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 )