P. 139 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13801-13900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ghmnsgpreima 13801	The inverse image of a normal subgroup under a homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- ( ( F e. ( S GrpHom T ) /\ V e. (NrmSGrp ` T ) ) -> ( `' F " V ) e. (NrmSGrp ` S ) )
Theorem	ghmker 13802	The kernel of a homomorphism is a normal subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- .0. = ( 0g ` T )   =>    |- ( F e. ( S GrpHom T ) -> ( `' F " { .0. } ) e. (NrmSGrp ` S ) )
Theorem	ghmeqker 13803	Two source points map to the same destination point under a group homomorphism iff their difference belongs to the kernel. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- B = ( Base ` S )   &    |- .0. = ( 0g ` T )   &    |- K = ( `' F " { .0. } )   &    |- .- = ( -g ` S )   =>    |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( ( F ` U ) = ( F ` V ) <-> ( U .- V ) e. K ) )
Theorem	f1ghm0to0 13804	If a group homomorphism F is injective, it maps the zero of one group (and only the zero) to the zero of the other group. (Contributed by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 13-May-2023.)
|- A = ( Base ` R )   &    |- B = ( Base ` S )   &    |- N = ( 0g ` R )   &    |- .0. = ( 0g ` S )   =>    |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( ( F ` X ) = .0. <-> X = N ) )
Theorem	ghmf1 13805*	Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 4-Apr-2025.)
|- A = ( Base ` R )   &    |- B = ( Base ` S )   &    |- N = ( 0g ` R )   &    |- .0. = ( 0g ` S )   =>    |- ( F e. ( R GrpHom S ) -> ( F : A -1-1-> B <-> A. x e. A ( ( F ` x ) = .0. -> x = N ) ) )
Theorem	kerf1ghm 13806	A group homomorphism F is injective if and only if its kernel is the singleton { N }. (Contributed by Thierry Arnoux, 27-Oct-2017.) (Proof shortened by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 13-May-2023.)
|- A = ( Base ` R )   &    |- B = ( Base ` S )   &    |- N = ( 0g ` R )   &    |- .0. = ( 0g ` S )   =>    |- ( F e. ( R GrpHom S ) -> ( F : A -1-1-> B <-> ( `' F " { .0. } ) = { N } ) )
Theorem	ghmf1o 13807	A bijective group homomorphism is an isomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.)
|- X = ( Base ` S )   &    |- Y = ( Base ` T )   =>    |- ( F e. ( S GrpHom T ) -> ( F : X -1-1-onto-> Y <-> `' F e. ( T GrpHom S ) ) )
Theorem	conjghm 13808*	Conjugation is an automorphism of the group. (Contributed by Mario Carneiro, 13-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- F = ( x e. X |-> ( ( A .+ x ) .- A ) )   =>    |- ( ( G e. Grp /\ A e. X ) -> ( F e. ( G GrpHom G ) /\ F : X -1-1-onto-> X ) )
Theorem	conjsubg 13809*	A conjugated subgroup is also a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- F = ( x e. S |-> ( ( A .+ x ) .- A ) )   =>    |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> ran F e. (SubGrp ` G ) )
Theorem	conjsubgen 13810*	A conjugated subgroup is equinumerous to the original subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- F = ( x e. S |-> ( ( A .+ x ) .- A ) )   =>    |- ( ( S e. (SubGrp ` G ) /\ A e. X ) -> S ~~ ran F )
Theorem	conjnmz 13811*	A subgroup is unchanged under conjugation by an element of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- F = ( x e. S |-> ( ( A .+ x ) .- A ) )   &    |- N = { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) }   =>    |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> S = ran F )
Theorem	conjnmzb 13812*	Alternative condition for elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- F = ( x e. S |-> ( ( A .+ x ) .- A ) )   &    |- N = { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) }   =>    |- ( S e. (SubGrp ` G ) -> ( A e. N <-> ( A e. X /\ S = ran F ) ) )
Theorem	conjnsg 13813*	A normal subgroup is unchanged under conjugation. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- F = ( x e. S |-> ( ( A .+ x ) .- A ) )   =>    |- ( ( S e. (NrmSGrp ` G ) /\ A e. X ) -> S = ran F )
Theorem	qusghm 13814*	If Y is a normal subgroup of G, then the "natural map" from elements to their cosets is a group homomorphism from G to G / Y. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 18-Sep-2015.)
