Theorem ghmabl

Description: The image of an abelian group G under a group homomorphism F is an abelian group. (Contributed by Mario Carneiro, 12-May-2014) (Revised 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 )
	ghmabl.3	|- ( ph -> G e. Abel )
Assertion	ghmabl	|- ( ph -> H e. Abel )

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		ghmabl.3	|- ( ph -> G e. Abel )
8		ablgrp	|- ( G e. Abel -> G e. Grp )
9	7 8	syl	|- ( ph -> G e. Grp )
10	5 1 2 3 4 6 9	ghmgrp	|- ( ph -> H e. Grp )
11		ablcmn	|- ( G e. Abel -> G e. CMnd )
12	7 11	syl	|- ( ph -> G e. CMnd )
13	1 2 3 4 5 6 12	ghmcmn	|- ( ph -> H e. CMnd )
14		isabl	|- ( H e. Abel <-> ( H e. Grp /\ H e. CMnd ) )
15	10 13 14	sylanbrc	|- ( ph -> H e. Abel )