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Metamath Proof Explorer

Theorem srg0cl

Description: The zero element of a semiring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	srg0cl.b	$⊢ B = {Base}_{R}$
	srg0cl.z	$⊢ \underset{˙}{0} = 0_{R}$
Assertion	srg0cl	$⊢ R \in SRing \to \underset{˙}{0} \in B$

Proof

Step	Hyp	Ref	Expression
1		srg0cl.b	$⊢ B = {Base}_{R}$
2		srg0cl.z	$⊢ \underset{˙}{0} = 0_{R}$
3		srgmnd	$⊢ R \in SRing \to R \in Mnd$
4	1 2	mndidcl	$⊢ R \in Mnd \to \underset{˙}{0} \in B$
5	3 4	syl	$⊢ R \in SRing \to \underset{˙}{0} \in B$