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Theorem List for Metamath Proof Explorer - 15501-15600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgsum2d 15501* Write a sum over a two-dimensional region as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                                 g g g

Theoremgsum2d2lem 15502* Lemma for gsum2d2 15503: show the function is finitely supported. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd

Theoremgsum2d2 15503* Write a group sum over a two-dimensional region as a double sum. (Note that is a function of .) (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                          g g g

Theoremgsumcom2 15504* Two-dimensional commutation of a group sum. Note that while and are constants w.r.t. , and are not. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                                        g g

Theoremgsumxp 15505* Write a group sum over a cartesian product as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                   g g g

Theoremgsumcom 15506* Commute the arguments of a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                          g g

Theoremprdsgsum 15507* Finite commutative sums in a product structure are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2015.)
s                                           CMnd                     g g

Theorempwsgsum 15508* Finite commutative sums in a power structure are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2015.)
s                                    CMnd                     g g

10.3.4  Internal direct products

Syntaxcdprd 15509 Internal direct product of a family of subgroups.
DProd

Syntaxcdpj 15510 Internal direct product of a family of subgroups.
dProj

Definitiondf-dprd 15511* Define the internal direct product of a family of subgroups. (Contributed by Mario Carneiro, 21-Apr-2016.)
DProd SubGrp Cntz mrClsSubGrp g

Definitiondf-dpj 15512* Define the projection operator for a direct product. (Contributed by Mario Carneiro, 21-Apr-2016.)
dProj DProd DProd

Theoremreldmdprd 15513 The domain of definition of the internal direct product, which states that is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdmdprd 15514* The domain of definition of the internal direct product, which states that is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz              mrClsSubGrp       DProd SubGrp

Theoremdmdprdd 15515* Show that a given family is a direct product decomposition. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz              mrClsSubGrp                     SubGrp                     DProd

Theoremdprdval 15516* The domain of definition of the internal direct product, which states that is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd DProd g

Theoremeldprd 15517* The domain of definition of the internal direct product, which states that is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd DProd g

Theoremdprdgrp 15518 Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdf 15519 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd SubGrp

Theoremdprdf2 15520 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd               SubGrp

Theoremdprdcntz 15521 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                                    Cntz

Theoremdprddisj 15522 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                             mrClsSubGrp

Theoremdprdw 15523* The property of being a finitely supported function in the family . (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdwd 15524* The property of being a finitely supported function in the family . (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdff 15525* A finitely supported function in is a function into the base. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdfcl 15526* A finitely supported function in has its -th element in . (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdffi 15527* The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdfcntz 15528* A function on the elements of an internal direct product has pairwise-commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      Cntz

Theoremdprdssv 15529 The internal direct product of a family of subgroups is a subset of the base. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdfid 15530* The zero function is the only function that sums two zero in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                                    g

Theoremeldprdi 15531* The domain of definition of the internal direct product, which states that is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      g DProd

Theoremdprdfinv 15532* Take the inverse of a group sum over a family of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                             g g

Theoremdprdfadd 15533* Take the sum of group sums over two families of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                                    g g g

Theoremdprdfsub 15534* Take the difference of group sums over two families of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                                    g g g

Theoremdprdfeq0 15535* The zero function is the only function that sums two zero in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      g

Theoremdprdf11 15536* Two group sums over a direct product that give the same value are equal as functions. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                             g g

Theoremdprdsubg 15537 The internal direct product of a family of subgroups is a subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd DProd SubGrp

Theoremdprdub 15538 Each factor is a subset of the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd

Theoremdprdlub 15539* The direct product is smaller than any subgroup which contains the factors. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd               SubGrp              DProd

Theoremdprdspan 15540 The direct product is the span of the union of the factors. (Contributed by Mario Carneiro, 25-Apr-2016.)
mrClsSubGrp       DProd DProd

Theoremdprdres 15541 Restriction of a direct product (dropping factors). (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd DProd DProd

Theoremdprdss 15542* Create a direct product by finding subgroups inside each factor of another direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd               SubGrp              DProd DProd DProd

Theoremdprdz 15543* A family consisting entirely of trivial groups is an internal direct product, the product of which is the trivial subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd DProd

