Theorem fprodnncl

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

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

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		fprodnncl.2	$$⊢\left(\phi \wedge k\in A\right)\to B\in \mathbb{N}$$
3		nnsscn	$$⊢\mathbb{N}\subseteq ℂ$$
4	3	a1i	$$⊢\phi \to \mathbb{N}\subseteq ℂ$$
5		nnmulcl	$$⊢\left(x\in \mathbb{N}\wedge y\in \mathbb{N}\right)\to xy\in \mathbb{N}$$
6	5	adantl	$$⊢\left(\phi \wedge \left(x\in \mathbb{N}\wedge y\in \mathbb{N}\right)\right)\to xy\in \mathbb{N}$$
7		1nn	$$⊢1\in \mathbb{N}$$
8	7	a1i	$$⊢\phi \to 1\in \mathbb{N}$$
9	4 6 1 2 8	fprodcllem	$$⊢\phi \to \prod _{k\in A}B\in \mathbb{N}$$