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Theorem List for Metamath Proof Explorer - 32601-32700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdedths2 32601 Generalization of dedths 32598 that is not useful unless we can separately prove . (Contributed by NM, 13-Jun-2019.)

Theorem19.9alt 32602 Version of 19.9t 1989 for universal quantifier. (Contributed by NM, 9-Nov-2020.)

Theoremnfcxfrdf 32603 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by NM, 19-Nov-2020.)

Theoremnfded 32604 A deduction theorem that converts a not-free inference directly to deduction form. The first hypothesis is the hypothesis of the deduction form. The second is an equality deduction (e.g. ) that starts from abidnf 3195. The last is assigned to the inference form (e.g. ) whose hypothesis is satisfied using nfaba1 2617. (Contributed by NM, 19-Nov-2020.)

Theoremnfded2 32605 A deduction theorem that converts a not-free inference directly to deduction form. The first 2 hypotheses are the hypotheses of the deduction form. The third is an equality deduction (e.g. for nfopd 4175) that starts from abidnf 3195. The last is assigned to the inference form (e.g. for nfop 4174) whose hypotheses are satisfied using nfaba1 2617. (Contributed by NM, 19-Nov-2020.)

TheoremnfunidALT2 32606 Deduction version of nfuni 4196. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremnfunidALT 32607 Deduction version of nfuni 4196. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremnfopdALT 32608 Deduction version of bound-variable hypothesis builder nfop 4174. This shows how the deduction version of a not-free theorem such as nfop 4174 can be created from the corresponding not-free inference theorem. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

21.21.5  Miscellanea

Theoremcnaddcom 32609 Recover the commutative law of addition for complex numbers from the Abelian group structure. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)

Theoremtoycom 32610* Show the commutative law for an operation on a toy structure class of commuatitive operations on . This illustrates how a structure class can be partially specialized. In practice, we would ordinarily define a new constant such as "CAbel" in place of . (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)

21.21.6  Atoms, hyperplanes, and covering in a left vector space (or module)

Syntaxclsa 32611 Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
LSAtoms

Syntaxclsh 32612 Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
LSHyp

Definitiondf-lsatoms 32613* Define the set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.)
LSAtoms

Definitiondf-lshyp 32614* Define the set of all hyperplanes of a left module or left vector space. Also called co-atoms, these are subspaces that are one dimension less that the full space. (Contributed by NM, 29-Jun-2014.)
LSHyp

Theoremlshpset 32615* The set of all hyperplanes of a left module or left vector space. The vector is called a generating vector for the hyperplane. (Contributed by NM, 29-Jun-2014.)
LSHyp

Theoremislshp 32616* The predicate "is a hyperplane" (of a left module or left vector space). (Contributed by NM, 29-Jun-2014.)
LSHyp

Theoremislshpsm 32617* Hyperplane properties expressed with subspace sum. (Contributed by NM, 3-Jul-2014.)
LSHyp

Theoremlshplss 32618 A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)
LSHyp

Theoremlshpne 32619 A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)
LSHyp

Theoremlshpnel 32620 A hyperplane's generating vector does not belong to the hyperplane. (Contributed by NM, 3-Jul-2014.)
LSHyp

Theoremlshpnelb 32621 The subspace sum of a hyperplane and the span of an element equals the vector space iff the element is not in the hyperplane. (Contributed by NM, 2-Oct-2014.)
LSHyp

Theoremlshpnel2N 32622 Condition that determines a hyperplane. (Contributed by NM, 3-Oct-2014.) (New usage is discouraged.)
LSHyp

Theoremlshpne0 32623 The member of the span in the hyperplane definition does not belong to the hyperplane. (Contributed by NM, 14-Jul-2014.)
LSHyp

Theoremlshpdisj 32624 A hyperplane and the span in the hyperplane definition are disjoint. (Contributed by NM, 3-Jul-2014.)
LSHyp

Theoremlshpcmp 32625 If two hyperplanes are comparable, they are equal. (Contributed by NM, 9-Oct-2014.)
LSHyp

TheoremlshpinN 32626 The intersection of two different hyperplanes is not a hyperplane. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
LSHyp

Theoremlsatset 32627* The set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
LSAtoms

