Theorem lmodacl

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

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

Proof

Step	Hyp	Ref	Expression
1		lmodacl.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		lmodacl.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
3		lmodacl.p	⊢ + = ( +g ‘ 𝐹 )
4	1	lmodfgrp	⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Grp )
5	2 3	grpcl	⊢ ( ( 𝐹 ∈ Grp ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 + 𝑌 ) ∈ 𝐾 )
6	4 5	syl3an1	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 + 𝑌 ) ∈ 𝐾 )