|- X = ( Base ` G )   &    |- H = ( G /.s ( G ~QG Y ) )   &    |- F = ( x e. X |-> [ x ] ( G ~QG Y ) )   =>    |- ( Y e. (NrmSGrp ` G ) -> F e. ( G GrpHom H ) )
Theorem	ghmpropd 13815*	Group homomorphism depends only on the group attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
|- ( ph -> B = ( Base ` J ) )   &    |- ( ph -> C = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ph -> C = ( Base ` M ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` J ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` M ) y ) )   =>    |- ( ph -> ( J GrpHom K ) = ( L GrpHom M ) )
7.2.5  Abelian groups
7.2.5.1  Definition and basic properties
Syntax	ccmn 13816	Extend class notation with class of all commutative monoids.
class CMnd
Syntax	cabl 13817	Extend class notation with class of all Abelian groups.
class Abel
Definition	df-cmn 13818*	Define class of all commutative monoids. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- CMnd = { g e. Mnd | A. a e. ( Base ` g ) A. b e. ( Base ` g ) ( a ( +g ` g ) b ) = ( b ( +g ` g ) a ) }
Definition	df-abl 13819	Define class of all Abelian groups. (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
|- Abel = ( Grp i^i CMnd )
Theorem	isabl 13820	The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.)
|- ( G e. Abel <-> ( G e. Grp /\ G e. CMnd ) )
Theorem	ablgrp 13821	An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
|- ( G e. Abel -> G e. Grp )
Theorem	ablgrpd 13822	An Abelian group is a group, deduction form of ablgrp 13821. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> G e. Abel )   =>    |- ( ph -> G e. Grp )
Theorem	ablcmn 13823	An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- ( G e. Abel -> G e. CMnd )
Theorem	ablcmnd 13824	An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. Abel )   =>    |- ( ph -> G e. CMnd )
Theorem	iscmn 13825*	The predicate "is a commutative monoid". (Contributed by Mario Carneiro, 6-Jan-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. CMnd <-> ( G e. Mnd /\ A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) ) )
Theorem	isabl2 13826*	The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. Abel <-> ( G e. Grp /\ A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) ) )
Theorem	cmnpropd 13827*	If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   =>    |- ( ph -> ( K e. CMnd <-> L e. CMnd ) )
Theorem	ablpropd 13828*	If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   =>    |- ( ph -> ( K e. Abel <-> L e. Abel ) )
Theorem	ablprop 13829	If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.)
