Theorem lmodvnegcl

Description: Closure of vector negative. (Contributed by NM, 18-Apr-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodvnegcl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lmodvnegcl.n	⊢ 𝑁 = ( invg ‘ 𝑊 )
Assertion	lmodvnegcl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		lmodvnegcl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lmodvnegcl.n	⊢ 𝑁 = ( invg ‘ 𝑊 )
3		lmodgrp	⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Grp )
4	1 2	grpinvcl	⊢ ( ( 𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝑉 )
5	3 4	sylan	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝑉 )