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Metamath Proof Explorer

Theorem lmodvnegid

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

Ref Expression
Hypotheses lmodvnegid.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
lmodvnegid.p ⊢ + = ( +g ‘ 𝑊 )
lmodvnegid.z ⊢ 0 = ( 0g ‘ 𝑊 )
lmodvnegid.n ⊢ 𝑁 = ( invg ‘ 𝑊 )
Assertion lmodvnegid ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 + ( 𝑁 ‘ 𝑋 ) ) = 0 )

Proof

Step Hyp Ref Expression
1 lmodvnegid.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 lmodvnegid.p ⊢ + = ( +g ‘ 𝑊 )
3 lmodvnegid.z ⊢ 0 = ( 0g ‘ 𝑊 )
4 lmodvnegid.n ⊢ 𝑁 = ( invg ‘ 𝑊 )
5 lmodgrp ⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Grp )
6 1 2 3 4 grprinv ⊢ ( ( 𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 + ( 𝑁 ‘ 𝑋 ) ) = 0 )
7 5 6 sylan ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 + ( 𝑁 ‘ 𝑋 ) ) = 0 )