Theorem ablcom

Description: An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011)

	Ref	Expression
Hypotheses	ablcom.b	|- B = ( Base ` G )
	ablcom.p	|- .+ = ( +g ` G )
Assertion	ablcom	|- ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )

Proof

Step	Hyp	Ref	Expression
1		ablcom.b	|- B = ( Base ` G )
2		ablcom.p	|- .+ = ( +g ` G )
3		ablcmn	|- ( G e. Abel -> G e. CMnd )
4	1 2	cmncom	|- ( ( G e. CMnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )
5	3 4	syl3an1	|- ( ( G e. Abel /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )