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Metamath Proof Explorer

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	$⊢ Z = ℤ_{\geq M}$
	serf.2	$⊢ φ \to M \in ℤ$
	serf.3	$⊢ (φ \land k \in Z) \to F (k) \in ℂ$
Assertion	serf	$⊢ φ \to seq M (+ F) : Z ⟶ ℂ$

Proof

Step	Hyp	Ref	Expression
1		serf.1	$⊢ Z = ℤ_{\geq M}$
2		serf.2	$⊢ φ \to M \in ℤ$
3		serf.3	$⊢ (φ \land k \in Z) \to F (k) \in ℂ$
4		addcl	$⊢ (k \in ℂ \land x \in ℂ) \to k + x \in ℂ$
5	4	adantl	$⊢ (φ \land (k \in ℂ \land x \in ℂ)) \to k + x \in ℂ$
6	1 2 3 5	seqf	$⊢ φ \to seq M (+ F) : Z ⟶ ℂ$