Home Metamath Proof ExplorerTheorem List (p. 274 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 - 27301-27400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempsgnpmtr 27301 All transpositions are odd. (Contributed by Stefan O'Rear, 29-Aug-2015.)
pmTrsp       pmSgn

Theoremcnmsgnsubg 27302 The signs form a multiplicative subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.)
mulGrpflds        SubGrp

Theoremcnmsgnbas 27303 The base set of the sign subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.)
mulGrpflds

Theoremcnmsgngrp 27304 The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.)
mulGrpflds

Theorempsgnghm 27305 The sign is a homomorphism from the finitary permutation group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmSgn       s        mulGrpflds

Theorempsgnghm2 27306 The sign is a homomorphism from the finite symmetric group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmSgn       mulGrpflds

19.16.57  The matrix algebra

Syntaxcmmul 27307 Syntax for the matrix multiplication operator.
maMul

Syntaxcmat 27308 Syntax for the square matrix algebra.
Mat

Definitiondf-mamu 27309* The operator which multiplies an MxN matrix with an NxP matrix. Note that it is not generally possible to recover the dimensions from the matrix, since all Nx0 and all 0xN matrices are represented by the empty set. (Contributed by Stefan O'Rear, 4-Sep-2015.)
maMul g

Definitiondf-mat 27310* The algebra of NxN matrices over a ring... (Contributed by Stefan O'Rear, 31-Aug-2015.)
Mat freeLMod sSet maMul

Theoremmamufval 27311* Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.)
maMul                                                  g

Theoremmamuval 27312* Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
maMul                                                                g

Theoremmamufv 27313* A cell in the multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
maMul                                                                              g

Theoremmndvcl 27314 Tuple-wise additive closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremmndvass 27315 Tuple-wise associativity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremmndvlid 27316 Tuple-wise left identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremmndvrid 27317 Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremgrpvlinv 27318 Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremgrpvrinv 27319 Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.)

Theoremmhmvlin 27320 Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015.)
MndHom

Theoremrngvcl 27321 Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremgsumcom3 27322* A commutative law for finitely supported iterated sums. (Contributed by Stefan O'Rear, 2-Nov-2015.)
CMnd                                          g g g g

Theoremgsumcom3fi 27323* A commutative law for finite iterated sums. (Contributed by Stefan O'Rear, 5-Sep-2015.)
CMnd                            g g g g

Theoremmamucl 27324 Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.)
maMul

Theoremmamudiagcl 27325* Diagonal matrices are matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)

Theoremmamulid 27326* Diagonal matrices are left identities. (Contributed by Stefan O'Rear, 3-Sep-2015.)
maMul

Theoremmamurid 27327* Diagonal matrices are right identities. (Contributed by Stefan O'Rear, 3-Sep-2015.)
maMul

Theoremmamuass 27328 Matrix multiplication is associative. (Contributed by Stefan O'Rear, 5-Sep-2015.)
maMul        maMul        maMul        maMul

Theoremmamudi 27329 Matrix multiplication distributes over addition on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
maMul

Theoremmamudir 27330 Matrix multiplication distributes over addition on the right. (Contributed by Stefan O'Rear, 5-Sep-2015.)
maMul

Theoremmamuvs1 27331 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
maMul

Theoremmamuvs2 27332 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
maMul

Theoremmatval 27333 Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Mat        freeLMod        maMul        sSet

Theoremmatrcl 27334 Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Mat

Theoremmatmulr 27335 Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Mat        maMul

Theoremmatbas 27336 The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Mat        freeLMod

Theoremmatplusg 27337 The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Mat        freeLMod

Theoremmatsca 27338 The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Mat        freeLMod        Scalar Scalar

Theoremmatvsca 27339 The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Mat        freeLMod

Theoremmat0 27340 The matrix ring has the same zero as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Mat        freeLMod

Theoremmatinvg 27341 The matrix ring has the same additive inverse as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Mat        freeLMod

Theoremmatsca2 27342 The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Mat        Scalar

Theoremmatbas2 27343 The base set of the matrix ring as a set exponential. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Mat

Theoremmatbas2i 27344 A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.)
Mat

Theoremmatplusg2 27345 Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Mat

Theoremmatvsca2 27346 Scalar multiplication in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Mat

Theoremmatlmod 27347 The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Mat

Theoremmatrng 27348 Existence of the matrix ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Mat

Theoremmatassa 27349 Existence of the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Mat        AssAlg

Theoremmat1 27350* Value of an identity matrix. (Contributed by Stefan O'Rear, 7-Sep-2015.)
Mat

19.16.58  The determinant

Syntaxcmdat 27351 Syntax for the matrix determinant function.

Definitiondf-mdet 27353* Determinant of a square matrix... (Contributed by Stefan O'Rear, 9-Sep-2015.)

Definitiondf-madu 27354* Define the adjunct (matrix of cofactors) of a square matrix. This definition gives the standard cofactors, however the internal minors are not the standard minors. (Contributed by Stefan O'Rear, 7-Sep-2015.)

Theoremmdetfval 27355* First substitution for the determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.)
maDet        Mat                      RHom       pmSgn              mulGrp       g g

Theoremmdetleib 27356* Full substitution of our determinant definition (also known as Leibniz' Formula). (Contributed by Stefan O'Rear, 3-Oct-2015.)
maDet        Mat                      RHom       pmSgn              mulGrp       g g

19.16.59  Endomorphism algebra

Syntaxcmend 27357 Syntax for module endomorphism algebra.
MEndo

Definitiondf-mend 27358* Define the endomorphism algebra of a module. (Contributed by Stefan O'Rear, 2-Sep-2015.)
MEndo LMHom Scalar Scalar Scalar

Theoremmendval 27359* Value of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
LMHom                      Scalar              MEndo Scalar

Theoremmendbas 27360 Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
MEndo       LMHom

Theoremmendplusgfval 27361* Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
MEndo

Theoremmendplusg 27362 A specific addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
MEndo

Theoremmendmulrfval 27363* Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
MEndo

Theoremmendmulr 27364 A specific multiplication in the module endormoprhism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
MEndo

Theoremmendsca 27365 The module endomorphism algebra has the same scalars as the underlying module. (Contributed by Stefan O'Rear, 2-Sep-2015.)
MEndo       Scalar       Scalar

Theoremmendvscafval 27366* Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
MEndo                     Scalar

Theoremmendvsca 27367 A specific scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
MEndo                     Scalar

Theoremmendrng 27368 The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
MEndo

Theoremmendlmod 27369 The module endomorphism algebra is a left module. (Contributed by Mario Carneiro, 22-Sep-2015.)
MEndo       Scalar

Theoremmendassa 27370 The module endomorphism algebra is an algebra. (Contributed by Mario Carneiro, 22-Sep-2015.)
MEndo       Scalar       AssAlg

19.16.60  Subfields

Syntaxcsdrg 27371 Syntax for subfields (sub-division-rings).
SubDRing

Definitiondf-sdrg 27372* A sub-division-ring is a subset of a division ring's set which is a division ring under the induced operation. If the overring is commutative this is a field; no special consideration is made of the fields in the center of a skew field. (Contributed by Stefan O'Rear, 3-Oct-2015.)
SubDRing SubRing s

Theoremissdrg 27373 Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.)
SubDRing SubRing s

Theoremissdrg2 27374* Property of a division subring (closure version). (Contributed by Mario Carneiro, 3-Oct-2015.)
SubDRing SubRing

Theoremacsfn1p 27375* Construction of a closure rule from a one-parameter partial operation. (Contributed by Stefan O'Rear, 12-Sep-2015.)
ACS

Theoremsubrgacs 27376 Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.)
SubRing ACS

Theoremsdrgacs 27377 Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
SubDRing ACS

Theoremcntzsdrg 27378 Centralizers in division rings/fields are subfields. (Contributed by Mario Carneiro, 3-Oct-2015.)
mulGrp       Cntz       SubDRing

19.16.61  Cyclic groups and order

Theoremidomrootle 27379* No element of an integral domain can have more than -th roots. (Contributed by Stefan O'Rear, 11-Sep-2015.)
.gmulGrp       IDomn

Theoremidomodle 27380* Limit on the number of -th roots of unity in an integral domain. (Contributed by Stefan O'Rear, 12-Sep-2015.)
mulGrps Unit                     IDomn

Theoremfiuneneq 27381 Two finite sets of equal size have a union of the same size iff they were equal. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Theoremidomsubgmo 27382* The units of an integral domain have at most one subgroup of any single finite cardinality. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Revised by NM, 17-Jun-2017.)
mulGrps Unit       IDomn SubGrp

Theoremproot1mul 27383 Any primitive -th root of unity is a multiple of any other. (Contributed by Stefan O'Rear, 2-Nov-2015.)
mulGrps Unit              mrClsSubGrp       IDomn

Theoremhashgcdlem 27384* A correspondence between elements of specific GCD and relative primes in a smaller ring. (Contributed by Stefan O'Rear, 12-Sep-2015.)
..^        ..^

Theoremhashgcdeq 27385* Number of initial natural numbers with specified divisors. (Contributed by Stefan O'Rear, 12-Sep-2015.)
..^

Theoremphisum 27386* The divisor sum identity of the totient function. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Theoremproot1hash 27387 If an integral domain has a primitive -th root of unity, it has exactly of them. (Contributed by Stefan O'Rear, 12-Sep-2015.)
mulGrps Unit              IDomn

Theoremproot1ex 27388 The complex field has primitive -th roots of unity for all . (Contributed by Stefan O'Rear, 12-Sep-2015.)
mulGrpflds

19.16.62  Cyclotomic polynomials

Syntaxccytp 27389 Syntax for the sequence of cyclotomic polynomials.
CytP

Definitiondf-cytp 27390* The Nth cyclotomic polynomial is the polynomial which has as its zeros precisely the primitive Nth roots of unity. (Contributed by Stefan O'Rear, 5-Sep-2015.)
CytP mulGrpPoly1fld g mulGrpflds var1fldPoly1fldalgScPoly1fld

Theoremisdomn3 27391 Nonzero elements form a multiplicative submonoid of any domain. (Contributed by Stefan O'Rear, 11-Sep-2015.)
mulGrp       Domn SubMnd

Theoremmon1pid 27392 Monicity and degree of the unit polynomial. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Poly1              Monic1p       deg1        NzRing

Theoremmon1psubm 27393 Monic polynomials are a multiplicative submonoid. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Poly1       Monic1p       mulGrp       NzRing SubMnd

Theoremdeg1mhm 27394 Homomorphic property of the polynomial degree. (Contributed by Stefan O'Rear, 12-Sep-2015.)
deg1               Poly1              mulGrps        flds        Domn MndHom

Theoremcytpfn 27395 Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
CytP

Theoremcytpval 27396* Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)
mulGrpflds               Poly1fld       var1fld       mulGrp              algSc       CytP g

19.16.63  Miscellaneous topology

Theoremfgraphopab 27397* Express a function as a subset of the cross product. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremfgraphxp 27398* Express a function as a subset of the cross product. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremhausgraph 27399 The graph of a continuous function into a Hausdorff space is closed. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Syntaxctopsep 27400 The class of separable toplogies.
TopSep

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 >