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Metamath Proof Explorer

Theorem slmd0cl

Description: The ring zero in a semimodule belongs to the ring base set. (Contributed by NM, 11-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

Ref Expression
Hypotheses slmd0cl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
slmd0cl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
slmd0cl.z ⊢ 0 = ( 0g ‘ 𝐹 )
Assertion slmd0cl ( 𝑊 ∈ SLMod → 0 ∈ 𝐾 )

Proof

Step Hyp Ref Expression
1 slmd0cl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2 slmd0cl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
3 slmd0cl.z ⊢ 0 = ( 0g ‘ 𝐹 )
4 1 slmdsrg ⊢ ( 𝑊 ∈ SLMod → 𝐹 ∈ SRing )
5 2 3 srg0cl ⊢ ( 𝐹 ∈ SRing → 0 ∈ 𝐾 )
6 4 5 syl ⊢ ( 𝑊 ∈ SLMod → 0 ∈ 𝐾 )