Home Metamath Proof ExplorerTheorem List (p. 219 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 - 21801-21900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgxnval 21801 The result of the group power operator when the exponent is negative. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgx0 21802 The result of the group power operator when the exponent is zero. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremgx1 21803 The result of the group power operator when the exponent is one. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgxnn0neg 21804 A negative group power is the inverse of the positive power (lemma with nonnegative exponent - use gxneg 21807 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgxnn0suc 21805 Induction on group power (lemma with nonnegative exponent - use gxsuc 21813 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremgxcl 21806 Closure of the group power operator. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxneg 21807 A negative group power is the inverse of the positive power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxneg2 21808 The inverse of a negative group power is the positive power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxm1 21809 The result of the group power operator when the exponent is minus one. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxcom 21810 The group power operator commutes with the group operation. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxinv 21811 The group power operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxinv2 21812 The group power operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxsuc 21813 Induction on group power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxid 21814 The identity element of a group to any power remains unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
GId

Theoremgxnn0add 21815 The group power of a sum is the group product of the powers (lemma with nonnegative exponent - use gxadd 21816 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxadd 21816 The group power of a sum is the group product of the powers. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxsub 21817 The group power of a difference is the group quotient of the powers. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxnn0mul 21818 The group power of a product is the composition of the powers (lemma with nonnegative exponent - use gxmul 21819 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxmul 21819 The group power of a product is the composition of the powers. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremgxmodid 21820 Casting out powers of the identity element leaves the group power unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
GId

Theoremresgrprn 21821 The underlying set of a group operation which is a restriction of a mapping. (Contributed by Paul Chapman, 25-Mar-2008.) (New usage is discouraged.)

16.1.2  Definition and basic properties of Abelian groups

Syntaxcablo 21822 Extend class notation with the class of all Abelian group operations.

Definitiondf-ablo 21823* Define the class of all Abelian group operations. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremisablo 21824* The predicate "is an Abelian (commutative) group operation." (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremablogrpo 21825 An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremablocom 21826 An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremablo32 21827 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremablo4 21828 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Theoremisabloi 21829* Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Theoremablomuldiv 21830 Law for group multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremablodivdiv 21831 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremablodivdiv4 21832 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremablodiv32 21833 Swap the second and third terms in a double division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremablonnncan 21834 Cancellation law for group division. (nnncan 9292 analog.) (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremablonncan 21835 Cancellation law for group division. (nncan 9286 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)

Theoremablonnncan1 21836 Cancellation law for group division. (nnncan1 9293 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)

Theoremgxdi 21837 Distribution of group power over group operation for abelian groups. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremisgrpda 21838* Properties that determine a group operation. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)

Theoremisgrpod 21839* Properties that determine a group operation. (Renamed from isgrpd 14785 to isgrpod 21839 to prevent naming conflict. -NM 5-Jun-2013) (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)

Theoremisabloda 21840* Properties that determine an Abelian group operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (New usage is discouraged.)

Theoremisablod 21841* Properties that determine an Abelian group operation. (Changed label from isabld 15380 to isablod 21841-NM 6-Aug-2013.) (Contributed by Jeff Madsen, 5-Dec-2009.) (New usage is discouraged.)

16.1.3  Subgroups

Syntaxcsubgo 21842 Extend class notation to include the class of subgroups.

Definitiondf-subgo 21843 Define the set of subgroups of . (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)

Theoremissubgo 21844 The predicate "is a subgroup of ." (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 12-Jul-2014.) (New usage is discouraged.)

Theoremsubgores 21845 A subgroup operation is the restriction of its parent group operation to its underlying set. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)

Theoremsubgoov 21846 The result of a subgroup operation is the same as the result of its parent operation. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 8-Jul-2014.) (New usage is discouraged.)

Theoremsubgornss 21847 The underlying set of a subgroup is a subset of its parent group's underlying set. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)

Theoremsubgoid 21848 The identity element of a subgroup is the same as its parent's. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
GId       GId

Theoremsubgoinv 21849 The inverse of a subgroup element is the same as its inverse in the parent group. (Contributed by Mario Carneiro, 8-Jul-2014.) (New usage is discouraged.)

Theoremissubgoilem 21850* Lemma for issubgoi 21851. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)

Theoremissubgoi 21851* Properties that determine a subgroup. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GId

Theoremsubgoablo 21852 A subgroup of an Abelian group is Abelian. (Contributed by Paul Chapman, 25-Apr-2008.) (New usage is discouraged.)

16.1.4  Operation properties

Syntaxcass 21853 Extend class notation with a device to add associativity to internal operations.

Definitiondf-ass 21854* A device to add associativity to various sorts of internal operations. The definition is meaningful when is a magma at least. (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)

Syntaxcexid 21855 Extend class notation with the class of all the internal operations with an identity element.

Definitiondf-exid 21856* A device to add an identity element to various sorts of internal operations. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremisass 21857* The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)

Theoremisexid 21858* The predicate has a left and right identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

16.1.5  Group-like structures

Syntaxcmagm 21859 Extend class notation with the class of all magmas.

Definitiondf-mgm 21860* A magma is a binary internal operation. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremismgm 21861 The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremclmgm 21862 Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.)

Theoremopidon 21863 An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Theoremrngopid 21864 Range of an operation with a left and right identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremopidon2 21865 An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremisexid2 21866* If , then it has a left and right identity element that belongs to the range of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Theoremexidu1 21867* Unicity of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Theoremidrval 21868* The value of the identity element. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
GId

Theoremiorlid 21869 A magma right and left identity belongs to the underlying set of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
GId

Theoremcmpidelt 21870 A magma right and left identity element keeps the other elements unchanged. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
GId

Syntaxcsem 21871 Extend class notation with the class of all semi-groups.

Definitiondf-sgr 21872 A semi-group is an associative magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpismgm 21873 A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpisass 21874 A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremissmgrp 21875* The predicate "is a semi-group". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpmgm 21876 A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpass 21877* A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Syntaxcmndo 21878 Extend class notation with the class of all monoids.
MndOp

Definitiondf-mndo 21879 A monoid is a semi-group with an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndoissmgrp 21880 A monoid is a semi-group. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndoisexid 21881 A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndoismgm 21882 A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndomgmid 21883 A monoid is a magma with an identity element. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
MndOp

Theoremismndo 21884* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

Theoremismndo1 21885* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

Theoremismndo2 21886* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

Theoremgrpomndo 21887 A group is a monoid. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

16.1.6  Examples of Abelian groups

Theoremablosn 21888 The Abelian group operation for the singleton group. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Theoremgidsn 21889 The identity element of the trivial group. (Contributed by FL, 21-Jun-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremginvsn 21890 The inverse function of the trivial group. (Contributed by FL, 21-Jun-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremcnaddablo 21891 Complex number addition is an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Theoremcnid 21892 The group identity element of complex number addition is zero. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
GId

Theoremaddinv 21893 Value of the group inverse of complex number addition. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremreaddsubgo 21894 The real numbers under addition comprise a subgroup of the complex numbers under addition. (Contributed by Paul Chapman, 25-Apr-2008.) (New usage is discouraged.)

Theoremzaddsubgo 21895 The integers under addition comprise a subgroup of the complex numbers under addition. (Contributed by Paul Chapman, 25-Apr-2008.) (New usage is discouraged.)

Theoremablomul 21896 Nonzero complex number multiplication is an Abelian group operation. (Contributed by Steve Rodriguez, 12-Feb-2007.) (New usage is discouraged.)

Theoremmulid 21897 The group identity element of nonzero complex number multiplication is one. (Contributed by Steve Rodriguez, 23-Feb-2007.) (Revised by Mario Carneiro, 17-Dec-2013.) (New usage is discouraged.)
GId

16.1.7  Group homomorphism and isomorphism

Syntaxcghom 21898 Extend class notation to include the class of group homomorphisms.
GrpOpHom

Definitiondf-ghom 21899* Define the set of group homomorphisms from to . (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom

Syntaxcgiso 21900 Extend class notation to include the class of group isomorphisms.

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 >