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 = ( 0g𝑅 )
Assertion srg0cl ( 𝑅 ∈ SRing → 0𝐵 )

Proof

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