Theorem ringlzd

Description: The zero of a unital ring is a left-absorbing element. (Contributed by SN, 7-Mar-2025)

	Ref	Expression
Hypotheses	ringz.b	|- B = ( Base ` R )
	ringz.t	|- .x. = ( .r ` R )
	ringz.z	|- .0. = ( 0g ` R )
	ringlzd.r	|- ( ph -> R e. Ring )
	ringlzd.x	|- ( ph -> X e. B )
Assertion	ringlzd	|- ( ph -> ( .0. .x. X ) = .0. )

Proof

Step	Hyp	Ref	Expression
1		ringz.b	|- B = ( Base ` R )
2		ringz.t	|- .x. = ( .r ` R )
3		ringz.z	|- .0. = ( 0g ` R )
4		ringlzd.r	|- ( ph -> R e. Ring )
5		ringlzd.x	|- ( ph -> X e. B )
6	1 2 3	ringlz	|- ( ( R e. Ring /\ X e. B ) -> ( .0. .x. X ) = .0. )
7	4 5 6	syl2anc	|- ( ph -> ( .0. .x. X ) = .0. )