|- ( Base ` K ) = ( Base ` L )   &    |- ( +g ` K ) = ( +g ` L )   =>    |- ( K e. Abel <-> L e. Abel )
Theorem	iscmnd 13830*	Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> .+ = ( +g ` G ) )   &    |- ( ph -> G e. Mnd )   &    |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) )   =>    |- ( ph -> G e. CMnd )
Theorem	isabld 13831*	Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> .+ = ( +g ` G ) )   &    |- ( ph -> G e. Grp )   &    |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) )   =>    |- ( ph -> G e. Abel )
Theorem	isabli 13832*	Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)
|- G e. Grp   &    |- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ( x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) )   =>    |- G e. Abel
Theorem	cmnmnd 13833	A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- ( G e. CMnd -> G e. Mnd )
Theorem	cmncom 13834	A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. CMnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )
Theorem	ablcom 13835	An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )
Theorem	cmn32 13836	Commutative/associative law for commutative monoids. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ Z ) .+ Y ) )
Theorem	cmn4 13837	Commutative/associative law for commutative monoids. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .+ Y ) .+ ( Z .+ W ) ) = ( ( X .+ Z ) .+ ( Y .+ W ) ) )
Theorem	cmn12 13838	Commutative/associative law for commutative monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .+ ( Y .+ Z ) ) = ( Y .+ ( X .+ Z ) ) )
Theorem	abl32 13839	Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ Z ) .+ Y ) )
Theorem	cmnmndd 13840	A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. CMnd )   =>    |- ( ph -> G e. Mnd )
Theorem	rinvmod 13841*	Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6198. (Contributed by AV, 31-Dec-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. B )   =>    |- ( ph -> E* w e. B ( A .+ w ) = .0. )
Theorem	ablinvadd 13842	The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- N = ( invg ` G )   =>    |- ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( N ` ( X .+ Y ) ) = ( ( N ` X ) .+ ( N ` Y ) ) )
Theorem	ablsub2inv 13843	Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- N = ( invg ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( Y .- X ) )
Theorem	ablsubadd 13844	Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Y ) = Z <-> ( Y .+ Z ) = X ) )
Theorem	ablsub4 13845	Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .+ Y ) .- ( Z .+ W ) ) = ( ( X .- Z ) .+ ( Y .- W ) ) )
Theorem	abladdsub4 13846	Abelian group addition/subtraction law. (Contributed by NM, 31-Mar-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .+ Y ) = ( Z .+ W ) <-> ( X .- Z ) = ( W .- Y ) ) )
Theorem	abladdsub 13847	Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .- Z ) = ( ( X .- Z ) .+ Y ) )
Theorem	ablpncan2 13848	Cancellation law for subtraction in an Abelian group. (Contributed by NM, 2-Oct-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( ( X .+ Y ) .- X ) = Y )
Theorem	ablpncan3 13849	A cancellation law for Abelian groups. (Contributed by NM, 23-Mar-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ ( X e. B /\ Y e. B ) ) -> ( X .+ ( Y .- X ) ) = Y )
Theorem	ablsubsub 13850	Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( X .- ( Y .- Z ) ) = ( ( X .- Y ) .+ Z ) )
Theorem	ablsubsub4 13851	Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .- Y ) .- Z ) = ( X .- ( Y .+ Z ) ) )
Theorem	ablpnpcan 13852	Cancellation law for mixed addition and subtraction. (pnpcan 8381 analog.) (Contributed by NM, 29-May-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( Y .- Z ) )
Theorem	ablnncan 13853	Cancellation law for group subtraction. (nncan 8371 analog.) (Contributed by NM, 7-Apr-2015.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .- ( X .- Y ) ) = Y )
Theorem	ablsub32 13854	Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .- Y ) .- Z ) = ( ( X .- Z ) .- Y ) )
Theorem	ablnnncan 13855	Cancellation law for group subtraction. (nnncan 8377 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .- ( Y .- Z ) ) .- Z ) = ( X .- Y ) )
Theorem	ablnnncan1 13856	Cancellation law for group subtraction. (nnncan1 8378 analog.) (Contributed by NM, 7-Apr-2015.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. Abel )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .- Y ) .- ( X .- Z ) ) = ( Z .- Y ) )
Theorem	ablsubsub23 13857	Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007.) (Revised by AV, 7-Oct-2021.)
|- V = ( Base ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( G e. Abel /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .- B ) = C <-> ( A .- C ) = B ) )
Theorem	ghmfghm 13858*	The function fulfilling the conditions of ghmgrp 13650 is a group homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.)
|- X = ( Base ` G )   &    |- Y = ( Base ` H )   &    |- .+ = ( +g ` G )   &    |- .+^ = ( +g ` H )   &    |- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )   &    |- ( ph -> F : X -onto-> Y )   &    |- ( ph -> G e. Grp )   =>    |- ( ph -> F e. ( G GrpHom H ) )
Theorem	ghmcmn 13859*	The image of a commutative monoid G under a group homomorphism F is a commutative monoid. (Contributed by Thierry Arnoux, 26-Jan-2020.)
