Theorem rng0cl

Description: The zero element of a non-unital ring belongs to its base set. (Contributed by AV, 16-Feb-2025)

	Ref	Expression
Hypotheses	rng0cl.b	$$⊢B={\mathrm{Base}}_R$$
	rng0cl.z	$$⊢\underset{\dot{}}{0}=0_R$$
Assertion	rng0cl	$$⊢R\in \mathrm{Rng}\to \underset{\dot{}}{0}\in B$$

Proof

Step	Hyp	Ref	Expression
1		rng0cl.b	$$⊢B={\mathrm{Base}}_R$$
2		rng0cl.z	$$⊢\underset{\dot{}}{0}=0_R$$
3		rnggrp	$$⊢R\in \mathrm{Rng}\to R\in \mathrm{Grp}$$
4	1 2	grpidcl	$$⊢R\in \mathrm{Grp}\to \underset{\dot{}}{0}\in B$$
5	3 4	syl	$$⊢R\in \mathrm{Rng}\to \underset{\dot{}}{0}\in B$$