P. 137 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13601-13700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	issubgrpd2 13601*	Prove a subgroup by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
|- ( ph -> S = ( I ↾s D ) )   &    |- ( ph -> .0. = ( 0g ` I ) )   &    |- ( ph -> .+ = ( +g ` I ) )   &    |- ( ph -> D C_ ( Base ` I ) )   &    |- ( ph -> .0. e. D )   &    |- ( ( ph /\ x e. D /\ y e. D ) -> ( x .+ y ) e. D )   &    |- ( ( ph /\ x e. D ) -> ( ( invg ` I ) ` x ) e. D )   &    |- ( ph -> I e. Grp )   =>    |- ( ph -> D e. (SubGrp ` I ) )
Theorem	issubgrpd 13602*	Prove a subgroup by closure. (Contributed by Stefan O'Rear, 7-Dec-2014.)
|- ( ph -> S = ( I ↾s D ) )   &    |- ( ph -> .0. = ( 0g ` I ) )   &    |- ( ph -> .+ = ( +g ` I ) )   &    |- ( ph -> D C_ ( Base ` I ) )   &    |- ( ph -> .0. e. D )   &    |- ( ( ph /\ x e. D /\ y e. D ) -> ( x .+ y ) e. D )   &    |- ( ( ph /\ x e. D ) -> ( ( invg ` I ) ` x ) e. D )   &    |- ( ph -> I e. Grp )   =>    |- ( ph -> S e. Grp )
Theorem	issubg3 13603*	A subgroup is a symmetric submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
|- I = ( invg ` G )   =>    |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> ( S e. (SubMnd ` G ) /\ A. x e. S ( I ` x ) e. S ) ) )
Theorem	issubg4m 13604*	A subgroup is an inhabited subset of the group closed under subtraction. (Contributed by Mario Carneiro, 17-Sep-2015.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   =>    |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> ( S C_ B /\ E. w w e. S /\ A. x e. S A. y e. S ( x .- y ) e. S ) ) )
Theorem	grpissubg 13605	If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the (base set of the) group is subgroup of the other group. (Contributed by AV, 14-Mar-2019.)
|- B = ( Base ` G )   &    |- S = ( Base ` H )   =>    |- ( ( G e. Grp /\ H e. Grp ) -> ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S e. (SubGrp ` G ) ) )
Theorem	resgrpisgrp 13606	If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the other group restricted to the base set of the group is a group. (Contributed by AV, 14-Mar-2019.)
|- B = ( Base ` G )   &    |- S = ( Base ` H )   =>    |- ( ( G e. Grp /\ H e. Grp ) -> ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> ( G ↾s S ) e. Grp ) )
Theorem	subgsubm 13607	A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- ( S e. (SubGrp ` G ) -> S e. (SubMnd ` G ) )
Theorem	subsubg 13608	A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
|- H = ( G ↾s S )   =>    |- ( S e. (SubGrp ` G ) -> ( A e. (SubGrp ` H ) <-> ( A e. (SubGrp ` G ) /\ A C_ S ) ) )
Theorem	subgintm 13609*	The intersection of an inhabited collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.)
|- ( ( S C_ (SubGrp ` G ) /\ E. w w e. S ) -> |^| S e. (SubGrp ` G ) )
Theorem	0subg 13610	The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof shortened by SN, 31-Jan-2025.)
|- .0. = ( 0g ` G )   =>    |- ( G e. Grp -> { .0. } e. (SubGrp ` G ) )
Theorem	trivsubgd 13611	The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = { .0. } )   &    |- ( ph -> A e. (SubGrp ` G ) )   =>    |- ( ph -> A = B )
Theorem	trivsubgsnd 13612	The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = { .0. } )   =>    |- ( ph -> (SubGrp ` G ) = { B } )
Theorem	isnsg 13613*	Property of being a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( S e. (NrmSGrp ` G ) <-> ( S e. (SubGrp ` G ) /\ A. x e. X A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) ) )
Theorem	isnsg2 13614*	Weaken the condition of isnsg 13613 to only one side of the implication. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( S e. (NrmSGrp ` G ) <-> ( S e. (SubGrp ` G ) /\ A. x e. X A. y e. X ( ( x .+ y ) e. S -> ( y .+ x ) e. S ) ) )
Theorem	nsgbi 13615	Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. X ) -> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) )
Theorem	nsgsubg 13616	A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
Theorem	nsgconj 13617	The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( A .+ B ) .- A ) e. S )
Theorem	isnsg3 13618*	A subgroup is normal iff the conjugation of all the elements of the subgroup is in the subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   =>    |- ( S e. (NrmSGrp ` G ) <-> ( S e. (SubGrp ` G ) /\ A. x e. X A. y e. S ( ( x .+ y ) .- x ) e. S ) )
Theorem	elnmz 13619*	Elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }   =>    |- ( A e. N <-> ( A e. X /\ A. z e. X ( ( A .+ z ) e. S <-> ( z .+ A ) e. S ) ) )
Theorem	nmzbi 13620*	Defining property of the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }   =>    |- ( ( A e. N /\ B e. X ) -> ( ( A .+ B ) e. S <-> ( B .+ A ) e. S ) )
Theorem	nmzsubg 13621*	The normalizer NG(S) of a subset S of the group is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }   &    |- X = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. Grp -> N e. (SubGrp ` G ) )
Theorem	ssnmz 13622*	A subgroup is a subset of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }   &    |- X = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( S e. (SubGrp ` G ) -> S C_ N )
Theorem	isnsg4 13623*	A subgroup is normal iff its normalizer is the entire group. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }   &    |- X = ( Base ` G )   &    |- .+ = ( +g ` G )   =>    |- ( S e. (NrmSGrp ` G ) <-> ( S e. (SubGrp ` G ) /\ N = X ) )
Theorem	nmznsg 13624*	Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-Jan-2015.)
|- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }   &    |- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- H = ( G ↾s N )   =>    |- ( S e. (SubGrp ` G ) -> S e. (NrmSGrp ` H ) )
Theorem	0nsg 13625	The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- .0. = ( 0g ` G )   =>    |- ( G e. Grp -> { .0. } e. (NrmSGrp ` G ) )
Theorem	nsgid 13626	The whole group is a normal subgroup of itself. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> B e. (NrmSGrp ` G ) )
Theorem	0idnsgd 13627	The whole group and the zero subgroup are normal subgroups of a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   =>    |- ( ph -> { { .0. } , B } C_ (NrmSGrp ` G ) )
Theorem	trivnsgd 13628	The only normal subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = { .0. } )   =>    |- ( ph -> (NrmSGrp ` G ) = { B } )
Theorem	triv1nsgd 13629	A trivial group has exactly one normal subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = { .0. } )   =>    |- ( ph -> (NrmSGrp ` G ) ~~ 1o )
Theorem	1nsgtrivd 13630	A group with exactly one normal subgroup is trivial. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> (NrmSGrp ` G ) ~~ 1o )   =>    |- ( ph -> B = { .0. } )
Theorem	releqgg 13631	The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- R = ( G ~QG S )   =>    |- ( ( G e. V /\ S e. W ) -> Rel R )
Theorem	eqgex 13632	The left coset equivalence relation exists. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( ( G e. V /\ S e. W ) -> ( G ~QG S ) e. _V )
Theorem	eqgfval 13633*	Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.)
|- X = ( Base ` G )   &    |- N = ( invg ` G )   &    |- .+ = ( +g ` G )   &    |- R = ( G ~QG S )   =>    |- ( ( G e. V /\ S C_ X ) -> R = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )
Theorem	eqgval 13634	Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
|- X = ( Base ` G )   &    |- N = ( invg ` G )   &    |- .+ = ( +g ` G )   &    |- R = ( G ~QG S )   =>    |- ( ( G e. V /\ S C_ X ) -> ( A R B <-> ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) )
Theorem	eqger 13635	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 13636*	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 13637	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 13638	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 13639	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 13640	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 13641*	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 13642	Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) Generalization of quseccl 13644 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 13643	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 13644	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 13645	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 13646	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 13647	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 13648	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 13649	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 13650	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 13651	Extend class notation with the generator of group hom-sets.
class GrpHom
Definition	df-ghm 13652*	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 13653	Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- Rel dom GrpHom
Theorem	isghm 13654*	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 13655*	Property of a group homomorphism, similar to ismhm 13368. (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 13656	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 13657	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 13658	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 13659	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 13660	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 13661	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 13662	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 13663*	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 13664	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 13665	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 13666	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 13667	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 13668	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 13669	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 13670	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 13671	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 13672	One direction of resghm2b 13673. (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 13673	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 13674	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 13675	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 13676	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 13677	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 13678	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 13679	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 13680	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 13681	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 13682	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 13683	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 13684*	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 13685	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 13686	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 13687*	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 13688*	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 13689*	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 13690*	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 13691*	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 13692*	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 13693*	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 13694*	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 13695	Extend class notation with class of all commutative monoids.
class CMnd
Syntax	cabl 13696	Extend class notation with class of all Abelian groups.
class Abel
Definition	df-cmn 13697*	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 13698	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 13699	The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.)
|- ( G e. Abel <-> ( G e. Grp /\ G e. CMnd ) )
Theorem	ablgrp 13700	An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
|- ( G e. Abel -> G e. Grp )