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Theorem List for Metamath Proof Explorer - 15201-15300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgsumadd 15201 The sum of two group sums. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 25-Apr-2016.)
CMnd                                          g g g

Theoremgsumzsplit 15202 Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                                        g g g

Theoremgsumsplit 15203 Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.)
CMnd                                          g g g

Theoremgsumsplit2 15204* Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.)
CMnd                                          g g g

Theoremgsumconst 15205* Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
.g       g

Theoremgsumzmhm 15206 Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                            MndHom                                    g g

Theoremgsummhm 15207 Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
CMnd                     MndHom                      g g

Theoremgsummhm2 15208* Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.)
CMnd                     MndHom                             g        g

Theoremgsummulglem 15209* Lemma for gsummulg 15210 and gsummulgz 15211. (Contributed by Mario Carneiro, 7-Jan-2015.)
.g                            CMnd                     g g

Theoremgsummulg 15210* Nonnegative multiple of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 7-Jan-2015.)
.g                            CMnd              g g

Theoremgsummulgz 15211* Integer multiple of a group sum. (Contributed by Mario Carneiro, 7-Jan-2015.)
.g                                          g g

Theoremgsumzoppg 15212 The opposite of a group sum is the same as the original. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz       oppg                                          g g

Theoremgsumzinv 15213 Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                                 g g

Theoremgsuminv 15214 Inverse of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 4-May-2015.)
g g

Theoremgsumsub 15215 The difference of two group sums. (Contributed by Mario Carneiro, 28-Dec-2014.)
g g g

Theoremgsumsn 15216* Group sum of a singleton. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
g

Theoremgsumunsn 15217* Append an element to a finite group sum. (Contributed by Mario Carneiro, 19-Dec-2014.)
CMnd                                                 g g

Theoremgsumpt 15218 Sum of a family that is nonzero at at most one point. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 25-Apr-2016.)
g

Theoremgsum2d 15219* Write a sum over a two-dimensional region as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                                 g g g

Theoremgsum2d2lem 15220* Lemma for gsum2d2 15221: show the function is finitely supported. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd

Theoremgsum2d2 15221* 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 15222* 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 15223* Write a group sum over a cartesian product as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                   g g g

Theoremgsumcom 15224* Commute the arguments of a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                          g g

Theoremprdsgsum 15225* 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 15226* 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 15227 Internal direct product of a family of subgroups.
DProd

Syntaxcdpj 15228 Internal direct product of a family of subgroups.
dProj

Definitiondf-dprd 15229* Define the internal direct product of a family of subgroups. (Contributed by Mario Carneiro, 21-Apr-2016.)
DProd SubGrp Cntz mrClsSubGrp g

Definitiondf-dpj 15230* Define the projection operator for a direct product. (Contributed by Mario Carneiro, 21-Apr-2016.)
dProj DProd DProd

Theoremreldmdprd 15231 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 15232* 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 15233* Show that a given family is a direct product decomposition. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz              mrClsSubGrp                     SubGrp                     DProd

Theoremdprdval 15234* 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 15235* 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 15236 Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdf 15237 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd SubGrp

Theoremdprdf2 15238 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd               SubGrp

Theoremdprdcntz 15239 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                                    Cntz

Theoremdprddisj 15240 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                             mrClsSubGrp

Theoremdprdw 15241* The property of being a finitely supported function in the family . (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdwd 15242* The property of being a finitely supported function in the family . (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdff 15243* A finitely supported function in is a function into the base. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdfcl 15244* A finitely supported function in has its -th element in . (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdffi 15245* The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdfcntz 15246* A function on the elements of an internal direct product has pairwise-commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      Cntz

Theoremdprdssv 15247 The internal direct product of a family of subgroups is a subset of the base. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdfid 15248* 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 15249* 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 15250* 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 15251* 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 15252* 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 15253* 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 15254* 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 15255 The internal direct product of a family of subgroups is a subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd DProd SubGrp

Theoremdprdub 15256 Each factor is a subset of the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd

Theoremdprdlub 15257* The direct product is smaller than any subgroup which contains the factors. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd               SubGrp              DProd

Theoremdprdspan 15258 The direct product is the span of the union of the factors. (Contributed by Mario Carneiro, 25-Apr-2016.)
mrClsSubGrp       DProd DProd

Theoremdprdres 15259 Restriction of a direct product (dropping factors). (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd DProd DProd

Theoremdprdss 15260* 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 15261* 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 15262 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 15263 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd DProd DProd

Theoremdprdf1 15264 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd DProd DProd

Theoremsubgdmdprd 15265 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
s        SubGrp DProd DProd

Theoremsubgdprd 15266 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
s        SubGrp       DProd               DProd DProd

Theoremdprdsn 15267 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 15268* Lemma for dmdprdsplit 15278. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                             g DProd

Theoremdprdcntz2 15269 The function is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                                    Cntz       DProd DProd

Theoremdprddisj2 15270 The function is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                                           DProd DProd

Theoremdprd2dlem2 15271* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp              DProd        DProd DProd        mrClsSubGrp       DProd

Theoremdprd2dlem1 15272* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp              DProd        DProd DProd        mrClsSubGrp              DProd DProd

Theoremdprd2da 15273* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp              DProd        DProd DProd        mrClsSubGrp       DProd

Theoremdprd2db 15274* 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 15275* 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 15276 Lemma for dmdprdsplit 15278. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp                     Cntz              DProd        DProd        DProd DProd        DProd DProd        mrClsSubGrp

Theoremdmdprdsplit2 15277 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 15278 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 15279 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 15280 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 15281 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 15282 Lemma for theorems about direct product projection. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                      DProd

Theoremdpjcntz 15283 The two subgroups that appear in dpjval 15287 commute. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                      Cntz       DProd

Theoremdpjdisj 15284 The two subgroups that appear in dpjval 15287 are disjoint. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                             DProd

Theoremdpjlsm 15285 The two subgroups that appear in dpjval 15287 add to the full direct product. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                             DProd DProd

Theoremdpjfval 15286* Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj              DProd

Theoremdpjval 15287 Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj                     DProd

Theoremdpjf 15288 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 15289* 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 15290* Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj       DProd                             g

Theoremdpjid 15291* 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 15292 The -th index projection acts as the identity on elements of the -th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj

Theoremdpjrid 15293 The -th index projection annihilates elements of other factors. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd               dProj

Theoremdpjghm 15294 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 15295 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 15296* Lemma for ablfacrp2 15298. (Contributed by Mario Carneiro, 19-Apr-2016.)

Theoremablfacrp 15297* 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 15298* The factors of ablfacrp 15297 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 15299* Lemma for ablfac1b 15301. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-Apr-2016.)

Theoremablfac1a 15300* The factors of ablfac1b 15301 are of prime power order. (Contributed by Mario Carneiro, 26-Apr-2016.)

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