Home Metamath Proof ExplorerTheorem List (p. 157 of 325) < Previous  Next > Browser slow? Try the Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

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

Theorem List for Metamath Proof Explorer - 15601-15700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremablfac 15601* The Fundamental Theorem of (finite) Abelian Groups. Any finite abelian group is a direct product of cyclic p-groups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp                      Word DProd DProd

Theoremablfac2 15602* Choose generators for each cyclic group in ablfac 15601. (Contributed by Mario Carneiro, 28-Apr-2016.)
SubGrp s CycGrp pGrp                      .g              Word DProd DProd

10.4  Rings

10.4.1  Multiplicative Group

Syntaxcmgp 15603 Multiplicative group.
mulGrp

Definitiondf-mgp 15604 Define a structure that puts the multiplication operation of a ring in the addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and unitgrp 15727 shows that we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" (rngmgp 15625) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 8896). (Contributed by Mario Carneiro, 21-Dec-2014.)
mulGrp sSet

Theoremfnmgp 15605 The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
mulGrp

Theoremmgpval 15606 Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
mulGrp              sSet

Theoremmgpplusg 15607 Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
mulGrp

Theoremmgplem 15608 Lemma for mgpbas 15609. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       Slot

Theoremmgpbas 15609 Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoremmgpsca 15610 The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 15679. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
mulGrp       Scalar       Scalar

Theoremmgptset 15611 Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       TopSet TopSet

Theoremmgptopn 15612 Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoremmgpds 15613 Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoremmgpress 15614 Subgroup commutes with the multiplication group operator. (Contributed by Mario Carneiro, 10-Jan-2015.)
s        mulGrp       s mulGrp

10.4.2  Definition and basic properties

Syntaxcrg 15615 Extend class notation with class of all (unital) rings.

Syntaxccrg 15616 Extend class notation with class of all (unital) commutative rings.

Syntaxcur 15617 Extend class notation with ring unit.

Definitiondf-rng 15618* Define class of all (unital) rings. A unital ring is a set equipped with two everywhere-defined internal operations, whose first one is an additive group structure and the second one is a multiplicative monoid structure, and where the addition is left- and right-distributive for the multiplication. So that the additive structure must be abelian (see rngcom 15647), care must be taken that in the case of a non-unital ring, the commutativity of addition must be postulated and cannot be proved from the other conditions. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 27-Dec-2014.)
mulGrp

Definitiondf-cring 15619 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
mulGrp CMnd

Definitiondf-ur 15620 Define the multiplicative neutral element of a ring. This definition works by extracting the element, i.e. the neutral element in a group or monoid, and transfering it to the multiplicative monoid via the mulGrp function (df-mgp 15604). See also dfur2 15622, which derives the "traditional" definition as the unique element of a ring which is left- and right-neutral under multiplication. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
mulGrp

Theoremrngidval 15621 The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
mulGrp

Theoremdfur2 15622* The multiplicative identity is the unique element of the ring that is left- and right-neutral on all elements under multiplication. (Contributed by Mario Carneiro, 10-Jan-2015.)

Theoremisrng 15623* The predicate "is a (unital) ring." Definition of ring with unit in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
mulGrp

Theoremrnggrp 15624 A ring is a group. (Contributed by NM, 15-Sep-2011.)

Theoremrngmgp 15625 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
mulGrp

Theoremiscrng 15626 A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
mulGrp       CMnd

Theoremcrngmgp 15627 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
mulGrp       CMnd

Theoremrngmnd 15628 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremcrngrng 15629 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremmgpf 15630 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
mulGrp

Theoremrngi 15631 Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremrngcl 15632 Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremcrngcom 15633 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremiscrng2 15634* A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)

Theoremrngass 15635 Associative law for the multiplication operation of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremrngideu 15636* The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremrngdi 15637 Distributive law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.)

Theoremrngdir 15638 Distributive law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.)

Theoremrngidcl 15639 The unit element of a ring belongs to the base set of the ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremrng0cl 15640 The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.)

Theoremrngidmlem 15641 Lemma for rnglidm 15642 and rngridm 15643. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremrnglidm 15642 The unit element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.)

Theoremrngridm 15643 The unit element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.)

Theoremisrngid 15644* Properties showing that an element is the unity element of a ring. (Contributed by NM, 7-Aug-2013.)

Theoremrngidss 15645 A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
mulGrps

Theoremrngacl 15646 Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.)

Theoremrngcom 15647 Commutativity of the additive group of a ring. (See also lmodcom 15945.) (Contributed by Gérard Lang, 4-Dec-2014.)

