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	$$⊢V={\mathrm{Base}}_W$$
	lmodvacl.a	$$⊢\underset{\dot{}}{+}={+}_W$$
Assertion	lmodvacl	$$⊢\left(W\in \mathrm{LMod}\wedge X\in V\wedge Y\in V\right)\to X\underset{\dot{}}{+}Y\in V$$

Proof

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