Theorem fsumcl

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

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

Proof

Step	Hyp	Ref	Expression
1		fsumcl.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		fsumcl.2	$$⊢\left(\phi \wedge k\in A\right)\to B\in ℂ$$
3		ssidd	$$⊢\phi \to ℂ\subseteq ℂ$$
4		addcl	$$⊢\left(x\in ℂ\wedge y\in ℂ\right)\to x+y\in ℂ$$
5	4	adantl	$$⊢\left(\phi \wedge \left(x\in ℂ\wedge y\in ℂ\right)\right)\to x+y\in ℂ$$
6		0cnd	$$⊢\phi \to 0\in ℂ$$
7	3 5 1 2 6	fsumcllem	$$⊢\phi \to \sum _{k\in A}B\in ℂ$$