Theoremislsat 32628* The predicate "is a 1-dim subspace (atom)" (of a left module or left vector space). (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
LSAtoms

Theoremlsatlspsn2 32629 The span of a non-zero singleton is an atom. TODO: make this obsolete and use lsatlspsn 32630 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
LSAtoms

Theoremlsatlspsn 32630 The span of a non-zero singleton is an atom. (Contributed by NM, 16-Jan-2015.)
LSAtoms

Theoremislsati 32631* A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014.)
LSAtoms

Theoremlsateln0 32632* A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.)
LSAtoms

Theoremlsatlss 32633 The set of 1-dim subspaces is a set of subspaces. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
LSAtoms

Theoremlsatlssel 32634 An atom is a subspace. (Contributed by NM, 25-Aug-2014.)
LSAtoms

Theoremlsatssv 32635 An atom is a set of vectors. (Contributed by NM, 27-Feb-2015.)
LSAtoms

Theoremlsatn0 32636 A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 28079 analog.) (Contributed by NM, 25-Aug-2014.)
LSAtoms

Theoremlsatspn0 32637 The span of a vector is an atom iff the vector is nonzero. (Contributed by NM, 4-Feb-2015.)
LSAtoms

Theoremlsator0sp 32638 The span of a vector is either an atom or the zero subspace. (Contributed by NM, 15-Mar-2015.)
LSAtoms

Theoremlsatssn0 32639 A subspace (or any class) including an atom is nonzero. (Contributed by NM, 3-Feb-2015.)
LSAtoms

Theoremlsatcmp 32640 If two atoms are comparable, they are equal. (atsseq 28081 analog.) TODO: can lspsncmp 18417 shorten this? (Contributed by NM, 25-Aug-2014.)
LSAtoms

Theoremlsatcmp2 32641 If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 32640. TODO: can lspsncmp 18417 shorten this? (Contributed by NM, 3-Feb-2015.)
LSAtoms

Theoremlsatel 32642 A nonzero vector in an atom determines the atom. (Contributed by NM, 25-Aug-2014.)
LSAtoms

TheoremlsatelbN 32643 A nonzero vector in an atom determines the atom. (Contributed by NM, 3-Feb-2015.) (New usage is discouraged.)
LSAtoms

Theoremlsat2el 32644 Two atoms sharing a nonzero vector are equal. (Contributed by NM, 8-Mar-2015.)
LSAtoms

Theoremlsmsat 32645* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 33441 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.)
LSAtoms

TheoremlsatfixedN 32646* Show equality with the span of the sum of two vectors, one of which () is fixed in advance. Compare lspfixed 18429. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
LSAtoms

Theoremlsmsatcv 32647 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 27386 analog.) Explicit atom version of lsmcv 18442. (Contributed by NM, 29-Oct-2014.)
LSAtoms

Theoremlssatomic 32648* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 28092 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlssats 32649* The lattice of subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. Hypothesis (shatomistici 28095 analog.) (Contributed by NM, 9-Apr-2014.)
LSAtoms

Theoremlpssat 32650* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 28097 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremlrelat 32651* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 28098 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremlssatle 32652* The ordering of two subspaces is determined by the atoms under them. (chrelat3 28105 analog.) (Contributed by NM, 29-Oct-2014.)
LSAtoms

Theoremlssat 32653* Two subspaces in a proper subset relationship imply the existence of a 1-dim subspace less than or equal to one but not the other. (chpssati 28097 analog.) (Contributed by NM, 9-Apr-2014.)
LSAtoms

Theoremislshpat 32654* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 32617. (Contributed by NM, 11-Jan-2015.)
LSHyp       LSAtoms

Syntaxclcv 32655 Extend class notation with the covering relation for a left module or left vector space.
L

Definitiondf-lcv 32656* Define the covering relation for subspaces of a left vector space. Similar to Definition 3.2.18 of [PtakPulmannova] p. 68. Ptak/Pulmannova's notation L is read " covers " or " is covered by " , and it means that is larger than and there is nothing in between. See lcvbr 32658 for binary relation. (df-cv 28013 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvfbr 32657* The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvbr 32658* The covers relation for a left vector space (or a left module). (cvbr 28016 analog.) (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvbr2 32659* The covers relation for a left vector space (or a left module). (cvbr2 28017 analog.) (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvbr3 32660* The covers relation for a left vector space (or a left module). (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvpss 32661 The covers relation implies proper subset. (cvpss 28019 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvnbtwn 32662 The covers relation implies no in-betweenness. (cvnbtwn 28020 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvntr 32663 The covers relation is not transitive. (cvntr 28026 analog.) (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvnbtwn2 32664 The covers relation implies no in-betweenness. (cvnbtwn2 28021 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvnbtwn3 32665 The covers relation implies no in-betweenness. (cvnbtwn3 28022 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlsmcv2 32666 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 28027 analog.) (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvat 32667* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 28100 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoremlsatcv0 32668 An atom covers the zero subspace. (atcv0 28076 analog.) (Contributed by NM, 7-Jan-2015.)
LSAtoms       L

Theoremlsatcveq0 32669 A subspace covered by an atom must be the zero subspace. (atcveq0 28082 analog.) (Contributed by NM, 7-Jan-2015.)
LSAtoms       L

Theoremlsat0cv 32670 A subspace is an atom iff it covers the zero subspace. This could serve as an alternate definition of an atom. TODO: this is a quick-and-dirty proof that could probably be more efficient. (Contributed by NM, 14-Mar-2015.)
LSAtoms       L

Theoremlcvexchlem1 32671 Lemma for lcvexch 32676. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem2 32672 Lemma for lcvexch 32676. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem3 32673 Lemma for lcvexch 32676. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem4 32674 Lemma for lcvexch 32676. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem5 32675 Lemma for lcvexch 32676. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexch 32676 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 28103 analog.) TODO: combine some lemmas. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvp 32677 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 28109 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlcv1 32678 Covering property of a subspace plus an atom. (chcv1 28089 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlcv2 32679 Covering property of a subspace plus an atom. (chcv2 28090 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatexch 32680 The atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem was originally proved by Hermann Grassmann in 1862. (atexch 28115 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatnle 32681 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 28110 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatnem0 32682 The meet of distinct atoms is the zero subspace. (atnemeq0 28111 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatexch1 32683 The atom exch1ange property. (hlatexch1 33031 analog.) (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremlsatcv0eq 32684 If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 28113 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcv1 32685 Two atoms covering the zero subspace are equal. (atcv1 28114 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcvatlem 32686 Lemma for lsatcvat 32687. (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatcvat 32687 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 28120 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatcvat2 32688 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 28121 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcvat3 32689 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 28130 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremislshpcv 32690 Hyperplane properties expressed with covers relation. (Contributed by NM, 11-Jan-2015.)
LSHyp       L

Theoreml1cvpat 32691 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 33111 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoreml1cvat 32692 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 33112 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoremlshpat 32693 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 33679 analog.) TODO: This changes in l1cvpat 32691 and l1cvat 32692 to , which in turn change in islshpcv 32690 to , with a couple of conversions of span to atom. Seems convoluted. Would a direct proof be better? (Contributed by NM, 11-Jan-2015.)
LSHyp       LSAtoms

21.21.7  Functionals and kernels of a left vector space (or module)

Syntaxclfn 32694 Extend class notation with all linear functionals of a left module or left vector space.
LFnl

Definitiondf-lfl 32695* Define the set of all linear functionals (maps from vectors to the ring) of a left module or left vector space. (Contributed by NM, 15-Apr-2014.)
LFnl Scalar Scalar Scalar Scalar

Theoremlflset 32696* The set of linear functionals in a left module or left vector space. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar                                   LFnl

Theoremislfl 32697* The predicate "is a linear functional". (Contributed by NM, 15-Apr-2014.)
Scalar                                   LFnl

Theoremlfli 32698 Property of a linear functional. (lnfnli 27774 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar                                   LFnl

Theoremislfld 32699* Properties that determine a linear functional. TODO: use this in place of islfl 32697 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)
Scalar                                   LFnl

Theoremlflf 32700 A linear functional is a function from vectors to scalars. (lnfnfi 27775 analog.) (Contributed by NM, 15-Apr-2014.)
Scalar                     LFnl

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41046
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