|- X = ( Base ` G )   &    |- Y = ( Base ` H )   &    |- .+ = ( +g ` G )   &    |- .+^ = ( +g ` H )   &    |- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )   &    |- ( ph -> F : X -onto-> Y )   &    |- ( ph -> G e. CMnd )   =>    |- ( ph -> H e. CMnd )
Theorem	ghmabl 13860*	The image of an abelian group G under a group homomorphism F is an abelian group. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
|- X = ( Base ` G )   &    |- Y = ( Base ` H )   &    |- .+ = ( +g ` G )   &    |- .+^ = ( +g ` H )   &    |- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )   &    |- ( ph -> F : X -onto-> Y )   &    |- ( ph -> G e. Abel )   =>    |- ( ph -> H e. Abel )
Theorem	invghm 13861	The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.)
|- B = ( Base ` G )   &    |- I = ( invg ` G )   =>    |- ( G e. Abel <-> I e. ( G GrpHom G ) )
Theorem	eqgabl 13862	Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- X = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- .~ = ( G ~QG S )   =>    |- ( ( G e. Abel /\ S C_ X ) -> ( A .~ B <-> ( A e. X /\ B e. X /\ ( B .- A ) e. S ) ) )
Theorem	qusecsub 13863	Two subgroup cosets are equal if and only if the difference of their representatives is a member of the subgroup. (Contributed by AV, 7-Mar-2025.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- .~ = ( G ~QG S )   =>    |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ ( X e. B /\ Y e. B ) ) -> ( [ X ] .~ = [ Y ] .~ <-> ( Y .- X ) e. S ) )
Theorem	subgabl 13864	A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.)
|- H = ( G ↾s S )   =>    |- ( ( G e. Abel /\ S e. (SubGrp ` G ) ) -> H e. Abel )
Theorem	subcmnd 13865	A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
|- ( ph -> H = ( G ↾s S ) )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> H e. Mnd )   &    |- ( ph -> S e. V )   =>    |- ( ph -> H e. CMnd )
Theorem	ablnsg 13866	Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- ( G e. Abel -> (NrmSGrp ` G ) = (SubGrp ` G ) )
Theorem	ablressid 13867	A commutative group restricted to its base set is a commutative group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 13099. (Contributed by Jim Kingdon, 5-May-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Abel -> ( G ↾s B ) e. Abel )
Theorem	imasabl 13868*	The image structure of an abelian group is an abelian group (imasgrp 13643 analog). (Contributed by AV, 22-Feb-2025.)
|- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> .+ = ( +g ` R ) )   &    |- ( ph -> F : V -onto-> B )   &    |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .+ b ) ) = ( F ` ( p .+ q ) ) ) )   &    |- ( ph -> R e. Abel )   &    |- .0. = ( 0g ` R )   =>    |- ( ph -> ( U e. Abel /\ ( F ` .0. ) = ( 0g ` U ) ) )
7.2.5.2  Group sum operation
Theorem	gsumfzreidx 13869	Re-index a finite group sum using a bijection. Corresponds to the first equation in [Lang] p. 5 with M = 1. (Contributed by AV, 26-Dec-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> F : ( M ... N ) --> B )   &    |- ( ph -> H : ( M ... N ) -1-1-onto-> ( M ... N ) )   =>    |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
Theorem	gsumfzsubmcl 13870	Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 30-Aug-2025.)
