Home Metamath Proof ExplorerTheorem List (p. 242 of 325) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22374) Hilbert Space Explorer (22375-23897) Users' Mathboxes (23898-32447)

Theorem List for Metamath Proof Explorer - 24101-24200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremssnnssfz 24101* For any finite subset of , find a superset in the form of a set of sequential integers. (Contributed by Thierry Arnoux, 13-Sep-2017.)

Theoremfzspl 24102 Split the last element of a finite set of sequential integers. (more generic than fzsuc 11052) (Contributed by Thierry Arnoux, 7-Nov-2016.)

Theoremfzsplit3 24103 Split a finite interval of integers into two parts. (Contributed by Thierry Arnoux, 2-May-2017.)

Theorembcm1n 24104 The proportion of one binomial coefficient to another with decreased by 1. (Contributed by Thierry Arnoux, 9-Nov-2016.)

19.3.5.6  Half-open integer ranges - misc additions

Theoremiundisjfi 24105* Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 19395 (Contributed by Thierry Arnoux, 15-Feb-2017.)
..^ ..^ ..^

Theoremiundisj2fi 24106* A disjoint union is disjoint, finite version. Cf. iundisj2 19396 (Contributed by Thierry Arnoux, 16-Feb-2017.)
Disj ..^ ..^

Theoremiundisjcnt 24107* Rewrite a countable union as a disjoint union. (Contributed by Thierry Arnoux, 16-Feb-2017.)
..^       ..^

Theoremiundisj2cnt 24108* A countable disjoint union is disjoint. Cf. iundisj2 19396 (Contributed by Thierry Arnoux, 16-Feb-2017.)
..^       Disj ..^

19.3.5.7  The ` # ` (finite set size) function - misc additions

Theoremhashresfn 24109 Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Theoremdmhashres 24110 Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 12-Jan-2017.)

Theoremhashunif 24111* The cardinality of a disjoint finite union of finite sets. Cf. hashuni 12559 (Contributed by Thierry Arnoux, 17-Feb-2017.)
Disj

Theoremishashinf 24112* Any set that is not finite contains subsets of arbitrarily large finite cardinality. Cf. isinf 7281 (Contributed by Thierry Arnoux, 5-Jul-2017.)

19.3.5.8  The greatest common divisor operator - misc. add

Theoremnumdenneg 24113 Numerator and denominator of the negative (Contributed by Thierry Arnoux, 27-Oct-2017.)
numer numer denom denom

Theoremdivnumden2 24114 Calculate the reduced form of a quotient using . This version extends divnumden 13095 for the negative integers. (Contributed by Thierry Arnoux, 25-Oct-2017.)
numer denom

19.3.5.9  Integers

Theoremltesubnnd 24115 Subtracting an integer number from another number decreases it. See ltsubrpd 10632 (Contributed by Thierry Arnoux, 18-Apr-2017.)

19.3.5.10  Division in the extended real number system

Syntaxcxdiv 24116 Extend class notation to include division of extended reals.
/𝑒

Definitiondf-xdiv 24117* Define division over extended real numbers. (Contributed by Thierry Arnoux, 17-Dec-2016.)
/𝑒

Theoremxdivval 24118* Value of division: the (unique) element such that . This is meaningful only when is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.)
/𝑒

Theoremxrecex 24119* Existence of reciprocal of nonzero real number. (Contributed by Thierry Arnoux, 17-Dec-2016.)

Theoremxmulcand 24120 Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.)

Theoremxreceu 24121* Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 17-Dec-2016.)

Theoremxdivcld 24122 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
/𝑒

Theoremxdivcl 24123 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
/𝑒

Theoremxdivmul 24124 Relationship between division and multiplication. (Contributed by Thierry Arnoux, 24-Dec-2016.)
/𝑒

Theoremrexdiv 24125 The extended real division operation when both arguments are real. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

Theoremxdivrec 24126 Relationship between division and reciprocal. (Contributed by Thierry Arnoux, 5-Jul-2017.)
/𝑒 /𝑒

Theoremxdivid 24127 A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

Theoremxdiv0 24128 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

Theoremxdiv0rp 24129 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

Theoremeliccioo 24130 Membership in a closed interval of extended reals vs. the same open interval (Contributed by Thierry Arnoux, 18-Dec-2016.)

