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Metamath Proof Explorer

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 )