Theorem mndlrid

Description: A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011)

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

Proof

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