Theorem lmodacl

Description: Closure of ring addition for a left module. (Contributed by NM, 14-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodacl.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
	lmodacl.k	$$⊢K={\mathrm{Base}}_F$$
	lmodacl.p	$$⊢\underset{\dot{}}{+}={+}_F$$
Assertion	lmodacl	$$⊢\left(W\in \mathrm{LMod}\wedge X\in K\wedge Y\in K\right)\to X\underset{\dot{}}{+}Y\in K$$

Proof

Step	Hyp	Ref	Expression
1		lmodacl.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		lmodacl.k	$$⊢K={\mathrm{Base}}_F$$
3		lmodacl.p	$$⊢\underset{\dot{}}{+}={+}_F$$
4	1	lmodfgrp	$$⊢W\in \mathrm{LMod}\to F\in \mathrm{Grp}$$
5	2 3	grpcl	$$⊢\left(F\in \mathrm{Grp}\wedge X\in K\wedge Y\in K\right)\to X\underset{\dot{}}{+}Y\in K$$
6	4 5	syl3an1	$$⊢\left(W\in \mathrm{LMod}\wedge X\in K\wedge Y\in K\right)\to X\underset{\dot{}}{+}Y\in K$$