Theorem fprodrpcl

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

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

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		fprodrpcl.2	$$⊢\left(\phi \wedge k\in A\right)\to B\in {ℝ}^{+}$$
3		rpssre	$$⊢{ℝ}^{+}\subseteq ℝ$$
4		ax-resscn	$$⊢ℝ\subseteq ℂ$$
5	3 4	sstri	$$⊢{ℝ}^{+}\subseteq ℂ$$
6	5	a1i	$$⊢\phi \to {ℝ}^{+}\subseteq ℂ$$
7		rpmulcl	$$⊢\left(x\in {ℝ}^{+}\wedge y\in {ℝ}^{+}\right)\to xy\in {ℝ}^{+}$$
8	7	adantl	$$⊢\left(\phi \wedge \left(x\in {ℝ}^{+}\wedge y\in {ℝ}^{+}\right)\right)\to xy\in {ℝ}^{+}$$
9		1rp	$$⊢1\in {ℝ}^{+}$$
10	9	a1i	$$⊢\phi \to 1\in {ℝ}^{+}$$
11	6 8 1 2 10	fprodcllem	$$⊢\phi \to \prod _{k\in A}B\in {ℝ}^{+}$$