Theorem lmodvsubcl

Description: Closure of vector subtraction. ( hvsubcl analog.) (Contributed by NM, 31-Mar-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodvsubcl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lmodvsubcl.m	⊢ − = ( -g ‘ 𝑊 )
Assertion	lmodvsubcl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 − 𝑌 ) ∈ 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		lmodvsubcl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lmodvsubcl.m	⊢ − = ( -g ‘ 𝑊 )
3		lmodgrp	⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Grp )
4	1 2	grpsubcl	⊢ ( ( 𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 − 𝑌 ) ∈ 𝑉 )
5	3 4	syl3an1	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 − 𝑌 ) ∈ 𝑉 )