P. 141 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14001-14100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eqgex 14001	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 14002*	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 14003	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 14004	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 14005*	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 14006	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 14007	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 14008	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 14009	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 14010*	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 14011	Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) Generalization of quseccl 14013 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 14012	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 14013	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 14014	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 14015	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 14016	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 14017	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 14018	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 14019	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 14020	Extend class notation with the generator of group hom-sets.
class GrpHom
Definition	df-ghm 14021*	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 14022	Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
|- Rel dom GrpHom
Theorem	isghm 14023*	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 14024*	Property of a group homomorphism, similar to ismhm 13745. (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 14025	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 14026	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 14027	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 14028	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 14029	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 14030	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 14031	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 14032*	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 14033	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 14034	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 14035	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 14036	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 14037	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 14038	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 14039	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 14040	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 14041	One direction of resghm2b 14042. (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 14042	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 14043	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 14044	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 14045	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 14046	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 14047	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 14048	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 14049	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 14050	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 14051	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 14052	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 14053*	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 14054	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 14055	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 14056*	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 14057*	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 14058*	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 14059*	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 14060*	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 14061*	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 14062*	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 14063*	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 14064	Extend class notation with class of all commutative monoids.
class CMnd
Syntax	cabl 14065	Extend class notation with class of all Abelian groups.
class Abel
Definition	df-cmn 14066*	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 14067	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 14068	The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.)
|- ( G e. Abel <-> ( G e. Grp /\ G e. CMnd ) )
Theorem	ablgrp 14069	An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
|- ( G e. Abel -> G e. Grp )
Theorem	ablgrpd 14070	An Abelian group is a group, deduction form of ablgrp 14069. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> G e. Abel )   =>    |- ( ph -> G e. Grp )
Theorem	ablcmn 14071	An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- ( G e. Abel -> G e. CMnd )
Theorem	ablcmnd 14072	An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. Abel )   =>    |- ( ph -> G e. CMnd )
Theorem	iscmn 14073*	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 14074*	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 14075*	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 14076*	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 14077	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 14078*	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 14079*	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 14080*	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 14081	A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
|- ( G e. CMnd -> G e. Mnd )
Theorem	cmncom 14082	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 14083	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 14084	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 14085	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 14086	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 14087	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 14088	A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. CMnd )   =>    |- ( ph -> G e. Mnd )
Theorem	cmnsubm 14089	A submonoid of a commutative monoid is commutative. (Contributed by Jim Kingdon, 7-Jul-2026.)
|- ( ph -> S e. (SubMnd ` G ) )   &    |- ( ph -> G e. CMnd )   &    |- H = ( G ↾s S )   =>    |- ( ph -> H e. CMnd )
Theorem	rinvmod 14090*	Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6273. (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 14091	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 14092	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 14093	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 14094	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 14095	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 14096	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 14097	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 14098	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 14099	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 14100	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 ) ) )