Theoremelxrge02 24131 Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)

Theoremxdivpnfrp 24132 Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

Theoremrpxdivcld 24133 Closure law for extended division of positive reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

Theoremxrpxdivcld 24134 Closure law for extended division of positive extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

19.3.6  Structure builders

19.3.6.1  Structure builder restriction operator

Theoremress0g 24135 is unaffected by restriction. This is a bit more generic than submnd0 14680 (Contributed by Thierry Arnoux, 23-Oct-2017.)
s

Theoremressplusf 24136 The group operation function of a structure's restriction is the operation function's restriction to the new base. (Contributed by Thierry Arnoux, 26-Mar-2017.)
s

Theoremressnm 24137 The norm in a restricted structure. (Contributed by Thierry Arnoux, 8-Oct-2017.)
s

Theoremabvpropd2 24138 Weaker version of abvpropd 15885. (Contributed by Thierry Arnoux, 8-Nov-2017.)
AbsVal AbsVal

19.3.6.2  Posets

Theoremtospos 24139 A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Toset

Theoremresspos 24140 The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
s

Theoremresstos 24141 The restriction of a Toset is a Toset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Toset s Toset

Theoremtleile 24142 In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Toset

Theoremtltnle 24143 In a Toset, less-than is equivalent to not inverse less-than-or-equal see pltnle 14378. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Toset

Theoremtoslub 24144* In a toset, the lowest upper bound , defined for partial orders is the supremum, , defined for total orders, if this supremum exists (these are the set.mm definition: lowest upper bound and supremum are normally synonymous) Note that the two definitions do not have the same value when there is no such supremum. In that case, will take the value , but takes the value , therefore, we restrict this theorem only to the case where this supremum exists, which is expressed by the last assumption. (Contributed by Thierry Arnoux, 15-Feb-2018.)
Toset

Theoremtosglb 24145* Same theorem as toslub 24144, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.)
Toset

19.3.6.3  Complete lattices

Theoremclatp0ex 24146 The poset zero of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)

Theoremclatp1ex 24147 The poset one of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)

19.3.6.4  Extended reals Structure - misc additions

Axiomax-xrssca 24148 Assume the scalar component of the extended real structure is the field of the real numbers (this has to be defined in the main body of set.mm) (Contributed by Thierry Arnoux, 22-Oct-2017.)
flds Scalar

Axiomax-xrsvsca 24149 Assume the scalar product of the extended real structure is the extended real number multiplication operation (this has to be defined in the main body of set.mm) (Contributed by Thierry Arnoux, 22-Oct-2017.)

Theoremxrs0 24150 The zero of the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 10784 and df-xrs 13681), however it has a zero. (Contributed by Thierry Arnoux, 13-Jun-2017.)

Theoremxrslt 24151 The "strictly less than" relation for the extended real structure. (Contributed by Thierry Arnoux, 30-Jan-2018.)

Theoremxrsinvgval 24152 The inversion operation in the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 10784 and df-xrs 13681), however it has an inversion operation. (Contributed by Thierry Arnoux, 13-Jun-2017.)

Theoremxrsmulgzz 24153 The "multiple" function in the extended real numbers structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
.g

Theoremxrstos 24154 The extended real numbers form a toset. (Contributed by Thierry Arnoux, 15-Feb-2018.)
Toset

Theoremxrsclat 24155 The extended real numbers form a complete lattice. (Contributed by Thierry Arnoux, 15-Feb-2018.)

Theoremxrsp0 24156 The poset 0 of the extended real numbers is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)

Theoremxrsp1 24157 The poset 1 of the extended real numbers is plus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)

Theoremressmulgnn 24158 Values for the group multiple function in a restricted structure (Contributed by Thierry Arnoux, 12-Jun-2017.)
s               .g              .g

Theoremressmulgnn0 24159 Values for the group multiple function in a restricted structure (Contributed by Thierry Arnoux, 14-Jun-2017.)
s               .g                     .g

19.3.6.5  The extended non-negative real numbers monoid

Theoremxrge0base 24160 The base of the extended non-negative real numbers. (Contributed by Thierry Arnoux, 30-Jan-2017.)
s

Theoremxrge00 24161 The zero of the extended non-negative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
s

Theoremxrge0plusg 24162 The additive law of the extended non-negative real numbers monoid is the addition in the extended real numbers. (Contributed by Thierry Arnoux, 20-Mar-2017.)
s

Theoremxrge0mulgnn0 24163 The group multiple function in the extended non-negative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.)
.gs

Theoremxrge0addass 24164 Associativity of extended non-negative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.)

Theoremxrge0neqmnf 24165 An extended non-negative real cannot be minus infinity. (Contributed by Thierry Arnoux, 9-Jun-2017.)

Theoremxrge0nre 24166 An extended real which is not a real is plus infinity. (Contributed by Thierry Arnoux, 16-Oct-2017.)

Theoremxrge0addgt0 24167 The sum of nonnegative and positive numbers is positive. See addgtge0 9472 (Contributed by Thierry Arnoux, 6-Jul-2017.)

Theoremxrge0adddir 24168 Distributivity of extended non-negative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)

Theoremxrge0npcan 24169 Extended non-negative real version of npcan 9270. (Contributed by Thierry Arnoux, 9-Jun-2017.)

Theoremfsumrp0cl 24170* Closure of a finite sum of positive integers. (Contributed by Thierry Arnoux, 25-Jun-2017.)

19.3.7  Algebra

19.3.7.1  Finitely supported group sums - misc additions

Theoremsumpr 24171* A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.)

Theoremgsumsn2 24172* Group sum of a singleton. (Contributed by Thierry Arnoux, 30-Jan-2017.)
g

Theoremgsumpropd2lem 24173* Lemma for gsumpropd2 24174 (Contributed by Thierry Arnoux, 28-Jun-2017.)
g g

Theoremgsumpropd2 24174* A stronger version of gsumpropd 14731, working for magma, where only the closure of the addition operation on a common base is required. (Contributed by Thierry Arnoux, 28-Jun-2017.)
g g

Theoremgsumconstf 24175* Sum of a constant series (Contributed by Thierry Arnoux, 5-Jul-2017.)
.g       g

Theoremxrge0tsmsd 24176* Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 30-Jan-2017.)
s                      g        tsums

Theoremxrge0tsmsbi 24177 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 23-Jun-2017.)
s                      tsums tsums

Theoremxrge0tsmseq 24178 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 24-Mar-2017.)
s                      tsums        tsums

Theoremdvrdir 24179 Distributive law for the division operation of a ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
Unit              /r

Theoremrdivmuldivd 24180 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.)
Unit              /r

Theoremrnginvval 24181* The ring inverse expressed in terms of multiplication. (Contributed by Thierry Arnoux, 23-Oct-2017.)
Unit

Theoremdvrcan5 24182 Cancellation law for common factor in ratio. (divcan5 9672 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.)
Unit       /r

Theoremsubrgchr 24183 If is a subring of , then they have the same characteristic. (Contributed by Thierry Arnoux, 24-Feb-2018.)
SubRing chrs chr

19.3.7.3  Ordered groups

Syntaxcogrp 24184 Extend class notation with the class of all ordered groups.
oGrp

Definitiondf-ogrp 24185* Define class of all ordered groups. An ordered group is a group with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 30-Jan-2018.)
oGrp Toset

19.3.7.4  Ordered fields

Syntaxcofld 24186 Extend class notation with the class of all ordered fields.
oField

Definitiondf-ofld 24187* Define class of all ordered fields. An ordered field is a field with a total ordering compatible with the operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
oField Field Toset

Theoremisofld 24188* An ordered field is a field with a total ordering compatible with the operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
oField Field Toset

Theoremofldfld 24189 An ordered field is a field. (Contributed by Thierry Arnoux, 20-Jan-2018.)
oField Field

Theoremofldtos 24190 An ordered field is a totally ordered set. (Contributed by Thierry Arnoux, 20-Jan-2018.)
oField Toset

Theoremofldadd 24191 In an ordered field, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
oField

Theoremofldmul 24192 In an ordered field, the ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 20-Jan-2018.)
oField

Theoremofldsqr 24193 In an ordered field, all squares are positive. (Contributed by Thierry Arnoux, 20-Jan-2018.)
oField

Theoremofldaddlt 24194 In an ordered field, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
oField

Theoremofld0le1 24195 In an ordered field, the ring unit is positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
oField

Theoremofldlt1 24196 In an ordered field, the ring unit is strictly positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
oField

Theoremofldchr 24197 The characteristic of an ordered field is zero. (Contributed by Thierry Arnoux, 21-Jan-2018.)
oField chr

Theoremsubofld 24198 Every subfield of an ordered field is also an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
oField s Field s oField

19.3.7.5  The Archimedean property for generic algebraic structures

Syntaxcinftm 24199 Class notation for the infinitesimal relation.
<<<

Syntaxcarchi 24200 Class notation for the Archimedean property.
Archi

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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 32401-32447
 Copyright terms: Public domain < Previous  Next >