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Metamath Proof Explorer

Theorem lmod1cl

Description: The ring unity 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 lmod1cl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
lmod1cl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
lmod1cl.u ⊢ 1 = ( 1r ‘ 𝐹 )
Assertion lmod1cl ( 𝑊 ∈ LMod → 1 ∈ 𝐾 )

Proof

Step Hyp Ref Expression
1 lmod1cl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2 lmod1cl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
3 lmod1cl.u ⊢ 1 = ( 1r ‘ 𝐹 )
4 1 lmodring ⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Ring )
5 2 3 ringidcl ⊢ ( 𝐹 ∈ Ring → 1 ∈ 𝐾 )
6 4 5 syl ⊢ ( 𝑊 ∈ LMod → 1 ∈ 𝐾 )