Theorem prodrb

Description: Rebase the starting point of a product. (Contributed by Scott Fenton, 4-Dec-2017)

	Ref	Expression
Hypotheses	prodmo.1	⊢ 𝐹 = ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) )
	prodmo.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
	prodrb.4	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	prodrb.5	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
	prodrb.6	⊢ ( 𝜑 → 𝐴 ⊆ ( ℤ≥ ‘ 𝑀 ) )
	prodrb.7	⊢ ( 𝜑 → 𝐴 ⊆ ( ℤ≥ ‘ 𝑁 ) )
Assertion	prodrb	⊢ ( 𝜑 → ( seq 𝑀 ( · , 𝐹 ) ⇝ 𝐶 ↔ seq 𝑁 ( · , 𝐹 ) ⇝ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		prodmo.1	⊢ 𝐹 = ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) )
2		prodmo.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
3		prodrb.4	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
4		prodrb.5	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
5		prodrb.6	⊢ ( 𝜑 → 𝐴 ⊆ ( ℤ≥ ‘ 𝑀 ) )
6		prodrb.7	⊢ ( 𝜑 → 𝐴 ⊆ ( ℤ≥ ‘ 𝑁 ) )
7	1 2 3 4 5 6	prodrblem2	⊢ ( ( 𝜑 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( seq 𝑀 ( · , 𝐹 ) ⇝ 𝐶 ↔ seq 𝑁 ( · , 𝐹 ) ⇝ 𝐶 ) )
8	1 2 4 3 6 5	prodrblem2	⊢ ( ( 𝜑 ∧ 𝑀 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( seq 𝑁 ( · , 𝐹 ) ⇝ 𝐶 ↔ seq 𝑀 ( · , 𝐹 ) ⇝ 𝐶 ) )
9	8	bicomd	⊢ ( ( 𝜑 ∧ 𝑀 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( seq 𝑀 ( · , 𝐹 ) ⇝ 𝐶 ↔ seq 𝑁 ( · , 𝐹 ) ⇝ 𝐶 ) )
10		uztric	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∨ 𝑀 ∈ ( ℤ≥ ‘ 𝑁 ) ) )
11	3 4 10	syl2anc	⊢ ( 𝜑 → ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∨ 𝑀 ∈ ( ℤ≥ ‘ 𝑁 ) ) )
12	7 9 11	mpjaodan	⊢ ( 𝜑 → ( seq 𝑀 ( · , 𝐹 ) ⇝ 𝐶 ↔ seq 𝑁 ( · , 𝐹 ) ⇝ 𝐶 ) )