Theorem slmdmcl

Description: Closure of ring multiplication for a semimodule. (Contributed by NM, 14-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

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

Proof

Step	Hyp	Ref	Expression
1		slmdmcl.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		slmdmcl.k	$$⊢K={\mathrm{Base}}_F$$
3		slmdmcl.t	$$⊢\underset{\dot{}}{·}={\cdot }_F$$
4	1	slmdsrg	$$⊢W\in \mathrm{SLMod}\to F\in \mathrm{SRing}$$
5	2 3	srgcl	$$⊢\left(F\in \mathrm{SRing}\wedge X\in K\wedge Y\in K\right)\to X\underset{\dot{}}{·}Y\in K$$
6	4 5	syl3an1	$$⊢\left(W\in \mathrm{SLMod}\wedge X\in K\wedge Y\in K\right)\to X\underset{\dot{}}{·}Y\in K$$