Theorem grplid

Description: The identity element of a group is a left identity. (Contributed by NM, 18-Aug-2011)

	Ref	Expression
Hypotheses	grpbn0.b	|- B = ( Base ` G )
	grplid.p	|- .+ = ( +g ` G )
	grplid.o	|- .0. = ( 0g ` G )
Assertion	grplid	|- ( ( G e. Grp /\ X e. B ) -> ( .0. .+ X ) = X )

Proof

Step	Hyp	Ref	Expression
1		grpbn0.b	|- B = ( Base ` G )
2		grplid.p	|- .+ = ( +g ` G )
3		grplid.o	|- .0. = ( 0g ` G )
4		grpmnd	|- ( G e. Grp -> G e. Mnd )
5	1 2 3	mndlid	|- ( ( G e. Mnd /\ X e. B ) -> ( .0. .+ X ) = X )
6	4 5	sylan	|- ( ( G e. Grp /\ X e. B ) -> ( .0. .+ X ) = X )