Theorem fprodnn0cl

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

	Ref	Expression
Hypotheses	fprodcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
	fprodnn0cl.2	$$⊢\left(\phi \wedge k\in A\right)\to B\in {\mathbb{N}}_0$$
Assertion	fprodnn0cl	$$⊢\phi \to \prod _{k\in A}B\in {\mathbb{N}}_0$$

Proof

Step	Hyp	Ref	Expression
1		fprodcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		fprodnn0cl.2	$$⊢\left(\phi \wedge k\in A\right)\to B\in {\mathbb{N}}_0$$
3		nn0sscn	$$⊢{\mathbb{N}}_0\subseteq ℂ$$
4	3	a1i	$$⊢\phi \to {\mathbb{N}}_0\subseteq ℂ$$
5		nn0mulcl	$$⊢\left(x\in {\mathbb{N}}_0\wedge y\in {\mathbb{N}}_0\right)\to xy\in {\mathbb{N}}_0$$
6	5	adantl	$$⊢\left(\phi \wedge \left(x\in {\mathbb{N}}_0\wedge y\in {\mathbb{N}}_0\right)\right)\to xy\in {\mathbb{N}}_0$$
7		1nn0	$$⊢1\in {\mathbb{N}}_0$$
8	7	a1i	$$⊢\phi \to 1\in {\mathbb{N}}_0$$
9	4 6 1 2 8	fprodcllem	$$⊢\phi \to \prod _{k\in A}B\in {\mathbb{N}}_0$$