P. 140 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13901-14000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	subgcl 13901	A subgroup is closed under group operation. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- .+ = ( +g ` G )   =>    |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .+ Y ) e. S )
Theorem	subgsubcl 13902	A subgroup is closed under group subtraction. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- .- = ( -g ` G )   =>    |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .- Y ) e. S )
Theorem	subgsub 13903	The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.)
|- .- = ( -g ` G )   &    |- H = ( G ↾s S )   &    |- N = ( -g ` H )   =>    |- ( ( S e. (SubGrp ` G ) /\ X e. S /\ Y e. S ) -> ( X .- Y ) = ( X N Y ) )
Theorem	subgmulgcl 13904	Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
|- .x. = (.g ` G )   =>    |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) e. S )
Theorem	subgmulg 13905	A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
|- .x. = (.g ` G )   &    |- H = ( G ↾s S )   &    |- .xb = (.g ` H )   =>    |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) = ( N .xb X ) )
Theorem	issubg2m 13906*	Characterize the subgroups of a group by closure properties. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- I = ( invg ` G )   =>    |- ( G e. Grp -> ( S e. (SubGrp ` G ) <-> ( S C_ B /\ E. u u e. S /\ A. x e. S ( A. y e. S ( x .+ y ) e. S /\ ( I ` x ) e. S ) ) ) )
Theorem	issubgrpd2 13907*	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 13908*	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 13909*	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 13910*	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 13911	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 13912	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 13913	A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- ( S e. (SubGrp ` G ) -> S e. (SubMnd ` G ) )
Theorem	subsubg 13914	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 13915*	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 13916	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 13917	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 13918	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 13919*	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 13920*	Weaken the condition of isnsg 13919 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 13921	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 13922	A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
|- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
Theorem	nsgconj 13923	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 13924*	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 13925*	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 13926*	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 13927*	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 13928*	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 13929*	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 13930*	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 13931	The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
|- .0. = ( 0g ` G )   =>    |- ( G e. Grp -> { .0. } e. (NrmSGrp ` G ) )
Theorem	nsgid 13932	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 13933	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 13934	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 13935	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 13936	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 13937	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 13938	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 13939*	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 13940	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 13941	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 13942*	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 13943	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 13944	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 13945	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 13946	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 13947*	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 13948	Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) Generalization of quseccl 13950 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 13949	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 13950	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 13951	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 13952	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 13953	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 13954	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 13955	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 13956	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 13957	Extend class notation with the generator of group hom-sets.
class GrpHom
Definition	df-ghm 13958*	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 13959	Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- Rel dom GrpHom
Theorem	isghm 13960*	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 13961*	Property of a group homomorphism, similar to ismhm 13674. (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 13962	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 13963	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 13964	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 13965	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 13966	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 13967	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 13968	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 13969*	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 13970	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 13971	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 13972	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 13973	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 13974	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 13975	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 13976	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 13977	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 13978	One direction of resghm2b 13979. (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 13979	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 13980	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 13981	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 13982	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 13983	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 13984	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 13985	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 13986	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 13987	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 13988	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 13989	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 13990*	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 13991	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 13992	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 13993*	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 13994*	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 13995*	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 13996*	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 13997*	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 13998*	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 13999*	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 14000*	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 ) )