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	eqger 13801	The subgroup coset equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 13-Jan-2015.)
|- X = ( Base ` G )   &    |- .~ = ( G ~QG Y )   =>    |- ( Y e. (SubGrp ` G ) -> .~ Er X )
Theorem	eqglact 13802*	A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)
|- X = ( Base ` G )   &    |- .~ = ( G ~QG Y )   &    |- .+ = ( +g ` G )   =>    |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> [ A ] .~ = ( ( x e. X |-> ( A .+ x ) ) " Y ) )
Theorem	eqgid 13803	The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.)
|- X = ( Base ` G )   &    |- .~ = ( G ~QG Y )   &    |- .0. = ( 0g ` G )   =>    |- ( Y e. (SubGrp ` G ) -> [ .0. ] .~ = Y )
Theorem	eqgen 13804	Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.)
|- X = ( Base ` G )   &    |- .~ = ( G ~QG Y )   =>    |- ( ( Y e. (SubGrp ` G ) /\ A e. ( X /. .~ ) ) -> Y ~~ A )
Theorem	eqgcpbl 13805	The subgroup coset equivalence relation is compatible with addition when the subgroup is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- X = ( Base ` G )   &    |- .~ = ( G ~QG Y )   &    |- .+ = ( +g ` G )   =>    |- ( Y e. (NrmSGrp ` G ) -> ( ( A .~ C /\ B .~ D ) -> ( A .+ B ) .~ ( C .+ D ) ) )
Theorem	eqg0el 13806	Equivalence class of a quotient group for a subgroup. (Contributed by Thierry Arnoux, 15-Jan-2024.)
|- .~ = ( G ~QG H )   =>    |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( [ X ] .~ = H <-> X e. H ) )
Theorem	quselbasg 13807*	Membership in the base set of a quotient group. (Contributed by AV, 1-Mar-2025.)
|- .~ = ( G ~QG S )   &    |- U = ( G /.s .~ )   &    |- B = ( Base ` G )   =>    |- ( ( G e. V /\ X e. W /\ S e. Z ) -> ( X e. ( Base ` U ) <-> E. x e. B X = [ x ] .~ ) )
Theorem	quseccl0g 13808	Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) Generalization of quseccl 13810 for arbitrary sets G. (Revised by AV, 24-Feb-2025.)
|- .~ = ( G ~QG S )   &    |- H = ( G /.s .~ )   &    |- C = ( Base ` G )   &    |- B = ( Base ` H )   =>    |- ( ( G e. V /\ X e. C /\ S e. Z ) -> [ X ] .~ e. B )
Theorem	qusgrp 13809	If Y is a normal subgroup of G, then H = G / Y is a group, called the quotient of G by Y. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- H = ( G /.s ( G ~QG S ) )   =>    |- ( S e. (NrmSGrp ` G ) -> H e. Grp )
Theorem	quseccl 13810	Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 9-Mar-2025.)
|- H = ( G /.s ( G ~QG S ) )   &    |- V = ( Base ` G )   &    |- B = ( Base ` H )   =>    |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ X ] ( G ~QG S ) e. B )
Theorem	qusadd 13811	Value of the group operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- H = ( G /.s ( G ~QG S ) )   &    |- V = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .+b = ( +g ` H )   =>    |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( [ X ] ( G ~QG S ) .+b [ Y ] ( G ~QG S ) ) = [ ( X .+ Y ) ] ( G ~QG S ) )
Theorem	qus0 13812	Value of the group identity operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- H = ( G /.s ( G ~QG S ) )   &    |- .0. = ( 0g ` G )   =>    |- ( S e. (NrmSGrp ` G ) -> [ .0. ] ( G ~QG S ) = ( 0g ` H ) )
Theorem	qusinv 13813	Value of the group inverse operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- H = ( G /.s ( G ~QG S ) )   &    |- V = ( Base ` G )   &    |- I = ( invg ` G )   &    |- N = ( invg ` H )   =>    |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( N ` [ X ] ( G ~QG S ) ) = [ ( I ` X ) ] ( G ~QG S ) )
Theorem	qussub 13814	Value of the group subtraction operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- H = ( G /.s ( G ~QG S ) )   &    |- V = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- N = ( -g ` H )   =>    |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( [ X ] ( G ~QG S ) N [ Y ] ( G ~QG S ) ) = [ ( X .- Y ) ] ( G ~QG S ) )
Theorem	ecqusaddd 13815	Addition of equivalence classes in a quotient group. (Contributed by AV, 25-Feb-2025.)
|- ( ph -> I e. (NrmSGrp ` R ) )   &    |- B = ( Base ` R )   &    |- .~ = ( R ~QG I )   &    |- Q = ( R /.s .~ )   =>    |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> [ ( A ( +g ` R ) C ) ] .~ = ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) )
Theorem	ecqusaddcl 13816	Closure of the addition in a quotient group. (Contributed by AV, 24-Feb-2025.)
