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Metamath Proof Explorer

Theorem isumclim

Description: An infinite sum equals the value its series converges to. (Contributed by NM, 25-Dec-2005) (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
isumclim.6 ⊢ φ → seq M + F ⇝ B
Assertion isumclim ⊢ φ → ∑ k ∈ Z A = B

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 isumclim.6 ⊢ φ → seq M + F ⇝ B
6 1 2 3 4 isum ⊢ φ → ∑ k ∈ Z A = ⇝ ⁡ seq M + F
7 fclim ⊢ ⇝ : dom ⁡ ⇝ ⟶ ℂ
8 ffun ⊢ ⇝ : dom ⁡ ⇝ ⟶ ℂ → Fun ⁡ ⇝
9 7 8 ax-mp ⊢ Fun ⁡ ⇝
10 funbrfv ⊢ Fun ⁡ ⇝ → seq M + F ⇝ B → ⇝ ⁡ seq M + F = B
11 9 5 10 mpsyl ⊢ φ → ⇝ ⁡ seq M + F = B
12 6 11 eqtrd ⊢ φ → ∑ k ∈ Z A = B