| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | mullid 11260 | . . 3
⊢ (𝑛 ∈ ℂ → (1
· 𝑛) = 𝑛) | 
| 2 | 1 | adantl 481 | . 2
⊢ (((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ ℂ) → (1 · 𝑛) = 𝑛) | 
| 3 |  | 1cnd 11256 | . 2
⊢ ((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) → 1 ∈
ℂ) | 
| 4 |  | prodrb.3 | . . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | 
| 5 | 4 | adantr 480 | . 2
⊢ ((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) → 𝑁 ∈ (ℤ≥‘𝑀)) | 
| 6 |  | iftrue 4531 | . . . . . . . . 9
⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) = 𝐵) | 
| 7 | 6 | adantl 481 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) = 𝐵) | 
| 8 |  | prodmo.2 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | 
| 9 | 8 | adantlr 715 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | 
| 10 | 7, 9 | eqeltrd 2841 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) | 
| 11 | 10 | ex 412 | . . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)) | 
| 12 |  | iffalse 4534 | . . . . . . 7
⊢ (¬
𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) = 1) | 
| 13 |  | ax-1cn 11213 | . . . . . . 7
⊢ 1 ∈
ℂ | 
| 14 | 12, 13 | eqeltrdi 2849 | . . . . . 6
⊢ (¬
𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) | 
| 15 | 11, 14 | pm2.61d1 180 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) | 
| 16 |  | prodmo.1 | . . . . 5
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) | 
| 17 | 15, 16 | fmptd 7134 | . . . 4
⊢ (𝜑 → 𝐹:ℤ⟶ℂ) | 
| 18 |  | uzssz 12899 | . . . . 5
⊢
(ℤ≥‘𝑀) ⊆ ℤ | 
| 19 | 18, 4 | sselid 3981 | . . . 4
⊢ (𝜑 → 𝑁 ∈ ℤ) | 
| 20 | 17, 19 | ffvelcdmd 7105 | . . 3
⊢ (𝜑 → (𝐹‘𝑁) ∈ ℂ) | 
| 21 | 20 | adantr 480 | . 2
⊢ ((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) → (𝐹‘𝑁) ∈ ℂ) | 
| 22 |  | elfzelz 13564 | . . . . 5
⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) → 𝑛 ∈ ℤ) | 
| 23 | 22 | adantl 481 | . . . 4
⊢ (((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → 𝑛 ∈ ℤ) | 
| 24 |  | simplr 769 | . . . . . 6
⊢ (((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → 𝐴 ⊆ (ℤ≥‘𝑁)) | 
| 25 | 19 | zcnd 12723 | . . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℂ) | 
| 26 | 25 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) → 𝑁 ∈ ℂ) | 
| 27 | 26 | adantr 480 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → 𝑁 ∈ ℂ) | 
| 28 |  | 1cnd 11256 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → 1 ∈
ℂ) | 
| 29 | 27, 28 | npcand 11624 | . . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁) | 
| 30 | 29 | fveq2d 6910 | . . . . . 6
⊢ (((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) →
(ℤ≥‘((𝑁 − 1) + 1)) =
(ℤ≥‘𝑁)) | 
| 31 | 24, 30 | sseqtrrd 4021 | . . . . 5
⊢ (((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → 𝐴 ⊆
(ℤ≥‘((𝑁 − 1) + 1))) | 
| 32 |  | fznuz 13649 | . . . . . 6
⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) → ¬ 𝑛 ∈ (ℤ≥‘((𝑁 − 1) +
1))) | 
| 33 | 32 | adantl 481 | . . . . 5
⊢ (((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ¬ 𝑛 ∈
(ℤ≥‘((𝑁 − 1) + 1))) | 
| 34 | 31, 33 | ssneldd 3986 | . . . 4
⊢ (((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → ¬ 𝑛 ∈ 𝐴) | 
| 35 | 23, 34 | eldifd 3962 | . . 3
⊢ (((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → 𝑛 ∈ (ℤ ∖ 𝐴)) | 
| 36 |  | fveqeq2 6915 | . . . 4
⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) = 1 ↔ (𝐹‘𝑛) = 1)) | 
| 37 |  | eldifi 4131 | . . . . . 6
⊢ (𝑘 ∈ (ℤ ∖ 𝐴) → 𝑘 ∈ ℤ) | 
| 38 |  | eldifn 4132 | . . . . . . . 8
⊢ (𝑘 ∈ (ℤ ∖ 𝐴) → ¬ 𝑘 ∈ 𝐴) | 
| 39 | 38, 12 | syl 17 | . . . . . . 7
⊢ (𝑘 ∈ (ℤ ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) = 1) | 
| 40 | 39, 13 | eqeltrdi 2849 | . . . . . 6
⊢ (𝑘 ∈ (ℤ ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) | 
| 41 | 16 | fvmpt2 7027 | . . . . . 6
⊢ ((𝑘 ∈ ℤ ∧ if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) | 
| 42 | 37, 40, 41 | syl2anc 584 | . . . . 5
⊢ (𝑘 ∈ (ℤ ∖ 𝐴) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) | 
| 43 | 42, 39 | eqtrd 2777 | . . . 4
⊢ (𝑘 ∈ (ℤ ∖ 𝐴) → (𝐹‘𝑘) = 1) | 
| 44 | 36, 43 | vtoclga 3577 | . . 3
⊢ (𝑛 ∈ (ℤ ∖ 𝐴) → (𝐹‘𝑛) = 1) | 
| 45 | 35, 44 | syl 17 | . 2
⊢ (((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑛) = 1) | 
| 46 | 2, 3, 5, 21, 45 | seqid 14088 | 1
⊢ ((𝜑 ∧ 𝐴 ⊆ (ℤ≥‘𝑁)) → (seq𝑀( · , 𝐹) ↾
(ℤ≥‘𝑁)) = seq𝑁( · , 𝐹)) |