Theorem slmdvacl

Description: Closure of vector addition for a semiring left module. (Contributed by NM, 8-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	slmdvacl.v	$$⊢V={\mathrm{Base}}_W$$
	slmdvacl.a	$$⊢\underset{\dot{}}{+}={+}_W$$
Assertion	slmdvacl	$$⊢\left(W\in \mathrm{SLMod}\wedge X\in V\wedge Y\in V\right)\to X\underset{\dot{}}{+}Y\in V$$

Proof

Step	Hyp	Ref	Expression
1		slmdvacl.v	$$⊢V={\mathrm{Base}}_W$$
2		slmdvacl.a	$$⊢\underset{\dot{}}{+}={+}_W$$
3		slmdmnd	$$⊢W\in \mathrm{SLMod}\to W\in \mathrm{Mnd}$$
4	1 2	mndcl	$$⊢\left(W\in \mathrm{Mnd}\wedge X\in V\wedge Y\in V\right)\to X\underset{\dot{}}{+}Y\in V$$
5	3 4	syl3an1	$$⊢\left(W\in \mathrm{SLMod}\wedge X\in V\wedge Y\in V\right)\to X\underset{\dot{}}{+}Y\in V$$