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Metamath Proof Explorer

Theorem slmdvacl

Description: Closure of vector addition for a semiring left module. (Contributed by NM, 8-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	slmdvacl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	slmdvacl.a	⊢ + = ( +_g ‘ 𝑊 )
Assertion	slmdvacl	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 + 𝑌 ) ∈ 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		slmdvacl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		slmdvacl.a	⊢ + = ( +_g ‘ 𝑊 )
3		slmdmnd	⊢ ( 𝑊 ∈ SLMod → 𝑊 ∈ Mnd )
4	1 2	mndcl	⊢ ( ( 𝑊 ∈ Mnd ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 + 𝑌 ) ∈ 𝑉 )
5	3 4	syl3an1	⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 + 𝑌 ) ∈ 𝑉 )