Theoremrngabl 15648 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)

Theoremrngcmn 15649 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
CMnd

Theoremrngpropd 15650* If two structures have the same group components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 6-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremcrngpropd 15651* If two structures have the same group components (properties), one is a commutative ring iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theoremrngprop 15652 If two structures have the same ring components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)

Theoremisrngd 15653* Properties that determine a ring. (Contributed by NM, 2-Aug-2013.)

Theoremiscrngd 15654* Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremrnglz 15655 The zero of a unital ring is a left absorbing element. (Contributed by FL, 31-Aug-2009.)

Theoremrngrz 15656 The zero of a unital ring is a right absorbing element. (Contributed by FL, 31-Aug-2009.)

Theoremrng1eq0 15657 If one and zero are equal, then any two elements of a ring are equal. Alternatively, every ring has one distinct from zero except the zero ring containing the single element . (Contributed by Mario Carneiro, 10-Sep-2014.)

Theoremrngnegl 15658 Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 26455 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngnegr 15659 Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 26456 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngmneg1 15660 Negation of a product in a ring. (mulneg1 9426 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngmneg2 15661 Negation of a product in a ring. (mulneg2 9427 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngm2neg 15662 Double negation of a product in a ring. (mul2neg 9429 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)

Theoremrngsubdi 15663 Ring multiplication distributes over subtraction. (subdi 9423 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngsubdir 15664 Ring multiplication distributes over subtraction. (subdir 9424 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremmulgass2 15665 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
.g

Theoremrnglghm 15666* Left-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.)

Theoremrngrghm 15667* Right-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.)

Theoremgsummulc1 15668* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.)
g g

Theoremgsummulc2 15669* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.)
g g

Theoremgsumdixp 15670* Distribute a binary product of sums to a sum of binary products in a ring. (Contributed by Mario Carneiro, 8-Mar-2015.)
g g g

Theoremprdsmgp 15671 The multiplicative monoid of a product is the product of the multiplicative monoids of the factors. (Contributed by Mario Carneiro, 11-Mar-2015.)
s       mulGrp       smulGrp

Theoremprdsmulrcl 15672 A structure product of rings has closed binary operation. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theoremprdsrngd 15673 A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theoremprdscrngd 15674 A product of commutative rings is a commutative ring. Since the resulting ring will have zero divisors in all nontrivial cases, this cannot be strengthened much further. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theoremprds1 15675 Value of the ring unit in a structure family product. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theorempwsrng 15676 A structure power of a ring is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theorempws1 15677 Value of the ring unit in a structure power. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theorempwscrng 15678 A structure power of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theorempwsmgp 15679 The multiplicative group of the power structure resembles the power of the multiplicative group. (Contributed by Mario Carneiro, 12-Mar-2015.)
s        mulGrp       s        mulGrp

Theoremimasrng 15680* The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
s

Theoremdivsrng2 15681* The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
s

10.4.3  Opposite ring

Syntaxcoppr 15682 The opposite ring operation.
oppr

Definitiondf-oppr 15683 Define an opposite ring, which is the same as the original ring but with multiplication written the other way around. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr sSet tpos

Theoremopprval 15684 Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr       sSet tpos

Theoremopprmulfval 15685 Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr              tpos

Theoremopprmul 15686 Value of the multiplication operation of an opposite ring. Hypotheses eliminated by a suggestion of Stefan O'Rear, 30-Aug-2015. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
oppr

Theoremcrngoppr 15687 In a commutative ring, the opposite ring is equivalent to the original ring (for theorems like unitpropd 15757). (Contributed by Mario Carneiro, 14-Jun-2015.)
oppr

Theoremopprlem 15688 Lemma for opprbas 15689 and oppradd 15690. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr       Slot

Theoremopprbas 15689 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr

Theoremoppradd 15690 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr

Theoremopprrng 15691 An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
oppr

Theoremopprrngb 15692 Bidirectional form of opprrng 15691. (Contributed by Mario Carneiro, 6-Dec-2014.)
oppr

Theoremoppr0 15693 Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr

Theoremoppr1 15694 Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr

Theoremopprneg 15695 The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
oppr

Theoremopprsubg 15696 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
oppr       SubGrp SubGrp

Theoremmulgass3 15697 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
.g

10.4.4  Divisibility

Syntaxcdsr 15698 Ring divides relation.
r

Syntaxcui 15699 Ring unit.
Unit

Syntaxcir 15700 Ring irreducibles.
Irred

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 >