Step | Hyp | Ref
| Expression |
1 | | eqid 2738 |
. 2
⊢
(ℤ≥‘𝑁) = (ℤ≥‘𝑁) |
2 | | prodrb.3 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
3 | | eluzelz 12334 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
4 | 2, 3 | syl 17 |
. 2
⊢ (𝜑 → 𝑁 ∈ ℤ) |
5 | | seqex 13462 |
. . 3
⊢ seq𝑀( · , 𝐹) ∈ V |
6 | 5 | a1i 11 |
. 2
⊢ (𝜑 → seq𝑀( · , 𝐹) ∈ V) |
7 | | eqid 2738 |
. . . 4
⊢
(ℤ≥‘𝑀) = (ℤ≥‘𝑀) |
8 | | eluzel2 12329 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
9 | 2, 8 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℤ) |
10 | | eluzelz 12334 |
. . . . . . 7
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → 𝑘 ∈ ℤ) |
11 | 10 | adantl 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℤ) |
12 | | iftrue 4420 |
. . . . . . . . . 10
⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) = 𝐵) |
13 | 12 | adantl 485 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) = 𝐵) |
14 | | prodmo.2 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
15 | 14 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
16 | 13, 15 | eqeltrd 2833 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) |
17 | 16 | ex 416 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)) |
18 | | iffalse 4423 |
. . . . . . . 8
⊢ (¬
𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) = 1) |
19 | | ax-1cn 10673 |
. . . . . . . 8
⊢ 1 ∈
ℂ |
20 | 18, 19 | eqeltrdi 2841 |
. . . . . . 7
⊢ (¬
𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) |
21 | 17, 20 | pm2.61d1 183 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) |
22 | | prodmo.1 |
. . . . . . 7
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) |
23 | 22 | fvmpt2 6786 |
. . . . . 6
⊢ ((𝑘 ∈ ℤ ∧ if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) |
24 | 11, 21, 23 | syl2anc 587 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) |
25 | 24, 21 | eqeltrd 2833 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
26 | 7, 9, 25 | prodf 15335 |
. . 3
⊢ (𝜑 → seq𝑀( · , 𝐹):(ℤ≥‘𝑀)⟶ℂ) |
27 | 26, 2 | ffvelrnd 6862 |
. 2
⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ) |
28 | | mulid1 10717 |
. . . . 5
⊢ (𝑚 ∈ ℂ → (𝑚 · 1) = 𝑚) |
29 | 28 | adantl 485 |
. . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) ∧ 𝑚 ∈ ℂ) → (𝑚 · 1) = 𝑚) |
30 | 2 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ (ℤ≥‘𝑀)) |
31 | | simpr 488 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑛 ∈ (ℤ≥‘𝑁)) |
32 | 9 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℤ) |
33 | 25 | adantlr 715 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
34 | 7, 32, 33 | prodf 15335 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → seq𝑀( · , 𝐹):(ℤ≥‘𝑀)⟶ℂ) |
35 | 34, 30 | ffvelrnd 6862 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ) |
36 | | elfzuz 12994 |
. . . . . 6
⊢ (𝑚 ∈ ((𝑁 + 1)...𝑛) → 𝑚 ∈ (ℤ≥‘(𝑁 + 1))) |
37 | | eluzelz 12334 |
. . . . . . . . 9
⊢ (𝑚 ∈
(ℤ≥‘(𝑁 + 1)) → 𝑚 ∈ ℤ) |
38 | 37 | adantl 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑚 ∈
ℤ) |
39 | | fprodcvg.4 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ⊆ (𝑀...𝑁)) |
40 | 39 | sseld 3876 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑚 ∈ 𝐴 → 𝑚 ∈ (𝑀...𝑁))) |
41 | | fznuz 13080 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ (𝑀...𝑁) → ¬ 𝑚 ∈ (ℤ≥‘(𝑁 + 1))) |
42 | 40, 41 | syl6 35 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑚 ∈ 𝐴 → ¬ 𝑚 ∈ (ℤ≥‘(𝑁 + 1)))) |
43 | 42 | con2d 136 |
. . . . . . . . 9
⊢ (𝜑 → (𝑚 ∈ (ℤ≥‘(𝑁 + 1)) → ¬ 𝑚 ∈ 𝐴)) |
44 | 43 | imp 410 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘(𝑁 + 1))) → ¬ 𝑚 ∈ 𝐴) |
45 | 38, 44 | eldifd 3854 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑚 ∈ (ℤ ∖ 𝐴)) |
46 | | fveqeq2 6683 |
. . . . . . . 8
⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) = 1 ↔ (𝐹‘𝑚) = 1)) |
47 | | eldifi 4017 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (ℤ ∖ 𝐴) → 𝑘 ∈ ℤ) |
48 | | eldifn 4018 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ (ℤ ∖ 𝐴) → ¬ 𝑘 ∈ 𝐴) |
49 | 48, 18 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (ℤ ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) = 1) |
50 | 49, 19 | eqeltrdi 2841 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (ℤ ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) |
51 | 47, 50, 23 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝑘 ∈ (ℤ ∖ 𝐴) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) |
52 | 51, 49 | eqtrd 2773 |
. . . . . . . 8
⊢ (𝑘 ∈ (ℤ ∖ 𝐴) → (𝐹‘𝑘) = 1) |
53 | 46, 52 | vtoclga 3478 |
. . . . . . 7
⊢ (𝑚 ∈ (ℤ ∖ 𝐴) → (𝐹‘𝑚) = 1) |
54 | 45, 53 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘(𝑁 + 1))) → (𝐹‘𝑚) = 1) |
55 | 36, 54 | sylan2 596 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → (𝐹‘𝑚) = 1) |
56 | 55 | adantlr 715 |
. . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) ∧ 𝑚 ∈ ((𝑁 + 1)...𝑛)) → (𝐹‘𝑚) = 1) |
57 | 29, 30, 31, 35, 56 | seqid2 13508 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (seq𝑀( · , 𝐹)‘𝑁) = (seq𝑀( · , 𝐹)‘𝑛)) |
58 | 57 | eqcomd 2744 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) = (seq𝑀( · , 𝐹)‘𝑁)) |
59 | 1, 4, 6, 27, 58 | climconst 14990 |
1
⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq𝑀( · , 𝐹)‘𝑁)) |