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	$$⊢B={\mathrm{Base}}_R$$
	ringz.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	ringz.z	$$⊢\underset{\dot{}}{0}=0_R$$
Assertion	ringlz	$$⊢\left(R\in \mathrm{Ring}\wedge X\in B\right)\to \underset{\dot{}}{0}\underset{\dot{}}{·}X=\underset{\dot{}}{0}$$

Proof

Step	Hyp	Ref	Expression
1		ringz.b	$$⊢B={\mathrm{Base}}_R$$
2		ringz.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		ringz.z	$$⊢\underset{\dot{}}{0}=0_R$$
4		ringrng	$$⊢R\in \mathrm{Ring}\to R\in \mathrm{Rng}$$
5	1 2 3	rnglz	$$⊢\left(R\in \mathrm{Rng}\wedge X\in B\right)\to \underset{\dot{}}{0}\underset{\dot{}}{·}X=\underset{\dot{}}{0}$$
6	4 5	sylan	$$⊢\left(R\in \mathrm{Ring}\wedge X\in B\right)\to \underset{\dot{}}{0}\underset{\dot{}}{·}X=\underset{\dot{}}{0}$$