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 ) )