Theorem ringlidm

Description: The unity element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011)

	Ref	Expression
Hypotheses	ringidm.b	|- B = ( Base ` R )
	ringidm.t	|- .x. = ( .r ` R )
	ringidm.u	|- .1. = ( 1r ` R )
Assertion	ringlidm	|- ( ( R e. Ring /\ X e. B ) -> ( .1. .x. X ) = X )

Proof

Step	Hyp	Ref	Expression
1		ringidm.b	|- B = ( Base ` R )
2		ringidm.t	|- .x. = ( .r ` R )
3		ringidm.u	|- .1. = ( 1r ` R )
4	1 2 3	ringidmlem	|- ( ( R e. Ring /\ X e. B ) -> ( ( .1. .x. X ) = X /\ ( X .x. .1. ) = X ) )
5	4	simpld	|- ( ( R e. Ring /\ X e. B ) -> ( .1. .x. X ) = X )