Theorem iprodn0

Description: Nonzero series product with an upper integer index set (i.e. an infinite product.) (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 )
	iprodn0.5	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐵 )
	iprodn0.6	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐵 ∈ ℂ )
Assertion	iprodn0	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑍 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		iprodn0.5	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐵 )
6		iprodn0.6	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐵 ∈ ℂ )
7		ssidd	⊢ ( 𝜑 → 𝑍 ⊆ 𝑍 )
8		iftrue	⊢ ( 𝑘 ∈ 𝑍 → if ( 𝑘 ∈ 𝑍 , 𝐵 , 1 ) = 𝐵 )
9	8	adantl	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → if ( 𝑘 ∈ 𝑍 , 𝐵 , 1 ) = 𝐵 )
10	5 9	eqtr4d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ 𝑍 , 𝐵 , 1 ) )
11	1 2 3 4 7 10 6	zprodn0	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑍 𝐵 = 𝑋 )