|- ( ph -> I e. (NrmSGrp ` R ) )   &    |- B = ( Base ` R )   &    |- .~ = ( R ~QG I )   &    |- Q = ( R /.s .~ )   =>    |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) e. ( Base ` Q ) )
7.2.4  Elementary theory of group homomorphisms
Syntax	cghm 13817	Extend class notation with the generator of group hom-sets.
class GrpHom
Definition	df-ghm 13818*	A homomorphism of groups is a map between two structures which preserves the group operation. Requiring both sides to be groups simplifies most theorems at the cost of complicating the theorem which pushes forward a group structure. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- GrpHom = ( s e. Grp , t e. Grp |-> { g | [. ( Base ` s ) / w ]. ( g : w --> ( Base ` t ) /\ A. x e. w A. y e. w ( g ` ( x ( +g ` s ) y ) ) = ( ( g ` x ) ( +g ` t ) ( g ` y ) ) ) } )
Theorem	reldmghm 13819	Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- Rel dom GrpHom
Theorem	isghm 13820*	Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- X = ( Base ` S )   &    |- Y = ( Base ` T )   &    |- .+ = ( +g ` S )   &    |- .+^ = ( +g ` T )   =>    |- ( F e. ( S GrpHom T ) <-> ( ( S e. Grp /\ T e. Grp ) /\ ( F : X --> Y /\ A. u e. X A. v e. X ( F ` ( u .+ v ) ) = ( ( F ` u ) .+^ ( F ` v ) ) ) ) )
Theorem	isghm3 13821*	Property of a group homomorphism, similar to ismhm 13534. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- X = ( Base ` S )   &    |- Y = ( Base ` T )   &    |- .+ = ( +g ` S )   &    |- .+^ = ( +g ` T )   =>    |- ( ( S e. Grp /\ T e. Grp ) -> ( F e. ( S GrpHom T ) <-> ( F : X --> Y /\ A. u e. X A. v e. X ( F ` ( u .+ v ) ) = ( ( F ` u ) .+^ ( F ` v ) ) ) ) )
Theorem	ghmgrp1 13822	A group homomorphism is only defined when the domain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- ( F e. ( S GrpHom T ) -> S e. Grp )
Theorem	ghmgrp2 13823	A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- ( F e. ( S GrpHom T ) -> T e. Grp )
Theorem	ghmf 13824	A group homomorphism is a function. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- X = ( Base ` S )   &    |- Y = ( Base ` T )   =>    |- ( F e. ( S GrpHom T ) -> F : X --> Y )
Theorem	ghmlin 13825	A homomorphism of groups is linear. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- X = ( Base ` S )   &    |- .+ = ( +g ` S )   &    |- .+^ = ( +g ` T )   =>    |- ( ( F e. ( S GrpHom T ) /\ U e. X /\ V e. X ) -> ( F ` ( U .+ V ) ) = ( ( F ` U ) .+^ ( F ` V ) ) )
Theorem	ghmid 13826	A homomorphism of groups preserves the identity. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- Y = ( 0g ` S )   &    |- .0. = ( 0g ` T )   =>    |- ( F e. ( S GrpHom T ) -> ( F ` Y ) = .0. )
Theorem	ghminv 13827	A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- B = ( Base ` S )   &    |- M = ( invg ` S )   &    |- N = ( invg ` T )   =>    |- ( ( F e. ( S GrpHom T ) /\ X e. B ) -> ( F ` ( M ` X ) ) = ( N ` ( F ` X ) ) )
Theorem	ghmsub 13828	Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- B = ( Base ` S )   &    |- .- = ( -g ` S )   &    |- N = ( -g ` T )   =>    |- ( ( F e. ( S GrpHom T ) /\ U e. B /\ V e. B ) -> ( F ` ( U .- V ) ) = ( ( F ` U ) N ( F ` V ) ) )
Theorem	isghmd 13829*	Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
|- X = ( Base ` S )   &    |- Y = ( Base ` T )   &    |- .+ = ( +g ` S )   &    |- .+^ = ( +g ` T )   &    |- ( ph -> S e. Grp )   &    |- ( ph -> T e. Grp )   &    |- ( ph -> F : X --> Y )   &    |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )   =>    |- ( ph -> F e. ( S GrpHom T ) )
Theorem	ghmmhm 13830	A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
|- ( F e. ( S GrpHom T ) -> F e. ( S MndHom T ) )
Theorem	ghmmhmb 13831	Group homomorphisms and monoid homomorphisms coincide. (Thus, GrpHom is somewhat redundant, although its stronger reverse closure properties are sometimes useful.) (Contributed by Stefan O'Rear, 7-Mar-2015.)
|- ( ( S e. Grp /\ T e. Grp ) -> ( S GrpHom T ) = ( S MndHom T ) )
Theorem	ghmex 13832	The set of group homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
|- ( ( S e. Grp /\ T e. Grp ) -> ( S GrpHom T ) e. _V )
Theorem	ghmmulg 13833	A group homomorphism preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- .X. = (.g ` H )   =>    |- ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
Theorem	ghmrn 13834	The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- ( F e. ( S GrpHom T ) -> ran F e. (SubGrp ` T ) )
Theorem	0ghm 13835	The constant zero linear function between two groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- .0. = ( 0g ` N )   &    |- B = ( Base ` M )   =>    |- ( ( M e. Grp /\ N e. Grp ) -> ( B X. { .0. } ) e. ( M GrpHom N ) )
Theorem	idghm 13836	The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> ( _I |` B ) e. ( G GrpHom G ) )
Theorem	resghm 13837	Restriction of a homomorphism to a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- U = ( S ↾s X )   =>    |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> ( F |` X ) e. ( U GrpHom T ) )
Theorem	resghm2 13838	One direction of resghm2b 13839. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
|- U = ( T ↾s X )   =>    |- ( ( F e. ( S GrpHom U ) /\ X e. (SubGrp ` T ) ) -> F e. ( S GrpHom T ) )
Theorem	resghm2b 13839	Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
|- U = ( T ↾s X )   =>    |- ( ( X e. (SubGrp ` T ) /\ ran F C_ X ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) )
Theorem	ghmghmrn 13840	A group homomorphism from G to H is also a group homomorphism from G to its image in H. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by AV, 26-Aug-2021.)
|- U = ( T ↾s ran F )   =>    |- ( F e. ( S GrpHom T ) -> F e. ( S GrpHom U ) )
Theorem	ghmco 13841	The composition of group homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
|- ( ( F e. ( T GrpHom U ) /\ G e. ( S GrpHom T ) ) -> ( F o. G ) e. ( S GrpHom U ) )
Theorem	ghmima 13842	The image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- ( ( F e. ( S GrpHom T ) /\ U e. (SubGrp ` S ) ) -> ( F " U ) e. (SubGrp ` T ) )
Theorem	ghmpreima 13843	The inverse image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- ( ( F e. ( S GrpHom T ) /\ V e. (SubGrp ` T ) ) -> ( `' F " V ) e. (SubGrp ` S ) )
Theorem	ghmeql 13844	The equalizer of two group homomorphisms is a subgroup. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
|- ( ( F e. ( S GrpHom T ) /\ G e. ( S GrpHom T ) ) -> dom ( F i^i G ) e. (SubGrp ` S ) )
Theorem	ghmnsgima 13845	The image of a normal subgroup under a surjective homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- Y = ( Base ` T )   =>    |- ( ( F e. ( S GrpHom T ) /\ U e. (NrmSGrp ` S ) /\ ran F = Y ) -> ( F " U ) e. (NrmSGrp ` T ) )
Theorem	ghmnsgpreima 13846	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 13847	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 13848	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 13849	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 13850*	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 13851	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 13852	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 13853*	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 13854*	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 13855*	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 13856*	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 13857*	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 13858*	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 13859*	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 13860*	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 13861	Extend class notation with class of all commutative monoids.
class CMnd
Syntax	cabl 13862	Extend class notation with class of all Abelian groups.
class Abel
Definition	df-cmn 13863*	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 13864	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 13865	The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.)
|- ( G e. Abel <-> ( G e. Grp /\ G e. CMnd ) )
Theorem	ablgrp 13866	An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
|- ( G e. Abel -> G e. Grp )
Theorem	ablgrpd 13867	An Abelian group is a group, deduction form of ablgrp 13866. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> G e. Abel )   =>    |- ( ph -> G e. Grp )
Theorem	ablcmn 13868	An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- ( G e. Abel -> G e. CMnd )
Theorem	ablcmnd 13869	An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. Abel )   =>    |- ( ph -> G e. CMnd )
Theorem	iscmn 13870*	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 13871*	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 13872*	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 13873*	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 13874	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 13875*	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 13876*	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 13877*	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 13878	A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- ( G e. CMnd -> G e. Mnd )
Theorem	cmncom 13879	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 13880	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 13881	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 13882	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 13883	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 13884	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 13885	A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. CMnd )   =>    |- ( ph -> G e. Mnd )
Theorem	rinvmod 13886*	Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6211. (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 13887	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 13888	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 13889	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 13890	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 13891	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 13892	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 13893	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 13894	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 13895	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 13896	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 13897	Cancellation law for mixed addition and subtraction. (pnpcan 8408 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 13898	Cancellation law for group subtraction. (nncan 8398 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 13899	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 13900	Cancellation law for group subtraction. (nnncan 8404 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 ) )