-pair class abst...
resresdm 40319 A restriction by an arbitr...
resisresindm 40320 The restriction of a relat...
fvifeq 40321 Equality of function value...
2f1fvneq 40322 If two one-to-one function...
f1cofveqaeq 40323 If the values of a composi...
f1cofveqaeqALT 40324 Alternate proof of ~ f1cof...
rnfdmpr 40325 The range of a one-to-one ...
imarnf1pr 40326 The image of the range of ...
funop1 40327 A function is an ordered p...
f1ssf1 40328 A subset of an injective f...
fun2dmnopgexmpl 40329 A function with a domain c...
opabresex0d 40330 A collection of ordered pa...
opabbrfex0d 40331 A collection of ordered pa...
opabresexd 40332 A collection of ordered pa...
opabbrfexd 40333 A collection of ordered pa...
opabresex2d 40334 Restrictions of a collecti...
mptmpt2opabbrd 40335 The operation value of a f...
mptmpt2opabovd 40336 The operation value of a f...
fpropnf1 40337 A function, given by an un...
riotaeqimp 40338 If two restricted iota des...
resfnfinfin 40339 The restriction of a funct...
residfi 40340 A restricted identity func...
cnambpcma 40341 ((a-b)+c)-a = c-a holds fo...
cnapbmcpd 40342 ((a+b)-c)+d = ((a+d)+b)-c ...
leaddsuble 40343 Addition and subtraction o...
2leaddle2 40344 If two real numbers are le...
ltnltne 40345 Variant of trichotomy law ...
p1lep2 40346 A real number increasd by ...
ltsubsubaddltsub 40347 If the result of subtracti...
zm1nn 40348 An integer minus 1 is posi...
nn0resubcl 40349 Closure law for subtractio...
eluzge0nn0 40350 If an integer is greater t...
ssfz12 40351 Subset relationship for fi...
elfz2z 40352 Membership of an integer i...
2elfz3nn0 40353 If there are two elements ...
fz0addcom 40354 The addition of two member...
2elfz2melfz 40355 If the sum of two integers...
fz0addge0 40356 The sum of two integers in...
elfzlble 40357 Membership of an integer i...
elfzelfzlble 40358 Membership of an element o...
subsubelfzo0 40359 Subtracting a difference f...
fzoopth 40360 A half-open integer range ...
2ffzoeq 40361 Two functions over a half-...
fzosplitpr 40362 Extending a half-open inte...
prinfzo0 40363 The intersection of a half...
elfzr 40364 A member of a finite inter...
elfzo0l 40365 A member of a half-open ra...
elfzlmr 40366 A member of a finite inter...
elfz0lmr 40367 A member of a finite inter...
resunimafz0 40368 Formerly part of proof of ...
nfile 40369 The size of any infinite s...
hash1n0 40370 If the size of a set is 1 ...
fsummsndifre 40371 A finite sum with one of i...
fsumsplitsndif 40372 Separate out a term in a f...
fsummmodsndifre 40373 A finite sum of summands m...
fsummmodsnunz 40374 A finite sum of summands m...
uhgruhgra 40375 Equivalence of the definit...
uhgrauhgr 40376 Equivalence of the definit...
uhgrauhgrbi 40377 Equivalence of the definit...
isuspgr 40382 The property of being a si...
isusgr 40383 The property of being a si...
uspgrf 40384 The edge function of a sim...
usgrf 40385 The edge function of a sim...
isusgrs 40386 The property of being a si...
usgrfs 40387 The edge function of a sim...
usgrfun 40388 The edge function of a sim...
usgrusgra 40389 A simple graph represented...
usgredgss 40390 The set of edges of a simp...
edgusgr 40391 An edge of a simple graph ...
isusgrop 40392 The property of being an u...
usgrop 40393 A simple graph represented...
isausgr 40394 The property of an unorder...
ausgrusgrb 40395 The equivalence of the def...
usgrausgri 40396 A simple graph represented...
ausgrumgri 40397 If an alternatively define...
ausgrusgri 40398 The equivalence of the def...
usgrausgrb 40399 The equivalence of the def...
usgredgop 40400 An edge of a simple graph ...
usgrf1o 40401 The edge function of a sim...
usgrf1 40402 The edge function of a sim...
uspgrf1oedg 40403 The edge function of a sim...
usgrss 40404 An edge is a subset of ver...
uspgrushgr 40405 A simple pseudograph is an...
uspgrupgr 40406 A simple pseudograph is an...
uspgrupgrushgr 40407 A graph is a simple pseudo...
usgruspgr 40408 A simple graph is a simple...
usgrumgr 40409 A simple graph is an undir...
usgrumgruspgr 40410 A graph is a simple graph ...
usgruspgrb 40411 A class is a simple graph ...
usgrupgr 40412 A simple graph is an undir...
usgruhgr 40413 A simple graph is an undir...
usgrislfuspgr 40414 A simple graph is a loop-f...
uspgrun 40415 The union ` U ` of two sim...
uspgrunop 40416 The union of two simple ps...
usgrun 40417 The union ` U ` of two sim...
usgrunop 40418 The union of two simple gr...
usgredg2 40419 The value of the "edge fun...
usgredg2ALT 40420 Alternate proof of ~ usgre...
usgredgprv 40421 In a simple graph, an edge...
usgredgprvALT 40422 Alternate proof of ~ usgre...
usgredgappr 40423 An edge of a simple graph ...
usgrpredgav 40424 An edge of a simple graph ...
edgassv2 40425 An edge of a simple graph ...
usgredg 40426 For each edge in a simple ...
usgrnloopv 40427 In a simple graph, there i...
usgrnloopvALT 40428 Alternate proof of ~ usgrn...
usgrnloop 40429 In a simple graph, there i...
usgrnloopALT 40430 Alternate proof of ~ usgrn...
usgrnloop0 40431 A simple graph has no loop...
usgrnloop0ALT 40432 Alternate proof of ~ usgrn...
usgredgne 40433 An edge of a simple graph ...
usgrf1oedg 40434 The edge function of a sim...
uhgr2edg 40435 If a vertex is adjacent to...
umgr2edg 40436 If a vertex is adjacent to...
usgr2edg 40437 If a vertex is adjacent to...
umgr2edg1 40438 If a vertex is adjacent to...
usgr2edg1 40439 If a vertex is adjacent to...
umgrvad2edg 40440 If a vertex is adjacent to...
umgr2edgneu 40441 If a vertex is adjacent to...
usgrsizedg 40442 In a simple graph, the siz...
usgredg3 40443 The value of the "edge fun...
usgredg4 40444 For a vertex incident to a...
usgredgreu 40445 For a vertex incident to a...
usgredg2vtx 40446 For a vertex incident to a...
uspgredg2vtxeu 40447 For a vertex incident to a...
usgredg2vtxeu 40448 For a vertex incident to a...
usgredg2vtxeuALT 40449 Alternate proof of ~ usgre...
uspgredg2vlem 40450 Lemma for ~ uspgredg2v . ...
uspgredg2v 40451 In a simple pseudograph, t...
usgredg2vlem1 40452 Lemma 1 for ~ usgredg2v . ...
usgredg2vlem2 40453 Lemma 2 for ~ usgredg2v . ...
usgredg2v 40454 In a simple graph, the map...
usgredgleord 40455 In a simple graph the numb...
ushgredgedga 40456 In a simple hypergraph the...
usgredgedga 40457 In a simple graph there is...
ushgredgedgaloop 40458 In a simple hypergraph the...
uspgredgaleord 40459 In a simple pseudograph th...
usgredgaleord 40460 In a simple graph the numb...
usgredgaleordALT 40461 In a simple graph the numb...
usgr0e 40462 The empty graph, with vert...
usgr0vb 40463 The null graph, with no ve...
uhgr0v0e 40464 The null graph, with no ve...
uhgr0vsize0 40465 The size of a hypergraph w...
uhgr0edgfi 40466 A graph of order 0 (i.e. w...
usgr0v 40467 The null graph, with no ve...
uhgr0vusgr 40468 The null graph, with no ve...
usgr0 40469 The null graph represented...
uspgr1e 40470 A simple pseudograph with ...
usgr1e 40471 A simple graph with one ed...
usgr0eop 40472 The empty graph, with vert...
uspgr1eop 40473 A simple pseudograph with ...
uspgr1ewop 40474 A simple pseudograph with ...
uspgr1v1eop 40475 A simple pseudograph with ...
usgr1eop 40476 A simple graph with (at le...
uspgr2v1e2w 40477 A simple pseudograph with ...
usgr2v1e2w 40478 A simple graph with two ve...
edg0usgr 40479 A class without edges is a...
lfuhgr1v0e 40480 A loop-free hypergraph wit...
usgr1vr 40481 A simple graph with one ve...
usgr1v 40482 A class with one (or no) v...
usgr1v0edg 40483 A class with one (or no) v...
usgrexmpllem 40484 Lemma for ~ usgrexmpl . (...
usgrexmplvtx 40485 The vertices ` 0 , 1 , 2 ,...
usgrexmpledg 40486 The edges ` { 0 , 1 } , { ...
usgrexmpl 40487 ` G ` is a simple graph of...
griedg0prc 40488 The class of empty graphs ...
griedg0ssusgr 40489 The class of all simple gr...
usgrprc 40490 The class of simple graphs...
relsubgr 40493 The class of the subgraph ...
subgrv 40494 If a class is a subgraph o...
issubgr 40495 The property of a set to b...
issubgr2 40496 The property of a set to b...
subgrprop 40497 The properties of a subgra...
subgrprop2 40498 The properties of a subgra...
uhgrissubgr 40499 The property of a hypergra...
subgrprop3 40500 The properties of a subgra...
egrsubgr 40501 An empty graph consisting ...
0grsubgr 40502 The null graph (represente...
0uhgrsubgr 40503 The null graph (as hypergr...
uhgrsubgrself 40504 A hypergraph is a subgraph...
subgrfun 40505 The edge function of a sub...
subgruhgrfun 40506 The edge function of a sub...
subgreldmiedg 40507 An element of the domain o...
subgruhgredgd 40508 An edge of a subgraph of a...
subumgredg2 40509 An edge of a subgraph of a...
subuhgr 40510 A subgraph of a hypergraph...
subupgr 40511 A subgraph of a pseudograp...
subumgr 40512 A subgraph of a multigraph...
subusgr 40513 A subgraph of a simple gra...
uhgrspansubgrlem 40514 Lemma for ~ uhgrspansubgr ...
uhgrspansubgr 40515 A spanning subgraph ` S ` ...
uhgrspan 40516 A spanning subgraph ` S ` ...
upgrspan 40517 A spanning subgraph ` S ` ...
umgrspan 40518 A spanning subgraph ` S ` ...
usgrspan 40519 A spanning subgraph ` S ` ...
uhgrspanop 40520 A spanning subgraph of a h...
upgrspanop 40521 A spanning subgraph of a p...
umgrspanop 40522 A spanning subgraph of a m...
usgrspanop 40523 A spanning subgraph of a s...
uhgrspan1lem1 40524 Lemma 1 for ~ uhgrspan1 . ...
uhgrspan1lem2 40525 Lemma 2 for ~ uhgrspan1 . ...
uhgrspan1lem3 40526 Lemma 3 for ~ uhgrspan1 . ...
uhgrspan1 40527 The induced subgraph ` S `...
upgrres1lem1 40528 Lemma 1 for ~ upgrres1 . ...
umgrres1lem 40529 Lemma for ~ umgrres1 . (C...
upgrres1lem2 40530 Lemma 2 for ~ upgrres1 . ...
upgrres1lem3 40531 Lemma 3 for ~ upgrres1 . ...
upgrres1 40532 A pseudograph obtained by ...
umgrres1 40533 A multigraph obtained by r...
usgrres1 40534 Restricting a simple graph...
isfusgr 40537 The property of being a fi...
fusgrvtxfi 40538 A finite simple graph has ...
isfusgrf1 40539 The property of being a fi...
isfusgrcl 40540 The property of being a fi...
fusgrusgr 40541 A finite simple graph is a...
opfusgr 40542 A finite simple graph repr...
usgredgffibi 40543 The number of edges in a s...
fusgredgfi 40544 In a finite simple graph t...
usgr1v0e 40545 The size of a (finite) sim...
usgrfilem 40546 In a finite simple graph, ...
fusgrfisbase 40547 Induction base for ~ fusgr...
fusgrfisstep 40548 Induction step in ~ fusgrf...
fusgrfis 40549 A finite simple graph is o...
nbgrprc0 40555 The set of neighbors is em...
nbgrcl 40559 If a class has at least on...
nbgrval 40560 The set of neighbors of a ...
dfnbgr2 40561 Alternate definition of th...
dfnbgr3 40562 Alternate definition of th...
nbgrnvtx0 40563 There are no neighbors of ...
nbgrel 40564 Characterization of a neig...
nbuhgr 40565 The set of neighbors of a ...
nbupgr 40566 The set of neighbors of a ...
nbupgrel 40567 A neighbor of a vertex in ...
nbumgrvtx 40568 The set of neighbors of a ...
nbumgr 40569 The set of neighbors of an...
nbusgrvtx 40570 The set of neighbors of a ...
nbusgr 40571 The set of neighbors of an...
nbgr2vtx1edg 40572 If a graph has two vertice...
nbuhgr2vtx1edgblem 40573 Lemma for ~ nbuhgr2vtx1edg...
nbuhgr2vtx1edgb 40574 If a hypergraph has two ve...
nbusgreledg 40575 A class/vertex is a neighb...
uhgrnbgr0nb 40576 A vertex which is not endp...
nbgr0vtxlem 40577 Lemma for ~ nbgr0vtx and ~...
nbgr0vtx 40578 In a null graph (with no v...
nbgr0edg 40579 In an empty graph (with no...
nbgr1vtx 40580 In a graph with one vertex...
nbgrisvtx 40581 Every neighbor of a class/...
nbgrssvtx 40582 The neighbors of a vertex ...
nbgrnself 40583 A vertex in a graph is not...
usgrnbnself 40584 A vertex in a simple graph...
nbgrnself2 40585 A class is not a neighbor ...
nbgrssovtx 40586 The neighbors of a vertex ...
nbgrssvwo2 40587 The neighbors of a vertex ...
usgrnbnself2 40588 In a simple graph, a class...
usgrnbssovtx 40589 The neighbors of a vertex ...
usgrnbssvwo2 40590 The neighbors of a vertex ...
nbgrsym 40591 A vertex in a graph is a n...
nbupgrres 40592 The neighborhood of a vert...
usgrnbcnvfv 40593 Applying the edge function...
nbusgredgeu 40594 For each neighbor of a ver...
edgnbusgreu 40595 For each edge incident to ...
nbusgredgeu0 40596 For each neighbor of a ver...
nbusgrf1o0 40597 The mapping of neighbors o...
nbusgrf1o1 40598 The set of neighbors of a ...
nbusgrf1o 40599 The set of neighbors of a ...
nbedgusgr 40600 The number of neighbors of...
edgusgrnbfin 40601 The number of neighbors of...
nbusgrfi 40602 The class of neighbors of ...
nbfiusgrfi 40603 The class of neighbors of ...
hashnbusgrnn0 40604 The number of neighbors of...
nbfusgrlevtxm1 40605 The number of neighbors of...
nbfusgrlevtxm2 40606 If there is a vertex which...
nbusgrvtxm1 40607 If the number of neighbors...
nb3grprlem1 40608 Lemma 1 for ~ nb3grapr . ...
nb3grprlem2 40609 Lemma 2 for ~ nb3grapr . ...
nb3grpr 40610 The neighbors of a vertex ...
nb3grpr2 40611 The neighbors of a vertex ...
nb3gr2nb 40612 If the neighbors of two ve...
uvtxaval 40613 The set of all universal v...
uvtxael 40614 A universal vertex, i.e. a...
uvtxaisvtx 40615 A universal vertex is a ve...
uvtxassvtx 40616 The set of the universal v...
vtxnbuvtx 40617 A universal vertex has all...
uvtxanbgr 40618 A universal vertex has all...
uvtxanbgrvtx 40619 A universal vertex is neig...
uvtxa0 40620 There is no universal vert...
isuvtxa 40621 The set of all universal v...
uvtxael1 40622 A universal vertex, i.e. a...
uvtxa01vtx0 40623 If a graph/class has no ed...
uvtxa01vtx 40624 If a graph/class has no ed...
uvtx2vtx1edg 40625 If a graph has two vertice...
uvtx2vtx1edgb 40626 If a hypergraph has two ve...
uvtxnbgr 40627 A universal vertex has all...
uvtxnbgrb 40628 A vertex is universal iff ...
uvtxusgr 40629 The set of all universal v...
uvtxusgrel 40630 A universal vertex, i.e. a...
uvtxanm1nbgr 40631 A universal vertex has ` n...
nbusgrvtxm1uvtx 40632 If the number of neighbors...
uvtxnbvtxm1 40633 A universal vertex has ` n...
nbupgruvtxres 40634 The neighborhood of a univ...
uvtxupgrres 40635 A universal vertex is univ...
iscplgr 40636 The property of being a co...
cplgruvtxb 40637 An graph is complete iff e...
iscplgrnb 40638 A graph is complete iff al...
iscplgredg 40639 A graph is complete iff al...
iscusgr 40640 The property of being a co...
cusgrusgr 40641 A complete simple graph is...
cusgrcplgr 40642 A complete simple graph is...
iscusgrvtx 40643 A simple graph is complete...
cusgruvtxb 40644 A simple graph is complete...
iscusgredg 40645 A simple graph is complete...
cusgredg 40646 In a complete simple graph...
cplgr0 40647 The null graph (with no ve...
cusgr0 40648 The null graph (with no ve...
cplgr0v 40649 A graph with no vertices (...
cusgr0v 40650 A graph with no vertices (...
cplgr1vlem 40651 Lemma for ~ cplgr1v and ~ ...
cplgr1v 40652 A graph with one vertex is...
cusgr1v 40653 A graph with one vertex an...
cplgr2v 40654 An undirected hypergraph w...
cplgr2vpr 40655 An undirected hypergraph w...
nbcplgr 40656 In a complete graph, each ...
cplgr3v 40657 A pseudograph with three (...
cusgr3vnbpr 40658 The neighbors of a vertex ...
cplgrop 40659 A complete graph represent...
cusgrop 40660 A complete simple graph re...
usgrexi 40661 An arbitrary set regarded ...
cusgrexi 40662 An arbitrary set regarded ...
cusgrexg 40663 For each set there is a se...
cusgrres 40664 Restricting a complete sim...
cusgrsizeindb0 40665 Base case of the induction...
cusgrsizeindb1 40666 Base case of the induction...
cusgrsizeindslem 40667 Lemma for ~ cusgrsizeinds ...
cusgrsizeinds 40668 Part 1 of induction step i...
cusgrsize2inds 40669 Induction step in ~ cusgra...
cusgrsize 40670 The size of a finite compl...
cusgrfilem1 40671 Lemma 1 for ~ cusgrfi . (...
cusgrfilem2 40672 Lemma 2 for ~ cusgrfi . (...
cusgrfilem3 40673 Lemma 3 for ~ cusgrfi . (...
cusgrfi 40674 If the size of a complete ...
usgredgsscusgredg 40675 A simple graph is a subgra...
usgrsscusgr 40676 A simple graph is a subgra...
sizusglecusglem1 40677 Lemma 1 for ~ sizusglecusg...
sizusglecusglem2 40678 Lemma 2 for ~ sizusglecusg...
sizusglecusg 40679 The size of a simple graph...
fusgrmaxsize 40680 The maximum size of a fini...
vtxdgfval 40683 The value of the vertex de...
vtxdgval 40684 The degree of a vertex. (...
vtxdgfival 40685 The degree of a vertex for...
vtxdgf 40686 The vertex degree function...
vtxdgelxnn0 40687 The degree of a vertex is ...
vtxdg0v 40688 The degree of a vertex in ...
vtxdg0e 40689 The degree of a vertex in ...
vtxdgfisnn0 40690 The degree of a vertex in ...
vtxdgfisf 40691 The vertex degree function...
vtxdeqd 40692 Equality theorem for the v...
vtxduhgr0e 40693 The degree of a vertex in ...
vtxdlfuhgr1v 40694 The degree of the vertex i...
vdumgr0 40695 A vertex in a multigraph h...
vtxdun 40696 The degree of a vertex in ...
vtxdfiun 40697 The degree of a vertex in ...
vtxduhgrun 40698 The degree of a vertex in ...
vtxduhgrfiun 40699 The degree of a vertex in ...
vtxdlfgrval 40700 The value of the vertex de...
vtxdumgrval 40701 The value of the vertex de...
vtxdusgrval 40702 The value of the vertex de...
vtxd0nedgb 40703 A vertex has degree 0 iff ...
vtxdushgrfvedglem 40704 Lemma for ~ vtxdushgrfvedg...
vtxdushgrfvedg 40705 The value of the vertex de...
vtxdusgrfvedg 40706 The value of the vertex de...
vtxduhgr0nedg 40707 If a vertex in a hypergrap...
vtxdumgr0nedg 40708 If a vertex in a multigrap...
vtxduhgr0edgnel 40709 A vertex in a hypergraph h...
vtxdusgr0edgnel 40710 A vertex in a simple graph...
vtxdusgr0edgnelALT 40711 Alternate proof of ~ vtxdu...
vtxdgfusgrf 40712 The vertex degree function...
vtxdgfusgr 40713 In a finite simple graph, ...
fusgrn0degnn0 40714 In a nonempty, finite grap...
1loopgruspgr 40715 A graph with one edge whic...
1loopgredg 40716 The set of edges in a grap...
1loopgrnb0 40717 In a graph (simple pseudog...
1loopgrvd2 40718 The vertex degree of a one...
1loopgrvd0 40719 The vertex degree of a one...
1hevtxdg0 40720 The vertex degree of verte...
1hevtxdg1 40721 The vertex degree of verte...
1hegrlfgr 40722 --- TODO-AV: not used anym...
1hegrvtxdg1 40723 The vertex degree of a gra...
1hegrvtxdg1r 40724 The vertex degree of a gra...
1egrvtxdg1 40725 The vertex degree of a one...
1egrvtxdg1r 40726 The vertex degree of a one...
1egrvtxdg0 40727 The vertex degree of a one...
p1evtxdeqlem 40728 Lemma for ~ p1evtxdeq and ...
p1evtxdeq 40729 If an edge ` E ` which doe...
p1evtxdp1 40730 If an edge ` E ` (not bein...
uspgrloopvtx 40731 The set of vertices in a g...
uspgrloopvtxel 40732 A vertex in a graph (simpl...
uspgrloopiedg 40733 The set of edges in a grap...
uspgrloopedg 40734 The set of edges in a grap...
uspgrloopnb0 40735 In a graph (simple pseudog...
uspgrloopvd2 40736 The vertex degree of a one...
umgr2v2evtx 40737 The set of vertices in a m...
umgr2v2evtxel 40738 A vertex in a multigraph w...
umgr2v2eiedg 40739 The edge function in a mul...
umgr2v2eedg 40740 The set of edges in a mult...
umgr2v2e 40741 A multigraph with two edge...
umgr2v2enb1 40742 In a multigraph with two e...
umgr2v2evd2 40743 In a multigraph with two e...
hashnbusgrvd 40744 In a simple graph, the num...
usgruvtxvdb 40745 In a finite simple graph w...
vdiscusgrb 40746 A finite simple graph with...
vdiscusgr 40747 In a finite complete simpl...
vtxdusgradjvtx 40748 The degree of a vertex in ...
usgrvd0nedg 40749 If a vertex in a simple gr...
uhgrvd00 40750 If every vertex in a hyper...
usgrvd00 40751 If every vertex in a simpl...
vdegp1ai-av 40752 The induction step for a v...
vdegp1bi-av 40753 The induction step for a v...
vdegp1ci-av 40754 The induction step for a v...
isrgr 40759 The property of a class be...
rgrprop 40760 The properties of a k-regu...
isrusgr 40761 The property of being a k-...
rusgrprop 40762 The properties of a k-regu...
rusgrrgr 40763 A k-regular simple graph i...
rusgrusgr 40764 A k-regular simple graph i...
finrusgrfusgr 40765 A finite regular simple gr...
isrusgr0 40766 The property of being a k-...
rusgrprop0 40767 The properties of a k-regu...
usgreqdrusgr 40768 If all vertices in a simpl...
fusgrregdegfi 40769 In a nonempty finite simpl...
fusgrn0eqdrusgr 40770 If all vertices in a nonem...
frusgrnn0 40771 In a nonempty finite k-reg...
0edg0rgr 40772 A graph is 0-regular if it...
uhgr0edg0rgr 40773 A hypergraph is 0-regular ...
uhgr0edg0rgrb 40774 A hypergraph is 0-regular ...
usgr0edg0rusgr 40775 A simple graph is 0-regula...
0vtxrgr 40776 A graph with no vertices i...
0vtxrusgr 40777 A graph with no vertices a...
0uhgrrusgr 40778 The null graph as hypergra...
0grrusgr 40779 The null graph represented...
0grrgr 40780 The null graph represented...
cusgrrusgr 40781 A complete simple graph wi...
cusgrm1rusgr 40782 A finite simple graph with...
rusgrpropnb 40783 The properties of a k-regu...
rusgrpropedg 40784 The properties of a k-regu...
rusgrpropadjvtx 40785 The properties of a k-regu...
rusgrnumwrdl2 40786 In a k-regular simple grap...
rusgr1vtxlem 40787 Lemma for ~ rusgr1vtx . (...
rusgr1vtx 40788 If a k-regular simple grap...
rgrusgrprc 40789 The class of 0-regular sim...
rusgrprc 40790 The class of 0-regular sim...
rgrprc 40791 The class of 0-regular gra...
rgrprcx 40792 The class of 0-regular gra...
rgrx0ndm 40793 0 is not in the domain of ...
rgrx0nd 40794 The potentially alternativ...
ewlksfval 40803 The set of s-walks of edge...
isewlk 40804 Conditions for a function ...
ewlkprop 40805 Properties of an s-walk of...
ewlkinedg 40806 The intersection (common v...
ewlkle 40807 An s-walk of edges is also...
upgrewlkle2 40808 In a pseudograph, there is...
1wlkslem1 40809 Lemma 1 for 1-walks to sub...
1wlkslem2 40810 Lemma 2 for 1-walks to sub...
1wlksfval 40811 The set of 1-walks (in an ...
wlksfval 40812 The set of walks (in an un...
is1wlk 40813 Properties of a pair of fu...
isWlk 40814 Properties of a pair of fu...
wlkv 40815 The classes involved in a ...
is1wlkg 40816 Generalisation of ~ is1wlk...
wlkbProp 40817 Basic properties of a walk...
2m1wlk 40818 The two mappings determini...
1wlkf 40819 The mapping enumerating th...
1wlkcl 40820 A 1-walk has length ` # ( ...
1wlkp 40821 The mapping enumerating th...
1wlkpwrd 40822 The sequence of vertices o...
1wlklenvp1 40823 The number of vertices of ...
1wlksv 40824 The class of 1-walks is a ...
1wlkn0 40825 The sequence of vertices o...
1wlklenvm1 40826 The number of edges of a w...
1wlkvtxeledglem 40827 Lemma for ~ 1wlkvtxeledg :...
1wlkvtxeledg 40828 Each pair of adjacent vert...
1wlkvtxiedg 40829 The vertices of a walk are...
rel1wlk 40830 The set ` ( 1Walks `` G ) ...
1wlkvv 40831 If there is at least one w...
1wlkop 40832 A walk is an ordered pair....
1wlkcpr 40833 A walk as class with two c...
1wlk2f 40834 If there is a 1-walk ` W `...
1wlkcomp 40835 A walk expressed by proper...
1wlkcompim 40836 Implications for the prope...
1wlkelwrd 40837 The components of a walk a...
1wlkeq 40838 Conditions for two walks (...
edginwlk 40839 The value of the edge func...
upgredginwlk 40840 The value of the edge func...
iedginwlk 40841 The value of the edge func...
1wlkl1loop 40842 A 1-walk of length 1 from ...
1wlk1walk 40843 A 1-walk is a 1-walk "on t...
1wlk1ewlk 40844 A 1-walk is an s-walk "on ...
ifpprsnss 40845 An unordered pair is a sin...
wlk1wlk 40846 A walk is a 1-walk. (Cont...
upgr1wlkwlk 40847 In a pseudograph, a 1-walk...
upgr1wlkwlkb 40848 In a pseudograph, the defi...
upgriswlk 40849 Properties of a pair of fu...
upgrwlkedg 40850 The edges of a walk in a p...
upgr1wlkcompim 40851 Implications for the prope...
1wlkvtxedg 40852 The vertices of a walk are...
upgr1wlkvtxedg 40853 The pairs of connected ver...
uspgr2wlkeq 40854 Conditions for two walks w...
uspgr2wlkeq2 40855 Conditions for two walks w...
uspgr2wlkeqi 40856 Conditions for two walks w...
umgr1wlknloop 40857 In a multigraph, each walk...
wlkRes 40858 Restrictions of walks (i.e...
1wlkv0 40859 If there is a walk in the ...
g01wlk0 40860 There is no walk in a null...
01wlk0 40861 There is no walk for the e...
1wlk0prc 40862 There is no walk in a null...
1wlklenvclwlk 40863 The number of vertices in ...
wlkson 40864 The set of walks between t...
iswlkOn 40865 Properties of a pair of fu...
wlkOnprop 40866 Properties of a walk betwe...
1wlkpvtx 40867 A 1-walk connects vertices...
1wlkepvtx 40868 The endpoints of a walk ar...
wlkOniswlk 40869 A walk between two vertice...
wlkOnwlk 40870 A walk is a walk between i...
wlkOnwlk1l 40871 A walk is a walk from its ...
wlksoneq1eq2 40872 Two walks with identical s...
wlkOnl1iedg 40873 If there is a walk between...
wlkOn2n0 40874 The length of a walk betwe...
2Wlklem 40875 Lemma for ~ upgr2wlk and ~...
upgr2wlk 40876 Properties of a pair of fu...
1wlkreslem0 40877 Lemma for ~ 1wlkres . TOD...
1wlkreslem 40878 Lemma for ~ 1wlkres . (Co...
1wlkres 40879 The restriction ` <. H , Q...
red1wlklem 40880 Lemma for ~ red1wlk . (Co...
red1wlk 40881 A 1-walk ending at the las...
1wlkp1lem1 40882 Lemma for ~ 1wlkp1 . (Con...
1wlkp1lem2 40883 Lemma for ~ 1wlkp1 . (Con...
1wlkp1lem3 40884 Lemma for ~ 1wlkp1 . (Con...
1wlkp1lem4 40885 Lemma for ~ 1wlkp1 . (Con...
1wlkp1lem5 40886 Lemma for ~ 1wlkp1 . (Con...
1wlkp1lem6 40887 Lemma for ~ 1wlkp1 . (Con...
1wlkp1lem7 40888 Lemma for ~ 1wlkp1 . (Con...
1wlkp1lem8 40889 Lemma for ~ 1wlkp1 . (Con...
1wlkp1 40890 Append one path segment (e...
1wlkdlem1 40891 Lemma 1 for ~ 1wlkd . (Co...
1wlkdlem2 40892 Lemma 2 for ~ 1wlkd . (Co...
1wlkdlem3 40893 Lemma 3 for ~ 1wlkd . (Co...
1wlkdlem4 40894 Lemma 4 for ~ 1wlkd . (Co...
1wlkd 40895 Two words representing a w...
lfgrwlkprop 40896 Two adjacent vertices in a...
lfgriswlk 40897 Conditions for a pair of f...
lfgr1wlknloop 40898 In a loop-free graph, each...
trlsfval 40903 The set of trails (in an u...
isTrl 40904 Conditions for a pair of f...
trlis1wlk 40905 A trail is a walk. (Contr...
trlf1 40906 The enumeration ` F ` of a...
trlreslem 40907 Lemma for ~ trlres . Form...
trlres 40908 The restriction ` <. H , Q...
upgrtrls 40909 The set of trails in a pse...
upgristrl 40910 Properties of a pair of fu...
upgrf1istrl 40911 Properties of a pair of a ...
1wlksonproplem 40912 Lemma for theorems for pro...
trlsonfval 40913 The set of trails between ...
istrlson 40914 Properties of a pair of fu...
trlsonprop 40915 Properties of a trail betw...
trlsonistrl 40916 A trail between two vertic...
trlsonwlkon 40917 A trail between two vertic...
trlOntrl 40918 A trail is a trail between...
pthsfval 40927 The set of paths (in an un...
spthsfval 40928 The set of simple paths (i...
isPth 40929 Conditions for a pair of f...
issPth 40930 Conditions for a pair of f...
PthisTrl 40931 A path is a trail (in an u...
sPthisPth 40932 A simple path is a path (i...
pthis1wlk 40933 A path is a 1-walk (in an ...
sPthis1wlk 40934 A simple path is a 1-walk ...
pthdivtx 40935 The inner vertices of a pa...
pthdadjvtx 40936 The adjacent vertices of a...
2pthnloop 40937 A path of length at least ...
upgr2pthnlp 40938 A path of length at least ...
sPthdifv 40939 The vertices of a simple p...
spthdep 40940 A simple path (at least of...
pthdepissPth 40941 A path with different star...
upgrwlkdvdelem 40942 Lemma for ~ upgrwlkdvde . ...
upgrwlkdvde 40943 In a pseudograph, all edge...
upgrspths1wlk 40944 The set of simple paths in...
upgrwlkdvspth 40945 A walk consisting of diffe...
pthsonfval 40946 The set of paths between t...
spthson 40947 The set of simple paths be...
ispthson 40948 Properties of a pair of fu...
isspthson 40949 Properties of a pair of fu...
pthsonprop 40950 Properties of a path betwe...
spthonprop 40951 Properties of a simple pat...
pthonispth-av 40952 A path between two vertice...
pthontrlon 40953 A path between two vertice...
pthOnpth 40954 A path is a path between i...
isspthonpth-av 40955 A pair of functions is a s...
spthonisspth-av 40956 A simple path between to v...
spthonpthon 40957 A simple path between two ...
spthonepeq-av 40958 The endpoints of a simple ...
uhgr1wlkspthlem1 40959 Lemma 1 for ~ uhgr1wlkspth...
uhgr1wlkspthlem2 40960 Lemma 2 for ~ uhgr1wlkspth...
uhgr1wlkspth 40961 Any walk of length 1 betwe...
usgr2wlkneq 40962 The vertices and edges are...
usgr2wlkspthlem1 40963 Lemma 1 for ~ usgr2wlkspth...
usgr2wlkspthlem2 40964 Lemma 2 for ~ usgr2wlkspth...
usgr2wlkspth 40965 In a simple graph, any wal...
usgr2trlncl 40966 In a simple graph, any tra...
usgr2trlspth 40967 In a simple graph, any tra...
usgr2pthspth 40968 In a simple graph, any pat...
usgr2pthlem 40969 Lemma for ~ usgr2pth . (C...
usgr2pth 40970 In a simple graph, there i...
usgr2pth0 40971 In a simply graph, there i...
pthdlem1 40972 Lemma 1 for ~ pthd . (Con...
pthdlem2lem 40973 Lemma for ~ pthdlem2 . (C...
pthdlem2 40974 Lemma 2 for ~ pthd . (Con...
pthd 40975 Two words representing a t...
clwlkS 40978 The set of closed walks (i...
isclWlk 40979 Properties of a pair of fu...
isclWlkb 40980 Generalisation of ~ isclwl...
clwlkis1wlk 40981 A closed walk is a walk (i...
clwlk1wlk 40982 Closed walks are walks (in...
clwlks1wlks 40983 Closed walks are walks (in...
isclWlke 40984 Properties of a pair of fu...
isclWlkupgr 40985 Properties of a pair of fu...
clWlkcomp 40986 A closed walk expressed by...
clWlkcompim 40987 Implications for the prope...
upgrclwlkcompim 40988 Implications for the prope...
clwlkl1loop 40989 A closed walk of length 1 ...
crctS 40994 The set of circuits (in an...
cyclS 40995 The set of cycles (in an u...
isCrct 40996 Sufficient and necessary c...
isCycl 40997 Sufficient and necessary c...
crctprop 40998 The properties of a circui...
cyclprop 40999 The properties of a cycle:...
crctisclwlk 41000 A circuit is a closed walk...
crctisTrl 41001 A circuit is a trail. (Co...
crctis1wlk 41002 A circuit is a walk. (Con...
cyclisPth 41003 A cycle is a path. (Contr...
cyclisWlk 41004 A cycle is a walk. (Contr...
cyclisCrct 41005 A cycle is a circuit. (Co...
cyclnsPth 41006 A (non trivial) cycle is n...
cyclisPthon 41007 A cycle is a path starting...
lfgrn1cycl 41008 In a loop-free graph there...
usgr2trlncrct 41009 In a simple graph, any tra...
umgrn1cycl 41010 In a multigraph graph (wit...
uspgrn2crct 41011 In a simple pseudograph th...
usgrn2cycl 41012 In a simple graph there ar...
crctcsh1wlkn0lem1 41013 Lemma for ~ crctcsh1wlkn0 ...
crctcsh1wlkn0lem2 41014 Lemma for ~ crctcsh1wlkn0 ...
crctcsh1wlkn0lem3 41015 Lemma for ~ crctcsh1wlkn0 ...
crctcsh1wlkn0lem4 41016 Lemma for ~ crctcsh1wlkn0 ...
crctcsh1wlkn0lem5 41017 Lemma for ~ crctcsh1wlkn0 ...
crctcsh1wlkn0lem6 41018 Lemma for ~ crctcsh1wlkn0 ...
crctcsh1wlkn0lem7 41019 Lemma for ~ crctcsh1wlkn0 ...
crctcshlem1 41020 Lemma for ~ crctcsh . (Co...
crctcshlem2 41021 Lemma for ~ crctcsh . (Co...
crctcshlem3 41022 Lemma for ~ crctcsh . (Co...
crctcshlem4 41023 Lemma for ~ crctcsh . (Co...
crctcsh1wlkn0 41024 Cyclically shifting the in...
crctcsh1wlk 41025 Cyclically shifting the in...
crctcshtrl 41026 Cyclically shifting the in...
crctcsh 41027 Cyclically shifting the in...
wwlks 41038 The set of walks (in an un...
iswwlks 41039 A word over the set of ver...
wwlksn 41040 The set of walks (in an un...
iswwlksn 41041 A word over the set of ver...
iswwlksnx 41042 Properties of a word to re...
wwlkbp 41043 Basic properties of a walk...
wwlknbp 41044 Basic properties of a walk...
wwlknp 41045 Properties of a set being ...
wspthsn 41046 The set of simple paths of...
iswspthn 41047 An element of the set of s...
wspthnp 41048 Properties of a set being ...
wwlksnon 41049 The set of walks of a fixe...
wspthsnon 41050 The set of simple paths of...
iswwlksnon 41051 The set of walks of a fixe...
iswspthsnon 41052 The set of simple paths of...
wwlknon 41053 An element of the set of w...
wspthnon 41054 An element of the set of s...
wspthnonp 41055 Properties of a set being ...
wspthneq1eq2 41056 Two simple paths with iden...
wwlksn0s 41057 The set of all walks as wo...
wwlkssswrd 41058 Walks (represented by word...
wwlksn0 41059 A walk of length 0 is repr...
0enwwlksnge1 41060 In graphs without edges, t...
wwlkswwlksn 41061 A walk of a fixed length a...
wwlkssswwlksn 41062 The walks of a fixed lengt...
wwlknbp2 41063 Other basic properties of ...
1wlkiswwlks1 41064 The sequence of vertices i...
1wlklnwwlkln1 41065 The sequence of vertices i...
1wlkiswwlks2lem1 41066 Lemma 1 for ~ 1wlkiswwlks2...
1wlkiswwlks2lem2 41067 Lemma 2 for ~ 1wlkiswwlks2...
1wlkiswwlks2lem3 41068 Lemma 3 for ~ 1wlkiswwlks2...
1wlkiswwlks2lem4 41069 Lemma 4 for ~ 1wlkiswwlks2...
1wlkiswwlks2lem5 41070 Lemma 5 for ~ 1wlkiswwlks2...
1wlkiswwlks2lem6 41071 Lemma 6 for ~ 1wlkiswwlks2...
1wlkiswwlks2 41072 A walk as word corresponds...
1wlkiswwlks 41073 A walk as word corresponds...
1wlkiswwlksupgr2 41074 A walk as word corresponds...
1wlkiswwlkupgr 41075 A walk as word corresponds...
1wlkpwwlkf1ouspgr 41076 The mapping of (ordinary) ...
1wlkisowwlkupgr 41077 The set of walks as words ...
wwlksm1edg 41078 Removing the trailing edge...
1wlklnwwlkln2lem 41079 Lemma for ~ 1wlklnwwlkln2 ...
1wlklnwwlkln2 41080 A walk of length ` N ` as ...
1wlklnwwlkn 41081 A walk of length ` N ` as ...
1wlklnwwlklnupgr2 41082 A walk of length ` N ` as ...
1wlklnwwlknupgr 41083 A walk of length ` N ` as ...
wlknewwlksn 41084 If a walk in a pseudograph...
wlknwwlksnfun 41085 Lemma 1 for ~ wlknwwlksnbi...
wlknwwlksninj 41086 Lemma 2 for ~ wlknwwlksnbi...
wlknwwlksnsur 41087 Lemma 3 for ~ wlknwwlksnbi...
wlknwwlksnbij 41088 Lemma 4 for ~ wlknwwlksnbi...
wlknwwlksnbij2 41089 There is a bijection betwe...
wlknwwlksnen 41090 In a simple pseudograph, t...
wlknwwlksneqs 41091 The set of walks of a fixe...
wlkwwlkfun 41092 Lemma 1 for ~ wlkwwlkbij2 ...
wlkwwlkinj 41093 Lemma 2 for ~ wlkwwlkbij2 ...
wlkwwlksur 41094 Lemma 3 for ~ wlkwwlkbij2 ...
wlkwwlkbij 41095 Lemma 4 for ~ wlkwwlkbij2 ...
wlkwwlkbij2 41096 There is a bijection betwe...
wwlkseq 41097 Equality of two walks (as ...
wwlksnred 41098 Reduction of a walk (as wo...
wwlksnext 41099 Extension of a walk (as wo...
wwlksnextbi 41100 Extension of a walk (as wo...
wwlksnredwwlkn 41101 For each walk (as word) of...
wwlksnredwwlkn0 41102 For each walk (as word) of...
wwlksnextwrd 41103 Lemma for ~ wwlksnextbij ....
wwlksnextfun 41104 Lemma for ~ wwlksnextbij ....
wwlksnextinj 41105 Lemma for ~ wwlksnextbij ....
wwlksnextsur 41106 Lemma for ~ wwlksnextbij ....
wwlksnextbij0 41107 Lemma for ~ wwlksnextbij ....
wwlksnextbij 41108 There is a bijection betwe...
wwlksnexthasheq 41109 The number of the extensio...
disjxwwlksn 41110 Sets of walks (as words) e...
wwlksnndef 41111 Conditions for ` WWalkSN `...
wwlksnfi 41112 The number of walks repres...
wlksnfi 41113 The number of walks of fix...
wlksnwwlknvbij 41114 There is a bijection betwe...
wwlksnextproplem1 41115 Lemma 1 for ~ wwlkextprop ...
wwlksnextproplem2 41116 Lemma 2 for ~ wwlkextprop ...
wwlksnextproplem3 41117 Lemma 3 for ~ wwlkextprop ...
wwlksnextprop 41118 Adding additional properti...
av-disjxwwlkn 41119 Sets of walks (as words) e...
hashwwlksnext 41120 Number of walks (as words)...
wwlksnwwlksnon 41121 A walk of fixed length is ...
wspthsnwspthsnon 41122 A simple path of fixed len...
wwlksnon0 41123 Conditions for a set of wa...
wspthsnonn0vne 41124 If the set of simple paths...
wspthsswwlkn 41125 The set of simple paths of...
wspthnfi 41126 In a finite graph, the set...
wwlksnonfi 41127 In a finite graph, the set...
wspthsswwlknon 41128 The set of simple paths of...
wspthnonfi 41129 In a finite graph, the set...
wspniunwspnon 41130 The set of nonempty simple...
wspn0 41131 If there are no vertices, ...
21wlkdlem1 41132 Lemma 1 for ~ 21wlkd . (C...
21wlkdlem2 41133 Lemma 2 for ~ 21wlkd . (C...
21wlkdlem3 41134 Lemma 3 for ~ 21wlkd . (C...
21wlkdlem4 41135 Lemma 4 for ~ 21wlkd . (C...
21wlkdlem5 41136 Lemma 5 for ~ 21wlkd . (C...
2pthdlem1 41137 Lemma 1 for ~ 2pthd . (Co...
21wlkdlem6 41138 Lemma 6 for ~ 21wlkd . (C...
21wlkdlem7 41139 Lemma 7 for ~ 21wlkd . (C...
21wlkdlem8 41140 Lemma 8 for ~ 21wlkd . (C...
21wlkdlem9 41141 Lemma 9 for ~ 21wlkd . (C...
21wlkdlem10 41142 Lemma 10 for ~ 31wlkd . (...
21wlkd 41143 Construction of a walk fro...
21wlkond 41144 A 1-walk of length 2 from ...
2trld 41145 Construction of a trail fr...
2trlond 41146 A trail of length 2 from o...
2pthd 41147 A path of length 2 from on...
2spthd 41148 A simple path of length 2 ...
2pthond 41149 A simple path of length 2 ...
2pthon3v-av 41150 For a vertex adjacent to t...
umgr2adedgwlklem 41151 Lemma for ~ umgr2adedgwlk ...
umgr2adedgwlk 41152 In a multigraph, two adjac...
umgr2adedgwlkon 41153 In a multigraph, two adjac...
umgr2adedgwlkonALT 41154 Alternate proof for ~ umgr...
umgr2adedgspth 41155 In a multigraph, two adjac...
umgr2wlk 41156 In a multigraph, there is ...
umgr2wlkon 41157 For each pair of adjacent ...
wwlks2onv 41158 If a length 3 string repre...
elwwlks2ons3 41159 For each walk of length 2 ...
s3wwlks2on 41160 A length 3 string which re...
umgrwwlks2on 41161 A walk of length 2 between...
elwwlks2on 41162 A walk of length 2 between...
elwspths2on 41163 A simple path of length 2 ...
wpthswwlks2on 41164 For two different vertices...
2wspdisj 41165 All simple paths of length...
2wspiundisj 41166 All simple paths of length...
usgr2wspthons3 41167 A simple path of length 2 ...
usgr2wspthon 41168 A simple path of length 2 ...
elwwlks2s3 41169 A walk of length 2 between...
elwwlks2 41170 A walk of length 2 between...
elwspths2spth 41171 A simple path of length 2 ...
rusgrnumwwlkl1 41172 In a k-regular graph, ther...
rusgrnumwwlklem 41173 Lemma for ~ rusgrnumwwlk e...
rusgrnumwwlkb0 41174 Induction base 0 for ~ rus...
rusgrnumwwlkb1 41175 Induction base 1 for ~ rus...
rusgr0edg 41176 Special case for graphs wi...
rusgrnumwwlks 41177 Induction step for ~ rusgr...
rusgrnumwwlk 41178 In a ` K `-regular graph, ...
rusgrnumwwlkg 41179 In a ` K `-regular graph, ...
rusgrnumwlkg 41180 In a k-regular graph, the ...
clwwlknclwwlkdifs 41181 The set of walks of length...
clwwlknclwwlkdifnum 41182 In a k-regular graph, the ...
clwwlks 41187 The set of closed walks (i...
isclwwlks 41188 Properties of a word to re...
clwwlksn 41189 The set of closed walks (i...
isclwwlksn 41190 A word over the set of ver...
clwwlkbp 41191 Basic properties of a clos...
clwwlknbp0 41192 Basic properties of a clos...
clwwlknbp 41193 Basic properties of a clos...
clwwlksnwrd 41194 A closed walk of a fixed l...
clwwlknp 41195 Properties of a set being ...
isclwwlksng 41196 Properties of a word to re...
isclwwlksnx 41197 Properties of a word to re...
clwwlksnndef 41198 Conditions for ` ClWWalkSN...
clwwlkclwwlkn 41199 A closed walk of a fixed l...
clwwlkssclwwlksn 41200 The closed walks of a fixe...
clwlkclwwlklem2a1 41201 Lemma 1 for ~ clwlkclwwlkl...
clwlkclwwlklem2a2 41202 Lemma 2 for ~ clwlkclwwlkl...
clwlkclwwlklem2a3 41203 Lemma 3 for ~ clwlkclwwlkl...
clwlkclwwlklem2fv1 41204 Lemma 4a for ~ clwlkclwwlk...
clwlkclwwlklem2fv2 41205 Lemma 4b for ~ clwlkclwwlk...
clwlkclwwlklem2a4 41206 Lemma 4 for ~ clwlkclwwlkl...
clwlkclwwlklem2a 41207 Lemma for ~ clwlkclwwlklem...
clwlkclwwlklem1 41208 Lemma 1 for ~ clwlkclwwlk ...
clwlkclwwlklem2 41209 Lemma 2 for ~ clwlkclwwlk ...
clwlkclwwlklem3 41210 Lemma 3 for ~ clwlkclwwlk ...
clwlkclwwlk 41211 A closed walk as word of l...
clwlkclwwlk2 41212 A closed walk corresponds ...
clwwlksgt0 41213 There is no empty closed w...
clwwlksn0 41214 There is no closed walk of...
clwwlks1loop 41215 A closed walk of length 1 ...
clwwlksn1loop 41216 A closed walk of length 1 ...
clwwlksn2 41217 A closed walk of length 2 ...
clwwlkssswrd 41218 Closed walks (represented ...
umgrclwwlksge2 41219 A closed walk in a multigr...
clwwlksnfi 41220 If there is only a finite ...
clwwlksel 41221 Obtaining a closed walk (a...
clwwlksf 41222 Lemma 1 for ~ clwwlksbij :...
clwwlksfv 41223 Lemma 2 for ~ clwwlksbij :...
clwwlksf1 41224 Lemma 3 for ~ clwwlksbij :...
clwwlksfo 41225 Lemma 4 for ~ clwwlksbij :...
clwwlksf1o 41226 Lemma 5 for ~ clwwlksbij :...
clwwlksbij 41227 There is a bijection betwe...
clwwlksnwwlkncl 41228 Obtaining a closed walk (a...
clwwlksvbij 41229 There is a bijection betwe...
clwwlksext2edg 41230 If a word concatenated wit...
wwlksext2clwwlk 41231 If a word represents a wal...
wwlksubclwwlks 41232 Any prefix of a word repre...
clwwisshclwwslemlem 41233 Lemma for ~ clwwisshclwwle...
clwwisshclwwslem 41234 Lemma for ~ clwwisshclww ....
clwwisshclwws 41235 Cyclically shifting a clos...
clwwisshclwwsn 41236 Cyclically shifting a clos...
clwwnisshclwwsn 41237 Cyclically shifting a clos...
erclwwlksrel 41238 ` .~ ` is a relation. (Co...
erclwwlkseq 41239 Two classes are equivalent...
erclwwlkseqlen 41240 If two classes are equival...
erclwwlksref 41241 ` .~ ` is a reflexive rela...
erclwwlkssym 41242 ` .~ ` is a symmetric rela...
erclwwlkstr 41243 ` .~ ` is a transitive rel...
erclwwlks 41244 ` .~ ` is an equivalence r...
eleclclwwlksnlem1 41245 Lemma 1 for ~ eleclclwwlks...
eleclclwwlksnlem2 41246 Lemma 2 for ~ eleclclwwlks...
clwwlksnscsh 41247 The set of cyclical shifts...
umgr2cwwk2dif 41248 If a word represents a clo...
umgr2cwwkdifex 41249 If a word represents a clo...
erclwwlksnrel 41250 ` .~ ` is a relation. (Co...
erclwwlksneq 41251 Two classes are equivalent...
erclwwlksneqlen 41252 If two classes are equival...
erclwwlksnref 41253 ` .~ ` is a reflexive rela...
erclwwlksnsym 41254 ` .~ ` is a symmetric rela...
erclwwlksntr 41255 ` .~ ` is a transitive rel...
erclwwlksn 41256 ` .~ ` is an equivalence r...
qerclwwlksnfi 41257 The quotient set of the se...
hashclwwlksn0 41258 The number of closed walks...
eclclwwlksn1 41259 An equivalence class accor...
eleclclwwlksn 41260 A member of an equivalence...
hashecclwwlksn1 41261 The size of every equivale...
umgrhashecclwwlk 41262 The size of every equivale...
fusgrhashclwwlkn 41263 The size of the set of clo...
clwwlksndivn 41264 The size of the set of clo...
clwlksfclwwlk2wrd 41265 The second component of a ...
clwlksfclwwlk1hashn 41266 The size of the first comp...
clwlksfclwwlk1hash 41267 The size of the first comp...
clwlksfclwwlk2sswd 41268 The size of a subword of t...
clwlksfclwwlk 41269 There is a function betwee...
clwlksfoclwwlk 41270 There is an onto function ...
clwlksf1clwwlklem0 41271 Lemma 1 for ~ clwlksf1clww...
clwlksf1clwwlklem1 41272 Lemma 1 for ~ clwlksf1clww...
clwlksf1clwwlklem2 41273 Lemma 2 for ~ clwlksf1clww...
clwlksf1clwwlklem3 41274 Lemma 3 for ~ clwlksf1clww...
clwlksf1clwwlklem 41275 Lemma for ~ clwlksf1clwwlk...
clwlksf1clwwlk 41276 There is a one-to-one func...
clwlksf1oclwwlk 41277 There is a one-to-one onto...
clwlkssizeeq 41278 The size of the set of clo...
clwlksndivn 41279 The size of the set of clo...
clwwlksndisj 41280 The sets of closed walks s...
clwwlksnun 41281 The set of closed walks of...
0ewlk 41282 The empty set (empty seque...
1ewlk 41283 A sequence of 1 edge is an...
01wlk 41284 A pair of an empty set (of...
is01wlk 41285 A pair of an empty set (of...
0wlkOnlem1 41286 Lemma 1 for ~ 0wlkOn and ~...
0wlkOnlem2 41287 Lemma 2 for ~ 0wlkOn and ~...
0wlkOn 41288 A walk of length 0 from a ...
0wlkOns1 41289 A walk of length 0 from a ...
0Trl 41290 A pair of an empty set (of...
is0Trl 41291 A pair of an empty set (of...
0TrlOn 41292 A trail of length 0 from a...
0pth-av 41293 A pair of an empty set (of...
0spth-av 41294 A pair of an empty set (of...
0pthon-av 41295 A path of length 0 from a ...
0pthon1-av 41296 A path of length 0 from a ...
0pthonv-av 41297 For each vertex there is a...
0clWlk 41298 A pair of an empty set (of...
0clwlk0 41299 There is no closed walk in...
0Crct 41300 A pair of an empty set (of...
0Cycl 41301 A pair of an empty set (of...
1pthdlem1 41302 Lemma 1 for ~ 1pthd . (Co...
1pthdlem2 41303 Lemma 2 for ~ 1pthd . (Co...
