Theorem ghmfghm

Description: The function fulfilling the conditions of ghmgrp is a group homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020)

	Ref	Expression
Hypotheses	ghmabl.x	|- X = ( Base ` G )
	ghmabl.y	|- Y = ( Base ` H )
	ghmabl.p	|- .+ = ( +g ` G )
	ghmabl.q	|- .+^ = ( +g ` H )
	ghmabl.f	|- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
	ghmabl.1	|- ( ph -> F : X -onto-> Y )
	ghmfghm.3	|- ( ph -> G e. Grp )
Assertion	ghmfghm	|- ( ph -> F e. ( G GrpHom H ) )

Proof

Step	Hyp	Ref	Expression
1		ghmabl.x	|- X = ( Base ` G )
2		ghmabl.y	|- Y = ( Base ` H )
3		ghmabl.p	|- .+ = ( +g ` G )
4		ghmabl.q	|- .+^ = ( +g ` H )
5		ghmabl.f	|- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
6		ghmabl.1	|- ( ph -> F : X -onto-> Y )
7		ghmfghm.3	|- ( ph -> G e. Grp )
8	5 1 2 3 4 6 7	ghmgrp	|- ( ph -> H e. Grp )
9		fof	|- ( F : X -onto-> Y -> F : X --> Y )
10	6 9	syl	|- ( ph -> F : X --> Y )
11	5	3expb	|- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
12	1 2 3 4 7 8 10 11	isghmd	|- ( ph -> F e. ( G GrpHom H ) )