Theorem abl32

Description: Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015) (Revised by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Hypotheses	ablcom.b	|- B = ( Base ` G )
	ablcom.p	|- .+ = ( +g ` G )
	abl32.g	|- ( ph -> G e. Abel )
	abl32.x	|- ( ph -> X e. B )
	abl32.y	|- ( ph -> Y e. B )
	abl32.z	|- ( ph -> Z e. B )
Assertion	abl32	|- ( ph -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ Z ) .+ Y ) )

Proof

Step	Hyp	Ref	Expression
1		ablcom.b	|- B = ( Base ` G )
2		ablcom.p	|- .+ = ( +g ` G )
3		abl32.g	|- ( ph -> G e. Abel )
4		abl32.x	|- ( ph -> X e. B )
5		abl32.y	|- ( ph -> Y e. B )
6		abl32.z	|- ( ph -> Z e. B )
7		ablcmn	|- ( G e. Abel -> G e. CMnd )
8	3 7	syl	|- ( ph -> G e. CMnd )
9	1 2	cmn32	|- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ Z ) .+ Y ) )
10	8 4 5 6 9	syl13anc	|- ( ph -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ Z ) .+ Y ) )