11wlkdlem1 41304 Lemma 1 for ~ 11wlkd . (C...
11wlkdlem2 41305 Lemma 2 for ~ 11wlkd . (C...
11wlkdlem3 41306 Lemma 3 for ~ 11wlkd . (C...
11wlkdlem4 41307 Lemma 4 for ~ 11wlkd . (C...
11wlkd 41308 In a graph with two vertic...
1trld 41309 In a graph with two vertic...
1pthd 41310 In a graph with two vertic...
1pthond 41311 In a graph with two vertic...
upgr11wlkdlem1 41312 Lemma 1 for ~ upgr11wlkd ....
upgr11wlkdlem2 41313 Lemma 2 for ~ upgr11wlkd ....
upgr11wlkd 41314 In a pseudograph with two ...
upgr1trld 41315 In a pseudograph with two ...
upgr1pthd 41316 In a pseudograph with two ...
upgr1pthond 41317 In a pseudograph with two ...
lppthon 41318 A loop (which is an edge a...
lp1cycl 41319 A loop (which is an edge a...
1pthon2v-av 41320 For each pair of adjacent ...
1pthon2ve 41321 For each pair of adjacent ...
1wlk2v2elem1 41322 Lemma 1 for ~ 1wlk2v2e : `...
1wlk2v2elem2 41323 Lemma 2 for ~ 1wlk2v2e : ...
1wlk2v2e 41324 In a graph with two vertic...
ntrl2v2e 41325 A walk which is not a trai...
31wlkdlem1 41326 Lemma 1 for ~ 31wlkd . (C...
31wlkdlem2 41327 Lemma 2 for ~ 31wlkd . (C...
31wlkdlem3 41328 Lemma 3 for ~ 31wlkd . (C...
31wlkdlem4 41329 Lemma 4 for ~ 31wlkd . (C...
31wlkdlem5 41330 Lemma 5 for ~ 31wlkd . (C...
3pthdlem1 41331 Lemma 1 for ~ 3pthd . (Co...
31wlkdlem6 41332 Lemma 6 for ~ 31wlkd . (C...
31wlkdlem7 41333 Lemma 7 for ~ 31wlkd . (C...
31wlkdlem8 41334 Lemma 8 for ~ 31wlkd . (C...
31wlkdlem9 41335 Lemma 9 for ~ 31wlkd . (C...
31wlkdlem10 41336 Lemma 10 for ~ 31wlkd . (...
31wlkd 41337 Construction of a walk fro...
31wlkond 41338 A 1-walk of length 3 from ...
3trld 41339 Construction of a trail fr...
3trlond 41340 A trail of length 3 from o...
3pthd 41341 A path of length 3 from on...
3pthond 41342 A path of length 3 from on...
3spthd 41343 A simple path of length 3 ...
3spthond 41344 A simple path of length 3 ...
3cycld 41345 Construction of a 3-cycle ...
3cyclpd 41346 Construction of a 3-cycle ...
upgr3v3e3cycl 41347 If there is a cycle of len...
uhgr3cyclexlem 41348 Lemma for ~ uhgr3cyclex . ...
uhgr3cyclex 41349 If there are three differe...
umgr3cyclex 41350 If there are three (differ...
umgr3v3e3cycl 41351 If and only if there is a ...
upgr4cycl4dv4e 41352 If there is a cycle of len...
dfconngr1 41355 Alternative definition of ...
isconngr 41356 The property of being a co...
isconngr1 41357 The property of being a co...
cusconngr 41358 A complete hypergraph is c...
0conngr 41359 A graph without vertices i...
0vconngr 41360 A graph without vertices i...
1conngr 41361 A graph with (at most) one...
conngrv2edg 41362 A vertex in a connected gr...
vdn0conngrumgrv2 41363 A vertex in a connected mu...
releupth 41366 The set ` ( EulerPaths `` ...
eupths 41367 The Eulerian paths on the ...
iseupth 41368 The property " ` <. F , P ...
iseupthf1o 41369 The property " ` <. F , P ...
eupthtrli 41370 Properties of an Eulerian ...
eupthi 41371 Properties of an Eulerian ...
eupthf1o 41372 The ` F ` function in an E...
eupthfi 41373 Any graph with an Eulerian...
eupthseg 41374 The ` N ` -th edge in an e...
upgriseupth 41375 The property " ` <. F , P ...
upgreupthi 41376 Properties of an Eulerian ...
upgreupthseg 41377 The ` N ` -th edge in an e...
eupthcl 41378 An Eulerian path has lengt...
eupthistrl 41379 An Eulerian path is a trai...
eupthis1wlk 41380 An Eulerian path is a walk...
eupthpf 41381 The ` P ` function in an E...
eupth0 41382 There is an Eulerian path ...
eupthres 41383 The restriction ` <. H , Q...
eupthp1 41384 Append one path segment to...
eupth2eucrct 41385 Append one path segment to...
eupth2lem1 41386 TODO-AV: Duplicate of ~ e...
eupth2lem2 41387 TODO-AV: Duplicate of ~ e...
trlsegvdeglem1 41388 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem2 41389 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem3 41390 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem4 41391 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem5 41392 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem6 41393 Lemma for ~ trlsegvdeg . ...
trlsegvdeglem7 41394 Lemma for ~ trlsegvdeg . ...
trlsegvdeg 41395 Formerly part of proof of ...
eupth2lem3lem1 41396 Lemma for ~ eupth2lem3 . ...
eupth2lem3lem2 41397 Lemma for ~ eupth2lem3 . ...
eupth2lem3lem3 41398 Lemma for ~ eupth2lem3 , f...
eupth2lem3lem4 41399 Lemma for ~ eupth2lem3 , f...
eupth2lem3lem5 41400 Lemma for ~ eupath2 . (Co...
eupth2lem3lem6 41401 Formerly part of proof of ...
eupth2lem3lem7 41402 Lemma for ~ eupath2lem3 : ...
eupthvdres 41403 Formerly part of proof of ...
eupth2lem3 41404 Lemma for ~ eupath2 . (Co...
eupth2lemb 41405 Lemma for ~ eupth2 (induct...
eupth2lems 41406 Lemma for ~ eupth2 (induct...
eupth2 41407 The only vertices of odd d...
eulerpathpr 41408 A graph with an Eulerian p...
eulerpath 41409 A pseudograph with an Eule...
eulercrct 41410 A pseudograph with an Eule...
eucrctshift 41411 Cyclically shifting the in...
eucrct2eupth1 41412 Removing one edge ` ( I ``...
eucrct2eupth 41413 Removing one edge ` ( I ``...
konigsbergvtx 41414 The set of vertices of the...
konigsbergiedg 41415 The indexed edges of the K...
konigsbergiedgw 41416 The indexed edges of the K...
konigsbergiedgwOLD 41417 The indexed edges of the K...
konigsbergssiedgwpr 41418 Each subset of the indexed...
konigsbergssiedgw 41419 Each subset of the indexed...
konigsbergumgr 41420 The Königsberg graph ...
konigsbergupgrOLD 41421 The Königsberg graph ...
konigsberglem1 41422 Lemma 1 for ~ konigsberg-a...
konigsberglem2 41423 Lemma 2 for ~ konigsberg-a...
konigsberglem3 41424 Lemma 3 for ~ konigsberg-a...
konigsberglem4 41425 Lemma 4 for ~ konigsberg-a...
konigsberglem5 41426 Lemma 5 for ~ konigsberg-a...
konigsberg-av 41427 The Königsberg Bridge...
isfrgr 41430 The property of being a fr...
frgrusgrfrcond 41431 A friendship graph is a si...
frgrusgr 41432 A friendship graph is a si...
frgr0v 41433 Any null graph (set with n...
frgr0vb 41434 Any null graph (without ve...
frgruhgr0v 41435 Any null graph (without ve...
frgr0 41436 The null graph (graph with...
rspc2vd 41437 Deduction version of 2-var...
frcond1 41438 The friendship condition: ...
frcond2 41439 The friendship condition: ...
frcond3 41440 The friendship condition, ...
frgr1v 41441 Any graph with (at most) o...
nfrgr2v 41442 Any graph with two (differ...
frgr3vlem1 41443 Lemma 1 for ~ frgra3v . (...
frgr3vlem2 41444 Lemma 2 for ~ frgra3v . (...
frgr3v 41445 Any graph with three verti...
1vwmgr 41446 Every graph with one verte...
3vfriswmgrlem 41447 Lemma for ~ 3vfriswmgra . ...
3vfriswmgr 41448 Every friendship graph wit...
1to2vfriswmgr 41449 Every friendship graph wit...
1to3vfriswmgr 41450 Every friendship graph wit...
1to3vfriendship-av 41451 The friendship theorem for...
2pthfrgrrn 41452 Between any two (different...
2pthfrgrrn2 41453 Between any two (different...
2pthfrgr 41454 Between any two (different...
3cyclfrgrrn1 41455 Every vertex in a friendsh...
3cyclfrgrrn 41456 Every vertex in a friendsh...
3cyclfrgrrn2 41457 Every vertex in a friendsh...
3cyclfrgr 41458 Every vertex in a friendsh...
4cycl2v2nb-av 41459 In a (maybe degenerated) 4...
4cycl2vnunb-av 41460 In a 4-cycle, two distinct...
n4cyclfrgr 41461 There is no 4-cycle in a f...
4cyclusnfrgr 41462 A graph with a 4-cycle is ...
frgrnbnb 41463 If two neighbors ` U ` and...
frgrconngr 41464 A friendship graph is conn...
vdgn0frgrv2 41465 A vertex in a friendship g...
vdgn1frgrv2 41466 Any vertex in a friendship...
vdgn1frgrv3 41467 Any vertex in a friendship...
vdgfrgrgt2 41468 Any vertex in a friendship...
frgrncvvdeqlem1 41469 Lemma 1 for ~ frgrncvvdeq ...
frgrncvvdeqlem2 41470 Lemma 2 for ~ frgrncvvdeq ...
frgrncvvdeqlem3 41471 Lemma 3 for ~ frgrncvvdeq ...
frgrncvvdeqlem4 41472 Lemma 4 for ~ frgrncvvdeq ...
frgrncvvdeqlem5 41473 Lemma 5 for ~ frgrncvvdeq ...
frgrncvvdeqlem6 41474 Lemma 6 for ~ frgrncvvdeq ...
frgrncvvdeqlem7 41475 Lemma 7 for ~ frgrncvvdeq ...
frgrncvvdeqlemA 41476 Lemma A for ~ frgrncvvdeq ...
frgrncvvdeqlemB 41477 Lemma B for ~ frgrncvvdeq ...
frgrncvvdeqlemC 41478 Lemma C for ~ frgrncvvdeq ...
frgrncvvdeqlem8 41479 Lemma 8 for ~ frgrncvvdeq ...
frgrncvvdeq 41480 In a friendship graph, two...
frgrwopreglem1 41481 Lemma 1 for ~ frgrwopreg :...
frgrwopreglem2 41482 Lemma 2 for ~ frgrwopreg ....
frgrwopreglem3 41483 Lemma 3 for ~ frgrwopreg ....
frgrwopreglem4 41484 Lemma 4 for ~ frgrwopreg ....
frgrwopreglem5 41485 Lemma 5 for ~ frgrwopreg ....
frgrwopreg 41486 In a friendship graph ther...
frgrwopreg1 41487 According to statement 5 i...
frgrwopreg2 41488 According to statement 5 i...
frgrregorufr0 41489 In a friendship graph ther...
frgrregorufr 41490 If there is a vertex havin...
frgreu 41491 Any two (different) vertic...
frgr2wwlkeu 41492 For two different vertices...
frgr2wwlkn0 41493 In a friendship graph, the...
frgr2wwlk1 41494 In a friendship graph, the...
frgr2wsp1 41495 In a friendship graph, the...
frgr2wwlkeqm 41496 If there is a (simple) pat...
frgrhash2wsp 41497 The number of simple paths...
fusgr2wsp2nb 41498 The set of paths of length...
fusgreghash2wspv 41499 According to statement 7 i...
fusgreg2wsp 41500 In a finite simple graph, ...
2wspmdisj 41501 The sets of paths of lengt...
fusgreghash2wsp 41502 In a finite k-regular grap...
frrusgrord0 41503 If a nonempty finite frien...
frrusgrord 41504 If a nonempty finite frien...
frgrregorufrg 41505 If there is a vertex havin...
av-numclwlk3lem3 41506 Lemma 3 for ~ numclwwlk3 ....
av-extwwlkfablem2lem 41507 Lemma for ~ extwwlkfablem2...
av-extwwlkfablem1 41508 Lemma 1 for ~ av-extwwlkfa...
av-clwwlkextfrlem1 41509 Lemma for ~ av-numclwwlk2l...
av-extwwlkfablem2 41510 Lemma 2 for ~ av-extwwlkfa...
av-numclwwlkovf 41511 Value of operation ` F ` ,...
av-numclwwlkffin 41512 In a finite graph, the val...
av-numclwwlkffin0 41513 In a finite graph, the val...
av-numclwwlkovfel2 41514 Properties of an element o...
av-numclwwlkovf2 41515 Value of operation ` F ` f...
av-numclwwlkovf2num 41516 In a ` K `-regular graph, ...
av-numclwwlkovf2ex 41517 Extending a closed walk st...
av-numclwwlkovg 41518 Value of operation ` C ` ,...
av-numclwwlkovgel 41519 Properties of an element o...
av-extwwlkfab 41520 The set ` ( X C N ) ` of c...
av-numclwlk1lem2foa 41521 Going forth and back form ...
av-numclwlk1lem2f 41522 ` T ` is a function, mappi...
av-numclwlk1lem2fv 41523 Value of the function ` T ...
av-numclwlk1lem2f1 41524 ` T ` is a 1-1 function. ...
av-numclwlk1lem2fo 41525 ` T ` is an onto function....
av-numclwlk1lem2f1o 41526 ` T ` is a 1-1 onto functi...
av-numclwlk1lem2 41527 There is a bijection betwe...
av-numclwwlk1 41528 Statement 9 in [Huneke] p....
av-numclwwlkovq 41529 Value of operation ` Q ` ,...
av-numclwwlkqhash 41530 In a ` K `-regular graph, ...
av-numclwwlkovh 41531 Value of operation ` H ` ,...
av-numclwwlk2lem1 41532 In a friendship graph, for...
av-numclwlk2lem2f 41533 ` R ` is a function mappin...
av-numclwlk2lem2fv 41534 Value of the function R. (...
av-numclwlk2lem2f1o 41535 R is a 1-1 onto function. ...
av-numclwwlk2lem3 41536 In a friendship graph, the...
av-numclwwlk2 41537 Statement 10 in [Huneke] p...
av-numclwwlk3lem 41538 Lemma for ~ av-numclwwlk3 ...
av-numclwwlk3 41539 Statement 12 in [Huneke] p...
av-numclwwlk4 41540 The total number of closed...
av-numclwwlk5lem 41541 Lemma for ~ av-numclwwlk5 ...
av-numclwwlk5 41542 Statement 13 in [Huneke] p...
av-numclwwlk7lem 41543 Lemma for ~ av-numclwwlk7 ...
av-numclwwlk6 41544 For a prime divisor ` P ` ...
av-numclwwlk7 41545 Statement 14 in [Huneke] p...
av-numclwwlk8 41546 The size of the set of clo...
av-frgrareggt1 41547 If a finite nonempty frien...
av-frgrareg 41548 If a finite nonempty frien...
av-frgraregord013 41549 If a finite friendship gra...
av-frgraregord13 41550 If a nonempty finite frien...
av-frgraogt3nreg 41551 If a finite friendship gra...
av-friendshipgt3 41552 The friendship theorem for...
av-friendship 41553 The friendship theorem: I...
ovn0dmfun 41554 If a class' operation valu...
xpsnopab 41555 A Cartesian product with a...
xpiun 41556 A Cartesian product expres...
ovn0ssdmfun 41557 If a class' operation valu...
fnxpdmdm 41558 The domain of the domain o...
cnfldsrngbas 41559 The base set of a subring ...
cnfldsrngadd 41560 The group addition operati...
cnfldsrngmul 41561 The ring multiplication op...
plusfreseq 41562 If the empty set is not co...
mgmplusfreseq 41563 If the empty set is not co...
0mgm 41564 A set with an empty base s...
mgmpropd 41565 If two structures have the...
ismgmd 41566 Deduce a magma from its pr...
mgmhmrcl 41571 Reverse closure of a magma...
submgmrcl 41572 Reverse closure for submag...
ismgmhm 41573 Property of a magma homomo...
mgmhmf 41574 A magma homomorphism is a ...
mgmhmpropd 41575 Magma homomorphism depends...
mgmhmlin 41576 A magma homomorphism prese...
mgmhmf1o 41577 A magma homomorphism is bi...
idmgmhm 41578 The identity homomorphism ...
issubmgm 41579 Expand definition of a sub...
issubmgm2 41580 Submagmas are subsets that...
rabsubmgmd 41581 Deduction for proving that...
submgmss 41582 Submagmas are subsets of t...
submgmid 41583 Every magma is trivially a...
submgmcl 41584 Submagmas are closed under...
submgmmgm 41585 Submagmas are themselves m...
submgmbas 41586 The base set of a submagma...
subsubmgm 41587 A submagma of a submagma i...
resmgmhm 41588 Restriction of a magma hom...
resmgmhm2 41589 One direction of ~ resmgmh...
resmgmhm2b 41590 Restriction of the codomai...
mgmhmco 41591 The composition of magma h...
mgmhmima 41592 The homomorphic image of a...
mgmhmeql 41593 The equalizer of two magma...
submgmacs 41594 Submagmas are an algebraic...
ismhm0 41595 Property of a monoid homom...
mhmismgmhm 41596 Each monoid homomorphism i...
opmpt2ismgm 41597 A structure with a group a...
copissgrp 41598 A structure with a constan...
copisnmnd 41599 A structure with a constan...
0nodd 41600 0 is not an odd integer. ...
1odd 41601 1 is an odd integer. (Con...
2nodd 41602 2 is not an odd integer. ...
oddibas 41603 Lemma 1 for ~ oddinmgm : ...
oddiadd 41604 Lemma 2 for ~ oddinmgm : ...
oddinmgm 41605 The structure of all odd i...
nnsgrpmgm 41606 The structure of positive ...
nnsgrp 41607 The structure of positive ...
nnsgrpnmnd 41608 The structure of positive ...
iscllaw 41615 The predicate "is a closed...
iscomlaw 41616 The predicate "is a commut...
clcllaw 41617 Closure of a closed operat...
isasslaw 41618 The predicate "is an assoc...
asslawass 41619 Associativity of an associ...
mgmplusgiopALT 41620 Slot 2 (group operation) o...
sgrpplusgaopALT 41621 Slot 2 (group operation) o...
intopval 41628 The internal (binary) oper...
intop 41629 An internal (binary) opera...
clintopval 41630 The closed (internal binar...
assintopval 41631 The associative (closed in...
assintopmap 41632 The associative (closed in...
isclintop 41633 The predicate "is a closed...
clintop 41634 A closed (internal binary)...
assintop 41635 An associative (closed int...
isassintop 41636 The predicate "is an assoc...
clintopcllaw 41637 The closure law holds for ...
assintopcllaw 41638 The closure low holds for ...
assintopasslaw 41639 The associative low holds ...
assintopass 41640 An associative (closed int...
ismgmALT 41649 The predicate "is a magma....
iscmgmALT 41650 The predicate "is a commut...
issgrpALT 41651 The predicate "is a semigr...
iscsgrpALT 41652 The predicate "is a commut...
mgm2mgm 41653 Equivalence of the two def...
sgrp2sgrp 41654 Equivalence of the two def...
idfusubc0 41655 The identity functor for a...
idfusubc 41656 The identity functor for a...
inclfusubc 41657 The "inclusion functor" fr...
lmod0rng 41658 If the scalar ring of a mo...
nzrneg1ne0 41659 The additive inverse of th...
0ringdif 41660 A zero ring is a ring whic...
0ringbas 41661 The base set of a zero rin...
0ring1eq0 41662 In a zero ring, a ring whi...
nrhmzr 41663 There is no ring homomorph...
isrng 41666 The predicate "is a non-un...
rngabl 41667 A non-unital ring is an (a...
rngmgp 41668 A non-unital ring is a sem...
ringrng 41669 A unital ring is a (non-un...
ringssrng 41670 The unital rings are (non-...
isringrng 41671 The predicate "is a unital...
rngdir 41672 Distributive law for the m...
rngcl 41673 Closure of the multiplicat...
rnglz 41674 The zero of a nonunital ri...
rnghmrcl 41679 Reverse closure of a non-u...
rnghmfn 41680 The mapping of two non-uni...
rnghmval 41681 The set of the non-unital ...
isrnghm 41682 A function is a non-unital...
isrnghmmul 41683 A function is a non-unital...
rnghmmgmhm 41684 A non-unital ring homomorp...
rnghmval2 41685 The non-unital ring homomo...
isrngisom 41686 An isomorphism of non-unit...
rngimrcl 41687 Reverse closure for an iso...
rnghmghm 41688 A non-unital ring homomorp...
rnghmf 41689 A ring homomorphism is a f...
rnghmmul 41690 A homomorphism of non-unit...
isrnghm2d 41691 Demonstration of non-unita...
isrnghmd 41692 Demonstration of non-unita...
rnghmf1o 41693 A non-unital ring homomorp...
isrngim 41694 An isomorphism of non-unit...
rngimf1o 41695 An isomorphism of non-unit...
rngimrnghm 41696 An isomorphism of non-unit...
rnghmco 41697 The composition of non-uni...
idrnghm 41698 The identity homomorphism ...
c0mgm 41699 The constant mapping to ze...
c0mhm 41700 The constant mapping to ze...
c0ghm 41701 The constant mapping to ze...
c0rhm 41702 The constant mapping to ze...
c0rnghm 41703 The constant mapping to ze...
c0snmgmhm 41704 The constant mapping to ze...
c0snmhm 41705 The constant mapping to ze...
c0snghm 41706 The constant mapping to ze...
zrrnghm 41707 The constant mapping to ze...
rhmfn 41708 The mapping of two rings t...
rhmval 41709 The ring homomorphisms bet...
rhmisrnghm 41710 Each unital ring homomorph...
lidldomn1 41711 If a (left) ideal (which i...
lidlssbas 41712 The base set of the restri...
lidlbas 41713 A (left) ideal of a ring i...
lidlabl 41714 A (left) ideal of a ring i...
lidlmmgm 41715 The multiplicative group o...
lidlmsgrp 41716 The multiplicative group o...
lidlrng 41717 A (left) ideal of a ring i...
zlidlring 41718 The zero (left) ideal of a...
uzlidlring 41719 Only the zero (left) ideal...
lidldomnnring 41720 A (left) ideal of a domain...
0even 41721 0 is an even integer. (Co...
1neven 41722 1 is not an even integer. ...
2even 41723 2 is an even integer. (Co...
2zlidl 41724 The even integers are a (l...
2zrng 41725 The ring of integers restr...
2zrngbas 41726 The base set of R is the s...
2zrngadd 41727 The group addition operati...
2zrng0 41728 The additive identity of R...
2zrngamgm 41729 R is an (additive) magma. ...
2zrngasgrp 41730 R is an (additive) semigro...
2zrngamnd 41731 R is an (additive) monoid....
2zrngacmnd 41732 R is a commutative (additi...
2zrngagrp 41733 R is an (additive) group. ...
2zrngaabl 41734 R is an (additive) abelian...
2zrngmul 41735 The ring multiplication op...
2zrngmmgm 41736 R is a (multiplicative) ma...
2zrngmsgrp 41737 R is a (multiplicative) se...
2zrngALT 41738 The ring of integers restr...
2zrngnmlid 41739 R has no multiplicative (l...
2zrngnmrid 41740 R has no multiplicative (r...
2zrngnmlid2 41741 R has no multiplicative (l...
2zrngnring 41742 R is not a unital ring. (...
plusgndxnmulrndx 41743 The slot for the group (ad...
basendxnmulrndx 41744 The slot for the base set ...
cznrnglem 41745 Lemma for ~ cznrng : The ...
cznabel 41746 The ring constructed from ...
cznrng 41747 The ring constructed from ...
cznnring 41748 The ring constructed from ...
rngcvalALTV 41753 Value of the category of n...
rngcval 41754 Value of the category of n...
rnghmresfn 41755 The class of non-unital ri...
rnghmresel 41756 An element of the non-unit...
rngcbas 41757 Set of objects of the cate...
rngchomfval 41758 Set of arrows of the categ...
rngchom 41759 Set of arrows of the categ...
elrngchom 41760 A morphism of non-unital r...
rngchomfeqhom 41761 The functionalized Hom-set...
rngccofval 41762 Composition in the categor...
rngcco 41763 Composition in the categor...
dfrngc2 41764 Alternate definition of th...
rnghmsscmap2 41765 The non-unital ring homomo...
rnghmsscmap 41766 The non-unital ring homomo...
rnghmsubcsetclem1 41767 Lemma 1 for ~ rnghmsubcset...
rnghmsubcsetclem2 41768 Lemma 2 for ~ rnghmsubcset...
rnghmsubcsetc 41769 The non-unital ring homomo...
rngccat 41770 The category of non-unital...
rngcid 41771 The identity arrow in the ...
rngcsect 41772 A section in the category ...
rngcinv 41773 An inverse in the category...
rngciso 41774 An isomorphism in the cate...
rngcbasALTV 41775 Set of objects of the cate...
rngchomfvalALTV 41776 Set of arrows of the categ...
rngchomALTV 41777 Set of arrows of the categ...
elrngchomALTV 41778 A morphism of non-unital r...
rngccofvalALTV 41779 Composition in the categor...
rngccoALTV 41780 Composition in the categor...
rngccatidALTV 41781 Lemma for ~ rngccatALTV . ...
rngccatALTV 41782 The category of non-unital...
rngcidALTV 41783 The identity arrow in the ...
rngcsectALTV 41784 A section in the category ...
rngcinvALTV 41785 An inverse in the category...
rngcisoALTV 41786 An isomorphism in the cate...
rngchomffvalALTV 41787 The value of the functiona...
rngchomrnghmresALTV 41788 The value of the functiona...
rngcifuestrc 41789 The "inclusion functor" fr...
funcrngcsetc 41790 The "natural forgetful fun...
funcrngcsetcALT 41791 Alternate proof of ~ funcr...
zrinitorngc 41792 The zero ring is an initia...
zrtermorngc 41793 The zero ring is a termina...
zrzeroorngc 41794 The zero ring is a zero ob...
ringcvalALTV 41799 Value of the category of r...
ringcval 41800 Value of the category of u...
rhmresfn 41801 The class of unital ring h...
rhmresel 41802 An element of the unital r...
ringcbas 41803 Set of objects of the cate...
ringchomfval 41804 Set of arrows of the categ...
ringchom 41805 Set of arrows of the categ...
elringchom 41806 A morphism of unital rings...
ringchomfeqhom 41807 The functionalized Hom-set...
ringccofval 41808 Composition in the categor...
ringcco 41809 Composition in the categor...
dfringc2 41810 Alternate definition of th...
rhmsscmap2 41811 The unital ring homomorphi...
rhmsscmap 41812 The unital ring homomorphi...
rhmsubcsetclem1 41813 Lemma 1 for ~ rhmsubcsetc ...
rhmsubcsetclem2 41814 Lemma 2 for ~ rhmsubcsetc ...
rhmsubcsetc 41815 The unital ring homomorphi...
ringccat 41816 The category of unital rin...
ringcid 41817 The identity arrow in the ...
rhmsscrnghm 41818 The unital ring homomorphi...
rhmsubcrngclem1 41819 Lemma 1 for ~ rhmsubcrngc ...
rhmsubcrngclem2 41820 Lemma 2 for ~ rhmsubcrngc ...
rhmsubcrngc 41821 The unital ring homomorphi...
rngcresringcat 41822 The restriction of the cat...
ringcsect 41823 A section in the category ...
ringcinv 41824 An inverse in the category...
ringciso 41825 An isomorphism in the cate...
ringcbasbas 41826 An element of the base set...
funcringcsetc 41827 The "natural forgetful fun...
funcringcsetcALTV2lem1 41828 Lemma 1 for ~ funcringcset...
funcringcsetcALTV2lem2 41829 Lemma 2 for ~ funcringcset...
funcringcsetcALTV2lem3 41830 Lemma 3 for ~ funcringcset...
funcringcsetcALTV2lem4 41831 Lemma 4 for ~ funcringcset...
funcringcsetcALTV2lem5 41832 Lemma 5 for ~ funcringcset...
funcringcsetcALTV2lem6 41833 Lemma 6 for ~ funcringcset...
funcringcsetcALTV2lem7 41834 Lemma 7 for ~ funcringcset...
funcringcsetcALTV2lem8 41835 Lemma 8 for ~ funcringcset...
funcringcsetcALTV2lem9 41836 Lemma 9 for ~ funcringcset...
funcringcsetcALTV2 41837 The "natural forgetful fun...
ringcbasALTV 41838 Set of objects of the cate...
ringchomfvalALTV 41839 Set of arrows of the categ...
ringchomALTV 41840 Set of arrows of the categ...
elringchomALTV 41841 A morphism of rings is a f...
ringccofvalALTV 41842 Composition in the categor...
ringccoALTV 41843 Composition in the categor...
ringccatidALTV 41844 Lemma for ~ ringccatALTV ....
ringccatALTV 41845 The category of rings is a...
ringcidALTV 41846 The identity arrow in the ...
ringcsectALTV 41847 A section in the category ...
ringcinvALTV 41848 An inverse in the category...
ringcisoALTV 41849 An isomorphism in the cate...
ringcbasbasALTV 41850 An element of the base set...
funcringcsetclem1ALTV 41851 Lemma 1 for ~ funcringcset...
funcringcsetclem2ALTV 41852 Lemma 2 for ~ funcringcset...
funcringcsetclem3ALTV 41853 Lemma 3 for ~ funcringcset...
funcringcsetclem4ALTV 41854 Lemma 4 for ~ funcringcset...
funcringcsetclem5ALTV 41855 Lemma 5 for ~ funcringcset...
funcringcsetclem6ALTV 41856 Lemma 6 for ~ funcringcset...
funcringcsetclem7ALTV 41857 Lemma 7 for ~ funcringcset...
funcringcsetclem8ALTV 41858 Lemma 8 for ~ funcringcset...
funcringcsetclem9ALTV 41859 Lemma 9 for ~ funcringcset...
funcringcsetcALTV 41860 The "natural forgetful fun...
irinitoringc 41861 The ring of integers is an...
zrtermoringc 41862 The zero ring is a termina...
zrninitoringc 41863 The zero ring is not an in...
nzerooringczr 41864 There is no zero object in...
srhmsubclem1 41865 Lemma 1 for ~ srhmsubc . ...
srhmsubclem2 41866 Lemma 2 for ~ srhmsubc . ...
srhmsubclem3 41867 Lemma 3 for ~ srhmsubc . ...
srhmsubc 41868 According to ~ df-subc , t...
sringcat 41869 The restriction of the cat...
crhmsubc 41870 According to ~ df-subc , t...
cringcat 41871 The restriction of the cat...
drhmsubc 41872 According to ~ df-subc , t...
drngcat 41873 The restriction of the cat...
fldcat 41874 The restriction of the cat...
fldc 41875 The restriction of the cat...
fldhmsubc 41876 According to ~ df-subc , t...
rngcrescrhm 41877 The category of non-unital...
rhmsubclem1 41878 Lemma 1 for ~ rhmsubc . (...
rhmsubclem2 41879 Lemma 2 for ~ rhmsubc . (...
rhmsubclem3 41880 Lemma 3 for ~ rhmsubc . (...
rhmsubclem4 41881 Lemma 4 for ~ rhmsubc . (...
rhmsubc 41882 According to ~ df-subc , t...
rhmsubccat 41883 The restriction of the cat...
srhmsubcALTVlem1 41884 Lemma 1 for ~ srhmsubcALTV...
srhmsubcALTVlem2 41885 Lemma 2 for ~ srhmsubcALTV...
srhmsubcALTVlem3 41886 Lemma 3 for ~ srhmsubcALTV...
srhmsubcALTV 41887 According to ~ df-subc , t...
sringcatALTV 41888 The restriction of the cat...
crhmsubcALTV 41889 According to ~ df-subc , t...
cringcatALTV 41890 The restriction of the cat...
drhmsubcALTV 41891 According to ~ df-subc , t...
drngcatALTV 41892 The restriction of the cat...
fldcatALTV 41893 The restriction of the cat...
fldcALTV 41894 The restriction of the cat...
fldhmsubcALTV 41895 According to ~ df-subc , t...
rngcrescrhmALTV 41896 The category of non-unital...
rhmsubcALTVlem1 41897 Lemma 1 for ~ rhmsubcALTV ...
rhmsubcALTVlem2 41898 Lemma 2 for ~ rhmsubcALTV ...
rhmsubcALTVlem3 41899 Lemma 3 for ~ rhmsubcALTV ...
rhmsubcALTVlem4 41900 Lemma 4 for ~ rhmsubcALTV ...
rhmsubcALTV 41901 According to ~ df-subc , t...
rhmsubcALTVcat 41902 The restriction of the cat...
xpprsng 41903 The Cartesian product of a...
opeliun2xp 41904 Membership of an ordered p...
eliunxp2 41905 Membership in a union of C...
mpt2mptx2 41906 Express a two-argument fun...
cbvmpt2x2 41907 Rule to change the bound v...
dmmpt2ssx2 41908 The domain of a mapping is...
mpt2exxg2 41909 Existence of an operation ...
ovmpt2rdxf 41910 Value of an operation give...
ovmpt2rdx 41911 Value of an operation give...
ovmpt2x2 41912 The value of an operation ...
fdmdifeqresdif 41913 The restriction of a condi...
offvalfv 41914 The function operation exp...
ofaddmndmap 41915 The function operation app...
mapsnop 41916 A singleton of an ordered ...
mapprop 41917 An unordered pair containi...
ztprmneprm 41918 A prime is not an integer ...
2t6m3t4e0 41919 2 times 6 minus 3 times 4 ...
ssnn0ssfz 41920 For any finite subset of `...
nn0sumltlt 41921 If the sum of two nonnegat...
bcpascm1 41922 Pascal's rule for the bino...
altgsumbc 41923 The sum of binomial coeffi...
altgsumbcALT 41924 Alternate proof of ~ altgs...
zlmodzxzlmod 41925 The ` ZZ `-module ` ZZ X. ...
zlmodzxzel 41926 An element of the (base se...
zlmodzxz0 41927 The ` 0 ` of the ` ZZ `-mo...
zlmodzxzscm 41928 The scalar multiplication ...
zlmodzxzadd 41929 The addition of the ` ZZ `...
zlmodzxzsubm 41930 The subtraction of the ` Z...
zlmodzxzsub 41931 The subtraction of the ` Z...
gsumpr 41932 Group sum of a pair. (Con...
mgpsumunsn 41933 Extract a summand/factor f...
mgpsumz 41934 If the group sum for the m...
mgpsumn 41935 If the group sum for the m...
gsumsplit2f 41936 Split a group sum into two...
gsumdifsndf 41937 Extract a summand from a f...
nn0le2is012 41938 A nonnegative integer whic...
exple2lt6 41939 A nonnegative integer to t...
pgrple2abl 41940 Every symmetric group on a...
pgrpgt2nabl 41941 Every symmetric group on a...
invginvrid 41942 Identity for a multiplicat...
rmsupp0 41943 The support of a mapping o...
domnmsuppn0 41944 The support of a mapping o...
rmsuppss 41945 The support of a mapping o...
mndpsuppss 41946 The support of a mapping o...
scmsuppss 41947 The support of a mapping o...
rmsuppfi 41948 The support of a mapping o...
rmfsupp 41949 A mapping of a multiplicat...
mndpsuppfi 41950 The support of a mapping o...
mndpfsupp 41951 A mapping of a scalar mult...
scmsuppfi 41952 The support of a mapping o...
scmfsupp 41953 A mapping of a scalar mult...
suppmptcfin 41954 The support of a mapping w...
mptcfsupp 41955 A mapping with value 0 exc...
fsuppmptdmf 41956 A mapping with a finite do...
lmodvsmdi 41957 Multiple distributive law ...
gsumlsscl 41958 Closure of a group sum in ...
ascl0 41959 The scalar 0 embedded into...
ascl1 41960 The scalar 1 embedded into...
assaascl0 41961 The scalar 0 embedded into...
assaascl1 41962 The scalar 1 embedded into...
ply1vr1smo 41963 The variable in a polynomi...
ply1ass23l 41964 Associative identity with ...
ply1sclrmsm 41965 The ring multiplication of...
coe1id 41966 Coefficient vector of the ...
coe1sclmulval 41967 The value of the coefficie...
ply1mulgsumlem1 41968 Lemma 1 for ~ ply1mulgsum ...
ply1mulgsumlem2 41969 Lemma 2 for ~ ply1mulgsum ...
ply1mulgsumlem3 41970 Lemma 3 for ~ ply1mulgsum ...
ply1mulgsumlem4 41971 Lemma 4 for ~ ply1mulgsum ...
ply1mulgsum 41972 The product of two polynom...
evl1at0 41973 Polynomial evaluation for ...
evl1at1 41974 Polynomial evaluation for ...
linply1 41975 A term of the form ` x - C...
lineval 41976 A term of the form ` x - C...
zringsubgval 41977 Subtraction in the ring of...
linevalexample 41978 The polynomial ` x - 3 ` o...
dmatALTval 41983 The algebra of ` N ` x ` N...
dmatALTbas 41984 The base set of the algebr...
dmatALTbasel 41985 An element of the base set...
dmatbas 41986 The set of all ` N ` x ` N...
lincop 41991 A linear combination as op...
lincval 41992 The value of a linear comb...
dflinc2 41993 Alternative definition of ...
lcoop 41994 A linear combination as op...
lcoval 41995 The value of a linear comb...
lincfsuppcl 41996 A linear combination of ve...
linccl 41997 A linear combination of ve...
lincval0 41998 The value of an empty line...
lincvalsng 41999 The linear combination ove...
lincvalsn 42000 The linear combination ove...
lincvalpr 42001 The linear combination ove...
lincval1 42002 The linear combination ove...
lcosn0 42003 Properties of a linear com...
lincvalsc0 42004 The linear combination whe...
lcoc0 42005 Properties of a linear com...
linc0scn0 42006 If a set contains the zero...
lincdifsn 42007 A vector is a linear combi...
linc1 42008 A vector is a linear combi...
lincellss 42009 A linear combination of a ...
lco0 42010 The set of empty linear co...
lcoel0 42011 The zero vector is always ...
lincsum 42012 The sum of two linear comb...
lincscm 42013 A linear combinations mult...
lincsumcl 42014 The sum of two linear comb...
lincscmcl 42015 The multiplication of a li...
lincsumscmcl 42016 The sum of a linear combin...
lincolss 42017 According to the statement...
ellcoellss 42018 Every linear combination o...
lcoss 42019 A set of vectors of a modu...
lspsslco 42020 Lemma for ~ lspeqlco . (C...
lcosslsp 42021 Lemma for ~ lspeqlco . (C...
lspeqlco 42022 Equivalence of a _span_ of...
rellininds 42026 The class defining the rel...
linindsv 42028 The classes of the module ...
islininds 42029 The property of being a li...
linindsi 42030 The implications of being ...
linindslinci 42031 The implications of being ...
islinindfis 42032 The property of being a li...
islinindfiss 42033 The property of being a li...
linindscl 42034 A linearly independent set...
lindepsnlininds 42035 A linearly dependent subse...
islindeps 42036 The property of being a li...
lincext1 42037 Property 1 of an extension...
lincext2 42038 Property 2 of an extension...
lincext3 42039 Property 3 of an extension...
lindslinindsimp1 42040 Implication 1 for ~ lindsl...
lindslinindimp2lem1 42041 Lemma 1 for ~ lindslininds...
lindslinindimp2lem2 42042 Lemma 2 for ~ lindslininds...
lindslinindimp2lem3 42043 Lemma 3 for ~ lindslininds...
lindslinindimp2lem4 42044 Lemma 4 for ~ lindslininds...
lindslinindsimp2lem5 42045 Lemma 5 for ~ lindslininds...
lindslinindsimp2 42046 Implication 2 for ~ lindsl...
lindslininds 42047 Equivalence of definitions...
linds0 42048 The empty set is always a ...
el0ldep 42049 A set containing the zero ...
el0ldepsnzr 42050 A set containing the zero ...
lindsrng01 42051 Any subset of a module is ...
lindszr 42052 Any subset of a module ove...
snlindsntorlem 42053 Lemma for ~ snlindsntor . ...
snlindsntor 42054 A singleton is linearly in...
ldepsprlem 42055 Lemma for ~ ldepspr . (Co...
ldepspr 42056 If a vector is a scalar mu...
lincresunit3lem3 42057 Lemma 3 for ~ lincresunit3...
lincresunitlem1 42058 Lemma 1 for properties of ...
lincresunitlem2 42059 Lemma for properties of a ...
lincresunit1 42060 Property 1 of a specially ...
lincresunit2 42061 Property 2 of a specially ...
lincresunit3lem1 42062 Lemma 1 for ~ lincresunit3...
lincresunit3lem2 42063 Lemma 2 for ~ lincresunit3...
lincresunit3 42064 Property 3 of a specially ...
lincreslvec3 42065 Property 3 of a specially ...
