Theorem ringlsmss

Description: Closure of the product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024)

	Ref	Expression
Hypotheses	ringlsmss.1	$$⊢B={\mathrm{Base}}_R$$
	ringlsmss.2	$$⊢G={\mathrm{mulGrp}}_R$$
	ringlsmss.3	$$⊢\underset{\dot{}}{\times }=\mathrm{LSSum}\left(G\right)$$
	ringlsmss.4	$$⊢\phi \to R\in \mathrm{Ring}$$
	ringlsmss.5	$$⊢\phi \to E\subseteq B$$
	ringlsmss.6	$$⊢\phi \to F\subseteq B$$
Assertion	ringlsmss	$$⊢\phi \to E\underset{\dot{}}{\times }F\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		ringlsmss.1	$$⊢B={\mathrm{Base}}_R$$
2		ringlsmss.2	$$⊢G={\mathrm{mulGrp}}_R$$
3		ringlsmss.3	$$⊢\underset{\dot{}}{\times }=\mathrm{LSSum}\left(G\right)$$
4		ringlsmss.4	$$⊢\phi \to R\in \mathrm{Ring}$$
5		ringlsmss.5	$$⊢\phi \to E\subseteq B$$
6		ringlsmss.6	$$⊢\phi \to F\subseteq B$$
7	2	ringmgp	$$⊢R\in \mathrm{Ring}\to G\in \mathrm{Mnd}$$
8	4 7	syl	$$⊢\phi \to G\in \mathrm{Mnd}$$
9	2 1	mgpbas	$$⊢B={\mathrm{Base}}_G$$
10	9 3	lsmssv	$$⊢\left(G\in \mathrm{Mnd}\wedge E\subseteq B\wedge F\subseteq B\right)\to E\underset{\dot{}}{\times }F\subseteq B$$
11	8 5 6 10	syl3anc	$$⊢\phi \to E\underset{\dot{}}{\times }F\subseteq B$$