Theorem lmodvacl

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

	Ref	Expression
Hypotheses	lmodvacl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lmodvacl.a	⊢ + = ( +g ‘ 𝑊 )
Assertion	lmodvacl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 + 𝑌 ) ∈ 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		lmodvacl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lmodvacl.a	⊢ + = ( +g ‘ 𝑊 )
3		lmodgrp	⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Grp )
4	1 2	grpcl	⊢ ( ( 𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 + 𝑌 ) ∈ 𝑉 )
5	3 4	syl3an1	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 + 𝑌 ) ∈ 𝑉 )