Theorem grpcl

Description: Closure of the operation of a group. (Contributed by NM, 14-Aug-2011)

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

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	mndcl	|- ( ( G e. Mnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
5	3 4	syl3an1	|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )