Theorem lmodmcl

Description: Closure of ring multiplication for a left module. (Contributed by NM, 14-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodmcl.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	lmodmcl.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
	lmodmcl.t	⊢ · = ( .r ‘ 𝐹 )
Assertion	lmodmcl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 · 𝑌 ) ∈ 𝐾 )

Proof

Step	Hyp	Ref	Expression
1		lmodmcl.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		lmodmcl.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
3		lmodmcl.t	⊢ · = ( .r ‘ 𝐹 )
4	1	lmodring	⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Ring )
5	2 3	ringcl	⊢ ( ( 𝐹 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 · 𝑌 ) ∈ 𝐾 )
6	4 5	syl3an1	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 · 𝑌 ) ∈ 𝐾 )