Theorem ringlz

Description: The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009) (Proof shortened by AV, 30-Mar-2025)

	Ref	Expression
Hypotheses	ringz.b	|- B = ( Base ` R )
	ringz.t	|- .x. = ( .r ` R )
	ringz.z	|- .0. = ( 0g ` R )
Assertion	ringlz	|- ( ( R e. Ring /\ X e. B ) -> ( .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		ringrng	|- ( R e. Ring -> R e. Rng )
5	1 2 3	rnglz	|- ( ( R e. Rng /\ X e. B ) -> ( .0. .x. X ) = .0. )
6	4 5	sylan	|- ( ( R e. Ring /\ X e. B ) -> ( .0. .x. X ) = .0. )