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

Theoremcygabl 15501 A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theoremcygctb 15502 A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theorem0cyg 15503 The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theoremprmcyg 15504 A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
CycGrp

Theoremlt6abl 15505 A group with fewer than elements is abelian. (Contributed by Mario Carneiro, 27-Apr-2016.)

Theoremghmcyg 15506 The image of a cyclic group under a surjective group homomorphism is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp CycGrp

Theoremcyggex2 15507 The exponent of a cyclic group is if the group is infinite, otherwise it equals the order of the group. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx       CycGrp

Theoremcyggex 15508 The exponent of a finite cyclic group is the order of the group. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx       CycGrp

Theoremcyggexb 15509 A finite abelian group is cyclic iff the exponent equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
gEx       CycGrp

Theoremgiccyg 15510 Cyclicity is a group property, i.e. it is preserved under isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑔 CycGrp CycGrp

Theoremcycsubgcyg 15511* The cyclic subgroup generated by is a cyclic group. (Contributed by Mario Carneiro, 24-Apr-2016.)
.g              s CycGrp

Theoremcycsubgcyg2 15512 The cyclic subgroup generated by is a cyclic group. (Contributed by Mario Carneiro, 27-Apr-2016.)
mrClsSubGrp       s CycGrp

10.3.3  Group sum operation

Theoremgsumval3a 15513* Value of the group sum operation over an index set with finite support. (Contributed by Mario Carneiro, 7-Dec-2014.)
Cntz                                                               g

Theoremgsumval3eu 15514* The group sum as defined in gsumval3a 15513 is uniquely defined. (Contributed by Mario Carneiro, 8-Dec-2014.)
Cntz

Theoremgsumval3 15515 Value of the group sum operation over an arbitrary finite set. (Contributed by Mario Carneiro, 15-Dec-2014.)
Cntz                                                               g

Theoremcntzcmnf 15516 Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz       CMnd

Theoremgsumcllem 15517* Lemma for gsumcl 15522 and related theorems. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)

Theoremgsumzres 15518 Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                                                 g g

Theoremgsumzcl 15519 Closure of a finite group sum. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                                          g

Theoremgsumzf1o 15520 Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                                                 g g

Theoremgsumres 15521 Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
CMnd                                   g g

Theoremgsumcl 15522 Closure of a finite group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
CMnd                            g

Theoremgsumf1o 15523 Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
CMnd                                   g g

Theoremgsumzsubmcl 15524 Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                     SubMnd                            g

Theoremgsumsubmcl 15525 Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by Mario Carneiro, 24-Apr-2016.)
CMnd              SubMnd                     g

Theoremgsumsubgcl 15526 Closure of a group sum in a subgroup. (Contributed by Mario Carneiro, 15-Dec-2014.)
SubGrp                     g

Theoremgsumzaddlem 15527* The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                                                             g        g g g

Theoremgsumzadd 15528 The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                   SubMnd                            g g g

Theoremgsumadd 15529 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 15530 Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                                        g g g

Theoremgsumsplit 15531 Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.)
CMnd                                          g g g

Theoremgsumsplit2 15532* Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.)
CMnd                                          g g g

Theoremgsumconst 15533* Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
.g       g

Theoremgsumzmhm 15534 Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                            MndHom                                    g g

Theoremgsummhm 15535 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 15536* 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 15537* Lemma for gsummulg 15538 and gsummulgz 15539. (Contributed by Mario Carneiro, 7-Jan-2015.)
.g                            CMnd                     g g

Theoremgsummulg 15538* 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 15539* Integer multiple of a group sum. (Contributed by Mario Carneiro, 7-Jan-2015.)
.g                                          g g

Theoremgsumzoppg 15540 The opposite of a group sum is the same as the original. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz       oppg                                          g g

Theoremgsumzinv 15541 Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                                 g g

Theoremgsuminv 15542 Inverse of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 4-May-2015.)
g g

Theoremgsumsub 15543 The difference of two group sums. (Contributed by Mario Carneiro, 28-Dec-2014.)
g g g

Theoremgsumsn 15544* Group sum of a singleton. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
g

Theoremgsumunsn 15545* Append an element to a finite group sum. (Contributed by Mario Carneiro, 19-Dec-2014.)
CMnd                                                 g g

Theoremgsumpt 15546 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 15547* Write a sum over a two-dimensional region as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                                 g g g

Theoremgsum2d2lem 15548* Lemma for gsum2d2 15549: show the function is finitely supported. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd

Theoremgsum2d2 15549* 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 15550* 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 15551* Write a group sum over a cartesian product as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                   g g g

Theoremgsumcom 15552* Commute the arguments of a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                          g g

Theoremprdsgsum 15553* 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 15554* 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 15555 Internal direct product of a family of subgroups.
DProd

Syntaxcdpj 15556 Internal direct product of a family of subgroups.
dProj

Definitiondf-dprd 15557* Define the internal direct product of a family of subgroups. (Contributed by Mario Carneiro, 21-Apr-2016.)
DProd SubGrp Cntz mrClsSubGrp g

Definitiondf-dpj 15558* Define the projection operator for a direct product. (Contributed by Mario Carneiro, 21-Apr-2016.)
dProj DProd DProd

Theoremreldmdprd 15559 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 15560* 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 15561* Show that a given family is a direct product decomposition. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz              mrClsSubGrp                     SubGrp                     DProd

Theoremdprdval 15562* 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 15563* 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 15564 Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdf 15565 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd SubGrp

Theoremdprdf2 15566 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd               SubGrp

Theoremdprdcntz 15567 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                                    Cntz

Theoremdprddisj 15568 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                             mrClsSubGrp

Theoremdprdw 15569* The property of being a finitely supported function in the family . (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdwd 15570* The property of being a finitely supported function in the family . (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdff 15571* A finitely supported function in is a function into the base. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdfcl 15572* A finitely supported function in has its -th element in . (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdffi 15573* The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdfcntz 15574* A function on the elements of an internal direct product has pairwise-commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      Cntz

Theoremdprdssv 15575 The internal direct product of a family of subgroups is a subset of the base. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdfid 15576* 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 15577* 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 15578* 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 15579* 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 15580* 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 15581* 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 15582* 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 15583 The internal direct product of a family of subgroups is a subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd DProd SubGrp

Theoremdprdub 15584 Each factor is a subset of the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd

Theoremdprdlub 15585* The direct product is smaller than any subgroup which contains the factors. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd               SubGrp              DProd

Theoremdprdspan 15586 The direct product is the span of the union of the factors. (Contributed by Mario Carneiro, 25-Apr-2016.)
mrClsSubGrp       DProd DProd

Theoremdprdres 15587 Restriction of a direct product (dropping factors). (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd DProd DProd

Theoremdprdss 15588* 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 15589* 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 15590 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 15591 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd DProd DProd

Theoremdprdf1 15592 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      DProd DProd DProd

Theoremsubgdmdprd 15593 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
s        SubGrp DProd DProd

Theoremsubgdprd 15594 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
s        SubGrp       DProd               DProd DProd

Theoremdprdsn 15595 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 15596* Lemma for dmdprdsplit 15606. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                             g DProd

Theoremdprdcntz2 15597 The function is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                                    Cntz       DProd DProd

Theoremdprddisj2 15598 The function is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
DProd                                           DProd DProd

Theoremdprd2dlem2 15599* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp              DProd        DProd DProd        mrClsSubGrp       DProd

Theoremdprd2dlem1 15600* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
SubGrp              DProd        DProd DProd        mrClsSubGrp              DProd DProd

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