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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	⊢ 0 = ( 0_g ‘ 𝑅 )
Assertion	srg0cl	⊢ ( 𝑅 ∈ SRing → 0 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		srg0cl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		srg0cl.z	⊢ 0 = ( 0_g ‘ 𝑅 )
3		srgmnd	⊢ ( 𝑅 ∈ SRing → 𝑅 ∈ Mnd )
4	1 2	mndidcl	⊢ ( 𝑅 ∈ Mnd → 0 ∈ 𝐵 )
5	3 4	syl	⊢ ( 𝑅 ∈ SRing → 0 ∈ 𝐵 )