Theorem serf

Description: An infinite series of complex terms is a function from NN to CC . (Contributed by NM, 18-Apr-2005) (Revised by Mario Carneiro, 27-May-2014)

	Ref	Expression
Hypotheses	serf.1	$$⊢Z={\mathbb{Z}}_{\ge M}$$
	serf.2	$$⊢\phi \to M\in \mathbb{Z}$$
	serf.3	$$⊢\left(\phi \wedge k\in Z\right)\to F\left(k\right)\in ℂ$$
Assertion	serf	$$⊢\phi \to seqM\left(+,F\right):Z⟶ℂ$$

Proof

Step	Hyp	Ref	Expression
1		serf.1	$$⊢Z={\mathbb{Z}}_{\ge M}$$
2		serf.2	$$⊢\phi \to M\in \mathbb{Z}$$
3		serf.3	$$⊢\left(\phi \wedge k\in Z\right)\to F\left(k\right)\in ℂ$$
4		addcl	$$⊢\left(k\in ℂ\wedge x\in ℂ\right)\to k+x\in ℂ$$
5	4	adantl	$$⊢\left(\phi \wedge \left(k\in ℂ\wedge x\in ℂ\right)\right)\to k+x\in ℂ$$
6	1 2 3 5	seqf	$$⊢\phi \to seqM\left(+,F\right):Z⟶ℂ$$