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

Theoremprdscmnd 15101 The product of a family of commutative monoids is commutative. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s                     CMnd       CMnd

Theoremprdsabld 15102 The product of a family of Abelian groups is an Abelian group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theorempwscmn 15103 The structure power on a commutative monoid is commutative. (Contributed by Mario Carneiro, 11-Jan-2015.)
s        CMnd CMnd

Theorempwsabl 15104 The structure power on an Abelian group is Abelian. (Contributed by Mario Carneiro, 21-Jan-2015.)
s

Theoremdivsabl 15105 If is a subgroup of the abelian group , then is an abelian group. (Contributed by Mario Carneiro, 26-Apr-2016.)
s ~QG        SubGrp

Theoremcnaddablx 15106 The complex numbers are an Abelian group under addition. This version of cnaddabl 15107 shows the explicit structure "scaffold" we chose for the definition for Abelian groups. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use cnaddabl 15107 instead. (Contributed by NM, 18-Oct-2012.)

Theoremcnaddabl 15107 The complex numbers are an Abelian group under addition. This version of cnaddablx 15106 hides the explicit structure indices i.e. is "scaffold-independent". Note that the proof also does not reference explicit structure indices. The actual structure is dependent on how and is defined. This theorem should not be referenced in any proof. For the group/ring properties of the complex numbers, see cnrng 16344. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.)

Theoremzaddablx 15108 The integers are an Abelian group under addition. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. Use zsubrg 16373 instead. (Contributed by NM, 4-Sep-2011.)

Theoremfrgpnabllem1 15109* Lemma for frgpnabl 15111. (Contributed by Mario Carneiro, 21-Apr-2016.)
freeGrp       Word        ~FG                      splice               varFGrp

Theoremfrgpnabllem2 15110* Lemma for frgpnabl 15111. (Contributed by Mario Carneiro, 21-Apr-2016.)
freeGrp       Word        ~FG                      splice               varFGrp

Theoremfrgpnabl 15111 The free group on two or more generators is not abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
freeGrp

10.3.2  Cyclic groups

Syntaxccyg 15112 Cyclic group.
CycGrp

Definitiondf-cyg 15113* Define a cyclic group, which is a group with an element , called the generator of the group, such that all elements in the group are multiples of . A generator is usually not unique. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp .g

Theoremiscyg 15114* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g       CycGrp

Theoremiscyggen 15115* The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g

Theoremiscyggen2 15116* The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g

Theoremiscyg2 15117* A cyclic group is a group which contains a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g              CycGrp

Theoremcyggeninv 15118* The inverse of a cyclic generator is a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g

Theoremcyggenod 15119* An element is the generator of a finite group iff the order of the generator equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g

Theoremcyggenod2 15120* In an infinite cyclic group, the generator must have infinite order, but this property no longer characterizes the generators. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g

Theoremiscyg3 15121* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g       CycGrp

Theoremiscygd 15122* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
.g                            CycGrp

Theoremiscygodd 15123 Show that a group with an element the same order as the group is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
CycGrp

Theoremcyggrp 15124 A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theoremcygabl 15125 A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theoremcygctb 15126 A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theorem0cyg 15127 The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp

Theoremprmcyg 15128 A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
CycGrp

Theoremlt6abl 15129 A group with fewer than elements is abelian. (Contributed by Mario Carneiro, 27-Apr-2016.)

Theoremghmcyg 15130 The image of a cyclic group under a surjective group homomorphism is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
CycGrp CycGrp

Theoremcyggex2 15131 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 15132 The exponent of a finite cyclic group is the order of the group. (Contributed by Mario Carneiro, 24-Apr-2016.)
gEx       CycGrp

Theoremcyggexb 15133 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 15134 Cyclicity is a group property, i.e. it is preserved under isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑔 CycGrp CycGrp

Theoremcycsubgcyg 15135* The cyclic subgroup generated by is a cyclic group. (Contributed by Mario Carneiro, 24-Apr-2016.)
.g              s CycGrp

Theoremcycsubgcyg2 15136 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 15137* Value of the group sum operation over an index set with finite support. (Contributed by Mario Carneiro, 7-Dec-2014.)
Cntz                                                               g

Theoremgsumval3eu 15138* The group sum as defined in gsumval3a 15137 is uniquely defined. (Contributed by Mario Carneiro, 8-Dec-2014.)
Cntz

Theoremgsumval3 15139 Value of the group sum operation over an arbitrary finite set. (Contributed by Mario Carneiro, 15-Dec-2014.)
Cntz                                                               g

Theoremcntzcmnf 15140 Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz       CMnd

Theoremgsumcllem 15141* Lemma for gsumcl 15146 and related theorems. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)

Theoremgsumzres 15142 Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                                                 g g

Theoremgsumzcl 15143 Closure of a finite group sum. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                                          g

Theoremgsumzf1o 15144 Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                                                 g g

Theoremgsumres 15145 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 15146 Closure of a finite group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
CMnd                            g

Theoremgsumf1o 15147 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 15148 Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                     SubMnd                            g

Theoremgsumsubmcl 15149 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 15150 Closure of a group sum in a subgroup. (Contributed by Mario Carneiro, 15-Dec-2014.)
SubGrp                     g

Theoremgsumzaddlem 15151* The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                                                             g        g g g

Theoremgsumzadd 15152 The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                   SubMnd                            g g g

Theoremgsumadd 15153 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 15154 Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                                        g g g

Theoremgsumsplit 15155 Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.)
CMnd                                          g g g

Theoremgsumsplit2 15156* Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.)
CMnd                                          g g g

Theoremgsumconst 15157* Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
.g       g

Theoremgsumzmhm 15158 Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.)
Cntz                            MndHom                                    g g

Theoremgsummhm 15159 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 15160* 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 15161* Lemma for gsummulg 15162 and gsummulgz 15163. (Contributed by Mario Carneiro, 7-Jan-2015.)
.g                            CMnd                     g g

Theoremgsummulg 15162* 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 15163* Integer multiple of a group sum. (Contributed by Mario Carneiro, 7-Jan-2015.)
.g                                          g g

Theoremgsumzoppg 15164 The opposite of a group sum is the same as the original. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz       oppg                                          g g

Theoremgsumzinv 15165 Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz                                                 g g

Theoremgsuminv 15166 Inverse of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 4-May-2015.)
g g

Theoremgsumsub 15167 The difference of two group sums. (Contributed by Mario Carneiro, 28-Dec-2014.)
g g g

Theoremgsumsn 15168* Group sum of a singleton. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
g

Theoremgsumunsn 15169* Append an element to a finite group sum. (Contributed by Mario Carneiro, 19-Dec-2014.)
CMnd                                                 g g

Theoremgsumpt 15170 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 15171* Write a sum over a two-dimensional region as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                                 g g g

Theoremgsum2d2lem 15172* Lemma for gsum2d2 15173: show the function is finitely supported. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd

Theoremgsum2d2 15173* 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 15174* 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 15175* Write a group sum over a cartesian product as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                   g g g

Theoremgsumcom 15176* Commute the arguments of a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
CMnd                                          g g

Theoremprdsgsum 15177* 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 15178* 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 15179 Internal direct product of a family of subgroups.
DProd

Syntaxcdpj 15180 Internal direct product of a family of subgroups.
dProj

Definitiondf-dprd 15181* Define the internal direct product of a family of subgroups. (Contributed by Mario Carneiro, 21-Apr-2016.)
DProd SubGrp Cntz mrClsSubGrp g

Definitiondf-dpj 15182* Define the projection operator for a direct product. (Contributed by Mario Carneiro, 21-Apr-2016.)
dProj DProd DProd

Theoremreldmdprd 15183 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 15184* 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 15185* Show that a given family is a direct product decomposition. (Contributed by Mario Carneiro, 25-Apr-2016.)
Cntz              mrClsSubGrp                     SubGrp                     DProd

Theoremdprdval 15186* 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 15187* 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 15188 Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdf 15189 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd SubGrp

Theoremdprdf2 15190 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd               SubGrp

Theoremdprdcntz 15191 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                                    Cntz

Theoremdprddisj 15192 The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                             mrClsSubGrp

Theoremdprdw 15193* The property of being a finitely supported function in the family . (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdwd 15194* The property of being a finitely supported function in the family . (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdff 15195* A finitely supported function in is a function into the base. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdfcl 15196* A finitely supported function in has its -th element in . (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdffi 15197* The function is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdfcntz 15198* A function on the elements of an internal direct product has pairwise-commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                      Cntz

Theoremdprdssv 15199 The internal direct product of a family of subgroups is a subset of the base. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd

Theoremdprdfid 15200* The zero function is the only function that sums two zero in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
DProd                                    g

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