Theorem ringlidmd

Description: The unit element of a ring is a left multiplicative identity. (Contributed by SN, 14-Aug-2024)

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

Proof

Step	Hyp	Ref	Expression
1		ringlidmd.b	|- B = ( Base ` R )
2		ringlidmd.t	|- .x. = ( .r ` R )
3		ringlidmd.u	|- .1. = ( 1r ` R )
4		ringlidmd.r	|- ( ph -> R e. Ring )
5		ringlidmd.x	|- ( ph -> X e. B )
6	1 2 3	ringlidm	|- ( ( R e. Ring /\ X e. B ) -> ( .1. .x. X ) = X )
7	4 5 6	syl2anc	|- ( ph -> ( .1. .x. X ) = X )