Metamath Proof Explorer

Theorem isumclim2

Description: A converging series converges to its infinite sum. (Contributed by NM, 2-Jan-2006) (Revised by Mario Carneiro, 23-Apr-2014)

Ref Expression
Hypotheses isumclim.1 ⊢ Z = ℤ ≥ M
isumclim.2 ⊢ φ → M ∈ ℤ
isumclim.3 ⊢ φ ∧ k ∈ Z → F ⁡ k = A
isumclim.4 ⊢ φ ∧ k ∈ Z → A ∈ ℂ
isumclim2.5 ⊢ φ → seq M + F ∈ dom ⁡ ⇝
Assertion isumclim2 ⊢ φ → seq M + F ⇝ ∑ k ∈ Z A

Proof

Step Hyp Ref Expression
1 isumclim.1 ⊢ Z = ℤ ≥ M
2 isumclim.2 ⊢ φ → M ∈ ℤ
3 isumclim.3 ⊢ φ ∧ k ∈ Z → F ⁡ k = A
4 isumclim.4 ⊢ φ ∧ k ∈ Z → A ∈ ℂ
5 isumclim2.5 ⊢ φ → seq M + F ∈ dom ⁡ ⇝
6 climdm ⊢ seq M + F ∈ dom ⁡ ⇝ ↔ seq M + F ⇝ ⇝ ⁡ seq M + F
7 5 6 sylib ⊢ φ → seq M + F ⇝ ⇝ ⁡ seq M + F
8 1 2 3 4 isum ⊢ φ → ∑ k ∈ Z A = ⇝ ⁡ seq M + F
9 7 8 breqtrrd ⊢ φ → seq M + F ⇝ ∑ k ∈ Z A