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Metamath Proof Explorer

Theorem lmodsubid

Description: Subtraction of a vector from itself. ( hvsubid analog.) (Contributed by NM, 16-Apr-2014) (Revised by Mario Carneiro, 19-Jun-2014)

Ref Expression
Hypotheses lmodsubeq0.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
lmodsubeq0.o ⊢ 0 = ( 0g ‘ 𝑊 )
lmodsubeq0.m ⊢ − = ( -g ‘ 𝑊 )
Assertion lmodsubid ( ( 𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 − 𝐴 ) = 0 )

Proof

Step Hyp Ref Expression
1 lmodsubeq0.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 lmodsubeq0.o ⊢ 0 = ( 0g ‘ 𝑊 )
3 lmodsubeq0.m ⊢ − = ( -g ‘ 𝑊 )
4 lmodgrp ⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Grp )
5 1 2 3 grpsubid ⊢ ( ( 𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 − 𝐴 ) = 0 )
6 4 5 sylan ⊢ ( ( 𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 − 𝐴 ) = 0 )