Theoremdprd0 15544 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 15545 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd DProd DProd

Theoremdprdf1 15546 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd DProd DProd

Theoremsubgdmdprd 15547 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
s        SubGrp DProd DProd

Theoremsubgdprd 15548 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
s        SubGrp       DProd               DProd DProd

Theoremdprdsn 15549 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 15550* Lemma for dmdprdsplit 15560. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                             g DProd

Theoremdprdcntz2 15551 The function is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                                    Cntz       DProd DProd

Theoremdprddisj2 15552 The function is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                                           DProd DProd

Theoremdprd2dlem2 15553* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp              DProd        DProd DProd        mrClsSubGrp       DProd

Theoremdprd2dlem1 15554* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp              DProd        DProd DProd        mrClsSubGrp              DProd DProd

Theoremdprd2da 15555* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp              DProd        DProd DProd        mrClsSubGrp       DProd

Theoremdprd2db 15556* 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 15557* 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 15558 Lemma for dmdprdsplit 15560. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp                     Cntz              DProd        DProd        DProd DProd        DProd DProd        mrClsSubGrp

Theoremdmdprdsplit2 15559 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 15560 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 15561 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 15562 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 15563 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 15564 Lemma for theorems about direct product projection. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                      DProd

Theoremdpjcntz 15565 The two subgroups that appear in dpjval 15569 commute. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                      Cntz       DProd

Theoremdpjdisj 15566 The two subgroups that appear in dpjval 15569 are disjoint. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                             DProd

Theoremdpjlsm 15567 The two subgroups that appear in dpjval 15569 add to the full direct product. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                             DProd DProd

Theoremdpjfval 15568* Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj              DProd

Theoremdpjval 15569 Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj                     DProd

Theoremdpjf 15570 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 15571* 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 15572* Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj       DProd                             g

Theoremdpjid 15573* 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 15574 The -th index projection acts as the identity on elements of the -th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj

Theoremdpjrid 15575 The -th index projection annihilates elements of other factors. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj

Theoremdpjghm 15576 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 15577 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 15578* Lemma for ablfacrp2 15580. (Contributed by Mario Carneiro, 19-Apr-2016.)

Theoremablfacrp 15579* 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 15580* The factors of ablfacrp 15579 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 15581* Lemma for ablfac1b 15583. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-Apr-2016.)

Theoremablfac1a 15582* The factors of ablfac1b 15583 are of prime power order. (Contributed by Mario Carneiro, 26-Apr-2016.)

Theoremablfac1b 15583* 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 15584* The factors of ablfac1b 15583 cover the entire group. (Contributed by Mario Carneiro, 21-Apr-2016.)
DProd

Theoremablfac1eulem 15585* Lemma for ablfac1eu 15586. (Contributed by Mario Carneiro, 27-Apr-2016.)
DProd DProd                                           DProd

Theoremablfac1eu 15586* The factorization of ablfac1b 15583 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 15587* Lemma for pgpfac1 15593. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp

Theorempgpfac1lem2 15588* Lemma for pgpfac1 15593. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp               .g

Theorempgpfac1lem3a 15589* Lemma for pgpfac1 15593. (Contributed by Mario Carneiro, 4-Jun-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp               .g

Theorempgpfac1lem3 15590* Lemma for pgpfac1 15593. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp               .g                            SubGrp

Theorempgpfac1lem4 15591* Lemma for pgpfac1 15593. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp                     SubGrp               .g       SubGrp

Theorempgpfac1lem5 15592* Lemma for pgpfac1 15593 (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp                            gEx                     pGrp                             SubGrp              SubGrp SubGrp        SubGrp

Theorempgpfac1 15593* 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 15594* Lemma for pgpfac 15597. (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 15595* Lemma for pgpfac 15597. (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 15596* Lemma for pgpfac 15597. (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 15597* Full factorization of a finite abelian p-group, by iterating pgpfac1 15593. 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 15598* Lemma for ablfac 15601. (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 15599* Lemma for ablfac 15601. (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 15600* Lemma for ablfac 15601. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
SubGrp s CycGrp pGrp                                           SubGrp Word DProd DProd

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