Theorem prdsmulrcl

Description: A structure product of rings has closed binary operation. (Contributed by Mario Carneiro, 11-Mar-2015) (Proof shortened by AV, 30-Mar-2025)

	Ref	Expression
Hypotheses	prdsmulrcl.y	$$⊢Y=S{⨉}_{𝑠}R$$
	prdsmulrcl.b	$$⊢B={\mathrm{Base}}_Y$$
	prdsmulrcl.t	$$⊢\underset{\dot{}}{·}={\cdot }_Y$$
	prdsmulrcl.s	$$⊢\phi \to S\in V$$
	prdsmulrcl.i	$$⊢\phi \to I\in W$$
	prdsmulrcl.r	$$⊢\phi \to R:I⟶\mathrm{Ring}$$
	prdsmulrcl.f	$$⊢\phi \to F\in B$$
	prdsmulrcl.g	$$⊢\phi \to G\in B$$
Assertion	prdsmulrcl	$$⊢\phi \to F\underset{\dot{}}{·}G\in B$$

Proof

Step	Hyp	Ref	Expression
1		prdsmulrcl.y	$$⊢Y=S{⨉}_{𝑠}R$$
2		prdsmulrcl.b	$$⊢B={\mathrm{Base}}_Y$$
3		prdsmulrcl.t	$$⊢\underset{\dot{}}{·}={\cdot }_Y$$
4		prdsmulrcl.s	$$⊢\phi \to S\in V$$
5		prdsmulrcl.i	$$⊢\phi \to I\in W$$
6		prdsmulrcl.r	$$⊢\phi \to R:I⟶\mathrm{Ring}$$
7		prdsmulrcl.f	$$⊢\phi \to F\in B$$
8		prdsmulrcl.g	$$⊢\phi \to G\in B$$
9		ringssrng	$$⊢\mathrm{Ring}\subseteq \mathrm{Rng}$$
10		fss	$$⊢\left(R:I⟶\mathrm{Ring}\wedge \mathrm{Ring}\subseteq \mathrm{Rng}\right)\to R:I⟶\mathrm{Rng}$$
11	6 9 10	sylancl	$$⊢\phi \to R:I⟶\mathrm{Rng}$$
12	1 2 3 4 5 11 7 8	prdsmulrngcl	$$⊢\phi \to F\underset{\dot{}}{·}G\in B$$