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	$$⊢V={\mathrm{Base}}_W$$
	lmodsubeq0.o	$$⊢\underset{\dot{}}{0}=0_W$$
	lmodsubeq0.m	$$⊢\underset{\dot{}}{-}={-}_W$$
Assertion	lmodsubeq0	$$⊢\left(W\in \mathrm{LMod}\wedge A\in V\wedge B\in V\right)\to \left(A\underset{\dot{}}{-}B=\underset{\dot{}}{0}\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		lmodsubeq0.v	$$⊢V={\mathrm{Base}}_W$$
2		lmodsubeq0.o	$$⊢\underset{\dot{}}{0}=0_W$$
3		lmodsubeq0.m	$$⊢\underset{\dot{}}{-}={-}_W$$
4		lmodgrp	$$⊢W\in \mathrm{LMod}\to W\in \mathrm{Grp}$$
5	1 2 3	grpsubeq0	$$⊢\left(W\in \mathrm{Grp}\wedge A\in V\wedge B\in V\right)\to \left(A\underset{\dot{}}{-}B=\underset{\dot{}}{0}\leftrightarrow A=B\right)$$
6	4 5	syl3an1	$$⊢\left(W\in \mathrm{LMod}\wedge A\in V\wedge B\in V\right)\to \left(A\underset{\dot{}}{-}B=\underset{\dot{}}{0}\leftrightarrow A=B\right)$$