Theorem grpass

Description: A group operation is associative. (Contributed by NM, 14-Aug-2011)

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

Proof

Step	Hyp	Ref	Expression
1		grpcl.b	|- B = ( Base ` G )
2		grpcl.p	|- .+ = ( +g ` G )
3		grpmnd	|- ( G e. Grp -> G e. Mnd )
4	1 2	mndass	|- ( ( G e. Mnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
5	3 4	sylan	|- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )