Theorem slmdmcl

Description: Closure of ring multiplication for a semimodule. (Contributed by NM, 14-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

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

Proof

Step	Hyp	Ref	Expression
1		slmdmcl.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		slmdmcl.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
3		slmdmcl.t	⊢ · = ( .r ‘ 𝐹 )
4	1	slmdsrg	⊢ ( 𝑊 ∈ SLMod → 𝐹 ∈ SRing )
5	2 3	srgcl	⊢ ( ( 𝐹 ∈ SRing ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 · 𝑌 ) ∈ 𝐾 )
6	4 5	syl3an1	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 · 𝑌 ) ∈ 𝐾 )