islindeps2 42066 Conditions for being a lin...
islininds2 42067 Implication of being a lin...
isldepslvec2 42068 Alternative definition of ...
lindssnlvec 42069 A singleton not containing...
lmod1lem1 42070 Lemma 1 for ~ lmod1 . (Co...
lmod1lem2 42071 Lemma 2 for ~ lmod1 . (Co...
lmod1lem3 42072 Lemma 3 for ~ lmod1 . (Co...
lmod1lem4 42073 Lemma 4 for ~ lmod1 . (Co...
lmod1lem5 42074 Lemma 5 for ~ lmod1 . (Co...
lmod1 42075 The (smallest) structure r...
lmod1zr 42076 The (smallest) structure r...
lmod1zrnlvec 42077 There is a (left) module (...
lmodn0 42078 Left modules exist. (Cont...
zlmodzxzequa 42079 Example of an equation wit...
zlmodzxznm 42080 Example of a linearly depe...
zlmodzxzldeplem 42081 A and B are not equal. (C...
zlmodzxzequap 42082 Example of an equation wit...
zlmodzxzldeplem1 42083 Lemma 1 for ~ zlmodzxzldep...
zlmodzxzldeplem2 42084 Lemma 2 for ~ zlmodzxzldep...
zlmodzxzldeplem3 42085 Lemma 3 for ~ zlmodzxzldep...
zlmodzxzldeplem4 42086 Lemma 4 for ~ zlmodzxzldep...
zlmodzxzldep 42087 { A , B } is a linearly de...
ldepsnlinclem1 42088 Lemma 1 for ~ ldepsnlinc ....
ldepsnlinclem2 42089 Lemma 2 for ~ ldepsnlinc ....
lvecpsslmod 42090 The class of all (left) ve...
ldepsnlinc 42091 The reverse implication of...
ldepslinc 42092 For (left) vector spaces, ...
offval0 42093 Value of an operation appl...
suppdm 42094 If the range of a function...
eluz2cnn0n1 42095 An integer greater than 1 ...
divge1b 42096 The ratio of a real number...
divgt1b 42097 The ratio of a real number...
ltsubaddb 42098 Equivalence for the "less ...
ltsubsubb 42099 Equivalence for the "less ...
ltsubadd2b 42100 Equivalence for the "less ...
divsub1dir 42101 Distribution of division o...
expnegico01 42102 An integer greater than 1 ...
elfzolborelfzop1 42103 An element of a half-open ...
pw2m1lepw2m1 42104 2 to the power of a positi...
zgtp1leeq 42105 If an integer is between a...
flsubz 42106 An integer can be moved in...
fldivmod 42107 Expressing the floor of a ...
mod0mul 42108 If an integer is 0 modulo ...
modn0mul 42109 If an integer is not 0 mod...
m1modmmod 42110 An integer decreased by 1 ...
difmodm1lt 42111 The difference between an ...
nn0onn0ex 42112 For each odd nonnegative i...
nn0enn0ex 42113 For each even nonnegative ...
nneop 42114 A positive integer is even...
nneom 42115 A positive integer is even...
nn0eo 42116 A nonnegative integer is e...
nnpw2even 42117 2 to the power of a positi...
zefldiv2 42118 The floor of an even integ...
zofldiv2 42119 The floor of an odd intege...
nn0ofldiv2 42120 The floor of an odd nonneg...
flnn0div2ge 42121 The floor of a positive in...
flnn0ohalf 42122 The floor of the half of a...
logge0b 42123 The logarithm of a number ...
loggt0b 42124 The logarithm of a number ...
logle1b 42125 The logarithm of a number ...
loglt1b 42126 The logarithm of a number ...
logcxp0 42127 Logarithm of a complex pow...
regt1loggt0 42128 The natural logarithm for ...
fdivval 42131 The quotient of two functi...
fdivmpt 42132 The quotient of two functi...
fdivmptf 42133 The quotient of two functi...
refdivmptf 42134 The quotient of two functi...
fdivpm 42135 The quotient of two functi...
refdivpm 42136 The quotient of two functi...
fdivmptfv 42137 The function value of a qu...
refdivmptfv 42138 The function value of a qu...
bigoval 42141 Set of functions of order ...
elbigofrcl 42142 Reverse closure of the "bi...
elbigo 42143 Properties of a function o...
elbigo2 42144 Properties of a function o...
elbigo2r 42145 Sufficient condition for a...
elbigof 42146 A function of order G(x) i...
elbigodm 42147 The domain of a function o...
elbigoimp 42148 The defining property of a...
elbigolo1 42149 A function (into the posit...
rege1logbrege0 42150 The general logarithm, wit...
rege1logbzge0 42151 The general logarithm, wit...
fllogbd 42152 A real number is between t...
relogbmulbexp 42153 The logarithm of the produ...
relogbdivb 42154 The logarithm of the quoti...
logbge0b 42155 The logarithm of a number ...
logblt1b 42156 The logarithm of a number ...
fldivexpfllog2 42157 The floor of a positive re...
nnlog2ge0lt1 42158 A positive integer is 1 if...
logbpw2m1 42159 The floor of the binary lo...
fllog2 42160 The floor of the binary lo...
blenval 42163 The binary length of an in...
blen0 42164 The binary length of 0. (...
blenn0 42165 The binary length of a "nu...
blenre 42166 The binary length of a pos...
blennn 42167 The binary length of a pos...
blennnelnn 42168 The binary length of a pos...
blennn0elnn 42169 The binary length of a non...
blenpw2 42170 The binary length of a pow...
blenpw2m1 42171 The binary length of a pow...
nnpw2blen 42172 A positive integer is betw...
nnpw2blenfzo 42173 A positive integer is betw...
nnpw2blenfzo2 42174 A positive integer is eith...
nnpw2pmod 42175 Every positive integer can...
blen1 42176 The binary length of 1. (...
blen2 42177 The binary length of 2. (...
nnpw2p 42178 Every positive integer can...
nnpw2pb 42179 A number is a positive int...
blen1b 42180 The binary length of a non...
blennnt2 42181 The binary length of a pos...
nnolog2flm1 42182 The floor of the binary lo...
blennn0em1 42183 The binary length of the h...
blennngt2o2 42184 The binary length of an od...
blengt1fldiv2p1 42185 The binary length of an in...
blennn0e2 42186 The binary length of an ev...
digfval 42189 Operation to obtain the ` ...
digval 42190 The ` K ` th digit of a no...
digvalnn0 42191 The ` K ` th digit of a no...
nn0digval 42192 The ` K ` th digit of a no...
dignn0fr 42193 The digits of the fraction...
dignn0ldlem 42194 Lemma for ~ dignnld . (Co...
dignnld 42195 The leading digits of a po...
dig2nn0ld 42196 The leading digits of a po...
dig2nn1st 42197 The first (relevant) digit...
dig0 42198 All digits of 0 are 0. (C...
digexp 42199 The ` K ` th digit of a po...
dig1 42200 All but one digits of 1 ar...
0dig1 42201 The ` 0 ` th digit of 1 is...
0dig2pr01 42202 The integers 0 and 1 corre...
dig2nn0 42203 A digit of a nonnegative i...
0dig2nn0e 42204 The last bit of an even in...
0dig2nn0o 42205 The last bit of an odd int...
dig2bits 42206 The ` K ` th digit of a no...
dignn0flhalflem1 42207 Lemma 1 for ~ dignn0flhalf...
dignn0flhalflem2 42208 Lemma 2 for ~ dignn0flhalf...
dignn0ehalf 42209 The digits of the half of ...
dignn0flhalf 42210 The digits of the rounded ...
nn0sumshdiglemA 42211 Lemma for ~ nn0sumshdig (i...
nn0sumshdiglemB 42212 Lemma for ~ nn0sumshdig (i...
nn0sumshdiglem1 42213 Lemma 1 for ~ nn0sumshdig ...
nn0sumshdiglem2 42214 Lemma 2 for ~ nn0sumshdig ...
nn0sumshdig 42215 A nonnegative integer can ...
nn0mulfsum 42216 Trivial algorithm to calcu...
nn0mullong 42217 Standard algorithm (also k...
nfintd 42218 Bound-variable hypothesis ...
nfiund 42219 Bound-variable hypothesis ...
iunord 42220 The indexed union of a col...
iunordi 42221 The indexed union of a col...
rspcdf 42222 Restricted specialization,...
spd 42223 Specialization deduction, ...
spcdvw 42224 A version of ~ spcdv where...
tfis2d 42225 Transfinite Induction Sche...
bnd2d 42226 Deduction form of ~ bnd2 ....
dffun3f 42227 Alternate definition of fu...
ssdifsn 42228 Subset of a set with an el...
setrecseq 42231 Equality theorem for set r...
nfsetrecs 42232 Bound-variable hypothesis ...
setrec1lem1 42233 Lemma for ~ setrec1 . Thi...
setrec1lem2 42234 Lemma for ~ setrec1 . If ...
setrec1lem3 42235 Lemma for ~ setrec1 . If ...
setrec1lem4 42236 Lemma for ~ setrec1 . If ...
setrec1 42237 This is the first of two f...
setrec2fun 42238 This is the second of two ...
setrec2lem1 42239 Lemma for ~ setrec2 . The...
setrec2lem2 42240 Lemma for ~ setrec2 . The...
setrec2 42241 This is the second of two ...
setrec2v 42242 Version of ~ setrec2 with ...
elsetrecslem 42243 Lemma for ~ elsetrecs . A...
elsetrecs 42244 A set ` A ` is an element ...
vsetrec 42245 Construct ` _V ` using set...
0setrec 42246 If a function sends the em...
onsetreclem1 42247 Lemma for ~ onsetrec . (C...
onsetreclem2 42248 Lemma for ~ onsetrec . (C...
onsetreclem3 42249 Lemma for ~ onsetrec . (C...
onsetrec 42250 Construct ` On ` using set...
elpglem1 42253 Lemma for ~ elpg . (Contr...
elpglem2 42254 Lemma for ~ elpg . (Contr...
elpglem3 42255 Lemma for ~ elpg . (Contr...
elpg 42256 Membership in the class of...
19.8ad 42257 If a wff is true, it is tr...
sbidd 42258 An identity theorem for su...
sbidd-misc 42259 An identity theorem for su...
gte-lte 42264 Simple relationship betwee...
gt-lt 42265 Simple relationship betwee...
gte-lteh 42266 Relationship between ` <_ ...
gt-lth 42267 Relationship between ` < `...
ex-gt 42268 Simple example of ` > ` , ...
ex-gte 42269 Simple example of ` >_ ` ,...
sinhval-named 42276 Value of the named sinh fu...
coshval-named 42277 Value of the named cosh fu...
tanhval-named 42278 Value of the named tanh fu...
sinh-conventional 42279 Conventional definition of...
sinhpcosh 42280 Prove that ` ( sinh `` A )...
secval 42287 Value of the secant functi...
cscval 42288 Value of the cosecant func...
cotval 42289 Value of the cotangent fun...
seccl 42290 The closure of the secant ...
csccl 42291 The closure of the cosecan...
cotcl 42292 The closure of the cotange...
reseccl 42293 The closure of the secant ...
recsccl 42294 The closure of the cosecan...
recotcl 42295 The closure of the cotange...
recsec 42296 The reciprocal of secant i...
reccsc 42297 The reciprocal of cosecant...
reccot 42298 The reciprocal of cotangen...
rectan 42299 The reciprocal of tangent ...
sec0 42300 The value of the secant fu...
onetansqsecsq 42301 Prove the tangent squared ...
cotsqcscsq 42302 Prove the tangent squared ...
ifnmfalse 42303 If A is not a member of B,...
dfdp2OLD 42307 Obsolete version of ~ df-d...
dp2cl 42309 Define the closure for the...
dpval 42310 Define the value of the de...
dpcl 42311 Prove that the closure of ...
dpfrac1 42312 Prove a simple equivalence...
dpfrac1OLD 42313 Obsolete version of ~ dpfr...
logb2aval 42314 Define the value of the ` ...
comraddi 42321 Commute RHS addition. See...
mvlladdd 42322 Move LHS left addition to ...
mvlraddi 42323 Move LHS right addition to...
mvrladdd 42324 Move RHS left addition to ...
mvrladdi 42325 Move RHS left addition to ...
assraddsubd 42326 Associate RHS addition-sub...
assraddsubi 42327 Associate RHS addition-sub...
joinlmuladdmuli 42328 Join AB+CB into (A+C) on L...
joinlmulsubmuld 42329 Join AB-CB into (A-C) on L...
joinlmulsubmuli 42330 Join AB-CB into (A-C) on L...
mvlrmuld 42331 Move LHS right multiplicat...
mvlrmuli 42332 Move LHS right multiplicat...
i2linesi 42333 Solve for the intersection...
i2linesd 42334 Solve for the intersection...
alimp-surprise 42335 Demonstrate that when usin...
alimp-no-surprise 42336 There is no "surprise" in ...
empty-surprise 42337 Demonstrate that when usin...
empty-surprise2 42338 "Prove" that false is true...
eximp-surprise 42339 Show what implication insi...
eximp-surprise2 42340 Show that "there exists" w...
alsconv 42345 There is an equivalence be...
alsi1d 42346 Deduction rule: Given "al...
alsi2d 42347 Deduction rule: Given "al...
alsc1d 42348 Deduction rule: Given "al...
alsc2d 42349 Deduction rule: Given "al...
alscn0d 42350 Deduction rule: Given "al...
alsi-no-surprise 42351 Demonstrate that there is ...
5m4e1 42352 Prove that 5 - 4 = 1. (Co...
2p2ne5 42353 Prove that ` 2 + 2 =/= 5 `...
resolution 42354 Resolution rule. This is ...
testable 42355 In classical logic all wff...
aacllem 42356 Lemma for other theorems a...
amgmwlem 42357 Weighted version of ~ amgm...
amgmlemALT 42358 Alternative proof of ~ amg...
amgmw2d 42359 Weighted arithmetic-geomet...
young2d 42360 Young's inequality for ` n...
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