Theorem fsumrecl

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

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

Proof

Step	Hyp	Ref	Expression
1		fsumcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		fsumrecl.2	$$⊢\left(\phi \wedge k\in A\right)\to B\in ℝ$$
3		ax-resscn	$$⊢ℝ\subseteq ℂ$$
4	3	a1i	$$⊢\phi \to ℝ\subseteq ℂ$$
5		readdcl	$$⊢\left(x\in ℝ\wedge y\in ℝ\right)\to x+y\in ℝ$$
6	5	adantl	$$⊢\left(\phi \wedge \left(x\in ℝ\wedge y\in ℝ\right)\right)\to x+y\in ℝ$$
7		0red	$$⊢\phi \to 0\in ℝ$$
8	4 6 1 2 7	fsumcllem	$$⊢\phi \to \sum _{k\in A}B\in ℝ$$