Step | Hyp | Ref
| Expression |
1 | | id 22 |
. 2
⊢ (𝜑 → 𝜑) |
2 | | dvnmul.n |
. . 3
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
3 | | nn0uz 12620 |
. . . . 5
⊢
ℕ0 = (ℤ≥‘0) |
4 | 2, 3 | eleqtrdi 2849 |
. . . 4
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘0)) |
5 | | eluzfz2 13264 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘0) → 𝑁 ∈ (0...𝑁)) |
6 | 4, 5 | syl 17 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) |
7 | | eleq1 2826 |
. . . . 5
⊢ (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁))) |
8 | | fveq2 6774 |
. . . . . . 7
⊢ (𝑛 = 𝑁 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁)) |
9 | | oveq2 7283 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁)) |
10 | 9 | sumeq1d 15413 |
. . . . . . . . 9
⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) |
11 | | oveq1 7282 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘)) |
12 | | fvoveq1 7298 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑁 → (𝐷‘(𝑛 − 𝑘)) = (𝐷‘(𝑁 − 𝑘))) |
13 | 12 | fveq1d 6776 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑁 → ((𝐷‘(𝑛 − 𝑘))‘𝑥) = ((𝐷‘(𝑁 − 𝑘))‘𝑥)) |
14 | 13 | oveq2d 7291 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑁 → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))) |
15 | 11, 14 | oveq12d 7293 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑁 → ((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = ((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))) |
16 | 15 | sumeq2sdv 15416 |
. . . . . . . . 9
⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))) |
17 | 10, 16 | eqtrd 2778 |
. . . . . . . 8
⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))) |
18 | 17 | mpteq2dv 5176 |
. . . . . . 7
⊢ (𝑛 = 𝑁 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))) |
19 | 8, 18 | eqeq12d 2754 |
. . . . . 6
⊢ (𝑛 = 𝑁 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))) |
20 | 19 | imbi2d 341 |
. . . . 5
⊢ (𝑛 = 𝑁 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))))) |
21 | 7, 20 | imbi12d 345 |
. . . 4
⊢ (𝑛 = 𝑁 → ((𝑛 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))))) ↔ (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))))) |
22 | | fveq2 6774 |
. . . . . . 7
⊢ (𝑚 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0)) |
23 | | simpl 483 |
. . . . . . . . . 10
⊢ ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 0) |
24 | 23 | oveq2d 7291 |
. . . . . . . . 9
⊢ ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...0)) |
25 | | simpll 764 |
. . . . . . . . . . 11
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → 𝑚 = 0) |
26 | 25 | oveq1d 7290 |
. . . . . . . . . 10
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (𝑚C𝑘) = (0C𝑘)) |
27 | 25 | fvoveq1d 7297 |
. . . . . . . . . . . 12
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(0 − 𝑘))) |
28 | 27 | fveq1d 6776 |
. . . . . . . . . . 11
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(0 − 𝑘))‘𝑥)) |
29 | 28 | oveq2d 7291 |
. . . . . . . . . 10
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) |
30 | 26, 29 | oveq12d 7293 |
. . . . . . . . 9
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))) |
31 | 24, 30 | sumeq12rdv 15419 |
. . . . . . . 8
⊢ ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))) |
32 | 31 | mpteq2dva 5174 |
. . . . . . 7
⊢ (𝑚 = 0 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))) |
33 | 22, 32 | eqeq12d 2754 |
. . . . . 6
⊢ (𝑚 = 0 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))) |
34 | 33 | imbi2d 341 |
. . . . 5
⊢ (𝑚 = 0 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))))) |
35 | | fveq2 6774 |
. . . . . . 7
⊢ (𝑚 = 𝑖 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)) |
36 | | simpl 483 |
. . . . . . . . . 10
⊢ ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 𝑖) |
37 | 36 | oveq2d 7291 |
. . . . . . . . 9
⊢ ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...𝑖)) |
38 | | simpll 764 |
. . . . . . . . . . 11
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → 𝑚 = 𝑖) |
39 | 38 | oveq1d 7290 |
. . . . . . . . . 10
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑚C𝑘) = (𝑖C𝑘)) |
40 | 38 | fvoveq1d 7297 |
. . . . . . . . . . . 12
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(𝑖 − 𝑘))) |
41 | 40 | fveq1d 6776 |
. . . . . . . . . . 11
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(𝑖 − 𝑘))‘𝑥)) |
42 | 41 | oveq2d 7291 |
. . . . . . . . . 10
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) |
43 | 39, 42 | oveq12d 7293 |
. . . . . . . . 9
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))) |
44 | 37, 43 | sumeq12rdv 15419 |
. . . . . . . 8
⊢ ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))) |
45 | 44 | mpteq2dva 5174 |
. . . . . . 7
⊢ (𝑚 = 𝑖 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) |
46 | 35, 45 | eqeq12d 2754 |
. . . . . 6
⊢ (𝑚 = 𝑖 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))) |
47 | 46 | imbi2d 341 |
. . . . 5
⊢ (𝑚 = 𝑖 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))))) |
48 | | fveq2 6774 |
. . . . . . 7
⊢ (𝑚 = (𝑖 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1))) |
49 | | simpl 483 |
. . . . . . . . . 10
⊢ ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → 𝑚 = (𝑖 + 1)) |
50 | 49 | oveq2d 7291 |
. . . . . . . . 9
⊢ ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...(𝑖 + 1))) |
51 | | simpll 764 |
. . . . . . . . . . 11
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑚 = (𝑖 + 1)) |
52 | 51 | oveq1d 7290 |
. . . . . . . . . 10
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑚C𝑘) = ((𝑖 + 1)C𝑘)) |
53 | 51 | fvoveq1d 7297 |
. . . . . . . . . . . 12
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘))) |
54 | 53 | fveq1d 6776 |
. . . . . . . . . . 11
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) |
55 | 54 | oveq2d 7291 |
. . . . . . . . . 10
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
56 | 52, 55 | oveq12d 7293 |
. . . . . . . . 9
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
57 | 50, 56 | sumeq12rdv 15419 |
. . . . . . . 8
⊢ ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
58 | 57 | mpteq2dva 5174 |
. . . . . . 7
⊢ (𝑚 = (𝑖 + 1) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
59 | 48, 58 | eqeq12d 2754 |
. . . . . 6
⊢ (𝑚 = (𝑖 + 1) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))) |
60 | 59 | imbi2d 341 |
. . . . 5
⊢ (𝑚 = (𝑖 + 1) → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))) |
61 | | fveq2 6774 |
. . . . . . 7
⊢ (𝑚 = 𝑛 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛)) |
62 | | simpl 483 |
. . . . . . . . . 10
⊢ ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 𝑛) |
63 | 62 | oveq2d 7291 |
. . . . . . . . 9
⊢ ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...𝑛)) |
64 | | simpll 764 |
. . . . . . . . . . 11
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → 𝑚 = 𝑛) |
65 | 64 | oveq1d 7290 |
. . . . . . . . . 10
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝑚C𝑘) = (𝑛C𝑘)) |
66 | 64 | fvoveq1d 7297 |
. . . . . . . . . . . 12
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(𝑛 − 𝑘))) |
67 | 66 | fveq1d 6776 |
. . . . . . . . . . 11
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(𝑛 − 𝑘))‘𝑥)) |
68 | 67 | oveq2d 7291 |
. . . . . . . . . 10
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) |
69 | 65, 68 | oveq12d 7293 |
. . . . . . . . 9
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) |
70 | 63, 69 | sumeq12rdv 15419 |
. . . . . . . 8
⊢ ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) |
71 | 70 | mpteq2dva 5174 |
. . . . . . 7
⊢ (𝑚 = 𝑛 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))) |
72 | 61, 71 | eqeq12d 2754 |
. . . . . 6
⊢ (𝑚 = 𝑛 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))))) |
73 | 72 | imbi2d 341 |
. . . . 5
⊢ (𝑚 = 𝑛 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))))) |
74 | | dvnmul.s |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
75 | | recnprss 25068 |
. . . . . . . . 9
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
76 | 74, 75 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
77 | | dvnmul.a |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) |
78 | | dvnmul.cc |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) |
79 | 77, 78 | mulcld 10995 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 · 𝐵) ∈ ℂ) |
80 | | restsspw 17142 |
. . . . . . . . . . 11
⊢
((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆 |
81 | | dvnmul.x |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
82 | 80, 81 | sselid 3919 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆) |
83 | | elpwi 4542 |
. . . . . . . . . 10
⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆) |
84 | 82, 83 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
85 | | cnex 10952 |
. . . . . . . . . 10
⊢ ℂ
∈ V |
86 | 85 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ℂ ∈
V) |
87 | 79, 84, 86, 74 | mptelpm 42712 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆)) |
88 | | dvn0 25088 |
. . . . . . . 8
⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))) |
89 | 76, 87, 88 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))) |
90 | | 0z 12330 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℤ |
91 | | fzsn 13298 |
. . . . . . . . . . . 12
⊢ (0 ∈
ℤ → (0...0) = {0}) |
92 | 90, 91 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (0...0) =
{0} |
93 | 92 | sumeq1i 15410 |
. . . . . . . . . 10
⊢
Σ𝑘 ∈
(0...0)((0C𝑘) ·
(((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) |
94 | 93 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))) |
95 | | nfcvd 2908 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Ⅎ𝑘(𝐴 · 𝐵)) |
96 | | nfv 1917 |
. . . . . . . . . 10
⊢
Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ 𝑋) |
97 | | oveq2 7283 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → (0C𝑘) = (0C0)) |
98 | | 0nn0 12248 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
ℕ0 |
99 | | bcn0 14024 |
. . . . . . . . . . . . . . . 16
⊢ (0 ∈
ℕ0 → (0C0) = 1) |
100 | 98, 99 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ (0C0) =
1 |
101 | 100 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → (0C0) =
1) |
102 | 97, 101 | eqtrd 2778 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 0 → (0C𝑘) = 1) |
103 | 102 | adantl 482 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (0C𝑘) = 1) |
104 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (𝐶‘𝑘) = (𝐶‘0)) |
105 | 104 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = (𝐶‘0)) |
106 | | dvnmul.c |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) |
107 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐹)‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑛)) |
108 | 107 | cbvmptv 5187 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛)) |
109 | 106, 108 | eqtri 2766 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛)) |
110 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 0 → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘0)) |
111 | | eluzfz1 13263 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ≥‘0) → 0 ∈ (0...𝑁)) |
112 | 4, 111 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 0 ∈ (0...𝑁)) |
113 | | fvexd 6789 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘0) ∈ V) |
114 | 109, 110,
112, 113 | fvmptd3 6898 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0)) |
115 | 114 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0)) |
116 | 105, 115 | eqtrd 2778 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘0)) |
117 | | dvnmulf |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 𝐴) |
118 | 77, 84, 86, 74 | mptelpm 42712 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆)) |
119 | 117, 118 | eqeltrid 2843 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
120 | | dvn0 25088 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ
↑pm 𝑆))
→ ((𝑆
D𝑛 𝐹)‘0) = 𝐹) |
121 | 76, 119, 120 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹) |
122 | 121 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹) |
123 | 116, 122 | eqtrd 2778 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = 𝐹) |
124 | 123 | fveq1d 6776 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = (𝐹‘𝑥)) |
125 | 124 | adantlr 712 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = (𝐹‘𝑥)) |
126 | | simpr 485 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
127 | 117 | fvmpt2 6886 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ℂ) → (𝐹‘𝑥) = 𝐴) |
128 | 126, 77, 127 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) = 𝐴) |
129 | 128 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (𝐹‘𝑥) = 𝐴) |
130 | 125, 129 | eqtrd 2778 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = 𝐴) |
131 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (0 − 𝑘) = (0 −
0)) |
132 | | 0m0e0 12093 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0
− 0) = 0 |
133 | 132 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (0 − 0) =
0) |
134 | 131, 133 | eqtrd 2778 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 0 → (0 − 𝑘) = 0) |
135 | 134 | fveq2d 6778 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 → (𝐷‘(0 − 𝑘)) = (𝐷‘0)) |
136 | 135 | fveq1d 6776 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥)) |
137 | 136 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥)) |
138 | 137 | adantlr 712 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥)) |
139 | | dvnmul.d |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) |
140 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑛)) |
141 | 140 | cbvmptv 5187 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) |
142 | 139, 141 | eqtri 2766 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) |
143 | 142 | fveq1i 6775 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0) |
144 | 143 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0)) |
145 | | eqidd 2739 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))) |
146 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 0 → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0)) |
147 | 146 | adantl 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0)) |
148 | | dvnmul.f |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ 𝐵) |
149 | 78, 84, 86, 74 | mptelpm 42712 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (ℂ ↑pm 𝑆)) |
150 | 148, 149 | eqeltrid 2843 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝐺 ∈ (ℂ ↑pm 𝑆)) |
151 | | dvn0 25088 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ
↑pm 𝑆))
→ ((𝑆
D𝑛 𝐺)‘0) = 𝐺) |
152 | 76, 150, 151 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑆 D𝑛 𝐺)‘0) = 𝐺) |
153 | 152 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺) |
154 | 147, 153 | eqtrd 2778 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = 𝐺) |
155 | 148 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝑋 ↦ 𝐵)) |
156 | | mptexg 7097 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑋 ∈ 𝒫 𝑆 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ V) |
157 | 82, 156 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ V) |
158 | 155, 157 | eqeltrd 2839 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐺 ∈ V) |
159 | 145, 154,
112, 158 | fvmptd 6882 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0) = 𝐺) |
160 | 144, 159 | eqtrd 2778 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐷‘0) = 𝐺) |
161 | 160 | fveq1d 6776 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐷‘0)‘𝑥) = (𝐺‘𝑥)) |
162 | 161 | ad2antrr 723 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘0)‘𝑥) = (𝐺‘𝑥)) |
163 | 155, 78 | fvmpt2d 6888 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) = 𝐵) |
164 | 163 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (𝐺‘𝑥) = 𝐵) |
165 | 138, 162,
164 | 3eqtrd 2782 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = 𝐵) |
166 | 130, 165 | oveq12d 7293 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)) = (𝐴 · 𝐵)) |
167 | 103, 166 | oveq12d 7293 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (1 · (𝐴 · 𝐵))) |
168 | 79 | mulid2d 10993 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵)) |
169 | 168 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵)) |
170 | 167, 169 | eqtrd 2778 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵)) |
171 | | 0re 10977 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
172 | 171 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ) |
173 | 95, 96, 170, 172, 79 | sumsnd 42569 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵)) |
174 | 94, 173 | eqtr2d 2779 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))) |
175 | 174 | mpteq2dva 5174 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))) |
176 | 89, 175 | eqtrd 2778 |
. . . . . 6
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))) |
177 | 176 | a1i 11 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘0) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))) |
178 | | simp3 1137 |
. . . . . . 7
⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → 𝜑) |
179 | | simp1 1135 |
. . . . . . 7
⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → 𝑖 ∈ (0..^𝑁)) |
180 | | simp2 1136 |
. . . . . . . 8
⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))) |
181 | | pm3.35 800 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) |
182 | 178, 180,
181 | syl2anc 584 |
. . . . . . 7
⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) |
183 | 76 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆 ⊆ ℂ) |
184 | 87 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆)) |
185 | | elfzonn0 13432 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0) |
186 | 185 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0) |
187 | | dvnp1 25089 |
. . . . . . . . . 10
⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆) ∧ 𝑖 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖))) |
188 | 183, 184,
186, 187 | syl3anc 1370 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖))) |
189 | 188 | adantr 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖))) |
190 | | simpr 485 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) |
191 | 190 | oveq2d 7291 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))) |
192 | | eqid 2738 |
. . . . . . . . . . 11
⊢
((TopOpen‘ℂfld) ↾t 𝑆) =
((TopOpen‘ℂfld) ↾t 𝑆) |
193 | | eqid 2738 |
. . . . . . . . . . 11
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
194 | 74 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆 ∈ {ℝ, ℂ}) |
195 | 81 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
196 | | fzfid 13693 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (0...𝑖) ∈ Fin) |
197 | 185 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℕ0) |
198 | | elfzelz 13256 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℤ) |
199 | 198 | adantl 482 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℤ) |
200 | 197, 199 | bccld 42854 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈
ℕ0) |
201 | 200 | nn0cnd 12295 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
202 | 201 | adantll 711 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
203 | 202 | 3adant3 1131 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝑖C𝑘) ∈ ℂ) |
204 | | simpll 764 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝜑) |
205 | | 0zd 12331 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℤ) |
206 | | elfzoel2 13386 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℤ) |
207 | 206 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℤ) |
208 | | elfzle1 13259 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (0...𝑖) → 0 ≤ 𝑘) |
209 | 208 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ 𝑘) |
210 | 199 | zred 12426 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℝ) |
211 | 206 | zred 12426 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℝ) |
212 | 211 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℝ) |
213 | 185 | nn0red 12294 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℝ) |
214 | 213 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℝ) |
215 | | elfzle2 13260 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ≤ 𝑖) |
216 | 215 | adantl 482 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ 𝑖) |
217 | | elfzolt2 13396 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 < 𝑁) |
218 | 217 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 < 𝑁) |
219 | 210, 214,
212, 216, 218 | lelttrd 11133 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 < 𝑁) |
220 | 210, 212,
219 | ltled 11123 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ 𝑁) |
221 | 205, 207,
199, 209, 220 | elfzd 13247 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁)) |
222 | 221 | adantll 711 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁)) |
223 | | dvnmul.dvnf |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ) |
224 | 106 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))) |
225 | | fvexd 6789 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘) ∈ V) |
226 | 224, 225 | fvmpt2d 6888 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘)) |
227 | 226 | feq1d 6585 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐶‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)) |
228 | 223, 227 | mpbird 256 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) |
229 | 204, 222,
228 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘):𝑋⟶ℂ) |
230 | 229 | 3adant3 1131 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝐶‘𝑘):𝑋⟶ℂ) |
231 | | simp3 1137 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
232 | 230, 231 | ffvelrnd 6962 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
233 | 185 | nn0zd 12424 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℤ) |
234 | 233 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℤ) |
235 | 234, 199 | zsubcld 12431 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ ℤ) |
236 | | elfzel2 13254 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℤ) |
237 | 236 | zred 12426 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℝ) |
238 | 198 | zred 12426 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℝ) |
239 | 237, 238 | subge0d 11565 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0...𝑖) → (0 ≤ (𝑖 − 𝑘) ↔ 𝑘 ≤ 𝑖)) |
240 | 215, 239 | mpbird 256 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑖 − 𝑘)) |
241 | 240 | adantl 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑖 − 𝑘)) |
242 | 214, 210 | resubcld 11403 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ ℝ) |
243 | 212, 210 | resubcld 11403 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ∈ ℝ) |
244 | 171 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℝ) |
245 | 212, 244 | jca 512 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 ∈ ℝ ∧ 0 ∈
ℝ)) |
246 | | resubcl 11285 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑁 ∈ ℝ ∧ 0 ∈
ℝ) → (𝑁 −
0) ∈ ℝ) |
247 | 245, 246 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) ∈ ℝ) |
248 | 214, 212,
210, 218 | ltsub1dd 11587 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < (𝑁 − 𝑘)) |
249 | 244, 210,
212, 209 | lesub2dd 11592 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ≤ (𝑁 − 0)) |
250 | 242, 243,
247, 248, 249 | ltletrd 11135 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < (𝑁 − 0)) |
251 | 211 | recnd 11003 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℂ) |
252 | 251 | subid1d 11321 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → (𝑁 − 0) = 𝑁) |
253 | 252 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) = 𝑁) |
254 | 250, 253 | breqtrd 5100 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < 𝑁) |
255 | 242, 212,
254 | ltled 11123 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ≤ 𝑁) |
256 | 205, 207,
235, 241, 255 | elfzd 13247 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ (0...𝑁)) |
257 | 256 | adantll 711 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ (0...𝑁)) |
258 | | ovex 7308 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 − 𝑘) ∈ V |
259 | | eleq1 2826 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑖 − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ (𝑖 − 𝑘) ∈ (0...𝑁))) |
260 | 259 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 − 𝑘) ∈ (0...𝑁)))) |
261 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑖 − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
262 | 261 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ)) |
263 | 260, 262 | imbi12d 345 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑖 − 𝑘) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ))) |
264 | | nfv 1917 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) |
265 | | eleq1 2826 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑗 → (𝑘 ∈ (0...𝑁) ↔ 𝑗 ∈ (0...𝑁))) |
266 | 265 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑗 ∈ (0...𝑁)))) |
267 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑗 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑗)) |
268 | 267 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑗 → (((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)) |
269 | 266, 268 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ))) |
270 | | dvnmul.dvng |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ) |
271 | 264, 269,
270 | chvarfv 2233 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) |
272 | 258, 263,
271 | vtocl 3498 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑖 − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ) |
273 | 204, 257,
272 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ) |
274 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = (𝑖 − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
275 | | fvexd 6789 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)) ∈ V) |
276 | 142, 274,
256, 275 | fvmptd3 6898 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
277 | 276 | adantll 711 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
278 | 277 | feq1d 6585 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ)) |
279 | 273, 278 | mpbird 256 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ) |
280 | 279 | 3adant3 1131 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ) |
281 | 280, 231 | ffvelrnd 6962 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 − 𝑘))‘𝑥) ∈ ℂ) |
282 | 232, 281 | mulcld 10995 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ) |
283 | 203, 282 | mulcld 10995 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ) |
284 | 203 | 3expa 1117 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (𝑖C𝑘) ∈ ℂ) |
285 | 234 | peano2zd 12429 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℤ) |
286 | 285, 199 | zsubcld 12431 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℤ) |
287 | | peano2re 11148 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑖 ∈ ℝ → (𝑖 + 1) ∈
ℝ) |
288 | 237, 287 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ (0...𝑖) → (𝑖 + 1) ∈ ℝ) |
289 | | peano2re 11148 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈
ℝ) |
290 | 238, 289 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ∈ ℝ) |
291 | 238 | ltp1d 11905 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑘 + 1)) |
292 | | 1red 10976 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 ∈ (0...𝑖) → 1 ∈ ℝ) |
293 | 238, 237,
292, 215 | leadd1dd 11589 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ≤ (𝑖 + 1)) |
294 | 238, 290,
288, 291, 293 | ltletrd 11135 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑖 + 1)) |
295 | 238, 288,
294 | ltled 11123 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ≤ (𝑖 + 1)) |
296 | 295 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ (𝑖 + 1)) |
297 | 214, 287 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℝ) |
298 | 297, 210 | subge0d 11565 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1))) |
299 | 296, 298 | mpbird 256 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ ((𝑖 + 1) − 𝑘)) |
300 | 297, 210 | resubcld 11403 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℝ) |
301 | | elfzop1le2 13400 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ≤ 𝑁) |
302 | 301 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ≤ 𝑁) |
303 | 297, 212,
210, 302 | lesub1dd 11591 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 𝑘)) |
304 | 249, 253 | breqtrd 5100 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ≤ 𝑁) |
305 | 300, 243,
212, 303, 304 | letrd 11132 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ 𝑁) |
306 | 205, 207,
286, 299, 305 | elfzd 13247 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
307 | 306 | adantll 711 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
308 | | ovex 7308 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 + 1) − 𝑘) ∈ V |
309 | | eleq1 2826 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))) |
310 | 309 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)))) |
311 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
312 | 311 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)) |
313 | 310, 312 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))) |
314 | 308, 313,
271 | vtocl 3498 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ) |
315 | 204, 307,
314 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ) |
316 | 142 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))) |
317 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → 𝑛 = ((𝑖 + 1) − 𝑘)) |
318 | 317 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
319 | | fvexd 6789 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) ∈ V) |
320 | 316, 318,
307, 319 | fvmptd 6882 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
321 | 320 | feq1d 6585 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)) |
322 | 315, 321 | mpbird 256 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ) |
323 | 322 | ffvelrnda 6961 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) |
324 | 232 | 3expa 1117 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
325 | 323, 324 | mulcomd 10996 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
326 | 325 | oveq2d 7291 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) = ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
327 | 199 | peano2zd 12429 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℤ) |
328 | 171 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ (0...𝑖) → 0 ∈ ℝ) |
329 | 328, 238,
290, 208, 291 | lelttrd 11133 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ (0...𝑖) → 0 < (𝑘 + 1)) |
330 | 328, 290,
329 | ltled 11123 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑘 + 1)) |
331 | 330 | adantl 482 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑘 + 1)) |
332 | 210, 289 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℝ) |
333 | 293 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ (𝑖 + 1)) |
334 | 332, 297,
212, 333, 302 | letrd 11132 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ 𝑁) |
335 | 205, 207,
327, 331, 334 | elfzd 13247 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁)) |
336 | 335 | adantll 711 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁)) |
337 | | ovex 7308 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 + 1) ∈ V |
338 | | eleq1 2826 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = (𝑘 + 1) → (𝑗 ∈ (0...𝑁) ↔ (𝑘 + 1) ∈ (0...𝑁))) |
339 | 338 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑘 + 1) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)))) |
340 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = (𝑘 + 1) → (𝐶‘𝑗) = (𝐶‘(𝑘 + 1))) |
341 | 340 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑘 + 1) → ((𝐶‘𝑗):𝑋⟶ℂ ↔ (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)) |
342 | 339, 341 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑘 + 1) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ))) |
343 | | nfv 1917 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (0...𝑁)) |
344 | | nfmpt1 5182 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) |
345 | 106, 344 | nfcxfr 2905 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘𝐶 |
346 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘𝑗 |
347 | 345, 346 | nffv 6784 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘(𝐶‘𝑗) |
348 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘𝑋 |
349 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘ℂ |
350 | 347, 348,
349 | nff 6596 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘(𝐶‘𝑗):𝑋⟶ℂ |
351 | 343, 350 | nfim 1899 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ) |
352 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑗 → (𝐶‘𝑘) = (𝐶‘𝑗)) |
353 | 352 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑗 → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘𝑗):𝑋⟶ℂ)) |
354 | 266, 353 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ))) |
355 | 351, 354,
228 | chvarfv 2233 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ) |
356 | 337, 342,
355 | vtocl 3498 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ) |
357 | 204, 336,
356 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ) |
358 | 357 | ffvelrnda 6961 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘(𝑘 + 1))‘𝑥) ∈ ℂ) |
359 | 281 | 3expa 1117 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 − 𝑘))‘𝑥) ∈ ℂ) |
360 | 358, 359 | mulcld 10995 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ) |
361 | 323, 324 | mulcld 10995 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥)) ∈ ℂ) |
362 | 360, 361 | addcld 10994 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) ∈ ℂ) |
363 | 326, 362 | eqeltrrd 2840 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
364 | 284, 363 | mulcld 10995 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ) |
365 | 364 | 3impa 1109 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ) |
366 | 204, 74 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ∈ {ℝ, ℂ}) |
367 | 171 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ) |
368 | 204, 81 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
369 | 366, 368,
202 | dvmptconst 43456 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝑖C𝑘))) = (𝑥 ∈ 𝑋 ↦ 0)) |
370 | 282 | 3expa 1117 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ) |
371 | 204, 222,
226 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘)) |
372 | 371 | eqcomd 2744 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘𝑘) = (𝐶‘𝑘)) |
373 | 229 | feqmptd 6837 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥))) |
374 | 372, 373 | eqtr2d 2779 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥)) = ((𝑆 D𝑛 𝐹)‘𝑘)) |
375 | 374 | oveq2d 7291 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘))) |
376 | 366, 75 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ⊆ ℂ) |
377 | 204, 119 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
378 | | elfznn0 13349 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ0) |
379 | 378 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ0) |
380 | | dvnp1 25089 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ
↑pm 𝑆)
∧ 𝑘 ∈
ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘))) |
381 | 376, 377,
379, 380 | syl3anc 1370 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘))) |
382 | 381 | eqcomd 2744 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1))) |
383 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = (𝑘 + 1) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1))) |
384 | | fvexd 6789 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) ∈ V) |
385 | 109, 383,
336, 384 | fvmptd3 6898 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1))) |
386 | 385 | eqcomd 2744 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝐶‘(𝑘 + 1))) |
387 | 357 | feqmptd 6837 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥))) |
388 | 386, 387 | eqtrd 2778 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥))) |
389 | 375, 382,
388 | 3eqtrd 2782 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥))) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥))) |
390 | 277 | eqcomd 2744 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)) = (𝐷‘(𝑖 − 𝑘))) |
391 | 279 | feqmptd 6837 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥))) |
392 | 390, 391 | eqtr2d 2779 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
393 | 392 | oveq2d 7291 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))) |
394 | 204, 150 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐺 ∈ (ℂ ↑pm 𝑆)) |
395 | | fznn0sub 13288 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...𝑖) → (𝑖 − 𝑘) ∈
ℕ0) |
396 | 395 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈
ℕ0) |
397 | | dvnp1 25089 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ
↑pm 𝑆)
∧ (𝑖 − 𝑘) ∈ ℕ0)
→ ((𝑆
D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))) |
398 | 376, 394,
396, 397 | syl3anc 1370 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))) |
399 | 398 | eqcomd 2744 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) = ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1))) |
400 | 214 | recnd 11003 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℂ) |
401 | | 1cnd 10970 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 1 ∈ ℂ) |
402 | 210 | recnd 11003 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℂ) |
403 | 400, 401,
402 | addsubd 11353 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) = ((𝑖 − 𝑘) + 1)) |
404 | 403 | eqcomd 2744 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 − 𝑘) + 1) = ((𝑖 + 1) − 𝑘)) |
405 | 404 | fveq2d 6778 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
406 | 405 | adantll 711 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
407 | 320 | eqcomd 2744 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘))) |
408 | 322 | feqmptd 6837 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
409 | 406, 407,
408 | 3eqtrd 2782 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
410 | 393, 399,
409 | 3eqtrd 2782 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
411 | 366, 324,
358, 389, 359, 323, 410 | dvmptmul 25125 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))))) |
412 | 366, 284,
367, 369, 370, 362, 411 | dvmptmul 25125 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ ((0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘))))) |
413 | 370 | mul02d 11173 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = 0) |
414 | 326 | oveq1d 7290 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘)) = (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) · (𝑖C𝑘))) |
415 | 363, 284 | mulcomd 10996 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) · (𝑖C𝑘)) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
416 | 414, 415 | eqtrd 2778 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘)) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
417 | 413, 416 | oveq12d 7293 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘))) = (0 + ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))) |
418 | 364 | addid2d 11176 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (0 + ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
419 | 417, 418 | eqtrd 2778 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘))) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
420 | 419 | mpteq2dva 5174 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥 ∈ 𝑋 ↦ ((0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘)))) = (𝑥 ∈ 𝑋 ↦ ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))) |
421 | 412, 420 | eqtrd 2778 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))) |
422 | 192, 193,
194, 195, 196, 283, 365, 421 | dvmptfsum 25139 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))) |
423 | 202 | adantlr 712 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
424 | 360 | an32s 649 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ) |
425 | | anass 469 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋))) |
426 | | ancom 461 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖))) |
427 | 426 | anbi2i 623 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋)) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖)))) |
428 | | anass 469 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖)))) |
429 | 428 | bicomi 223 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖))) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖))) |
430 | 427, 429 | bitri 274 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋)) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖))) |
431 | 425, 430 | bitri 274 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖))) |
432 | 431 | imbi1i 350 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) ↔ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ)) |
433 | 324, 432 | mpbi 229 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
434 | 431 | imbi1i 350 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) ↔ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)) |
435 | 323, 434 | mpbi 229 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) |
436 | 433, 435 | mulcld 10995 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ) |
437 | 423, 424,
436 | adddid 10999 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
438 | 437 | sumeq2dv 15415 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...𝑖)(((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
439 | 196 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (0...𝑖) ∈ Fin) |
440 | 423, 424 | mulcld 10995 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ) |
441 | 423, 436 | mulcld 10995 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
442 | 439, 440,
441 | fsumadd 15452 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)(((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
443 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = ℎ → (𝑖C𝑘) = (𝑖Cℎ)) |
444 | | fvoveq1 7298 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = ℎ → (𝐶‘(𝑘 + 1)) = (𝐶‘(ℎ + 1))) |
445 | 444 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = ℎ → ((𝐶‘(𝑘 + 1))‘𝑥) = ((𝐶‘(ℎ + 1))‘𝑥)) |
446 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = ℎ → (𝑖 − 𝑘) = (𝑖 − ℎ)) |
447 | 446 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = ℎ → (𝐷‘(𝑖 − 𝑘)) = (𝐷‘(𝑖 − ℎ))) |
448 | 447 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = ℎ → ((𝐷‘(𝑖 − 𝑘))‘𝑥) = ((𝐷‘(𝑖 − ℎ))‘𝑥)) |
449 | 445, 448 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = ℎ → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) = (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) |
450 | 443, 449 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = ℎ → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)))) |
451 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎℎ(0...𝑖) |
452 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘(0...𝑖) |
453 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎℎ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) |
454 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(𝑖Cℎ) |
455 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘
· |
456 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘(ℎ + 1) |
457 | 345, 456 | nffv 6784 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘(𝐶‘(ℎ + 1)) |
458 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘𝑥 |
459 | 457, 458 | nffv 6784 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝐶‘(ℎ + 1))‘𝑥) |
460 | | nfmpt1 5182 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) |
461 | 139, 460 | nfcxfr 2905 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘𝐷 |
462 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘(𝑖 − ℎ) |
463 | 461, 462 | nffv 6784 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘(𝐷‘(𝑖 − ℎ)) |
464 | 463, 458 | nffv 6784 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝐷‘(𝑖 − ℎ))‘𝑥) |
465 | 459, 455,
464 | nfov 7305 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)) |
466 | 454, 455,
465 | nfov 7305 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) |
467 | 450, 451,
452, 453, 466 | cbvsum 15407 |
. . . . . . . . . . . . . . . . 17
⊢
Σ𝑘 ∈
(0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = Σℎ ∈ (0...𝑖)((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) |
468 | 467 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = Σℎ ∈ (0...𝑖)((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)))) |
469 | | 1zzd 12351 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℤ) |
470 | 90 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℤ) |
471 | 233 | ad2antlr 724 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 𝑖 ∈ ℤ) |
472 | | nfv 1917 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) |
473 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘ℎ |
474 | 473, 452 | nfel 2921 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘 ℎ ∈ (0...𝑖) |
475 | 472, 474 | nfan 1902 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) |
476 | 466, 349 | nfel 2921 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ |
477 | 475, 476 | nfim 1899 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ) |
478 | | eleq1 2826 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = ℎ → (𝑘 ∈ (0...𝑖) ↔ ℎ ∈ (0...𝑖))) |
479 | 478 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = ℎ → ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)))) |
480 | 450 | eleq1d 2823 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = ℎ → (((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ ↔ ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ)) |
481 | 479, 480 | imbi12d 345 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = ℎ → (((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ) ↔ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ))) |
482 | 477, 481,
440 | chvarfv 2233 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ) |
483 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . 18
⊢ (ℎ = (𝑗 − 1) → (𝑖Cℎ) = (𝑖C(𝑗 − 1))) |
484 | | fvoveq1 7298 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℎ = (𝑗 − 1) → (𝐶‘(ℎ + 1)) = (𝐶‘((𝑗 − 1) + 1))) |
485 | 484 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℎ = (𝑗 − 1) → ((𝐶‘(ℎ + 1))‘𝑥) = ((𝐶‘((𝑗 − 1) + 1))‘𝑥)) |
486 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (ℎ = (𝑗 − 1) → (𝑖 − ℎ) = (𝑖 − (𝑗 − 1))) |
487 | 486 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℎ = (𝑗 − 1) → (𝐷‘(𝑖 − ℎ)) = (𝐷‘(𝑖 − (𝑗 − 1)))) |
488 | 487 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℎ = (𝑗 − 1) → ((𝐷‘(𝑖 − ℎ))‘𝑥) = ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)) |
489 | 485, 488 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . 18
⊢ (ℎ = (𝑗 − 1) → (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)) = (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) |
490 | 483, 489 | oveq12d 7293 |
. . . . . . . . . . . . . . . . 17
⊢ (ℎ = (𝑗 − 1) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) = ((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)))) |
491 | 469, 470,
471, 482, 490 | fsumshft 15492 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σℎ ∈ (0...𝑖)((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) = Σ𝑗 ∈ ((0 + 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)))) |
492 | 468, 491 | eqtrd 2778 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = Σ𝑗 ∈ ((0 + 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)))) |
493 | | 0p1e1 12095 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 + 1) =
1 |
494 | 493 | oveq1i 7285 |
. . . . . . . . . . . . . . . . 17
⊢ ((0 +
1)...(𝑖 + 1)) = (1...(𝑖 + 1)) |
495 | 494 | sumeq1i 15410 |
. . . . . . . . . . . . . . . 16
⊢
Σ𝑗 ∈ ((0
+ 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) |
496 | 495 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ ((0 + 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)))) |
497 | | elfzelz 13256 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℤ) |
498 | 497 | zcnd 12427 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℂ) |
499 | | 1cnd 10970 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈
ℂ) |
500 | 498, 499 | npcand 11336 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → ((𝑗 − 1) + 1) = 𝑗) |
501 | 500 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → (𝐶‘((𝑗 − 1) + 1)) = (𝐶‘𝑗)) |
502 | 501 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶‘𝑗)‘𝑥)) |
503 | 502 | adantl 482 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶‘𝑗)‘𝑥)) |
504 | 213 | recnd 11003 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℂ) |
505 | 504 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℂ) |
506 | 498 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℂ) |
507 | 499 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈
ℂ) |
508 | 505, 506,
507 | subsub3d 11362 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 − (𝑗 − 1)) = ((𝑖 + 1) − 𝑗)) |
509 | 508 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘(𝑖 − (𝑗 − 1))) = (𝐷‘((𝑖 + 1) − 𝑗))) |
510 | 509 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) |
511 | 503, 510 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)) = (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) |
512 | 511 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) |
513 | 512 | sumeq2dv 15415 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (0..^𝑁) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) |
514 | 513 | ad2antlr 724 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) |
515 | | nfv 1917 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑗((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) |
516 | | nfcv 2907 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑗((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) |
517 | | fzfid 13693 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...(𝑖 + 1)) ∈ Fin) |
518 | 185 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℕ0) |
519 | 497 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℤ) |
520 | | 1zzd 12351 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈
ℤ) |
521 | 519, 520 | zsubcld 12431 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑗 − 1) ∈ ℤ) |
522 | 518, 521 | bccld 42854 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈
ℕ0) |
523 | 522 | nn0cnd 12295 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
524 | 523 | adantll 711 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
525 | 524 | adantlr 712 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
526 | 1 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝜑) |
527 | | 0zd 12331 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ∈
ℤ) |
528 | 206 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℤ) |
529 | 171 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 ∈
ℝ) |
530 | 497 | zred 12426 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ) |
531 | | 1red 10976 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈
ℝ) |
532 | | 0lt1 11497 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ 0 <
1 |
533 | 532 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 1) |
534 | | elfzle1 13259 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 1 ≤ 𝑗) |
535 | 529, 531,
530, 533, 534 | ltletrd 11135 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 𝑗) |
536 | 529, 530,
535 | ltled 11123 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 ≤ 𝑗) |
537 | 536 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ 𝑗) |
538 | 530 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ) |
539 | 213 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℝ) |
540 | | 1red 10976 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈
ℝ) |
541 | 539, 540 | readdcld 11004 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ) |
542 | 211 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℝ) |
543 | | elfzle2 13260 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ≤ (𝑖 + 1)) |
544 | 543 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ (𝑖 + 1)) |
545 | 301 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁) |
546 | 538, 541,
542, 544, 545 | letrd 11132 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ 𝑁) |
547 | 527, 528,
519, 537, 546 | elfzd 13247 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁)) |
548 | 547 | adantll 711 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁)) |
549 | 526, 548,
355 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶‘𝑗):𝑋⟶ℂ) |
550 | 549 | adantlr 712 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶‘𝑗):𝑋⟶ℂ) |
551 | | simplr 766 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑥 ∈ 𝑋) |
552 | 550, 551 | ffvelrnd 6962 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘𝑗)‘𝑥) ∈ ℂ) |
553 | 233 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℤ) |
554 | 553 | peano2zd 12429 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ) |
555 | 554, 519 | zsubcld 12431 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℤ) |
556 | 541, 538 | subge0d 11565 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑗) ↔ 𝑗 ≤ (𝑖 + 1))) |
557 | 544, 556 | mpbird 256 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑗)) |
558 | 541, 538 | resubcld 11403 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℝ) |
559 | 558 | leidd 11541 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ ((𝑖 + 1) − 𝑗)) |
560 | 530, 535 | elrpd 12769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ+) |
561 | 560 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ+) |
562 | 541, 561 | ltsubrpd 12804 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < (𝑖 + 1)) |
563 | 558, 541,
542, 562, 545 | ltletrd 11135 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁) |
564 | 558, 558,
542, 559, 563 | lelttrd 11133 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁) |
565 | 558, 542,
564 | ltled 11123 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ 𝑁) |
566 | 527, 528,
555, 557, 565 | elfzd 13247 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) |
567 | 566 | adantll 711 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) |
568 | | nfv 1917 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) |
569 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑘((𝑖 + 1) − 𝑗) |
570 | 461, 569 | nffv 6784 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝐷‘((𝑖 + 1) − 𝑗)) |
571 | 570, 348,
349 | nff 6596 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ |
572 | 568, 571 | nfim 1899 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ) |
573 | | ovex 7308 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 + 1) − 𝑗) ∈ V |
574 | | eleq1 2826 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → (𝑘 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))) |
575 | 574 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)))) |
576 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → (𝐷‘𝑘) = (𝐷‘((𝑖 + 1) − 𝑗))) |
577 | 576 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)) |
578 | 575, 577 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ))) |
579 | 139 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))) |
580 | | fvexd 6789 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘) ∈ V) |
581 | 579, 580 | fvmpt2d 6888 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑘)) |
582 | 581 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ)) |
583 | 270, 582 | mpbird 256 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) |
584 | 572, 573,
578, 583 | vtoclf 3497 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ) |
585 | 526, 567,
584 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ) |
586 | 585 | adantlr 712 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ) |
587 | 586, 551 | ffvelrnd 6962 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ) |
588 | 552, 587 | mulcld 10995 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ) |
589 | 525, 588 | mulcld 10995 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ) |
590 | | 1zzd 12351 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 1 ∈ ℤ) |
591 | 233 | peano2zd 12429 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℤ) |
592 | 493 | eqcomi 2747 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 1 = (0 +
1) |
593 | 592 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → 1 = (0 + 1)) |
594 | 171 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 0 ∈ ℝ) |
595 | | 1red 10976 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 1 ∈ ℝ) |
596 | 185 | nn0ge0d 12296 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 0 ≤ 𝑖) |
597 | 594, 213,
595, 596 | leadd1dd 11589 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → (0 + 1) ≤ (𝑖 + 1)) |
598 | 593, 597 | eqbrtrd 5096 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 1 ≤ (𝑖 + 1)) |
599 | 590, 591,
598 | 3jca 1127 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1
≤ (𝑖 +
1))) |
600 | | eluz2 12588 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 + 1) ∈
(ℤ≥‘1) ↔ (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1
≤ (𝑖 +
1))) |
601 | 599, 600 | sylibr 233 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈
(ℤ≥‘1)) |
602 | | eluzfz2 13264 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 + 1) ∈
(ℤ≥‘1) → (𝑖 + 1) ∈ (1...(𝑖 + 1))) |
603 | 601, 602 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (1...(𝑖 + 1))) |
604 | 603 | ad2antlr 724 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝑖 + 1) ∈ (1...(𝑖 + 1))) |
605 | | oveq1 7282 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 + 1) → (𝑗 − 1) = ((𝑖 + 1) − 1)) |
606 | 605 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑖 + 1) → (𝑖C(𝑗 − 1)) = (𝑖C((𝑖 + 1) − 1))) |
607 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑖 + 1) → (𝐶‘𝑗) = (𝐶‘(𝑖 + 1))) |
608 | 607 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 + 1) → ((𝐶‘𝑗)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥)) |
609 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = (𝑖 + 1) → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − (𝑖 + 1))) |
610 | 609 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1)))) |
611 | 610 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) |
612 | 608, 611 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑖 + 1) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) |
613 | 606, 612 | oveq12d 7293 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = (𝑖 + 1) → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))) |
614 | 515, 516,
517, 589, 604, 613 | fsumsplit1 15457 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = (((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))) |
615 | | 1cnd 10970 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 1 ∈ ℂ) |
616 | 504, 615 | pncand 11333 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 1) = 𝑖) |
617 | 616 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = (𝑖C𝑖)) |
618 | | bcnn 14026 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ ℕ0
→ (𝑖C𝑖) = 1) |
619 | 185, 618 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C𝑖) = 1) |
620 | 617, 619 | eqtrd 2778 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = 1) |
621 | 504, 615 | addcld 10994 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℂ) |
622 | 621 | subidd 11320 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − (𝑖 + 1)) = 0) |
623 | 622 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0)) |
624 | 623 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) = ((𝐷‘0)‘𝑥)) |
625 | 624 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) |
626 | 620, 625 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))) |
627 | 626 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))) |
628 | | simpl 483 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝜑) |
629 | | fzofzp1 13484 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁)) |
630 | 629 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁)) |
631 | | nfv 1917 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) |
632 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑘(𝑖 + 1) |
633 | 345, 632 | nffv 6784 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝐶‘(𝑖 + 1)) |
634 | 633, 348,
349 | nff 6596 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐶‘(𝑖 + 1)):𝑋⟶ℂ |
635 | 631, 634 | nfim 1899 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ) |
636 | | ovex 7308 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 + 1) ∈ V |
637 | | eleq1 2826 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = (𝑖 + 1) → (𝑘 ∈ (0...𝑁) ↔ (𝑖 + 1) ∈ (0...𝑁))) |
638 | 637 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = (𝑖 + 1) → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)))) |
639 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = (𝑖 + 1) → (𝐶‘𝑘) = (𝐶‘(𝑖 + 1))) |
640 | 639 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = (𝑖 + 1) → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)) |
641 | 638, 640 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = (𝑖 + 1) → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ))) |
642 | 635, 636,
641, 228 | vtoclf 3497 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ) |
643 | 628, 630,
642 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ) |
644 | 643 | ffvelrnda 6961 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘(𝑖 + 1))‘𝑥) ∈ ℂ) |
645 | | nfv 1917 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝜑 ∧ 0 ∈ (0...𝑁)) |
646 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
Ⅎ𝑘0 |
647 | 461, 646 | nffv 6784 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑘(𝐷‘0) |
648 | 647, 348,
349 | nff 6596 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝐷‘0):𝑋⟶ℂ |
649 | 645, 648 | nfim 1899 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ) |
650 | | c0ex 10969 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 0 ∈
V |
651 | | eleq1 2826 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 = 0 → (𝑘 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁))) |
652 | 651 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = 0 → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁)))) |
653 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 = 0 → (𝐷‘𝑘) = (𝐷‘0)) |
654 | 653 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = 0 → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ)) |
655 | 652, 654 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = 0 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ))) |
656 | 649, 650,
655, 583 | vtoclf 3497 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ) |
657 | 1, 112, 656 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝐷‘0):𝑋⟶ℂ) |
658 | 657 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘0):𝑋⟶ℂ) |
659 | 658 | ffvelrnda 6961 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘0)‘𝑥) ∈ ℂ) |
660 | 644, 659 | mulcld 10995 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) ∈ ℂ) |
661 | 660 | mulid2d 10993 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) |
662 | 627, 661 | eqtrd 2778 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) |
663 | | 1m1e0 12045 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (1
− 1) = 0 |
664 | 663 | fveq2i 6777 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(ℤ≥‘(1 − 1)) =
(ℤ≥‘0) |
665 | 3 | eqcomi 2747 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(ℤ≥‘0) = ℕ0 |
666 | 664, 665 | eqtr2i 2767 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
ℕ0 = (ℤ≥‘(1 −
1)) |
667 | 666 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → ℕ0 =
(ℤ≥‘(1 − 1))) |
668 | 185, 667 | eleqtrd 2841 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ≥‘(1
− 1))) |
669 | | fzdifsuc2 42849 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈
(ℤ≥‘(1 − 1)) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})) |
670 | 668, 669 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})) |
671 | 670 | eqcomd 2744 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}) = (1...𝑖)) |
672 | 671 | sumeq1d 15413 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) |
673 | 672 | ad2antlr 724 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) |
674 | 662, 673 | oveq12d 7293 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))) |
675 | 514, 614,
674 | 3eqtrd 2782 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))) |
676 | 492, 496,
675 | 3eqtrd 2782 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))) |
677 | | nfcv 2907 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(𝑖C0) |
678 | 345, 646 | nffv 6784 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(𝐶‘0) |
679 | 678, 458 | nffv 6784 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝐶‘0)‘𝑥) |
680 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝑖 + 1) − 0) |
681 | 461, 680 | nffv 6784 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(𝐷‘((𝑖 + 1) − 0)) |
682 | 681, 458 | nffv 6784 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝐷‘((𝑖 + 1) − 0))‘𝑥) |
683 | 679, 455,
682 | nfov 7305 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) |
684 | 677, 455,
683 | nfov 7305 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) |
685 | 665 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → (ℤ≥‘0) =
ℕ0) |
686 | 185, 685 | eleqtrrd 2842 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈
(ℤ≥‘0)) |
687 | | eluzfz1 13263 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈
(ℤ≥‘0) → 0 ∈ (0...𝑖)) |
688 | 686, 687 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (0..^𝑁) → 0 ∈ (0...𝑖)) |
689 | 688 | ad2antlr 724 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ (0...𝑖)) |
690 | | oveq2 7283 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 → (𝑖C𝑘) = (𝑖C0)) |
691 | 104 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 0 → ((𝐶‘𝑘)‘𝑥) = ((𝐶‘0)‘𝑥)) |
692 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 0 → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − 0)) |
693 | 692 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 0))) |
694 | 693 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 0 → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) |
695 | 691, 694 | oveq12d 7293 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) |
696 | 690, 695 | oveq12d 7293 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))) |
697 | 472, 684,
439, 441, 689, 696 | fsumsplit1 15457 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
698 | 621 | subid1d 11321 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 0) = (𝑖 + 1)) |
699 | 698 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − 0)) = (𝐷‘(𝑖 + 1))) |
700 | 699 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − 0))‘𝑥) = ((𝐷‘(𝑖 + 1))‘𝑥)) |
701 | 700 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) |
702 | 701 | oveq2d 7291 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))) |
703 | 702 | oveq1d 7290 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (0..^𝑁) → (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
704 | 703 | ad2antlr 724 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
705 | | bcn0 14024 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ ℕ0
→ (𝑖C0) =
1) |
706 | 185, 705 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C0) = 1) |
707 | 706 | oveq1d 7290 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))) |
708 | 707 | ad2antlr 724 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))) |
709 | 678, 348,
349 | nff 6596 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐶‘0):𝑋⟶ℂ |
710 | 645, 709 | nfim 1899 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ) |
711 | 104 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = 0 → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘0):𝑋⟶ℂ)) |
712 | 652, 711 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = 0 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ))) |
713 | 710, 650,
712, 228 | vtoclf 3497 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ) |
714 | 1, 112, 713 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐶‘0):𝑋⟶ℂ) |
715 | 714 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐶‘0):𝑋⟶ℂ) |
716 | 715 | ffvelrnda 6961 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘0)‘𝑥) ∈ ℂ) |
717 | 461, 632 | nffv 6784 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐷‘(𝑖 + 1)) |
718 | 717, 348,
349 | nff 6596 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘(𝐷‘(𝑖 + 1)):𝑋⟶ℂ |
719 | 631, 718 | nfim 1899 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ) |
720 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = (𝑖 + 1) → (𝐷‘𝑘) = (𝐷‘(𝑖 + 1))) |
721 | 720 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = (𝑖 + 1) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)) |
722 | 638, 721 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = (𝑖 + 1) → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ))) |
723 | 719, 636,
722, 583 | vtoclf 3497 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ) |
724 | 628, 630,
723 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ) |
725 | 724 | ffvelrnda 6961 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 + 1))‘𝑥) ∈ ℂ) |
726 | 716, 725 | mulcld 10995 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) ∈ ℂ) |
727 | 726 | mulid2d 10993 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) |
728 | 708, 727 | eqtrd 2778 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) |
729 | | nfv 1917 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑗 𝑖 ∈ (0..^𝑁) |
730 | | 1zzd 12351 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ∈
ℤ) |
731 | 233 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑖 ∈ ℤ) |
732 | | eldifi 4061 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (0...𝑖)) |
733 | | elfzelz 13256 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ) |
734 | 732, 733 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℤ) |
735 | 734 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ ℤ) |
736 | | elfznn0 13349 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℕ0) |
737 | 732, 736 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ0) |
738 | | eldifsni 4723 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≠ 0) |
739 | 737, 738 | jca 512 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → (𝑗 ∈ ℕ0 ∧ 𝑗 ≠ 0)) |
740 | | elnnne0 12247 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ℕ ↔ (𝑗 ∈ ℕ0
