Theorem slmdacl

Description: Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018)

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

Proof

Step	Hyp	Ref	Expression
1		slmdacl.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		slmdacl.k	$$⊢K={\mathrm{Base}}_F$$
3		slmdacl.p	$$⊢\underset{\dot{}}{+}={+}_F$$
4	1	slmdsrg	$$⊢W\in \mathrm{SLMod}\to F\in \mathrm{SRing}$$
5		srgmnd	$$⊢F\in \mathrm{SRing}\to F\in \mathrm{Mnd}$$
6	4 5	syl	$$⊢W\in \mathrm{SLMod}\to F\in \mathrm{Mnd}$$
7	2 3	mndcl	$$⊢\left(F\in \mathrm{Mnd}\wedge X\in K\wedge Y\in K\right)\to X\underset{\dot{}}{+}Y\in K$$
8	6 7	syl3an1	$$⊢\left(W\in \mathrm{SLMod}\wedge X\in K\wedge Y\in K\right)\to X\underset{\dot{}}{+}Y\in K$$