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Metamath Proof Explorer

Theorem zprodn0

Description: Nonzero series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 6-Dec-2017)

Ref Expression
Hypotheses zprodn0.1 ⊢ Z = ℤ ≥ M
zprodn0.2 ⊢ φ → M ∈ ℤ
zprodn0.3 ⊢ φ → X ≠ 0
zprodn0.4 ⊢ φ → seq M × F ⇝ X
zprodn0.5 ⊢ φ → A ⊆ Z
zprodn0.6 ⊢ φ ∧ k ∈ Z → F ⁡ k = if k ∈ A B 1
zprodn0.7 ⊢ φ ∧ k ∈ A → B ∈ ℂ
Assertion zprodn0 ⊢ φ → ∏ k ∈ A B = X

Proof

Step Hyp Ref Expression
1 zprodn0.1 ⊢ Z = ℤ ≥ M
2 zprodn0.2 ⊢ φ → M ∈ ℤ
3 zprodn0.3 ⊢ φ → X ≠ 0
4 zprodn0.4 ⊢ φ → seq M × F ⇝ X
5 zprodn0.5 ⊢ φ → A ⊆ Z
6 zprodn0.6 ⊢ φ ∧ k ∈ Z → F ⁡ k = if k ∈ A B 1
7 zprodn0.7 ⊢ φ ∧ k ∈ A → B ∈ ℂ
8 1 2 4 3 ntrivcvgn0 ⊢ φ → ∃ m ∈ Z ∃ x x ≠ 0 ∧ seq m × F ⇝ x
9 1 2 8 5 6 7 zprod ⊢ φ → ∏ k ∈ A B = ⇝ ⁡ seq M × F
10 fclim ⊢ ⇝ : dom ⁡ ⇝ ⟶ ℂ
11 ffun ⊢ ⇝ : dom ⁡ ⇝ ⟶ ℂ → Fun ⁡ ⇝
12 10 11 ax-mp ⊢ Fun ⁡ ⇝
13 funbrfv ⊢ Fun ⁡ ⇝ → seq M × F ⇝ X → ⇝ ⁡ seq M × F = X
14 12 4 13 mpsyl ⊢ φ → ⇝ ⁡ seq M × F = X
15 9 14 eqtrd ⊢ φ → ∏ k ∈ A B = X