Theorem ringridmd

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

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

Proof

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