∧ 𝑗 ≠
0)) |
741 | 739, 740 | sylibr 233 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ) |
742 | | nnge1 12001 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ ℕ → 1 ≤
𝑗) |
743 | 741, 742 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 1 ≤ 𝑗) |
744 | 743 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ≤ 𝑗) |
745 | | elfzle2 13260 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ≤ 𝑖) |
746 | 732, 745 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≤ 𝑖) |
747 | 746 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ≤ 𝑖) |
748 | 730, 731,
735, 744, 747 | elfzd 13247 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ (1...𝑖)) |
749 | 748 | ex 413 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (1...𝑖))) |
750 | | 0zd 12331 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...𝑖) → 0 ∈ ℤ) |
751 | | elfzel2 13254 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...𝑖) → 𝑖 ∈ ℤ) |
752 | | elfzelz 13256 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℤ) |
753 | 171 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...𝑖) → 0 ∈ ℝ) |
754 | 752 | zred 12426 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℝ) |
755 | | 1red 10976 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 1 ∈ ℝ) |
756 | 532 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 0 < 1) |
757 | | elfzle1 13259 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 1 ≤ 𝑗) |
758 | 753, 755,
754, 756, 757 | ltletrd 11135 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...𝑖) → 0 < 𝑗) |
759 | 753, 754,
758 | ltled 11123 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...𝑖) → 0 ≤ 𝑗) |
760 | | elfzle2 13260 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ≤ 𝑖) |
761 | 750, 751,
752, 759, 760 | elfzd 13247 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (0...𝑖)) |
762 | 753, 758 | gtned 11110 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ≠ 0) |
763 | | nelsn 4601 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ≠ 0 → ¬ 𝑗 ∈ {0}) |
764 | 762, 763 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (1...𝑖) → ¬ 𝑗 ∈ {0}) |
765 | 761, 764 | eldifd 3898 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0})) |
766 | 765 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ((0...𝑖) ∖ {0})) |
767 | 766 | ex 413 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0}))) |
768 | 749, 767 | impbid 211 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖))) |
769 | 729, 768 | alrimi 2206 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖))) |
770 | | dfcleq 2731 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((0...𝑖) ∖
{0}) = (1...𝑖) ↔
∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖))) |
771 | 769, 770 | sylibr 233 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → ((0...𝑖) ∖ {0}) = (1...𝑖)) |
772 | 771 | sumeq1d 15413 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (0..^𝑁) → Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
773 | 772 | ad2antlr 724 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
774 | 728, 773 | oveq12d 7293 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
775 | 697, 704,
774 | 3eqtrd 2782 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
776 | 676, 775 | oveq12d 7293 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))) |
777 | | fzfid 13693 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...𝑖) ∈ Fin) |
778 | 185 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0) |
779 | 766, 734 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ℤ) |
780 | | 1zzd 12351 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 1 ∈ ℤ) |
781 | 779, 780 | zsubcld 12431 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑗 − 1) ∈ ℤ) |
782 | 778, 781 | bccld 42854 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈
ℕ0) |
783 | 782 | nn0cnd 12295 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
784 | 783 | adantll 711 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
785 | 784 | adantlr 712 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
786 | | simpl 483 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋)) |
787 | | fzelp1 13308 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (1...(𝑖 + 1))) |
788 | 787 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ (1...(𝑖 + 1))) |
789 | 786, 788,
552 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐶‘𝑗)‘𝑥) ∈ ℂ) |
790 | 788, 587 | syldan 591 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ) |
791 | 789, 790 | mulcld 10995 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ) |
792 | 785, 791 | mulcld 10995 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ) |
793 | 777, 792 | fsumcl 15445 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ) |
794 | 185 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0) |
795 | | elfzelz 13256 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ ℤ) |
796 | 795 | adantl 482 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ ℤ) |
797 | 794, 796 | bccld 42854 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈
ℕ0) |
798 | 797 | nn0cnd 12295 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
799 | 798 | adantll 711 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
800 | 799 | adantlr 712 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
801 | | simpll 764 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝜑 ∧ 𝑖 ∈ (0..^𝑁))) |
802 | | simplr 766 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑥 ∈ 𝑋) |
803 | 761 | ssriv 3925 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(1...𝑖) ⊆
(0...𝑖) |
804 | | id 22 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (1...𝑖)) |
805 | 803, 804 | sselid 3919 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (0...𝑖)) |
806 | 805 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ (0...𝑖)) |
807 | 801, 802,
806, 433 | syl21anc 835 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
808 | 806, 435 | syldan 591 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) |
809 | 807, 808 | mulcld 10995 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ) |
810 | 800, 809 | mulcld 10995 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
811 | 777, 810 | fsumcl 15445 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
812 | 660, 793,
726, 811 | add4d 11203 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))) |
813 | | oveq1 7282 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑘 → (𝑗 − 1) = (𝑘 − 1)) |
814 | 813 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑘 → (𝑖C(𝑗 − 1)) = (𝑖C(𝑘 − 1))) |
815 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 𝑘 → (𝐶‘𝑗) = (𝐶‘𝑘)) |
816 | 815 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑘 → ((𝐶‘𝑗)‘𝑥) = ((𝐶‘𝑘)‘𝑥)) |
817 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 = 𝑘 → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − 𝑘)) |
818 | 817 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 𝑘 → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − 𝑘))) |
819 | 818 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑘 → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) |
820 | 816, 819 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑘 → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
821 | 814, 820 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑘 → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = ((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
822 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(1...𝑖) |
823 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑗(1...𝑖) |
824 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘(𝑖C(𝑗 − 1)) |
825 | 347, 458 | nffv 6784 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘((𝐶‘𝑗)‘𝑥) |
826 | 570, 458 | nffv 6784 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) |
827 | 825, 455,
826 | nfov 7305 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘(((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) |
828 | 824, 455,
827 | nfov 7305 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) |
829 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑗((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
830 | 821, 822,
823, 828, 829 | cbvsum 15407 |
. . . . . . . . . . . . . . . . . 18
⊢
Σ𝑗 ∈
(1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
831 | 830 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
832 | 831 | oveq1d 7290 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
833 | | peano2zm 12363 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ ℤ → (𝑘 − 1) ∈
ℤ) |
834 | 796, 833 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑘 − 1) ∈ ℤ) |
835 | 794, 834 | bccld 42854 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈
ℕ0) |
836 | 835 | nn0cnd 12295 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ) |
837 | 836 | adantll 711 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ) |
838 | 837 | adantlr 712 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ) |
839 | 838, 809 | mulcld 10995 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
840 | 777, 839,
810 | fsumadd 15452 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
841 | 840 | eqcomd 2744 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (1...𝑖)(((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
842 | 836, 798 | addcomd 11177 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) = ((𝑖C𝑘) + (𝑖C(𝑘 − 1)))) |
843 | | bcpasc 14035 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘)) |
844 | 794, 796,
843 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘)) |
845 | 842, 844 | eqtr2d 2779 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) = ((𝑖C(𝑘 − 1)) + (𝑖C𝑘))) |
846 | 845 | oveq1d 7290 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
847 | 846 | adantll 711 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
848 | 847 | adantlr 712 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
849 | 838, 800,
809 | adddird 11000 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
850 | 848, 849 | eqtr2d 2779 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
851 | 850 | sumeq2dv 15415 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
852 | 832, 841,
851 | 3eqtrd 2782 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
853 | 852 | oveq2d 7291 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
854 | | peano2nn0 12273 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ ℕ0
→ (𝑖 + 1) ∈
ℕ0) |
855 | 794, 854 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖 + 1) ∈
ℕ0) |
856 | 855, 796 | bccld 42854 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈
ℕ0) |
857 | 856 | nn0cnd 12295 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
858 | 857 | adantll 711 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
859 | 858 | adantlr 712 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
860 | 859, 809 | mulcld 10995 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
861 | 777, 860 | fsumcl 15445 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
862 | 660, 726,
861 | addassd 10997 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))) |
863 | 185, 854 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈
ℕ0) |
864 | | bcn0 14024 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 + 1) ∈ ℕ0
→ ((𝑖 + 1)C0) =
1) |
865 | 863, 864 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C0) = 1) |
866 | 865, 701 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))) |
867 | 866 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))) |
868 | 867, 727 | eqtr2d 2779 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))) |
869 | 771 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((0...𝑖) ∖ {0}) = (1...𝑖)) |
870 | 869 | eqcomd 2744 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...𝑖) = ((0...𝑖) ∖ {0})) |
871 | 870 | sumeq1d 15413 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
872 | 868, 871 | oveq12d 7293 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
873 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝑖 + 1)C0) |
874 | 873, 455,
683 | nfov 7305 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) |
875 | 197, 854 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈
ℕ0) |
876 | 875, 199 | bccld 42854 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈
ℕ0) |
877 | 876 | nn0cnd 12295 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
878 | 877 | adantll 711 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
879 | 878 | adantlr 712 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
880 | 879, 436 | mulcld 10995 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
881 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 0 → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C0)) |
882 | 881, 695 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))) |
883 | 472, 874,
439, 880, 689, 882 | fsumsplit1 15457 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
884 | 883 | eqcomd 2744 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
885 | 872, 884 | eqtrd 2778 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
886 | 885 | oveq2d 7291 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
887 | | bcnn 14026 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 + 1) ∈ ℕ0
→ ((𝑖 + 1)C(𝑖 + 1)) = 1) |
888 | 863, 887 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C(𝑖 + 1)) = 1) |
889 | 888 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖 + 1)C(𝑖 + 1)) = 1) |
890 | 889 | oveq1d 7290 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))) |
891 | 623 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0)) |
892 | 891 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ)) |
893 | 658, 892 | mpbird 256 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ) |
894 | 893 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ) |
895 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
896 | 894, 895 | ffvelrnd 6962 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) ∈ ℂ) |
897 | 644, 896 | mulcld 10995 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) ∈ ℂ) |
898 | 897 | mulid2d 10993 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) |
899 | 625 | ad2antlr 724 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) |
900 | 890, 898,
899 | 3eqtrrd 2783 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) = (((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))) |
901 | | fzdifsuc 13316 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈
(ℤ≥‘0) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})) |
902 | 686, 901 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})) |
903 | 902 | sumeq1d 15413 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (0..^𝑁) → Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
904 | 903 | ad2antlr 724 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
905 | 900, 904 | oveq12d 7293 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
906 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝑖 + 1)C(𝑖 + 1)) |
907 | 633, 458 | nffv 6784 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝐶‘(𝑖 + 1))‘𝑥) |
908 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘((𝑖 + 1) − (𝑖 + 1)) |
909 | 461, 908 | nffv 6784 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘(𝐷‘((𝑖 + 1) − (𝑖 + 1))) |
910 | 909, 458 | nffv 6784 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) |
911 | 907, 455,
910 | nfov 7305 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘(((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) |
912 | 906, 455,
911 | nfov 7305 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) |
913 | | fzfid 13693 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (0...(𝑖 + 1)) ∈ Fin) |
914 | 863 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈
ℕ0) |
915 | | elfzelz 13256 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ∈ ℤ) |
916 | 915 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℤ) |
917 | 914, 916 | bccld 42854 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈
ℕ0) |
918 | 917 | nn0cnd 12295 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
919 | 918 | adantll 711 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
920 | 919 | adantlr 712 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
921 | 628 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑) |
922 | 90 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈
ℤ) |
923 | 206 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℤ) |
924 | | elfzle1 13259 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ (0...(𝑖 + 1)) → 0 ≤ 𝑘) |
925 | 924 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ 𝑘) |
926 | 916 | zred 12426 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℝ) |
927 | 914 | nn0red 12294 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ) |
928 | 211 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℝ) |
929 | | elfzle2 13260 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ≤ (𝑖 + 1)) |
930 | 929 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ (𝑖 + 1)) |
931 | 301 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁) |
932 | 926, 927,
928, 930, 931 | letrd 11132 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ 𝑁) |
933 | 922, 923,
916, 925, 932 | elfzd 13247 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁)) |
934 | 933 | adantll 711 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁)) |
935 | 921, 934,
228 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶‘𝑘):𝑋⟶ℂ) |
936 | 935 | adantlr 712 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶‘𝑘):𝑋⟶ℂ) |
937 | | simplr 766 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑥 ∈ 𝑋) |
938 | 936, 937 | ffvelrnd 6962 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
939 | 921 | adantlr 712 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑) |
940 | 591 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ) |
941 | 940, 916 | zsubcld 12431 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℤ) |
942 | 927, 926 | subge0d 11565 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1))) |
943 | 930, 942 | mpbird 256 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑘)) |
944 | 927, 926 | resubcld 11403 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℝ) |
945 | 928, 926 | resubcld 11403 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 𝑘) ∈ ℝ) |
946 | 928, 171,
246 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) ∈ ℝ) |
947 | 927, 928,
926, 931 | lesub1dd 11591 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 𝑘)) |
948 | 171 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈
ℝ) |
949 | 948, 926,
928, 925 | lesub2dd 11592 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 𝑘) ≤ (𝑁 − 0)) |
950 | 944, 945,
946, 947, 949 | letrd 11132 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 0)) |
951 | 252 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) = 𝑁) |
952 | 950, 951 | breqtrd 5100 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ 𝑁) |
953 | 922, 923,
941, 943, 952 | elfzd 13247 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
954 | 953 | adantll 711 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
955 | 954 | adantlr 712 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
956 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (𝐷‘𝑗) = (𝐷‘((𝑖 + 1) − 𝑘))) |
957 | 956 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝐷‘𝑗):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)) |
958 | 310, 957 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))) |
959 | 461, 346 | nffv 6784 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝐷‘𝑗) |
960 | 959, 348,
349 | nff 6596 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐷‘𝑗):𝑋⟶ℂ |
961 | 343, 960 | nfim 1899 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ) |
962 | | fveq2 6774 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = 𝑗 → (𝐷‘𝑘) = (𝐷‘𝑗)) |
963 | 962 | feq1d 6585 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = 𝑗 → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘𝑗):𝑋⟶ℂ)) |
964 | 266, 963 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ))) |
965 | 961, 964,
583 | chvarfv 2233 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ) |
966 | 308, 958,
965 | vtocl 3498 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ) |
967 | 939, 955,
966 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ) |
968 | 967, 937 | ffvelrnd 6962 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) |
969 | 938, 968 | mulcld 10995 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ) |
970 | 920, 969 | mulcld 10995 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
971 | 863, 685 | eleqtrrd 2842 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈
(ℤ≥‘0)) |
972 | | eluzfz2 13264 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 + 1) ∈
(ℤ≥‘0) → (𝑖 + 1) ∈ (0...(𝑖 + 1))) |
973 | 971, 972 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...(𝑖 + 1))) |
974 | 973 | ad2antlr 724 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝑖 + 1) ∈ (0...(𝑖 + 1))) |
975 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = (𝑖 + 1) → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C(𝑖 + 1))) |
976 | 639 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = (𝑖 + 1) → ((𝐶‘𝑘)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥)) |
977 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = (𝑖 + 1) → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − (𝑖 + 1))) |
978 | 977 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1)))) |
979 | 978 | fveq1d 6776 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) |
980 | 976, 979 | oveq12d 7293 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = (𝑖 + 1) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) |
981 | 975, 980 | oveq12d 7293 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = (𝑖 + 1) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))) |
982 | 472, 912,
913, 970, 974, 981 | fsumsplit1 15457 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
983 | 982 | eqcomd 2744 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
984 | 886, 905,
983 | 3eqtrd 2782 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
985 | 853, 862,
984 | 3eqtrd 2782 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
986 | 776, 812,
985 | 3eqtrd 2782 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
987 | 438, 442,
986 | 3eqtrd 2782 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
988 | 987 | mpteq2dva 5174 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
989 | 422, 988 | eqtrd 2778 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
990 | 989 | adantr 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
991 | 189, 191,
990 | 3eqtrd 2782 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
992 | 178, 179,
182, 991 | syl21anc 835 |
. . . . . 6
⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
993 | 992 | 3exp 1118 |
. . . . 5
⊢ (𝑖 ∈ (0..^𝑁) → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))) |
994 | 34, 47, 60, 73, 177, 993 | fzind2 13505 |
. . . 4
⊢ (𝑛 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))))) |
995 | 21, 994 | vtoclg 3505 |
. . 3
⊢ (𝑁 ∈ ℕ0
→ (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))))) |
996 | 2, 6, 995 | sylc 65 |
. 2
⊢ (𝜑 → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))) |
997 | 1, 996 | mpd 15 |
1
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))) |