Theorem fsumnn0cl

Description: Closure of a finite sum of nonnegative integers. (Contributed by Mario Carneiro, 23-Apr-2015)

	Ref	Expression
Hypotheses	fsumcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
	fsumnn0cl.2	$$⊢\left(\phi \wedge k\in A\right)\to B\in {\mathbb{N}}_0$$
Assertion	fsumnn0cl	$$⊢\phi \to \sum _{k\in A}B\in {\mathbb{N}}_0$$

Proof

Step	Hyp	Ref	Expression
1		fsumcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		fsumnn0cl.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		nn0addcl	$$⊢\left(x\in {\mathbb{N}}_0\wedge y\in {\mathbb{N}}_0\right)\to x+y\in {\mathbb{N}}_0$$
6	5	adantl	$$⊢\left(\phi \wedge \left(x\in {\mathbb{N}}_0\wedge y\in {\mathbb{N}}_0\right)\right)\to x+y\in {\mathbb{N}}_0$$
7		0nn0	$$⊢0\in {\mathbb{N}}_0$$
8	7	a1i	$$⊢\phi \to 0\in {\mathbb{N}}_0$$
9	4 6 1 2 8	fsumcllem	$$⊢\phi \to \sum _{k\in A}B\in {\mathbb{N}}_0$$