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Metamath Proof Explorer

Theorem fprodreclf

Description: Closure of a finite product of real numbers. A version of fprodrecl using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020)

	Ref	Expression
Hypotheses	fprodreclf.kph	$⊢ Ⅎ k φ$
	fprodcl.a	$⊢ φ \to A \in Fin$
	fprodrecl.b	$⊢ (φ \land k \in A) \to B \in ℝ$
Assertion	fprodreclf	$⊢ φ \to \prod_{k \in A} B \in ℝ$

Proof

Step	Hyp	Ref	Expression
1		fprodreclf.kph	$⊢ Ⅎ k φ$
2		fprodcl.a	$⊢ φ \to A \in Fin$
3		fprodrecl.b	$⊢ (φ \land k \in A) \to B \in ℝ$
4		ax-resscn	$⊢ ℝ \subseteq ℂ$
5	4	a1i	$⊢ φ \to ℝ \subseteq ℂ$
6		remulcl	$⊢ (x \in ℝ \land y \in ℝ) \to x y \in ℝ$
7	6	adantl	$⊢ (φ \land (x \in ℝ \land y \in ℝ)) \to x y \in ℝ$
8		1red	$⊢ φ \to 1 \in ℝ$
9	1 5 7 2 3 8	fprodcllemf	$⊢ φ \to \prod_{k \in A} B \in ℝ$