Theorem lmodvsubcl

Description: Closure of vector subtraction. ( hvsubcl analog.) (Contributed by NM, 31-Mar-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodvsubcl.v	$$⊢V={\mathrm{Base}}_W$$
	lmodvsubcl.m	$$⊢\underset{\dot{}}{-}={-}_W$$
Assertion	lmodvsubcl	$$⊢\left(W\in \mathrm{LMod}\wedge X\in V\wedge Y\in V\right)\to X\underset{\dot{}}{-}Y\in V$$

Proof

Step	Hyp	Ref	Expression
1		lmodvsubcl.v	$$⊢V={\mathrm{Base}}_W$$
2		lmodvsubcl.m	$$⊢\underset{\dot{}}{-}={-}_W$$
3		lmodgrp	$$⊢W\in \mathrm{LMod}\to W\in \mathrm{Grp}$$
4	1 2	grpsubcl	$$⊢\left(W\in \mathrm{Grp}\wedge X\in V\wedge Y\in V\right)\to X\underset{\dot{}}{-}Y\in V$$
5	3 4	syl3an1	$$⊢\left(W\in \mathrm{LMod}\wedge X\in V\wedge Y\in V\right)\to X\underset{\dot{}}{-}Y\in V$$