Step | Hyp | Ref
| Expression |
1 | | id 22 |
. 2
⊢ (𝜑 → 𝜑) |
2 | | dvnmul.n |
. . 3
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
3 | | nn0uz 11598 |
. . . . 5
⊢
ℕ0 = (ℤ≥‘0) |
4 | 2, 3 | syl6eleq 2698 |
. . . 4
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘0)) |
5 | | eluzfz2 12220 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘0) → 𝑁 ∈ (0...𝑁)) |
6 | 4, 5 | syl 17 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) |
7 | | eleq1 2676 |
. . . . 5
⊢ (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁))) |
8 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑛 = 𝑁 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁)) |
9 | | oveq2 6557 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁)) |
10 | 9 | sumeq1d 14279 |
. . . . . . . . 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 6556 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘)) |
12 | | oveq1 6556 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑁 → (𝑛 − 𝑘) = (𝑁 − 𝑘)) |
13 | 12 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑁 → (𝐷‘(𝑛 − 𝑘)) = (𝐷‘(𝑁 − 𝑘))) |
14 | 13 | fveq1d 6105 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑁 → ((𝐷‘(𝑛 − 𝑘))‘𝑥) = ((𝐷‘(𝑁 − 𝑘))‘𝑥)) |
15 | 14 | oveq2d 6565 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑁 → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))) |
16 | 11, 15 | oveq12d 6567 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑁 → ((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = ((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))) |
17 | 16 | sumeq2ad 38632 |
. . . . . . . . 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)))) |
18 | 10, 17 | eqtrd 2644 |
. . . . . . . 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)))) |
19 | 18 | mpteq2dv 4673 |
. . . . . . 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))))) |
20 | 8, 19 | eqeq12d 2625 |
. . . . . 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)))))) |
21 | 20 | imbi2d 329 |
. . . . 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))))))) |
22 | 7, 21 | imbi12d 333 |
. . . 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)))))))) |
23 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑚 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0)) |
24 | | simpl 472 |
. . . . . . . . . 10
⊢ ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 0) |
25 | 24 | oveq2d 6565 |
. . . . . . . . 9
⊢ ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...0)) |
26 | | simpll 786 |
. . . . . . . . . . 11
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → 𝑚 = 0) |
27 | 26 | oveq1d 6564 |
. . . . . . . . . 10
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (𝑚C𝑘) = (0C𝑘)) |
28 | 26 | oveq1d 6564 |
. . . . . . . . . . . . 13
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (𝑚 − 𝑘) = (0 − 𝑘)) |
29 | 28 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(0 − 𝑘))) |
30 | 29 | fveq1d 6105 |
. . . . . . . . . . 11
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(0 − 𝑘))‘𝑥)) |
31 | 30 | oveq2d 6565 |
. . . . . . . . . 10
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) |
32 | 27, 31 | oveq12d 6567 |
. . . . . . . . 9
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))) |
33 | 25, 32 | sumeq12rdv 14285 |
. . . . . . . 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)))) |
34 | 33 | mpteq2dva 4672 |
. . . . . . 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))))) |
35 | 23, 34 | eqeq12d 2625 |
. . . . . 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)))))) |
36 | 35 | imbi2d 329 |
. . . . 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))))))) |
37 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑚 = 𝑖 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)) |
38 | | simpl 472 |
. . . . . . . . . 10
⊢ ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 𝑖) |
39 | 38 | oveq2d 6565 |
. . . . . . . . 9
⊢ ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...𝑖)) |
40 | | simpll 786 |
. . . . . . . . . . 11
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → 𝑚 = 𝑖) |
41 | 40 | oveq1d 6564 |
. . . . . . . . . 10
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑚C𝑘) = (𝑖C𝑘)) |
42 | 40 | oveq1d 6564 |
. . . . . . . . . . . . 13
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑚 − 𝑘) = (𝑖 − 𝑘)) |
43 | 42 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(𝑖 − 𝑘))) |
44 | 43 | fveq1d 6105 |
. . . . . . . . . . 11
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(𝑖 − 𝑘))‘𝑥)) |
45 | 44 | oveq2d 6565 |
. . . . . . . . . 10
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) |
46 | 41, 45 | oveq12d 6567 |
. . . . . . . . 9
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))) |
47 | 39, 46 | sumeq12rdv 14285 |
. . . . . . . 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)))) |
48 | 47 | mpteq2dva 4672 |
. . . . . . 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))))) |
49 | 37, 48 | eqeq12d 2625 |
. . . . . 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)))))) |
50 | 49 | imbi2d 329 |
. . . . 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))))))) |
51 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑚 = (𝑖 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1))) |
52 | | simpl 472 |
. . . . . . . . . 10
⊢ ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → 𝑚 = (𝑖 + 1)) |
53 | 52 | oveq2d 6565 |
. . . . . . . . 9
⊢ ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...(𝑖 + 1))) |
54 | | simpll 786 |
. . . . . . . . . . 11
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑚 = (𝑖 + 1)) |
55 | 54 | oveq1d 6564 |
. . . . . . . . . 10
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑚C𝑘) = ((𝑖 + 1)C𝑘)) |
56 | 54 | oveq1d 6564 |
. . . . . . . . . . . . 13
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑚 − 𝑘) = ((𝑖 + 1) − 𝑘)) |
57 | 56 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘))) |
58 | 57 | fveq1d 6105 |
. . . . . . . . . . 11
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) |
59 | 58 | oveq2d 6565 |
. . . . . . . . . 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))) |
60 | 55, 59 | oveq12d 6567 |
. . . . . . . . 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)))) |
61 | 53, 60 | sumeq12rdv 14285 |
. . . . . . . 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)))) |
62 | 61 | mpteq2dva 4672 |
. . . . . . 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))))) |
63 | 51, 62 | eqeq12d 2625 |
. . . . . 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)))))) |
64 | 63 | imbi2d 329 |
. . . . 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))))))) |
65 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑚 = 𝑛 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛)) |
66 | | simpl 472 |
. . . . . . . . . 10
⊢ ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 𝑛) |
67 | 66 | oveq2d 6565 |
. . . . . . . . 9
⊢ ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...𝑛)) |
68 | | simpll 786 |
. . . . . . . . . . 11
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → 𝑚 = 𝑛) |
69 | 68 | oveq1d 6564 |
. . . . . . . . . 10
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝑚C𝑘) = (𝑛C𝑘)) |
70 | 68 | oveq1d 6564 |
. . . . . . . . . . . . 13
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝑚 − 𝑘) = (𝑛 − 𝑘)) |
71 | 70 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(𝑛 − 𝑘))) |
72 | 71 | fveq1d 6105 |
. . . . . . . . . . 11
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(𝑛 − 𝑘))‘𝑥)) |
73 | 72 | oveq2d 6565 |
. . . . . . . . . 10
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) |
74 | 69, 73 | oveq12d 6567 |
. . . . . . . . 9
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) |
75 | 67, 74 | sumeq12rdv 14285 |
. . . . . . . 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)))) |
76 | 75 | mpteq2dva 4672 |
. . . . . . 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))))) |
77 | 65, 76 | eqeq12d 2625 |
. . . . . 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)))))) |
78 | 77 | imbi2d 329 |
. . . . 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))))))) |
79 | | dvnmul.s |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
80 | | recnprss 23474 |
. . . . . . . . 9
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
81 | 79, 80 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
82 | | dvnmul.a |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) |
83 | | dvnmul.cc |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) |
84 | 82, 83 | mulcld 9939 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 · 𝐵) ∈ ℂ) |
85 | | restsspw 15915 |
. . . . . . . . . . 11
⊢
((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆 |
86 | | dvnmul.x |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
87 | 85, 86 | sseldi 3566 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆) |
88 | | elpwi 4117 |
. . . . . . . . . 10
⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆) |
89 | 87, 88 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
90 | | cnex 9896 |
. . . . . . . . . 10
⊢ ℂ
∈ V |
91 | 90 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ℂ ∈
V) |
92 | 84, 89, 91, 79 | mptelpm 38352 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ
↑pm 𝑆)) |
93 | | dvn0 23493 |
. . . . . . . 8
⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ
↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))) |
94 | 81, 92, 93 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))) |
95 | | 0z 11265 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℤ |
96 | | fzsn 12254 |
. . . . . . . . . . . 12
⊢ (0 ∈
ℤ → (0...0) = {0}) |
97 | 95, 96 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (0...0) =
{0} |
98 | 97 | sumeq1i 14276 |
. . . . . . . . . 10
⊢
Σ𝑘 ∈
(0...0)((0C𝑘) ·
(((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) |
99 | 98 | 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)))) |
100 | | nfcvd 2752 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Ⅎ𝑘(𝐴 · 𝐵)) |
101 | | nfv 1830 |
. . . . . . . . . 10
⊢
Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ 𝑋) |
102 | | oveq2 6557 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → (0C𝑘) = (0C0)) |
103 | | 0nn0 11184 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
ℕ0 |
104 | | bcn0 12959 |
. . . . . . . . . . . . . . . 16
⊢ (0 ∈
ℕ0 → (0C0) = 1) |
105 | 103, 104 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ (0C0) =
1 |
106 | 105 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → (0C0) =
1) |
107 | 102, 106 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 0 → (0C𝑘) = 1) |
108 | 107 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (0C𝑘) = 1) |
109 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (𝐶‘𝑘) = (𝐶‘0)) |
110 | 109 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = (𝐶‘0)) |
111 | | dvnmul.c |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) |
112 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐹)‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑛)) |
113 | 112 | cbvmptv 4678 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛)) |
114 | 111, 113 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛)) |
115 | 114 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))) |
116 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 0 → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘0)) |
117 | 116 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘0)) |
118 | | eluzfz1 12219 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ≥‘0) → 0 ∈ (0...𝑁)) |
119 | 4, 118 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 0 ∈ (0...𝑁)) |
120 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑆 D𝑛 𝐹)‘0) ∈
V |
121 | 120 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘0) ∈ V) |
122 | 115, 117,
119, 121 | fvmptd 6197 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0)) |
123 | 122 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0)) |
124 | 110, 123 | eqtrd 2644 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘0)) |
125 | | dvnmulf |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 𝐴) |
126 | 82, 89, 91, 79 | mptelpm 38352 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm
𝑆)) |
127 | 125, 126 | syl5eqel 2692 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm
𝑆)) |
128 | | dvn0 23493 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ
↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹) |
129 | 81, 127, 128 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹) |
130 | 129 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹) |
131 | 124, 130 | eqtrd 2644 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = 𝐹) |
132 | 131 | fveq1d 6105 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = (𝐹‘𝑥)) |
133 | 132 | adantlr 747 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = (𝐹‘𝑥)) |
134 | | simpr 476 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
135 | 125 | fvmpt2 6200 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ℂ) → (𝐹‘𝑥) = 𝐴) |
136 | 134, 82, 135 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) = 𝐴) |
137 | 136 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (𝐹‘𝑥) = 𝐴) |
138 | 133, 137 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = 𝐴) |
139 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (0 − 𝑘) = (0 −
0)) |
140 | | 0m0e0 11007 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0
− 0) = 0 |
141 | 140 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (0 − 0) =
0) |
142 | 139, 141 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 0 → (0 − 𝑘) = 0) |
143 | 142 | fveq2d 6107 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 → (𝐷‘(0 − 𝑘)) = (𝐷‘0)) |
144 | 143 | fveq1d 6105 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥)) |
145 | 144 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥)) |
146 | 145 | adantlr 747 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥)) |
147 | | dvnmul.d |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) |
148 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑛)) |
149 | 148 | cbvmptv 4678 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) |
150 | 147, 149 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) |
151 | 150 | fveq1i 6104 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0) |
152 | 151 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0)) |
153 | | eqidd 2611 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))) |
154 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 0 → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0)) |
155 | 154 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0)) |
156 | | dvnmul.f |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ 𝐵) |
157 | 83, 89, 91, 79 | mptelpm 38352 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (ℂ ↑pm
𝑆)) |
158 | 156, 157 | syl5eqel 2692 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝐺 ∈ (ℂ ↑pm
𝑆)) |
159 | | dvn0 23493 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ
↑pm 𝑆)) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺) |
160 | 81, 158, 159 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑆 D𝑛 𝐺)‘0) = 𝐺) |
161 | 160 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺) |
162 | 155, 161 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = 𝐺) |
163 | 156 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝑋 ↦ 𝐵)) |
164 | | mptexg 6389 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑋 ∈ 𝒫 𝑆 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ V) |
165 | 87, 164 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ V) |
166 | 163, 165 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐺 ∈ V) |
167 | 153, 162,
119, 166 | fvmptd 6197 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0) = 𝐺) |
168 | 152, 167 | eqtrd 2644 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐷‘0) = 𝐺) |
169 | 168 | fveq1d 6105 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐷‘0)‘𝑥) = (𝐺‘𝑥)) |
170 | 169 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘0)‘𝑥) = (𝐺‘𝑥)) |
171 | 163, 83 | fvmpt2d 6202 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) = 𝐵) |
172 | 171 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (𝐺‘𝑥) = 𝐵) |
173 | 146, 170,
172 | 3eqtrd 2648 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = 𝐵) |
174 | 138, 173 | oveq12d 6567 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)) = (𝐴 · 𝐵)) |
175 | 108, 174 | oveq12d 6567 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (1 · (𝐴 · 𝐵))) |
176 | 84 | mulid2d 9937 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵)) |
177 | 176 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵)) |
178 | 175, 177 | eqtrd 2644 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵)) |
179 | | 0re 9919 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
180 | 179 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ) |
181 | 100, 101,
178, 180, 84 | sumsnd 38208 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵)) |
182 | 99, 181 | eqtr2d 2645 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))) |
183 | 182 | mpteq2dva 4672 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))) |
184 | 94, 183 | eqtrd 2644 |
. . . . . 6
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))) |
185 | 184 | 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)))))) |
186 | | simp3 1056 |
. . . . . . 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) |
187 | | simp1 1054 |
. . . . . . 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)) |
188 | | simp2 1055 |
. . . . . . . 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)))))) |
189 | | pm3.35 609 |
. . . . . . . 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))))) |
190 | 186, 188,
189 | syl2anc 691 |
. . . . . . 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))))) |
191 | 81 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆 ⊆ ℂ) |
192 | 92 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ
↑pm 𝑆)) |
193 | | elfzonn0 12380 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0) |
194 | 193 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0) |
195 | | dvnp1 23494 |
. . . . . . . . . 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))) |
196 | 191, 192,
194, 195 | syl3anc 1318 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖))) |
197 | 196 | adantr 480 |
. . . . . . . 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))) |
198 | | simpr 476 |
. . . . . . . . 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))))) |
199 | 198 | oveq2d 6565 |
. . . . . . . 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)))))) |
200 | | eqid 2610 |
. . . . . . . . . . 11
⊢
((TopOpen‘ℂfld) ↾t 𝑆) =
((TopOpen‘ℂfld) ↾t 𝑆) |
201 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
202 | 79 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆 ∈ {ℝ, ℂ}) |
203 | 86 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
204 | | fzfid 12634 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (0...𝑖) ∈ Fin) |
205 | 193 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℕ0) |
206 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℤ) |
207 | 206 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℤ) |
208 | 205, 207 | bccld 38472 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈
ℕ0) |
209 | 208 | nn0cnd 11230 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
210 | 209 | adantll 746 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
211 | 210 | 3adant3 1074 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝑖C𝑘) ∈ ℂ) |
212 | | simpll 786 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝜑) |
213 | | 0zd 11266 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℤ) |
214 | | elfzoel2 12338 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℤ) |
215 | 214 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℤ) |
216 | 213, 215,
207 | 3jca 1235 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈
ℤ)) |
217 | | elfzle1 12215 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (0...𝑖) → 0 ≤ 𝑘) |
218 | 217 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ 𝑘) |
219 | 207 | zred 11358 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℝ) |
220 | 214 | zred 11358 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℝ) |
221 | 220 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℝ) |
222 | 193 | nn0red 11229 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℝ) |
223 | 222 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℝ) |
224 | | elfzle2 12216 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ≤ 𝑖) |
225 | 224 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ 𝑖) |
226 | | elfzolt2 12348 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 < 𝑁) |
227 | 226 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 < 𝑁) |
228 | 219, 223,
221, 225, 227 | lelttrd 10074 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 < 𝑁) |
229 | 219, 221,
228 | ltled 10064 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ 𝑁) |
230 | 216, 218,
229 | jca32 556 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤
𝑘 ∧ 𝑘 ≤ 𝑁))) |
231 | | elfz2 12204 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤
𝑘 ∧ 𝑘 ≤ 𝑁))) |
232 | 230, 231 | sylibr 223 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁)) |
233 | 232 | adantll 746 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁)) |
234 | | dvnmul.dvnf |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ) |
235 | 111 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))) |
236 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑆 D𝑛 𝐹)‘𝑘) ∈ V |
237 | 236 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘) ∈ V) |
238 | 235, 237 | fvmpt2d 6202 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘)) |
239 | 238 | feq1d 5943 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐶‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)) |
240 | 234, 239 | mpbird 246 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) |
241 | 212, 233,
240 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘):𝑋⟶ℂ) |
242 | 241 | 3adant3 1074 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝐶‘𝑘):𝑋⟶ℂ) |
243 | | simp3 1056 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
244 | 242, 243 | ffvelrnd 6268 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
245 | 193 | nn0zd 11356 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℤ) |
246 | 245 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℤ) |
247 | 246, 207 | zsubcld 11363 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ ℤ) |
248 | 213, 215,
247 | 3jca 1235 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖 − 𝑘) ∈ ℤ)) |
249 | | elfzel2 12211 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℤ) |
250 | 249 | zred 11358 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℝ) |
251 | 206 | zred 11358 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℝ) |
252 | 250, 251 | subge0d 10496 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ (0...𝑖) → (0 ≤ (𝑖 − 𝑘) ↔ 𝑘 ≤ 𝑖)) |
253 | 224, 252 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑖 − 𝑘)) |
254 | 253 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑖 − 𝑘)) |
255 | 223, 219 | resubcld 10337 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ ℝ) |
256 | 221, 219 | resubcld 10337 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ∈ ℝ) |
257 | 179 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℝ) |
258 | 221, 257 | jca 553 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 ∈ ℝ ∧ 0 ∈
ℝ)) |
259 | | resubcl 10224 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑁 ∈ ℝ ∧ 0 ∈
ℝ) → (𝑁 −
0) ∈ ℝ) |
260 | 258, 259 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) ∈ ℝ) |
261 | 223, 221,
219, 227 | ltsub1dd 10518 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < (𝑁 − 𝑘)) |
262 | 257, 219,
221, 218 | lesub2dd 10523 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ≤ (𝑁 − 0)) |
263 | 255, 256,
260, 261, 262 | ltletrd 10076 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < (𝑁 − 0)) |
264 | 220 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℂ) |
265 | 264 | subid1d 10260 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → (𝑁 − 0) = 𝑁) |
266 | 265 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) = 𝑁) |
267 | 263, 266 | breqtrd 4609 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < 𝑁) |
268 | 255, 221,
267 | ltled 10064 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ≤ 𝑁) |
269 | 248, 254,
268 | jca32 556 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝑖 − 𝑘) ∧ (𝑖 − 𝑘) ≤ 𝑁))) |
270 | | elfz2 12204 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 − 𝑘) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝑖 − 𝑘) ∧ (𝑖 − 𝑘) ≤ 𝑁))) |
271 | 269, 270 | sylibr 223 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ (0...𝑁)) |
272 | 271 | adantll 746 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ (0...𝑁)) |
273 | | ovex 6577 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 − 𝑘) ∈ V |
274 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑖 − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ (𝑖 − 𝑘) ∈ (0...𝑁))) |
275 | 274 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 − 𝑘) ∈ (0...𝑁)))) |
276 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑖 − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
277 | 276 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ)) |
278 | 275, 277 | imbi12d 333 |
. . . . . . . . . . . . . . . . . 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))) |
279 | | nfv 1830 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) |
280 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑗 → (𝑘 ∈ (0...𝑁) ↔ 𝑗 ∈ (0...𝑁))) |
281 | 280 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑗 ∈ (0...𝑁)))) |
282 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑗 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑗)) |
283 | 282 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑗 → (((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)) |
284 | 281, 283 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ))) |
285 | | dvnmul.dvng |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ) |
286 | 279, 284,
285 | chvar 2250 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) |
287 | 273, 278,
286 | vtocl 3232 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑖 − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ) |
288 | 212, 272,
287 | syl2anc 691 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ) |
289 | 150 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))) |
290 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = (𝑖 − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
291 | 290 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = (𝑖 − 𝑘)) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
292 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)) ∈ V |
293 | 292 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)) ∈ V) |
294 | 289, 291,
271, 293 | fvmptd 6197 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
295 | 294 | adantll 746 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
296 | 295 | feq1d 5943 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ)) |
297 | 288, 296 | mpbird 246 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ) |
298 | 297 | 3adant3 1074 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ) |
299 | 298, 243 | ffvelrnd 6268 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 − 𝑘))‘𝑥) ∈ ℂ) |
300 | 244, 299 | mulcld 9939 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ) |
301 | 211, 300 | mulcld 9939 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ) |
302 | 211 | 3expa 1257 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (𝑖C𝑘) ∈ ℂ) |
303 | 246 | peano2zd 11361 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℤ) |
304 | 303, 207 | zsubcld 11363 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℤ) |
305 | 213, 215,
304 | 3jca 1235 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ)) |
306 | | peano2re 10088 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑖 ∈ ℝ → (𝑖 + 1) ∈
ℝ) |
307 | 250, 306 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ (0...𝑖) → (𝑖 + 1) ∈ ℝ) |
308 | | peano2re 10088 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈
ℝ) |
309 | 251, 308 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ∈ ℝ) |
310 | 251 | ltp1d 10833 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑘 + 1)) |
311 | | 1red 9934 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑘 ∈ (0...𝑖) → 1 ∈ ℝ) |
312 | 251, 250,
311, 224 | leadd1dd 10520 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ≤ (𝑖 + 1)) |
313 | 251, 309,
307, 310, 312 | ltletrd 10076 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑖 + 1)) |
314 | 251, 307,
313 | ltled 10064 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ≤ (𝑖 + 1)) |
315 | 314 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ (𝑖 + 1)) |
316 | 223, 306 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℝ) |
317 | 316, 219 | subge0d 10496 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1))) |
318 | 315, 317 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ ((𝑖 + 1) − 𝑘)) |
319 | 316, 219 | resubcld 10337 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℝ) |
320 | | elfzop1le2 38443 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ≤ 𝑁) |
321 | 320 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ≤ 𝑁) |
322 | 316, 221,
219, 321 | lesub1dd 10522 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 𝑘)) |
323 | 262, 266 | breqtrd 4609 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ≤ 𝑁) |
324 | 319, 256,
221, 322, 323 | letrd 10073 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ 𝑁) |
325 | 305, 318,
324 | jca32 556 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑘) ∧ ((𝑖 + 1) − 𝑘) ≤ 𝑁))) |
326 | | elfz2 12204 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑖 + 1) − 𝑘) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑘) ∧ ((𝑖 + 1) − 𝑘) ≤ 𝑁))) |
327 | 325, 326 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
328 | 327 | adantll 746 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
329 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 + 1) − 𝑘) ∈ V |
330 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))) |
331 | 330 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)))) |
332 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
333 | 332 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)) |
334 | 331, 333 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . 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))) |
335 | 329, 334,
286 | vtocl 3232 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ) |
336 | 212, 328,
335 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ) |
337 | 150 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))) |
338 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → 𝑛 = ((𝑖 + 1) − 𝑘)) |
339 | 338 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
340 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) ∈ V |
341 | 340 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) ∈ V) |
342 | 337, 339,
328, 341 | fvmptd 6197 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
343 | 342 | feq1d 5943 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)) |
344 | 336, 343 | mpbird 246 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ) |
345 | 344 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) |
346 | 244 | 3expa 1257 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
347 | 345, 346 | mulcomd 9940 |
. . . . . . . . . . . . . . 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))) |
348 | 347 | oveq2d 6565 |
. . . . . . . . . . . . . 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)))) |
349 | 207 | peano2zd 11361 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℤ) |
350 | 213, 215,
349 | 3jca 1235 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ∈
ℤ)) |
351 | 179 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...𝑖) → 0 ∈ ℝ) |
352 | 351, 251,
309, 217, 310 | lelttrd 10074 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...𝑖) → 0 < (𝑘 + 1)) |
353 | 351, 309,
352 | ltled 10064 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑘 + 1)) |
354 | 353 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑘 + 1)) |
355 | 219, 308 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℝ) |
356 | 312 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ (𝑖 + 1)) |
357 | 355, 316,
221, 356, 321 | letrd 10073 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ 𝑁) |
358 | 350, 354,
357 | jca32 556 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) ∧ (0
≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑁))) |
359 | | elfz2 12204 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 + 1) ∈ (0...𝑁) ↔ ((0 ∈ ℤ
∧ 𝑁 ∈ ℤ
∧ (𝑘 + 1) ∈
ℤ) ∧ (0 ≤ (𝑘 +
1) ∧ (𝑘 + 1) ≤ 𝑁))) |
360 | 358, 359 | sylibr 223 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁)) |
361 | 360 | adantll 746 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁)) |
362 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 + 1) ∈ V |
363 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = (𝑘 + 1) → (𝑗 ∈ (0...𝑁) ↔ (𝑘 + 1) ∈ (0...𝑁))) |
364 | 363 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑘 + 1) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)))) |
365 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = (𝑘 + 1) → (𝐶‘𝑗) = (𝐶‘(𝑘 + 1))) |
366 | 365 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑘 + 1) → ((𝐶‘𝑗):𝑋⟶ℂ ↔ (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)) |
367 | 364, 366 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑘 + 1) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ))) |
368 | | nfv 1830 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (0...𝑁)) |
369 | | nfmpt1 4675 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) |
370 | 111, 369 | nfcxfr 2749 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘𝐶 |
371 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘𝑗 |
372 | 370, 371 | nffv 6110 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘(𝐶‘𝑗) |
373 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘𝑋 |
374 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘ℂ |
375 | 372, 373,
374 | nff 5954 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘(𝐶‘𝑗):𝑋⟶ℂ |
376 | 368, 375 | nfim 1813 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ) |
377 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑗 → (𝐶‘𝑘) = (𝐶‘𝑗)) |
378 | 377 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑗 → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘𝑗):𝑋⟶ℂ)) |
379 | 281, 378 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ))) |
380 | 376, 379,
240 | chvar 2250 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ) |
381 | 362, 367,
380 | vtocl 3232 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ) |
382 | 212, 361,
381 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ) |
383 | 382 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘(𝑘 + 1))‘𝑥) ∈ ℂ) |
384 | 299 | 3expa 1257 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 − 𝑘))‘𝑥) ∈ ℂ) |
385 | 383, 384 | mulcld 9939 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ) |
386 | 345, 346 | mulcld 9939 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥)) ∈ ℂ) |
387 | 385, 386 | addcld 9938 |
. . . . . . . . . . . . . 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) |
388 | 348, 387 | eqeltrrd 2689 |
. . . . . . . . . . . . 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) |
389 | 302, 388 | mulcld 9939 |
. . . . . . . . . . . 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) |
390 | 389 | 3impa 1251 |
. . . . . . . . . . 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) |
391 | 212, 79 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ∈ {ℝ, ℂ}) |
392 | 179 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ) |
393 | 212, 86 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
394 | 391, 393,
210 | dvmptconst 38803 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝑖C𝑘))) = (𝑥 ∈ 𝑋 ↦ 0)) |
395 | 300 | 3expa 1257 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ) |
396 | 212, 233,
238 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘)) |
397 | 396 | eqcomd 2616 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘𝑘) = (𝐶‘𝑘)) |
398 | 241 | feqmptd 6159 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥))) |
399 | 397, 398 | eqtr2d 2645 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥)) = ((𝑆 D𝑛 𝐹)‘𝑘)) |
400 | 399 | oveq2d 6565 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘))) |
401 | 391, 80 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ⊆ ℂ) |
402 | 212, 127 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐹 ∈ (ℂ ↑pm
𝑆)) |
403 | | elfznn0 12302 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ0) |
404 | 403 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ0) |
405 | | dvnp1 23494 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ
↑pm 𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘))) |
406 | 401, 402,
404, 405 | syl3anc 1318 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘))) |
407 | 406 | eqcomd 2616 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1))) |
408 | 114 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))) |
409 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = (𝑘 + 1) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1))) |
410 | 409 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = (𝑘 + 1)) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1))) |
411 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) ∈ V |
412 | 411 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) ∈ V) |
413 | 408, 410,
361, 412 | fvmptd 6197 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1))) |
414 | 413 | eqcomd 2616 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝐶‘(𝑘 + 1))) |
415 | 382 | feqmptd 6159 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥))) |
416 | 414, 415 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥))) |
417 | 400, 407,
416 | 3eqtrd 2648 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥))) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥))) |
418 | 295 | eqcomd 2616 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)) = (𝐷‘(𝑖 − 𝑘))) |
419 | 297 | feqmptd 6159 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥))) |
420 | 418, 419 | eqtr2d 2645 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
421 | 420 | oveq2d 6565 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))) |
422 | 212, 158 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐺 ∈ (ℂ ↑pm
𝑆)) |
423 | | fznn0sub 12244 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...𝑖) → (𝑖 − 𝑘) ∈
ℕ0) |
424 | 423 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈
ℕ0) |
425 | | dvnp1 23494 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ
↑pm 𝑆) ∧ (𝑖 − 𝑘) ∈ ℕ0) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))) |
426 | 401, 422,
424, 425 | syl3anc 1318 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))) |
427 | 426 | eqcomd 2616 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) = ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1))) |
428 | 223 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℂ) |
429 | | 1cnd 9935 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 1 ∈ ℂ) |
430 | 219 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℂ) |
431 | 428, 429,
430 | addsubd 10292 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) = ((𝑖 − 𝑘) + 1)) |
432 | 431 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 − 𝑘) + 1) = ((𝑖 + 1) − 𝑘)) |
433 | 432 | fveq2d 6107 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
434 | 433 | adantll 746 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
435 | 342 | eqcomd 2616 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘))) |
436 | 344 | feqmptd 6159 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
437 | 434, 435,
436 | 3eqtrd 2648 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
438 | 421, 427,
437 | 3eqtrd 2648 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
439 | 391, 346,
383, 417, 384, 345, 438 | dvmptmul 23530 |
. . . . . . . . . . . . 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))))) |
440 | 391, 302,
392, 394, 395, 387, 439 | dvmptmul 23530 |
. . . . . . . . . . . 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))))) |
441 | 395 | mul02d 10113 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = 0) |
442 | 348 | oveq1d 6564 |
. . . . . . . . . . . . . . . 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))) |
443 | 388, 302 | mulcomd 9940 |
. . . . . . . . . . . . . . . 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))))) |
444 | 442, 443 | eqtrd 2644 |
. . . . . . . . . . . . . . 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))))) |
445 | 441, 444 | oveq12d 6567 |
. . . . . . . . . . . . . 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)))))) |
446 | 389 | addid2d 10116 |
. . . . . . . . . . . . . 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))))) |
447 | 445, 446 | eqtrd 2644 |
. . . . . . . . . . . . 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))))) |
448 | 447 | mpteq2dva 4672 |
. . . . . . . . . . . 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)))))) |
449 | 440, 448 | eqtrd 2644 |
. . . . . . . . . . 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)))))) |
450 | 200, 201,
202, 203, 204, 301, 390, 449 | dvmptfsum 23542 |
. . . . . . . . . 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)))))) |
451 | 210 | adantlr 747 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
452 | 385 | an32s 842 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ) |
453 | | anass 679 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋))) |
454 | | ancom 465 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖))) |
455 | 454 | anbi2i 726 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋)) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖)))) |
456 | | anass 679 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖)))) |
457 | 456 | bicomi 213 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖))) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖))) |
458 | 455, 457 | bitri 263 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋)) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖))) |
459 | 453, 458 | bitri 263 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖))) |
460 | 459 | imbi1i 338 |
. . . . . . . . . . . . . . . 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)) |
461 | 346, 460 | mpbi 219 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
462 | 459 | imbi1i 338 |
. . . . . . . . . . . . . . . 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)) |
463 | 345, 462 | mpbi 219 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) |
464 | 461, 463 | mulcld 9939 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ) |
465 | 451, 452,
464 | adddid 9943 |
. . . . . . . . . . . . 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))))) |
466 | 465 | sumeq2dv 14281 |
. . . . . . . . . . . 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))))) |
467 | 204 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (0...𝑖) ∈ Fin) |
468 | 451, 452 | mulcld 9939 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ) |
469 | 451, 464 | mulcld 9939 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
470 | 467, 468,
469 | fsumadd 14317 |
. . . . . . . . . . . 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))))) |
471 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = ℎ → (𝑖C𝑘) = (𝑖Cℎ)) |
472 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = ℎ → (𝑘 + 1) = (ℎ + 1)) |
473 | 472 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = ℎ → (𝐶‘(𝑘 + 1)) = (𝐶‘(ℎ + 1))) |
474 | 473 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = ℎ → ((𝐶‘(𝑘 + 1))‘𝑥) = ((𝐶‘(ℎ + 1))‘𝑥)) |
475 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = ℎ → (𝑖 − 𝑘) = (𝑖 − ℎ)) |
476 | 475 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = ℎ → (𝐷‘(𝑖 − 𝑘)) = (𝐷‘(𝑖 − ℎ))) |
477 | 476 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = ℎ → ((𝐷‘(𝑖 − 𝑘))‘𝑥) = ((𝐷‘(𝑖 − ℎ))‘𝑥)) |
478 | 474, 477 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = ℎ → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) = (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) |
479 | 471, 478 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = ℎ → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)))) |
480 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎℎ(0...𝑖) |
481 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘(0...𝑖) |
482 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎℎ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) |
483 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(𝑖Cℎ) |
484 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘
· |
485 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘(ℎ + 1) |
486 | 370, 485 | nffv 6110 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘(𝐶‘(ℎ + 1)) |
487 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘𝑥 |
488 | 486, 487 | nffv 6110 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝐶‘(ℎ + 1))‘𝑥) |
489 | | nfmpt1 4675 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) |
490 | 147, 489 | nfcxfr 2749 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘𝐷 |
491 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘(𝑖 − ℎ) |
492 | 490, 491 | nffv 6110 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘(𝐷‘(𝑖 − ℎ)) |
493 | 492, 487 | nffv 6110 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝐷‘(𝑖 − ℎ))‘𝑥) |
494 | 488, 484,
493 | nfov 6575 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)) |
495 | 483, 484,
494 | nfov 6575 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) |
496 | 479, 480,
481, 482, 495 | cbvsum 14273 |
. . . . . . . . . . . . . . . . 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))) |
497 | 496 | 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)))) |
498 | | 1zzd 11285 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℤ) |
499 | 95 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℤ) |
500 | 245 | ad2antlr 759 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 𝑖 ∈ ℤ) |
501 | | nfv 1830 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) |
502 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘ℎ |
503 | 502, 481 | nfel 2763 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘 ℎ ∈ (0...𝑖) |
504 | 501, 503 | nfan 1816 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) |
505 | 495, 374 | nfel 2763 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ |
506 | 504, 505 | nfim 1813 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ) |
507 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = ℎ → (𝑘 ∈ (0...𝑖) ↔ ℎ ∈ (0...𝑖))) |
508 | 507 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = ℎ → ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)))) |
509 | 479 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = ℎ → (((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ ↔ ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ)) |
510 | 508, 509 | imbi12d 333 |
. . . . . . . . . . . . . . . . . 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))) |
511 | 506, 510,
468 | chvar 2250 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ) |
512 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . 18
⊢ (ℎ = (𝑗 − 1) → (𝑖Cℎ) = (𝑖C(𝑗 − 1))) |
513 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (ℎ = (𝑗 − 1) → (ℎ + 1) = ((𝑗 − 1) + 1)) |
514 | 513 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℎ = (𝑗 − 1) → (𝐶‘(ℎ + 1)) = (𝐶‘((𝑗 − 1) + 1))) |
515 | 514 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℎ = (𝑗 − 1) → ((𝐶‘(ℎ + 1))‘𝑥) = ((𝐶‘((𝑗 − 1) + 1))‘𝑥)) |
516 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (ℎ = (𝑗 − 1) → (𝑖 − ℎ) = (𝑖 − (𝑗 − 1))) |
517 | 516 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℎ = (𝑗 − 1) → (𝐷‘(𝑖 − ℎ)) = (𝐷‘(𝑖 − (𝑗 − 1)))) |
518 | 517 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℎ = (𝑗 − 1) → ((𝐷‘(𝑖 − ℎ))‘𝑥) = ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)) |
519 | 515, 518 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . 18
⊢ (ℎ = (𝑗 − 1) → (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)) = (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) |
520 | 512, 519 | oveq12d 6567 |
. . . . . . . . . . . . . . . . 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)))) |
521 | 498, 499,
500, 511, 520 | fsumshft 14354 |
. . . . . . . . . . . . . . . 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)))) |
522 | 497, 521 | eqtrd 2644 |
. . . . . . . . . . . . . . 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)))) |
523 | | 0p1e1 11009 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 + 1) =
1 |
524 | 523 | oveq1i 6559 |
. . . . . . . . . . . . . . . . 17
⊢ ((0 +
1)...(𝑖 + 1)) = (1...(𝑖 + 1)) |
525 | 524 | sumeq1i 14276 |
. . . . . . . . . . . . . . . 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))) |
526 | 525 | 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)))) |
527 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℤ) |
528 | 527 | zcnd 11359 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℂ) |
529 | | 1cnd 9935 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈
ℂ) |
530 | 528, 529 | npcand 10275 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → ((𝑗 − 1) + 1) = 𝑗) |
531 | 530 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → (𝐶‘((𝑗 − 1) + 1)) = (𝐶‘𝑗)) |
532 | 531 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶‘𝑗)‘𝑥)) |
533 | 532 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶‘𝑗)‘𝑥)) |
534 | 222 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℂ) |
535 | 534 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℂ) |
536 | 528 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℂ) |
537 | 529 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈
ℂ) |
538 | 535, 536,
537 | subsub3d 10301 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 − (𝑗 − 1)) = ((𝑖 + 1) − 𝑗)) |
539 | 538 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘(𝑖 − (𝑗 − 1))) = (𝐷‘((𝑖 + 1) − 𝑗))) |
540 | 539 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) |
541 | 533, 540 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . 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))) |
542 | 541 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . 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)))) |
543 | 542 | sumeq2dv 14281 |
. . . . . . . . . . . . . . . . 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)))) |
544 | 543 | ad2antlr 759 |
. . . . . . . . . . . . . . . 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)))) |
545 | | nfv 1830 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑗((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) |
546 | | nfcv 2751 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑗((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) |
547 | | fzfid 12634 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...(𝑖 + 1)) ∈ Fin) |
548 | 193 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℕ0) |
549 | 527 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℤ) |
550 | | 1zzd 11285 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈
ℤ) |
551 | 549, 550 | zsubcld 11363 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑗 − 1) ∈ ℤ) |
552 | 548, 551 | bccld 38472 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈
ℕ0) |
553 | 552 | nn0cnd 11230 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
554 | 553 | adantll 746 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
555 | 554 | adantlr 747 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
556 | 1 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝜑) |
557 | | 0zd 11266 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ∈
ℤ) |
558 | 214 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℤ) |
559 | 557, 558,
549 | 3jca 1235 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈
ℤ)) |
560 | 179 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 ∈
ℝ) |
561 | 527 | zred 11358 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ) |
562 | | 1red 9934 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈
ℝ) |
563 | | 0lt1 10429 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ 0 <
1 |
564 | 563 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 1) |
565 | | elfzle1 12215 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 1 ≤ 𝑗) |
566 | 560, 562,
561, 564, 565 | ltletrd 10076 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 𝑗) |
567 | 560, 561,
566 | ltled 10064 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 ≤ 𝑗) |
568 | 567 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ 𝑗) |
569 | 561 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ) |
570 | 222 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℝ) |
571 | | 1red 9934 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈
ℝ) |
572 | 570, 571 | readdcld 9948 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ) |
573 | 220 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℝ) |
574 | | elfzle2 12216 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ≤ (𝑖 + 1)) |
575 | 574 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ (𝑖 + 1)) |
576 | 320 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁) |
577 | 569, 572,
573, 575, 576 | letrd 10073 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ 𝑁) |
578 | 559, 568,
577 | jca32 556 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤
𝑗 ∧ 𝑗 ≤ 𝑁))) |
579 | | elfz2 12204 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤
𝑗 ∧ 𝑗 ≤ 𝑁))) |
580 | 578, 579 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁)) |
581 | 580 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁)) |
582 | 556, 581,
380 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶‘𝑗):𝑋⟶ℂ) |
583 | 582 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶‘𝑗):𝑋⟶ℂ) |
584 | | simplr 788 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑥 ∈ 𝑋) |
585 | 583, 584 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘𝑗)‘𝑥) ∈ ℂ) |
586 | 245 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℤ) |
587 | 586 | peano2zd 11361 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ) |
588 | 587, 549 | zsubcld 11363 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℤ) |
589 | 557, 558,
588 | 3jca 1235 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ)) |
590 | 572, 569 | subge0d 10496 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑗) ↔ 𝑗 ≤ (𝑖 + 1))) |
591 | 575, 590 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑗)) |
592 | 572, 569 | resubcld 10337 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℝ) |
593 | 592 | leidd 10473 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ ((𝑖 + 1) − 𝑗)) |
594 | 561, 566 | elrpd 11745 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ+) |
595 | 594 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ+) |
596 | 572, 595 | ltsubrpd 11780 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < (𝑖 + 1)) |
597 | 592, 572,
573, 596, 576 | ltletrd 10076 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁) |
598 | 592, 592,
573, 593, 597 | lelttrd 10074 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁) |
599 | 592, 573,
598 | ltled 10064 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ 𝑁) |
600 | 589, 591,
599 | jca32 556 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑗) ∧ ((𝑖 + 1) − 𝑗) ≤ 𝑁))) |
601 | | elfz2 12204 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑖 + 1) − 𝑗) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑗) ∧ ((𝑖 + 1) − 𝑗) ≤ 𝑁))) |
602 | 600, 601 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) |
603 | 602 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) |
604 | | nfv 1830 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) |
605 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑘((𝑖 + 1) − 𝑗) |
606 | 490, 605 | nffv 6110 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝐷‘((𝑖 + 1) − 𝑗)) |
607 | 606, 373,
374 | nff 5954 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ |
608 | 604, 607 | nfim 1813 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ) |
609 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 + 1) − 𝑗) ∈ V |
610 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → (𝑘 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))) |
611 | 610 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)))) |
612 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → (𝐷‘𝑘) = (𝐷‘((𝑖 + 1) − 𝑗))) |
613 | 612 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)) |
614 | 611, 613 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . 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))) |
615 | 147 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))) |
616 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑆 D𝑛 𝐺)‘𝑘) ∈ V |
617 | 616 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘) ∈ V) |
618 | 615, 617 | fvmpt2d 6202 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑘)) |
619 | 618 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ)) |
620 | 285, 619 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) |
621 | 608, 609,
614, 620 | vtoclf 3231 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ) |
622 | 556, 603,
621 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ) |
623 | 622 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ) |
624 | 623, 584 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ) |
625 | 585, 624 | mulcld 9939 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ) |
626 | 555, 625 | mulcld 9939 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ) |
