Theorem lmodvacl

Description: Closure of vector addition for a left module. (Contributed by NM, 8-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodvacl.v	|- V = ( Base ` W )
	lmodvacl.a	|- .+ = ( +g ` W )
Assertion	lmodvacl	|- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .+ Y ) e. V )

Proof

Step	Hyp	Ref	Expression
1		lmodvacl.v	|- V = ( Base ` W )
2		lmodvacl.a	|- .+ = ( +g ` W )
3		lmodgrp	|- ( W e. LMod -> W e. Grp )
4	1 2	grpcl	|- ( ( 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 )