Theorem ring0cl

Description: The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014)

	Ref	Expression
Hypotheses	ring0cl.b	$$⊢B={\mathrm{Base}}_R$$
	ring0cl.z	$$⊢\underset{\dot{}}{0}=0_R$$
Assertion	ring0cl	$$⊢R\in \mathrm{Ring}\to \underset{\dot{}}{0}\in B$$

Proof

Step	Hyp	Ref	Expression
1		ring0cl.b	$$⊢B={\mathrm{Base}}_R$$
2		ring0cl.z	$$⊢\underset{\dot{}}{0}=0_R$$
3		ringgrp	$$⊢R\in \mathrm{Ring}\to R\in \mathrm{Grp}$$
4	1 2	grpidcl	$$⊢R\in \mathrm{Grp}\to \underset{\dot{}}{0}\in B$$
5	3 4	syl	$$⊢R\in \mathrm{Ring}\to \underset{\dot{}}{0}\in B$$