|- ( ph -> G e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> S e. (SubMnd ` G ) )   &    |- ( ph -> F : ( M ... N ) --> S )   =>    |- ( ph -> ( G gsumg F ) e. S )
Theorem	gsumfzmptfidmadd 13871*	The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.) (Revised by Jim Kingdon, 31-Aug-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> C e. B )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> D e. B )   &    |- F = ( x e. ( M ... N ) |-> C )   &    |- H = ( x e. ( M ... N ) |-> D )   =>    |- ( ph -> ( G gsumg ( x e. ( M ... N ) |-> ( C .+ D ) ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )
Theorem	gsumfzmptfidmadd2 13872*	The sum of two group sums expressed as mappings with finite domain, using a function operation. (Contributed by AV, 23-Jul-2019.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> C e. B )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> D e. B )   &    |- F = ( x e. ( M ... N ) |-> C )   &    |- H = ( x e. ( M ... N ) |-> D )   =>    |- ( ph -> ( G gsumg ( F oF .+ H ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )
Theorem	gsumfzconst 13873*	Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Jim Kingdon, 6-Sep-2025.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( ( G e. Mnd /\ N e. ( ZZ>= ` M ) /\ X e. B ) -> ( G gsumg ( k e. ( M ... N ) |-> X ) ) = ( ( ( N - M ) + 1 ) .x. X ) )
Theorem	gsumfzconstf 13874*	Sum of a constant series. (Contributed by Thierry Arnoux, 5-Jul-2017.)
|- F/_ k X   &    |- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( ( G e. Mnd /\ N e. ( ZZ>= ` M ) /\ X e. B ) -> ( G gsumg ( k e. ( M ... N ) |-> X ) ) = ( ( ( N - M ) + 1 ) .x. X ) )
Theorem	gsumfzmhm 13875	Apply a monoid homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 8-Sep-2025.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> H e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> K e. ( G MndHom H ) )   &    |- ( ph -> F : ( M ... N ) --> B )   =>    |- ( ph -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
Theorem	gsumfzmhm2 13876*	Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 9-Sep-2025.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> H e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> ( x e. B |-> C ) e. ( G MndHom H ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> X e. B )   &    |- ( x = X -> C = D )   &    |- ( x = ( G gsumg ( k e. ( M ... N ) |-> X ) ) -> C = E )   =>    |- ( ph -> ( H gsumg ( k e. ( M ... N ) |-> D ) ) = E )
Theorem	gsumfzsnfd 13877*	Group sum of a singleton, deduction form, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Thierry Arnoux, 28-Mar-2018.) (Revised by AV, 11-Dec-2019.)
|- B = ( Base ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> C e. B )   &    |- ( ( ph /\ k = M ) -> A = C )   &    |- F/ k ph   &    |- F/_ k C   =>    |- ( ph -> ( G gsumg ( k e. { M } |-> A ) ) = C )
7.3  Rings
7.3.1  Multiplicative Group
Syntax	cmgp 13878	Multiplicative group.
class mulGrp
Definition	df-mgp 13879	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 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" or "the multiplicative identity" in terms of the identity of a monoid (df-ur 13918). (Contributed by Mario Carneiro, 21-Dec-2014.)
|- mulGrp = ( w e. _V |-> ( w sSet <. ( +g ` ndx ) , ( .r ` w ) >. ) )
Theorem	fnmgp 13880	The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
|- mulGrp Fn _V
Theorem	mgpvalg 13881	Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
|- M = (mulGrp ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. V -> M = ( R sSet <. ( +g ` ndx ) , .x. >. ) )
Theorem	mgpplusgg 13882	Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
|- M = (mulGrp ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. V -> .x. = ( +g ` M ) )
Theorem	mgpex 13883	Existence of the multiplication group. If R is known to be a semiring, see srgmgp 13926. (Contributed by Jim Kingdon, 10-Jan-2025.)
