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Metamath Proof Explorer

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	$⊢ B = {Base}_{Y}$
		prdsmulrcl.t	$⊢ \underset{˙}{\cdot} = \cdot_{Y}$
		prdsmulrcl.s	$⊢ φ \to S \in V$
		prdsmulrcl.i	$⊢ φ \to I \in W$
		prdsmulrcl.r	$⊢ φ \to R : I ⟶ Ring$
		prdsmulrcl.f	$⊢ φ \to F \in B$
		prdsmulrcl.g	$⊢ φ \to G \in B$
	Assertion	prdsmulrcl	$⊢ φ \to F \underset{˙}{\cdot} G \in B$

Proof

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