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 = ( Base ` W )
	lmodvsubcl.m	|- .- = ( -g ` W )
Assertion	lmodvsubcl	|- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .- Y ) e. V )

Proof

Step	Hyp	Ref	Expression
1		lmodvsubcl.v	|- V = ( Base ` W )
2		lmodvsubcl.m	|- .- = ( -g ` W )
3		lmodgrp	|- ( W e. LMod -> W e. Grp )
4	1 2	grpsubcl	|- ( ( W e. Grp /\ X e. V /\ Y e. V ) -> ( X .- Y ) e. V )
5	3 4	syl3an1	|- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .- Y ) e. V )