|- M = (mulGrp ` R )   =>    |- ( R e. V -> M e. _V )
Theorem	mgpbasg 13884	Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   &    |- B = ( Base ` R )   =>    |- ( R e. V -> B = ( Base ` M ) )
Theorem	mgpscag 13885	The multiplication monoid has the same (if any) scalars as the original ring. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
|- M = (mulGrp ` R )   &    |- S = (Scalar ` R )   =>    |- ( R e. V -> S = (Scalar ` M ) )
Theorem	mgptsetg 13886	Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   =>    |- ( R e. V -> (TopSet ` R ) = (TopSet ` M ) )
Theorem	mgptopng 13887	Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   &    |- J = ( TopOpen ` R )   =>    |- ( R e. V -> J = ( TopOpen ` M ) )
Theorem	mgpdsg 13888	Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   &    |- B = ( dist ` R )   =>    |- ( R e. V -> B = ( dist ` M ) )
Theorem	mgpress 13889	Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)
|- S = ( R ↾s A )   &    |- M = (mulGrp ` R )   =>    |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = (mulGrp ` S ) )
7.3.2  Non-unital rings ("rngs") According to Wikipedia, "... in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a [unital] ring, without assuming the existence of a multiplicative identity. The term "rng" (pronounced rung) is meant to suggest that it is a "ring" without "i", i.e. without the requirement for an "identity element"." (see https://en.wikipedia.org/wiki/Rng_(algebra), 28-Mar-2025).
Syntax	crng 13890	Extend class notation with class of all non-unital rings.
class Rng
Definition	df-rng 13891*	Define the class of all non-unital rings. A non-unital ring (or rng, or pseudoring) is a set equipped with two everywhere-defined internal operations, whose first one is an additive abelian group operation and the second one is a multiplicative semigroup operation, and where the addition is left- and right-distributive for the multiplication. Definition of a pseudo-ring in section I.8.1 of [BourbakiAlg1] p. 93 or the definition of a ring in part Preliminaries of [Roman] p. 18. As almost always in mathematics, "non-unital" means "not necessarily unital". Therefore, by talking about a ring (in general) or a non-unital ring the "unital" case is always included. In contrast to a unital ring, the commutativity of addition must be postulated and cannot be proven from the other conditions. (Contributed by AV, 6-Jan-2020.)
|- Rng = { f e. Abel | ( (mulGrp ` f ) e. Smgrp /\ [. ( Base ` f ) / b ]. [. ( +g ` f ) / p ]. [. ( .r ` f ) / t ]. A. x e. b A. y e. b A. z e. b ( ( x t ( y p z ) ) = ( ( x t y ) p ( x t z ) ) /\ ( ( x p y ) t z ) = ( ( x t z ) p ( y t z ) ) ) ) }
Theorem	isrng 13892*	The predicate "is a non-unital ring." (Contributed by AV, 6-Jan-2020.)
|- B = ( Base ` R )   &    |- G = (mulGrp ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Rng <-> ( R e. Abel /\ G e. Smgrp /\ A. x e. B A. y e. B A. z e. B ( ( x .x. ( y .+ z ) ) = ( ( x .x. y ) .+ ( x .x. z ) ) /\ ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) ) ) )
Theorem	rngabl 13893	A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
|- ( R e. Rng -> R e. Abel )
Theorem	rngmgp 13894	A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
|- G = (mulGrp ` R )   =>    |- ( R e. Rng -> G e. Smgrp )
Theorem	rngmgpf 13895	Restricted functionality of the multiplicative group on non-unital rings (mgpf 13969 analog). (Contributed by AV, 22-Feb-2025.)
|- (mulGrp |` Rng ) :Rng -->Smgrp
Theorem	rnggrp 13896	A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
|- ( R e. Rng -> R e. Grp )
Theorem	rngass 13897	Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011.) (Revised by AV, 13-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )
Theorem	rngdi 13898	Distributive law for the multiplication operation of a non-unital ring (left-distributivity). (Contributed by AV, 14-Feb-2025.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) )
Theorem	rngdir 13899	Distributive law for the multiplication operation of a non-unital ring (right-distributivity). (Contributed by AV, 17-Apr-2020.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .x. Z ) = ( ( X .x. Z ) .+ ( Y .x. Z ) ) )
Theorem	rngacl 13900	Closure of the addition operation of a non-unital ring. (Contributed by AV, 16-Feb-2025.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   =>    |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )