Step | Hyp | Ref
| Expression |
1 | | fprodabs.2 |
. . 3
⊢ (𝜑 → 𝑁 ∈ 𝑍) |
2 | | fprodabs.1 |
. . 3
⊢ 𝑍 =
(ℤ≥‘𝑀) |
3 | 1, 2 | eleqtrdi 2263 |
. 2
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
4 | | oveq2 5861 |
. . . . . . 7
⊢ (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀)) |
5 | 4 | prodeq1d 11527 |
. . . . . 6
⊢ (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑀)𝐴) |
6 | 5 | fveq2d 5500 |
. . . . 5
⊢ (𝑎 = 𝑀 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴)) |
7 | 4 | prodeq1d 11527 |
. . . . 5
⊢ (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)) |
8 | 6, 7 | eqeq12d 2185 |
. . . 4
⊢ (𝑎 = 𝑀 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))) |
9 | 8 | imbi2d 229 |
. . 3
⊢ (𝑎 = 𝑀 → ((𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴)) ↔ (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)))) |
10 | | oveq2 5861 |
. . . . . . 7
⊢ (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛)) |
11 | 10 | prodeq1d 11527 |
. . . . . 6
⊢ (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑛)𝐴) |
12 | 11 | fveq2d 5500 |
. . . . 5
⊢ (𝑎 = 𝑛 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴)) |
13 | 10 | prodeq1d 11527 |
. . . . 5
⊢ (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) |
14 | 12, 13 | eqeq12d 2185 |
. . . 4
⊢ (𝑎 = 𝑛 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴))) |
15 | 14 | imbi2d 229 |
. . 3
⊢ (𝑎 = 𝑛 → ((𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴)) ↔ (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)))) |
16 | | oveq2 5861 |
. . . . . . 7
⊢ (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1))) |
17 | 16 | prodeq1d 11527 |
. . . . . 6
⊢ (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) |
18 | 17 | fveq2d 5500 |
. . . . 5
⊢ (𝑎 = (𝑛 + 1) → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴)) |
19 | 16 | prodeq1d 11527 |
. . . . 5
⊢ (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴)) |
20 | 18, 19 | eqeq12d 2185 |
. . . 4
⊢ (𝑎 = (𝑛 + 1) → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))) |
21 | 20 | imbi2d 229 |
. . 3
⊢ (𝑎 = (𝑛 + 1) → ((𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴)) ↔ (𝜑 → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴)))) |
22 | | oveq2 5861 |
. . . . . . 7
⊢ (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁)) |
23 | 22 | prodeq1d 11527 |
. . . . . 6
⊢ (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑁)𝐴) |
24 | 23 | fveq2d 5500 |
. . . . 5
⊢ (𝑎 = 𝑁 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴)) |
25 | 22 | prodeq1d 11527 |
. . . . 5
⊢ (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴)) |
26 | 24, 25 | eqeq12d 2185 |
. . . 4
⊢ (𝑎 = 𝑁 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))) |
27 | 26 | imbi2d 229 |
. . 3
⊢ (𝑎 = 𝑁 → ((𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴)) ↔ (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴)))) |
28 | | csbfv2g 5533 |
. . . . . 6
⊢ (𝑀 ∈ ℤ →
⦋𝑀 / 𝑘⦌(abs‘𝐴) =
(abs‘⦋𝑀
/ 𝑘⦌𝐴)) |
29 | 28 | adantl 275 |
. . . . 5
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌(abs‘𝐴) = (abs‘⦋𝑀 / 𝑘⦌𝐴)) |
30 | | fzsn 10022 |
. . . . . . . 8
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
31 | 30 | adantl 275 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀}) |
32 | 31 | prodeq1d 11527 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = ∏𝑘 ∈ {𝑀} (abs‘𝐴)) |
33 | | simpr 109 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ) |
34 | | uzid 9501 |
. . . . . . . . . . . 12
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) |
35 | 34, 2 | eleqtrrdi 2264 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ ℤ → 𝑀 ∈ 𝑍) |
36 | | fprodabs.3 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
37 | 36 | ralrimiva 2543 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ) |
38 | | nfcsb1v 3082 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐴 |
39 | 38 | nfel1 2323 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ |
40 | | csbeq1a 3058 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑀 → 𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
41 | 40 | eleq1d 2239 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ)) |
42 | 39, 41 | rspc 2828 |
. . . . . . . . . . . 12
⊢ (𝑀 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ)) |
43 | 37, 42 | mpan9 279 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑀 ∈ 𝑍) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) |
44 | 35, 43 | sylan2 284 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) |
45 | 44 | abscld 11145 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) →
(abs‘⦋𝑀
/ 𝑘⦌𝐴) ∈
ℝ) |
46 | 45 | recnd 7948 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) →
(abs‘⦋𝑀
/ 𝑘⦌𝐴) ∈
ℂ) |
47 | 29, 46 | eqeltrd 2247 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌(abs‘𝐴) ∈ ℂ) |
48 | | prodsns 11566 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧
⦋𝑀 / 𝑘⦌(abs‘𝐴) ∈ ℂ) →
∏𝑘 ∈ {𝑀} (abs‘𝐴) = ⦋𝑀 / 𝑘⦌(abs‘𝐴)) |
49 | 33, 47, 48 | syl2anc 409 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = ⦋𝑀 / 𝑘⦌(abs‘𝐴)) |
50 | 32, 49 | eqtrd 2203 |
. . . . 5
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = ⦋𝑀 / 𝑘⦌(abs‘𝐴)) |
51 | 30 | prodeq1d 11527 |
. . . . . . . 8
⊢ (𝑀 ∈ ℤ →
∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴) |
52 | 51 | adantl 275 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴) |
53 | | prodsns 11566 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧
⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
54 | 33, 44, 53 | syl2anc 409 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀}𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
55 | 52, 54 | eqtrd 2203 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
56 | 55 | fveq2d 5500 |
. . . . 5
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) →
(abs‘∏𝑘 ∈
(𝑀...𝑀)𝐴) = (abs‘⦋𝑀 / 𝑘⦌𝐴)) |
57 | 29, 50, 56 | 3eqtr4rd 2214 |
. . . 4
⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) →
(abs‘∏𝑘 ∈
(𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)) |
58 | 57 | expcom 115 |
. . 3
⊢ (𝑀 ∈ ℤ → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))) |
59 | | simp3 994 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) |
60 | | peano2uz 9542 |
. . . . . . . . . . 11
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (𝑛 + 1) ∈
(ℤ≥‘𝑀)) |
61 | | csbfv2g 5533 |
. . . . . . . . . . 11
⊢ ((𝑛 + 1) ∈
(ℤ≥‘𝑀) → ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴) = (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴)) |
62 | 60, 61 | syl 14 |
. . . . . . . . . 10
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴) = (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴)) |
63 | 62 | eqcomd 2176 |
. . . . . . . . 9
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴) = ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴)) |
64 | 63 | 3ad2ant2 1014 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴) = ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴)) |
65 | 59, 64 | oveq12d 5871 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴)) = (∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) · ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴))) |
66 | | simpr 109 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀)) |
67 | | elfzuz 9977 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘 ∈ (ℤ≥‘𝑀)) |
68 | 67, 2 | eleqtrrdi 2264 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘 ∈ 𝑍) |
69 | 68, 36 | sylan2 284 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ) |
70 | 69 | adantlr 474 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ) |
71 | 66, 70 | fprodp1s 11565 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴 = (∏𝑘 ∈ (𝑀...𝑛)𝐴 · ⦋(𝑛 + 1) / 𝑘⦌𝐴)) |
72 | 71 | fveq2d 5500 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) →
(abs‘∏𝑘 ∈
(𝑀...(𝑛 + 1))𝐴) = (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · ⦋(𝑛 + 1) / 𝑘⦌𝐴))) |
73 | | eluzel2 9492 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
74 | 73 | adantl 275 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℤ) |
75 | | eluzelz 9496 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → 𝑛 ∈ ℤ) |
76 | 75 | adantl 275 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ ℤ) |
77 | 74, 76 | fzfigd 10387 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝑀...𝑛) ∈ Fin) |
78 | | elfzuz 9977 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀)) |
79 | 78, 2 | eleqtrrdi 2264 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍) |
80 | 79, 36 | sylan2 284 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ) |
81 | 80 | adantlr 474 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ) |
82 | 77, 81 | fprodcl 11570 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∏𝑘 ∈ (𝑀...𝑛)𝐴 ∈ ℂ) |
83 | 60, 2 | eleqtrrdi 2264 |
. . . . . . . . . . 11
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (𝑛 + 1) ∈ 𝑍) |
84 | | nfcsb1v 3082 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘⦋(𝑛 + 1) / 𝑘⦌𝐴 |
85 | 84 | nfel1 2323 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ |
86 | | csbeq1a 3058 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = (𝑛 + 1) → 𝐴 = ⦋(𝑛 + 1) / 𝑘⦌𝐴) |
87 | 86 | eleq1d 2239 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ)) |
88 | 85, 87 | rspc 2828 |
. . . . . . . . . . . 12
⊢ ((𝑛 + 1) ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ)) |
89 | 37, 88 | mpan9 279 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ) |
90 | 83, 89 | sylan2 284 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ) |
91 | 82, 90 | absmuld 11158 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) →
(abs‘(∏𝑘 ∈
(𝑀...𝑛)𝐴 · ⦋(𝑛 + 1) / 𝑘⦌𝐴)) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴))) |
92 | 72, 91 | eqtrd 2203 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) →
(abs‘∏𝑘 ∈
(𝑀...(𝑛 + 1))𝐴) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴))) |
93 | 92 | 3adant3 1012 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘⦋(𝑛 + 1) / 𝑘⦌𝐴))) |
94 | 70 | abscld 11145 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℝ) |
95 | 94 | recnd 7948 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℂ) |
96 | 66, 95 | fprodp1s 11565 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴) = (∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) · ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴))) |
97 | 96 | 3adant3 1012 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴) = (∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) · ⦋(𝑛 + 1) / 𝑘⦌(abs‘𝐴))) |
98 | 65, 93, 97 | 3eqtr4d 2213 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴)) |
99 | 98 | 3exp 1197 |
. . . . 5
⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝑀) →
((abs‘∏𝑘 ∈
(𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴)))) |
100 | 99 | com12 30 |
. . . 4
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (𝜑 → ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴)))) |
101 | 100 | a2d 26 |
. . 3
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → ((𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴)))) |
102 | 9, 15, 21, 27, 58, 101 | uzind4 9547 |
. 2
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))) |
103 | 3, 102 | mpcom 36 |
1
⊢ (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴)) |