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Metamath Proof Explorer

Theorem lmod0cl

Description: The ring zero in a left module belongs to the set of scalars. (Contributed by NM, 11-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

Ref Expression
Hypotheses lmod0cl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
lmod0cl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
lmod0cl.z ⊢ 0 = ( 0g ‘ 𝐹 )
Assertion lmod0cl ( 𝑊 ∈ LMod → 0 ∈ 𝐾 )

Proof

Step Hyp Ref Expression
1 lmod0cl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2 lmod0cl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
3 lmod0cl.z ⊢ 0 = ( 0g ‘ 𝐹 )
4 1 lmodring ⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Ring )
5 2 3 ring0cl ⊢ ( 𝐹 ∈ Ring → 0 ∈ 𝐾 )
6 4 5 syl ⊢ ( 𝑊 ∈ LMod → 0 ∈ 𝐾 )