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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 )