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Theorem List for Metamath Proof Explorer - 15101-15200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdprd0 15101 The empty family is an internal direct product, the product of which is the trivial subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd DProd

Theoremdprdf1o 15102 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd DProd DProd

Theoremdprdf1 15103 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd DProd DProd

Theoremsubgdmdprd 15104 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
s        SubGrp DProd DProd

Theoremsubgdprd 15105 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
s        SubGrp       DProd               DProd DProd

Theoremdprdsn 15106 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
SubGrp DProd DProd

Theoremdmdprdsplitlem 15107* Lemma for dmdprdsplit 15117. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                             g DProd

Theoremdprdcntz2 15108 The function is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                                    Cntz       DProd DProd

Theoremdprddisj2 15109 The function is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                                           DProd DProd

Theoremdprd2dlem2 15110* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp              DProd        DProd DProd        mrClsSubGrp       DProd

Theoremdprd2dlem1 15111* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp              DProd        DProd DProd        mrClsSubGrp              DProd DProd

Theoremdprd2da 15112* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp              DProd        DProd DProd        mrClsSubGrp       DProd

Theoremdprd2db 15113* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 25-Apr-2016.)
SubGrp              DProd        DProd DProd        mrClsSubGrp       DProd DProd DProd

Theoremdprd2d2 15114* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp       DProd        DProd DProd        DProd DProd DProd DProd

Theoremdmdprdsplit2lem 15115 Lemma for dmdprdsplit 15117. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp                     Cntz              DProd        DProd        DProd DProd        DProd DProd        mrClsSubGrp

Theoremdmdprdsplit2 15116 The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)
SubGrp                     Cntz              DProd        DProd        DProd DProd        DProd DProd        DProd

Theoremdmdprdsplit 15117 The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)
SubGrp                     Cntz              DProd DProd DProd DProd DProd DProd DProd

Theoremdprdsplit 15118 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 25-Apr-2016.)
SubGrp                            DProd        DProd DProd DProd

Theoremdmdprdpr 15119 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz              SubGrp       SubGrp       DProd

Theoremdprdpr 15120 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 26-Apr-2016.)
Cntz              SubGrp       SubGrp                            DProd

Theoremdpjlem 15121 Lemma for theorems about direct product projection. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                      DProd

Theoremdpjcntz 15122 The two subgroups that appear in dpjval 15126 commute. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                      Cntz       DProd

Theoremdpjdisj 15123 The two subgroups that appear in dpjval 15126 are disjoint. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                             DProd

Theoremdpjlsm 15124 The two subgroups that appear in dpjval 15126 add to the full direct product. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                             DProd DProd

Theoremdpjfval 15125* Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj              DProd

Theoremdpjval 15126 Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj                     DProd

Theoremdpjf 15127 The -th index projection is a function from the direct product to the -th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj              DProd

Theoremdpjidcl 15128* The key property of projections: the sum of all the projections of is . (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj       DProd                      g

Theoremdpjeq 15129* Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj       DProd                             g

Theoremdpjid 15130* The key property of projections: the sum of all the projections of is . (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj       DProd        g

Theoremdpjlid 15131 The -th index projection acts as the identity on elements of the -th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj

Theoremdpjrid 15132 The -th index projection annihilates elements of other factors. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj

Theoremdpjghm 15133 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj              s DProd

Theoremdpjghm2 15134 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj              s DProd s

10.3.5  The Fundamental Theorem of Abelian Groups

Theoremablfacrplem 15135* Lemma for ablfacrp2 15137. (Contributed by Mario Carneiro, 19-Apr-2016.)

Theoremablfacrp 15136* A finite abelian group whose order factors into relatively prime integers, itself "factors" into two subgroups that have trivial intersection and whose product is the whole group. Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 19-Apr-2016.)

Theoremablfacrp2 15137* The factors of ablfacrp 15136 have the expected orders (which allows for repeated application to decompose into subgroups of prime-power order). Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 21-Apr-2016.)

Theoremablfac1lem 15138* Lemma for ablfac1b 15140. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-Apr-2016.)

Theoremablfac1a 15139* The factors of ablfac1b 15140 are of prime power order. (Contributed by Mario Carneiro, 26-Apr-2016.)

