Step | Hyp | Ref
| Expression |
1 | | simpr 485 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → 𝑛 ∈ (0...𝑀)) |
2 | | simpl 483 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → 𝜑) |
3 | | fveq2 6774 |
. . . . 5
⊢ (𝑘 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0)) |
4 | | csbeq1 3835 |
. . . . . . 7
⊢ (𝑘 = 0 → ⦋𝑘 / 𝑛⦌𝐵 = ⦋0 / 𝑛⦌𝐵) |
5 | 4 | oveq1d 7290 |
. . . . . 6
⊢ (𝑘 = 0 →
(⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋0 / 𝑛⦌𝐵 / 𝐶)) |
6 | 5 | mpteq2dv 5176 |
. . . . 5
⊢ (𝑘 = 0 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶))) |
7 | 3, 6 | eqeq12d 2754 |
. . . 4
⊢ (𝑘 = 0 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶)))) |
8 | 7 | imbi2d 341 |
. . 3
⊢ (𝑘 = 0 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶))))) |
9 | | fveq2 6774 |
. . . . 5
⊢ (𝑘 = 𝑗 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)) |
10 | | csbeq1 3835 |
. . . . . . 7
⊢ (𝑘 = 𝑗 → ⦋𝑘 / 𝑛⦌𝐵 = ⦋𝑗 / 𝑛⦌𝐵) |
11 | 10 | oveq1d 7290 |
. . . . . 6
⊢ (𝑘 = 𝑗 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)) |
12 | 11 | mpteq2dv 5176 |
. . . . 5
⊢ (𝑘 = 𝑗 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) |
13 | 9, 12 | eqeq12d 2754 |
. . . 4
⊢ (𝑘 = 𝑗 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)))) |
14 | 13 | imbi2d 341 |
. . 3
⊢ (𝑘 = 𝑗 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))))) |
15 | | fveq2 6774 |
. . . . 5
⊢ (𝑘 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1))) |
16 | | csbeq1 3835 |
. . . . . . 7
⊢ (𝑘 = (𝑗 + 1) → ⦋𝑘 / 𝑛⦌𝐵 = ⦋(𝑗 + 1) / 𝑛⦌𝐵) |
17 | 16 | oveq1d 7290 |
. . . . . 6
⊢ (𝑘 = (𝑗 + 1) → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)) |
18 | 17 | mpteq2dv 5176 |
. . . . 5
⊢ (𝑘 = (𝑗 + 1) → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))) |
19 | 15, 18 | eqeq12d 2754 |
. . . 4
⊢ (𝑘 = (𝑗 + 1) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))) |
20 | 19 | imbi2d 341 |
. . 3
⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))))) |
21 | | fveq2 6774 |
. . . . 5
⊢ (𝑘 = 𝑛 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛)) |
22 | | csbeq1a 3846 |
. . . . . . . . 9
⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵) |
23 | 22 | equcoms 2023 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵) |
24 | 23 | eqcomd 2744 |
. . . . . . 7
⊢ (𝑘 = 𝑛 → ⦋𝑘 / 𝑛⦌𝐵 = 𝐵) |
25 | 24 | oveq1d 7290 |
. . . . . 6
⊢ (𝑘 = 𝑛 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (𝐵 / 𝐶)) |
26 | 25 | mpteq2dv 5176 |
. . . . 5
⊢ (𝑘 = 𝑛 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶))) |
27 | 21, 26 | eqeq12d 2754 |
. . . 4
⊢ (𝑘 = 𝑛 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶)))) |
28 | 27 | imbi2d 341 |
. . 3
⊢ (𝑘 = 𝑛 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶))))) |
29 | | dvnmptdivc.s |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
30 | | recnprss 25068 |
. . . . . . 7
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
31 | 29, 30 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
32 | | cnex 10952 |
. . . . . . . 8
⊢ ℂ
∈ V |
33 | 32 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → ℂ ∈
V) |
34 | | dvnmptdivc.a |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) |
35 | | dvnmptdivc.c |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 ∈ ℂ) |
36 | 35 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ) |
37 | | dvnmptdivc.cne0 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 ≠ 0) |
38 | 37 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ≠ 0) |
39 | 34, 36, 38 | divcld 11751 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐶) ∈ ℂ) |
40 | 39 | fmpttd 6989 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ) |
41 | | dvnmptdivc.x |
. . . . . . 7
⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
42 | | elpm2r 8633 |
. . . . . . 7
⊢
(((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆)) |
43 | 33, 29, 40, 41, 42 | syl22anc 836 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆)) |
44 | | dvn0 25088 |
. . . . . 6
⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) |
45 | 31, 43, 44 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) |
46 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝜑) |
47 | | dvnmptdivc.8 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
48 | | nn0uz 12620 |
. . . . . . . . . . . . . 14
⊢
ℕ0 = (ℤ≥‘0) |
49 | 47, 48 | eleqtrdi 2849 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈
(ℤ≥‘0)) |
50 | | eluzfz1 13263 |
. . . . . . . . . . . . 13
⊢ (𝑀 ∈
(ℤ≥‘0) → 0 ∈ (0...𝑀)) |
51 | 49, 50 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ∈ (0...𝑀)) |
52 | | nfv 1917 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛(𝜑 ∧ 0 ∈ (0...𝑀)) |
53 | | nfcv 2907 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) |
54 | | nfcv 2907 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑛𝑋 |
55 | | nfcsb1v 3857 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑛⦋0 / 𝑛⦌𝐵 |
56 | 54, 55 | nfmpt 5181 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵) |
57 | 53, 56 | nfeq 2920 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵) |
58 | 52, 57 | nfim 1899 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)) |
59 | | c0ex 10969 |
. . . . . . . . . . . . 13
⊢ 0 ∈
V |
60 | | eleq1 2826 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 0 → (𝑛 ∈ (0...𝑀) ↔ 0 ∈ (0...𝑀))) |
61 | 60 | anbi2d 629 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 0 → ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 0 ∈ (0...𝑀)))) |
62 | | fveq2 6774 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)) |
63 | | csbeq1a 3846 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 0 → 𝐵 = ⦋0 / 𝑛⦌𝐵) |
64 | 63 | mpteq2dv 5176 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 0 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)) |
65 | 62, 64 | eqeq12d 2754 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 0 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))) |
66 | 61, 65 | imbi12d 345 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 0 → (((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ↔ ((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)))) |
67 | | dvnmptdivc.dvn |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵)) |
68 | 58, 59, 66, 67 | vtoclf 3497 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)) |
69 | 46, 51, 68 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)) |
70 | 69 | fveq1d 6776 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥)) |
71 | 70 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥)) |
72 | | simpr 485 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
73 | | simpl 483 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝜑) |
74 | 51 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ (0...𝑀)) |
75 | | 0re 10977 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℝ |
76 | | nfcv 2907 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛0 |
77 | | nfv 1917 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) |
78 | | nfcv 2907 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑛ℂ |
79 | 55, 78 | nfel 2921 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛⦋0 / 𝑛⦌𝐵 ∈ ℂ |
80 | 77, 79 | nfim 1899 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ) |
81 | 60 | 3anbi3d 1441 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 0 → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)))) |
82 | 63 | eleq1d 2823 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 0 → (𝐵 ∈ ℂ ↔ ⦋0 /
𝑛⦌𝐵 ∈
ℂ)) |
83 | 81, 82 | imbi12d 345 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 0 → (((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ))) |
84 | | dvnmptdivc.b |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) |
85 | 76, 80, 83, 84 | vtoclgf 3503 |
. . . . . . . . . . . 12
⊢ (0 ∈
ℝ → ((𝜑 ∧
𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ)) |
86 | 75, 85 | ax-mp 5 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ) |
87 | 73, 72, 74, 86 | syl3anc 1370 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ) |
88 | | eqid 2738 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵) |
89 | 88 | fvmpt2 6886 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝑋 ∧ ⦋0 / 𝑛⦌𝐵 ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥) = ⦋0 / 𝑛⦌𝐵) |
90 | 72, 87, 89 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥) = ⦋0 / 𝑛⦌𝐵) |
91 | 71, 90 | eqtr2d 2779 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋0 / 𝑛⦌𝐵 = (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥)) |
92 | 34 | fmpttd 6989 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) |
93 | | elpm2r 8633 |
. . . . . . . . . . . 12
⊢
(((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆)) |
94 | 33, 29, 92, 41, 93 | syl22anc 836 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆)) |
95 | | dvn0 25088 |
. . . . . . . . . . 11
⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ 𝐴)) |
96 | 31, 94, 95 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ 𝐴)) |
97 | 96 | fveq1d 6776 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) |
98 | 97 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) |
99 | | eqid 2738 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) |
100 | 99 | fvmpt2 6886 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴) |
101 | 72, 34, 100 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴) |
102 | 91, 98, 101 | 3eqtrrd 2783 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 = ⦋0 / 𝑛⦌𝐵) |
103 | 102 | oveq1d 7290 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐶) = (⦋0 / 𝑛⦌𝐵 / 𝐶)) |
104 | 103 | mpteq2dva 5174 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶))) |
105 | 45, 104 | eqtrd 2778 |
. . . 4
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶))) |
106 | 105 | a1i 11 |
. . 3
⊢ (𝑀 ∈
(ℤ≥‘0) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶)))) |
107 | | simp3 1137 |
. . . . 5
⊢ ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → 𝜑) |
108 | | simp1 1135 |
. . . . 5
⊢ ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → 𝑗 ∈ (0..^𝑀)) |
109 | | simpr 485 |
. . . . . . 7
⊢ (((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → 𝜑) |
110 | | simpl 483 |
. . . . . . 7
⊢ (((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)))) |
111 | 109, 110 | mpd 15 |
. . . . . 6
⊢ (((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) |
112 | 111 | 3adant1 1129 |
. . . . 5
⊢ ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) |
113 | 31 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → 𝑆 ⊆ ℂ) |
114 | 43 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆)) |
115 | | elfzofz 13403 |
. . . . . . . 8
⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀)) |
116 | | elfznn0 13349 |
. . . . . . . . 9
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0) |
117 | 116 | ad2antlr 724 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → 𝑗 ∈ ℕ0) |
118 | 115, 117 | sylanl2 678 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → 𝑗 ∈ ℕ0) |
119 | | dvnp1 25089 |
. . . . . . 7
⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆) ∧ 𝑗 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗))) |
120 | 113, 114,
118, 119 | syl3anc 1370 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗))) |
121 | | oveq2 7283 |
. . . . . . 7
⊢ (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)) → (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)))) |
122 | 121 | adantl 482 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)))) |
123 | 31 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑆 ⊆ ℂ) |
124 | 43 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆)) |
125 | | simpr 485 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀)) |
126 | 125, 116 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℕ0) |
127 | 115, 126 | sylan2 593 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ ℕ0) |
128 | 123, 124,
127, 119 | syl3anc 1370 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗))) |
129 | 128 | adantr 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗))) |
130 | 29 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑆 ∈ {ℝ, ℂ}) |
131 | | simplr 766 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝑗 ∈ (0...𝑀)) |
132 | 46 | ad2antrr 723 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝜑) |
133 | | simpr 485 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
134 | 132, 133,
131 | 3jca 1127 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀))) |
135 | | nfcv 2907 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛𝑗 |
136 | | nfv 1917 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)) |
137 | 135 | nfcsb1 3856 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑛⦋𝑗 / 𝑛⦌𝐵 |
138 | 137, 78 | nfel 2921 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑛⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ |
139 | 136, 138 | nfim 1899 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ) |
140 | | eleq1 2826 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑗 → (𝑛 ∈ (0...𝑀) ↔ 𝑗 ∈ (0...𝑀))) |
141 | 140 | 3anbi3d 1441 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑗 → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)))) |
142 | | csbeq1a 3846 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑛⦌𝐵) |
143 | 142 | eleq1d 2823 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑗 → (𝐵 ∈ ℂ ↔ ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ)) |
144 | 141, 143 | imbi12d 345 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑗 → (((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ))) |
145 | 135, 139,
144, 84 | vtoclgf 3503 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (0...𝑀) → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ)) |
146 | 131, 134,
145 | sylc 65 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ) |
147 | 115, 146 | sylanl2 678 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ) |
148 | | fzofzp1 13484 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀)) |
149 | 148 | ad2antlr 724 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → (𝑗 + 1) ∈ (0...𝑀)) |
150 | 115, 132 | sylanl2 678 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝜑) |
151 | | simpr 485 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
152 | 150, 151,
149 | 3jca 1127 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))) |
153 | | nfcv 2907 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛(𝑗 + 1) |
154 | | nfv 1917 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) |
155 | 153 | nfcsb1 3856 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑛⦋(𝑗 + 1) / 𝑛⦌𝐵 |
156 | 155, 78 | nfel 2921 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ |
157 | 154, 156 | nfim 1899 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ) |
158 | | eleq1 2826 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑗 + 1) → (𝑛 ∈ (0...𝑀) ↔ (𝑗 + 1) ∈ (0...𝑀))) |
159 | 158 | 3anbi3d 1441 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑗 + 1) → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)))) |
160 | | csbeq1a 3846 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑗 + 1) → 𝐵 = ⦋(𝑗 + 1) / 𝑛⦌𝐵) |
161 | 160 | eleq1d 2823 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑗 + 1) → (𝐵 ∈ ℂ ↔ ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ)) |
162 | 159, 161 | imbi12d 345 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑗 + 1) → (((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ))) |
