Theorem clmmcl

Description: Closure of ring multiplication for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	clm0.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	clmsub.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
Assertion	clmmcl	⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 · 𝑌 ) ∈ 𝐾 )

Proof

Step	Hyp	Ref	Expression
1		clm0.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		clmsub.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
3	1 2	clmsubrg	⊢ ( 𝑊 ∈ ℂMod → 𝐾 ∈ ( SubRing ‘ ℂfld ) )
4		cnfldmul	⊢ · = ( .r ‘ ℂfld )
5	4	subrgmcl	⊢ ( ( 𝐾 ∈ ( SubRing ‘ ℂfld ) ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 · 𝑌 ) ∈ 𝐾 )
6	3 5	syl3an1	⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 · 𝑌 ) ∈ 𝐾 )