This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lmodsubeq0

Description: If the difference between two vectors is zero, they are equal. ( hvsubeq0 analog.) (Contributed by NM, 31-Mar-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 lmodsubeq0 ( ( 𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )

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 grpsubeq0 ⊢ ( ( 𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )
6 4 5 syl3an1 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )