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

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	grpsubid	$$⊢\left(W\in \mathrm{Grp}\wedge A\in V\right)\to A\underset{\dot{}}{-}A=\underset{\dot{}}{0}$$
6	4 5	sylan	$$⊢\left(W\in \mathrm{LMod}\wedge A\in V\right)\to A\underset{\dot{}}{-}A=\underset{\dot{}}{0}$$