Theorem lmodvnegcl

Description: Closure of vector negative. (Contributed by NM, 18-Apr-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodvnegcl.v	|- V = ( Base ` W )
	lmodvnegcl.n	|- N = ( invg ` W )
Assertion	lmodvnegcl	|- ( ( W e. LMod /\ X e. V ) -> ( N ` X ) e. V )

Proof

Step	Hyp	Ref	Expression
1		lmodvnegcl.v	|- V = ( Base ` W )
2		lmodvnegcl.n	|- N = ( invg ` W )
3		lmodgrp	|- ( W e. LMod -> W e. Grp )
4	1 2	grpinvcl	|- ( ( W e. Grp /\ X e. V ) -> ( N ` X ) e. V )
5	3 4	sylan	|- ( ( W e. LMod /\ X e. V ) -> ( N ` X ) e. V )