Theoremablfac1b 15140* Any abelian group is the direct product of factors of prime power order (with the exact order further matching the prime factorization of the group order). (Contributed by Mario Carneiro, 21-Apr-2016.)
DProd

Theoremablfac1c 15141* The factors of ablfac1b 15140 cover the entire group. (Contributed by Mario Carneiro, 21-Apr-2016.)
DProd

Theoremablfac1eulem 15142* Lemma for ablfac1eu 15143. (Contributed by Mario Carneiro, 27-Apr-2016.)
DProd DProd                                           DProd

Theoremablfac1eu 15143* The factorization of ablfac1b 15140 is unique, in that any other factorization into prime power factors (even if the exponents are different) must be equal to . (Contributed by Mario Carneiro, 21-Apr-2016.)
DProd DProd

Theorempgpfac1lem1 15144* Lemma for pgpfac1 15150. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp

Theorempgpfac1lem2 15145* Lemma for pgpfac1 15150. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp               .g

Theorempgpfac1lem3a 15146* Lemma for pgpfac1 15150. (Contributed by Mario Carneiro, 4-Jun-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp               .g

Theorempgpfac1lem3 15147* Lemma for pgpfac1 15150. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp               .g                            SubGrp

Theorempgpfac1lem4 15148* Lemma for pgpfac1 15150. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp               .g       SubGrp

Theorempgpfac1lem5 15149* Lemma for pgpfac1 15150 (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp SubGrp        SubGrp

Theorempgpfac1 15150* Factorization of a finite abelian p-group. There is a direct product decomposition of any abelian group of prime-power order where one of the factors is cyclic and generated by an element of maximal order. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                                    SubGrp

Theorempgpfaclem1 15151* Lemma for pgpfac 15154. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp               pGrp               SubGrp       SubGrp Word DProd DProd        s        mrClsSubGrp              gEx                                          SubGrp                     Word        DProd        DProd        concat        Word DProd DProd

Theorempgpfaclem2 15152* Lemma for pgpfac 15154. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp               pGrp               SubGrp       SubGrp Word DProd DProd        s        mrClsSubGrp              gEx                                          SubGrp                     Word DProd DProd

Theorempgpfaclem3 15153* Lemma for pgpfac 15154. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp               pGrp               SubGrp       SubGrp Word DProd DProd        Word DProd DProd

Theorempgpfac 15154* Full factorization of a finite abelian p-group, by iterating pgpfac1 15150. There is a direct product decomposition of any abelian group of prime-power order into cyclic subgroups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp               pGrp               Word DProd DProd

Theoremablfaclem1 15155* Lemma for ablfac 15158. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp                                           SubGrp Word DProd DProd        SubGrp Word DProd DProd

Theoremablfaclem2 15156* Lemma for ablfac 15158. (Contributed by Mario Carneiro, 27-Apr-2016.) (Proof shortened by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp                                           SubGrp Word DProd DProd        Word                      ..^

Theoremablfaclem3 15157* Lemma for ablfac 15158. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp                                           SubGrp Word DProd DProd

Theoremablfac 15158* 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 15159* Choose generators for each cyclic group in ablfac 15158. (Contributed by Mario Carneiro, 28-Apr-2016.)
SubGrp s CycGrp pGrp                      .g              Word DProd DProd

10.4  Rings

10.4.1  Multiplicative Group

Syntaxcmgp 15160 Multiplicative group.
mulGrp

Definitiondf-mgp 15161 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 15284 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 15182) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 8567). (Contributed by Mario Carneiro, 21-Dec-2014.)
mulGrp sSet

Theoremfnmgp 15162 The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
mulGrp

Theoremmgpval 15163 Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
mulGrp              sSet

Theoremmgpplusg 15164 Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
mulGrp

Theoremmgplem 15165 Lemma for mgpbas 15166. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       Slot

Theoremmgpbas 15166 Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoremmgpsca 15167 The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 15236. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
mulGrp       Scalar       Scalar

Theoremmgptset 15168 Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       TopSet TopSet

Theoremmgptopn 15169 Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoremmgpds 15170 Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoremmgpress 15171 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 15172 Extend class notation with class of all (unital) rings.

Syntaxccrg 15173 Extend class notation with class of all (unital) commutative rings.

Syntaxcur 15174 Extend class notation with ring unit.

Definitiondf-ring 15175* 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 15204), 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 15176 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
mulGrp CMnd

Definitiondf-ur 15177 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 15161). See also dfur2 15179, 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 15178 The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
mulGrp

Theoremdfur2 15179* 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 15180* 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 15181 A ring is a group. (Contributed by NM, 15-Sep-2011.)

Theoremrngmgp 15182 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
mulGrp

Theoremiscrng 15183 A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
mulGrp       CMnd

Theoremcrngmgp 15184 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
mulGrp       CMnd

Theoremrngmnd 15185 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremcrngrng 15186 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremmgpf 15187 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
mulGrp

Theoremrngi 15188 Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremrngcl 15189 Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremcrngcom 15190 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremiscrng2 15191* A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)

Theoremrngass 15192 Associative law for the multiplication operation of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremrngideu 15193* The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremrngdi 15194 Distributive law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.)

Theoremrngdir 15195 Distributive law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.)

Theoremrngidcl 15196 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 15197 The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.)

Theoremrngidmlem 15198 Lemma for rnglidm 15199 and rngridm 15200. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremrnglidm 15199 The unit element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.)

Theoremrngridm 15200 The unit element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.)

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