Theorem mndidcl

Description: The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011) (Revised by Mario Carneiro, 27-Dec-2014)

	Ref	Expression
Hypotheses	mndidcl.b	|- B = ( Base ` G )
	mndidcl.o	|- .0. = ( 0g ` G )
Assertion	mndidcl	|- ( G e. Mnd -> .0. e. B )

Proof

Step	Hyp	Ref	Expression
1		mndidcl.b	|- B = ( Base ` G )
2		mndidcl.o	|- .0. = ( 0g ` G )
3		eqid	|- ( +g ` G ) = ( +g ` G )
4	1 3	mndid	|- ( G e. Mnd -> E. x e. B A. y e. B ( ( x ( +g ` G ) y ) = y /\ ( y ( +g ` G ) x ) = y ) )
5	1 2 3 4	mgmidcl	|- ( G e. Mnd -> .0. e. B )