Theorem fprodrecl

Description: Closure of a finite product of real numbers. (Contributed by Scott Fenton, 14-Dec-2017)

	Ref	Expression
Hypotheses	fprodcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
	fprodrecl.2	$$⊢\left(\phi \wedge k\in A\right)\to B\in ℝ$$
Assertion	fprodrecl	$$⊢\phi \to \prod _{k\in A}B\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		fprodrecl.2	$$⊢\left(\phi \wedge k\in A\right)\to B\in ℝ$$
3		ax-resscn	$$⊢ℝ\subseteq ℂ$$
4	3	a1i	$$⊢\phi \to ℝ\subseteq ℂ$$
5		remulcl	$$⊢\left(x\in ℝ\wedge y\in ℝ\right)\to xy\in ℝ$$
6	5	adantl	$$⊢\left(\phi \wedge \left(x\in ℝ\wedge y\in ℝ\right)\right)\to xy\in ℝ$$
7		1red	$$⊢\phi \to 1\in ℝ$$
8	4 6 1 2 7	fprodcllem	$$⊢\phi \to \prod _{k\in A}B\in ℝ$$