Theorem fsumzcl

Description: Closure of a finite sum of integers. (Contributed by NM, 9-Nov-2005) (Revised by Mario Carneiro, 22-Apr-2014)

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

Proof

Step	Hyp	Ref	Expression
1		fsumcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		fsumzcl.2	$$⊢\left(\phi \wedge k\in A\right)\to B\in \mathbb{Z}$$
3		zsscn	$$⊢\mathbb{Z}\subseteq ℂ$$
4	3	a1i	$$⊢\phi \to \mathbb{Z}\subseteq ℂ$$
5		zaddcl	$$⊢\left(x\in \mathbb{Z}\wedge y\in \mathbb{Z}\right)\to x+y\in \mathbb{Z}$$
6	5	adantl	$$⊢\left(\phi \wedge \left(x\in \mathbb{Z}\wedge y\in \mathbb{Z}\right)\right)\to x+y\in \mathbb{Z}$$
7		0zd	$$⊢\phi \to 0\in \mathbb{Z}$$
8	4 6 1 2 7	fsumcllem	$$⊢\phi \to \sum _{k\in A}B\in \mathbb{Z}$$