Step | Hyp | Ref
| Expression |
1 | | eqid 2821 |
. . 3
⊢
(ℤ≥‘(𝑁 + 1)) =
(ℤ≥‘(𝑁 + 1)) |
2 | | clim2div.2 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ 𝑍) |
3 | | eluzelz 12254 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
4 | | clim2div.1 |
. . . . . 6
⊢ 𝑍 =
(ℤ≥‘𝑀) |
5 | 3, 4 | eleq2s 2931 |
. . . . 5
⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
6 | 2, 5 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑁 ∈ ℤ) |
7 | 6 | peano2zd 12091 |
. . 3
⊢ (𝜑 → (𝑁 + 1) ∈ ℤ) |
8 | | clim2div.4 |
. . 3
⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝐴) |
9 | | eluzel2 12249 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
10 | 9, 4 | eleq2s 2931 |
. . . . . . 7
⊢ (𝑁 ∈ 𝑍 → 𝑀 ∈ ℤ) |
11 | 2, 10 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℤ) |
12 | | clim2div.3 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
13 | 4, 11, 12 | prodf 15243 |
. . . . 5
⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ) |
14 | 13, 2 | ffvelrnd 6852 |
. . . 4
⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ) |
15 | | clim2div.5 |
. . . 4
⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0) |
16 | 14, 15 | reccld 11409 |
. . 3
⊢ (𝜑 → (1 / (seq𝑀( · , 𝐹)‘𝑁)) ∈ ℂ) |
17 | | seqex 13372 |
. . . 4
⊢ seq(𝑁 + 1)( · , 𝐹) ∈ V |
18 | 17 | a1i 11 |
. . 3
⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ∈ V) |
19 | 2, 4 | eleqtrdi 2923 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
20 | | peano2uz 12302 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) |
21 | 19, 20 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) |
22 | 21, 4 | eleqtrrdi 2924 |
. . . . 5
⊢ (𝜑 → (𝑁 + 1) ∈ 𝑍) |
23 | 4 | uztrn2 12263 |
. . . . 5
⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑗 ∈ 𝑍) |
24 | 22, 23 | sylan 582 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑗 ∈ 𝑍) |
25 | 13 | ffvelrnda 6851 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( · , 𝐹)‘𝑗) ∈ ℂ) |
26 | 24, 25 | syldan 593 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘𝑗) ∈ ℂ) |
27 | | mulcl 10621 |
. . . . . . . 8
⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 · 𝑥) ∈ ℂ) |
28 | 27 | adantl 484 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ) |
29 | | mulass 10625 |
. . . . . . . 8
⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑘 · 𝑥) · 𝑦) = (𝑘 · (𝑥 · 𝑦))) |
30 | 29 | adantl 484 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑘 · 𝑥) · 𝑦) = (𝑘 · (𝑥 · 𝑦))) |
31 | | simpr 487 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑗 ∈
(ℤ≥‘(𝑁 + 1))) |
32 | 19 | adantr 483 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) |
33 | | elfzuz 12905 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀)) |
34 | 33, 4 | eleqtrrdi 2924 |
. . . . . . . . 9
⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍) |
35 | 34, 12 | sylan2 594 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) |
36 | 35 | adantlr 713 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) |
37 | 28, 30, 31, 32, 36 | seqsplit 13404 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘𝑗) = ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑗))) |
38 | 37 | eqcomd 2827 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑗)) = (seq𝑀( · , 𝐹)‘𝑗)) |
39 | 14 | adantr 483 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘𝑁) ∈ ℂ) |
40 | 4 | uztrn2 12263 |
. . . . . . . . . 10
⊢ (((𝑁 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍) |
41 | 22, 40 | sylan 582 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ 𝑍) |
42 | 41, 12 | syldan 593 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → (𝐹‘𝑘) ∈ ℂ) |
43 | 1, 7, 42 | prodf 15243 |
. . . . . . 7
⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹):(ℤ≥‘(𝑁 +
1))⟶ℂ) |
44 | 43 | ffvelrnda 6851 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘𝑗) ∈ ℂ) |
45 | 15 | adantr 483 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0) |
46 | 26, 39, 44, 45 | divmuld 11438 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (((seq𝑀( · , 𝐹)‘𝑗) / (seq𝑀( · , 𝐹)‘𝑁)) = (seq(𝑁 + 1)( · , 𝐹)‘𝑗) ↔ ((seq𝑀( · , 𝐹)‘𝑁) · (seq(𝑁 + 1)( · , 𝐹)‘𝑗)) = (seq𝑀( · , 𝐹)‘𝑗))) |
47 | 38, 46 | mpbird 259 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → ((seq𝑀( · , 𝐹)‘𝑗) / (seq𝑀( · , 𝐹)‘𝑁)) = (seq(𝑁 + 1)( · , 𝐹)‘𝑗)) |
48 | 26, 39, 45 | divrec2d 11420 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → ((seq𝑀( · , 𝐹)‘𝑗) / (seq𝑀( · , 𝐹)‘𝑁)) = ((1 / (seq𝑀( · , 𝐹)‘𝑁)) · (seq𝑀( · , 𝐹)‘𝑗))) |
49 | 47, 48 | eqtr3d 2858 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))) → (seq(𝑁 + 1)( · , 𝐹)‘𝑗) = ((1 / (seq𝑀( · , 𝐹)‘𝑁)) · (seq𝑀( · , 𝐹)‘𝑗))) |
50 | 1, 7, 8, 16, 18, 26, 49 | climmulc2 14993 |
. 2
⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ ((1 / (seq𝑀( · , 𝐹)‘𝑁)) · 𝐴)) |
51 | | climcl 14856 |
. . . 4
⊢ (seq𝑀( · , 𝐹) ⇝ 𝐴 → 𝐴 ∈ ℂ) |
52 | 8, 51 | syl 17 |
. . 3
⊢ (𝜑 → 𝐴 ∈ ℂ) |
53 | 52, 14, 15 | divrec2d 11420 |
. 2
⊢ (𝜑 → (𝐴 / (seq𝑀( · , 𝐹)‘𝑁)) = ((1 / (seq𝑀( · , 𝐹)‘𝑁)) · 𝐴)) |
54 | 50, 53 | breqtrrd 5094 |
1
⊢ (𝜑 → seq(𝑁 + 1)( · , 𝐹) ⇝ (𝐴 / (seq𝑀( · , 𝐹)‘𝑁))) |