Theorem clmring

Description: The scalar ring of a subcomplex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypothesis	clm0.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
Assertion	clmring	⊢ ( 𝑊 ∈ ℂMod → 𝐹 ∈ Ring )

Proof

Step	Hyp	Ref	Expression
1		clm0.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		clmlmod	⊢ ( 𝑊 ∈ ℂMod → 𝑊 ∈ LMod )
3	1	lmodring	⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Ring )
4	2 3	syl	⊢ ( 𝑊 ∈ ℂMod → 𝐹 ∈ Ring )