Theorem slmdacl

Description: Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	slmdacl.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	slmdacl.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
	slmdacl.p	⊢ + = ( +g ‘ 𝐹 )
Assertion	slmdacl	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 + 𝑌 ) ∈ 𝐾 )

Proof

Step	Hyp	Ref	Expression
1		slmdacl.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		slmdacl.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
3		slmdacl.p	⊢ + = ( +g ‘ 𝐹 )
4	1	slmdsrg	⊢ ( 𝑊 ∈ SLMod → 𝐹 ∈ SRing )
5		srgmnd	⊢ ( 𝐹 ∈ SRing → 𝐹 ∈ Mnd )
6	4 5	syl	⊢ ( 𝑊 ∈ SLMod → 𝐹 ∈ Mnd )
7	2 3	mndcl	⊢ ( ( 𝐹 ∈ Mnd ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 + 𝑌 ) ∈ 𝐾 )
8	6 7	syl3an1	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 + 𝑌 ) ∈ 𝐾 )