Theorem lmod1cl

Description: The ring unit in a left module belongs to the ring base set. (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 )