163 | 153, 157,
162, 84 | vtoclgf 3503 |
. . . . . . . . . . . 12
⊢ ((𝑗 + 1) ∈ (0...𝑀) → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ)) |
164 | 149, 152,
163 | sylc 65 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ) |
165 | | simpl 483 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝜑) |
166 | 115 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀)) |
167 | | nfv 1917 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑛(𝜑 ∧ 𝑗 ∈ (0...𝑀)) |
168 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) |
169 | 54, 137 | nfmpt 5181 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵) |
170 | 168, 169 | nfeq 2920 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵) |
171 | 167, 170 | nfim 1899 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑛((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)) |
172 | 140 | anbi2d 629 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑗 → ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑗 ∈ (0...𝑀)))) |
173 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 𝑗 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)) |
174 | 142 | mpteq2dv 5176 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 𝑗 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)) |
175 | 173, 174 | eqeq12d 2754 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑗 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))) |
176 | 172, 175 | imbi12d 345 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑗 → (((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)))) |
177 | 171, 176,
67 | chvarfv 2233 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)) |
178 | 165, 166,
177 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)) |
179 | 178 | eqcomd 2744 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)) |
180 | 179 | oveq2d 7291 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗))) |
181 | 165, 94 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆)) |
182 | | dvnp1 25089 |
. . . . . . . . . . . . . 14
⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆) ∧ 𝑗 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗))) |
183 | 123, 181,
127, 182 | syl3anc 1370 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗))) |
184 | 183 | eqcomd 2744 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1))) |
185 | 148 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀)) |
186 | 165, 185 | jca 512 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))) |
187 | | nfv 1917 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑛(𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) |
188 | | nfcv 2907 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) |
189 | 54, 155 | nfmpt 5181 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵) |
190 | 188, 189 | nfeq 2920 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵) |
191 | 187, 190 | nfim 1899 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)) |
192 | 158 | anbi2d 629 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑗 + 1) → ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)))) |
193 | | fveq2 6774 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1))) |
194 | 160 | mpteq2dv 5176 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = (𝑗 + 1) → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)) |
195 | 193, 194 | eqeq12d 2754 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑗 + 1) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))) |
196 | 192, 195 | imbi12d 345 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑗 + 1) → (((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)))) |
197 | 153, 191,
196, 67 | vtoclgf 3503 |
. . . . . . . . . . . . 13
⊢ ((𝑗 + 1) ∈ (0...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))) |
198 | 185, 186,
197 | sylc 65 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)) |
199 | 180, 184,
198 | 3eqtrd 2782 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)) |
200 | 35 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝐶 ∈ ℂ) |
201 | 37 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝐶 ≠ 0) |
202 | 130, 147,
164, 199, 200, 201 | dvmptdivc 25129 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))) |
203 | 202 | adantr 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))) |
204 | 129, 122,
203 | 3eqtrd 2782 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))) |
205 | 204 | eqcomd 2744 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1))) |
206 | 205, 120,
122 | 3eqtrrd 2783 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))) |
207 | 120, 122,
206 | 3eqtrd 2782 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))) |
208 | 107, 108,
112, 207 | syl21anc 835 |
. . . 4
⊢ ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))) |
209 | 208 | 3exp 1118 |
. . 3
⊢ (𝑗 ∈ (0..^𝑀) → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))))) |
210 | 8, 14, 20, 28, 106, 209 | fzind2 13505 |
. 2
⊢ (𝑛 ∈ (0...𝑀) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶)))) |
211 | 1, 2, 210 | sylc 65 |
1
⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶))) |