627 | | 1zzd 11285 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 1 ∈ ℤ) |
628 | 245 | peano2zd 11361 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℤ) |
629 | 523 | eqcomi 2619 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 1 = (0 +
1) |
630 | 629 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → 1 = (0 + 1)) |
631 | 179 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 0 ∈ ℝ) |
632 | | 1red 9934 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 1 ∈ ℝ) |
633 | 193 | nn0ge0d 11231 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 0 ≤ 𝑖) |
634 | 631, 222,
632, 633 | leadd1dd 10520 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → (0 + 1) ≤ (𝑖 + 1)) |
635 | 630, 634 | eqbrtrd 4605 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 1 ≤ (𝑖 + 1)) |
636 | 627, 628,
635 | 3jca 1235 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1
≤ (𝑖 +
1))) |
637 | | eluz2 11569 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 + 1) ∈
(ℤ≥‘1) ↔ (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1
≤ (𝑖 +
1))) |
638 | 636, 637 | sylibr 223 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈
(ℤ≥‘1)) |
639 | | eluzfz2 12220 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 + 1) ∈
(ℤ≥‘1) → (𝑖 + 1) ∈ (1...(𝑖 + 1))) |
640 | 638, 639 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (1...(𝑖 + 1))) |
641 | 640 | ad2antlr 759 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝑖 + 1) ∈ (1...(𝑖 + 1))) |
642 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 + 1) → (𝑗 − 1) = ((𝑖 + 1) − 1)) |
643 | 642 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑖 + 1) → (𝑖C(𝑗 − 1)) = (𝑖C((𝑖 + 1) − 1))) |
644 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑖 + 1) → (𝐶‘𝑗) = (𝐶‘(𝑖 + 1))) |
645 | 644 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 + 1) → ((𝐶‘𝑗)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥)) |
646 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = (𝑖 + 1) → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − (𝑖 + 1))) |
647 | 646 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1)))) |
648 | 647 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) |
649 | 645, 648 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑖 + 1) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) |
650 | 643, 649 | oveq12d 6567 |
. . . . . . . . . . . . . . . . 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)))) |
651 | 545, 546,
547, 626, 641, 650 | fsumsplit1 38639 |
. . . . . . . . . . . . . . . 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))))) |
652 | | 1cnd 9935 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 1 ∈ ℂ) |
653 | 534, 652 | pncand 10272 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 1) = 𝑖) |
654 | 653 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = (𝑖C𝑖)) |
655 | | bcnn 12961 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ ℕ0
→ (𝑖C𝑖) = 1) |
656 | 193, 655 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C𝑖) = 1) |
657 | 654, 656 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = 1) |
658 | 534, 652 | addcld 9938 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℂ) |
659 | 658 | subidd 10259 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − (𝑖 + 1)) = 0) |
660 | 659 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0)) |
661 | 660 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) = ((𝐷‘0)‘𝑥)) |
662 | 661 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) |
663 | 657, 662 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . 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)))) |
664 | 663 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . 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)))) |
665 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝜑) |
666 | | fzofzp1 12431 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁)) |
667 | 666 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁)) |
668 | | nfv 1830 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) |
669 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑘(𝑖 + 1) |
670 | 370, 669 | nffv 6110 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝐶‘(𝑖 + 1)) |
671 | 670, 373,
374 | nff 5954 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐶‘(𝑖 + 1)):𝑋⟶ℂ |
672 | 668, 671 | nfim 1813 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ) |
673 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 + 1) ∈ V |
674 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = (𝑖 + 1) → (𝑘 ∈ (0...𝑁) ↔ (𝑖 + 1) ∈ (0...𝑁))) |
675 | 674 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = (𝑖 + 1) → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)))) |
676 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = (𝑖 + 1) → (𝐶‘𝑘) = (𝐶‘(𝑖 + 1))) |
677 | 676 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = (𝑖 + 1) → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)) |
678 | 675, 677 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = (𝑖 + 1) → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ))) |
679 | 672, 673,
678, 240 | vtoclf 3231 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ) |
680 | 665, 667,
679 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ) |
681 | 680 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘(𝑖 + 1))‘𝑥) ∈ ℂ) |
682 | | nfv 1830 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝜑 ∧ 0 ∈ (0...𝑁)) |
683 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
Ⅎ𝑘0 |
684 | 490, 683 | nffv 6110 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑘(𝐷‘0) |
685 | 684, 373,
374 | nff 5954 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝐷‘0):𝑋⟶ℂ |
686 | 682, 685 | nfim 1813 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ) |
687 | | c0ex 9913 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 0 ∈
V |
688 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 = 0 → (𝑘 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁))) |
689 | 688 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = 0 → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁)))) |
690 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 = 0 → (𝐷‘𝑘) = (𝐷‘0)) |
691 | 690 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = 0 → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ)) |
692 | 689, 691 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = 0 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ))) |
693 | 686, 687,
692, 620 | vtoclf 3231 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ) |
694 | 1, 119, 693 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝐷‘0):𝑋⟶ℂ) |
695 | 694 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘0):𝑋⟶ℂ) |
696 | 695 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘0)‘𝑥) ∈ ℂ) |
697 | 681, 696 | mulcld 9939 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) ∈ ℂ) |
698 | 697 | mulid2d 9937 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) |
699 | 664, 698 | eqtrd 2644 |
. . . . . . . . . . . . . . . . 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))) |
700 | | 1m1e0 10966 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (1
− 1) = 0 |
701 | 700 | fveq2i 6106 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(ℤ≥‘(1 − 1)) =
(ℤ≥‘0) |
702 | 3 | eqcomi 2619 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(ℤ≥‘0) = ℕ0 |
703 | 701, 702 | eqtr2i 2633 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
ℕ0 = (ℤ≥‘(1 −
1)) |
704 | 703 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → ℕ0 =
(ℤ≥‘(1 − 1))) |
705 | 193, 704 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ≥‘(1
− 1))) |
706 | | fzdifsuc2 38466 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈
(ℤ≥‘(1 − 1)) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})) |
707 | 705, 706 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})) |
708 | 707 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}) = (1...𝑖)) |
709 | 708 | sumeq1d 14279 |
. . . . . . . . . . . . . . . . . 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)))) |
710 | 709 | ad2antlr 759 |
. . . . . . . . . . . . . . . . 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)))) |
711 | 699, 710 | oveq12d 6567 |
. . . . . . . . . . . . . . . 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))))) |
712 | 544, 651,
711 | 3eqtrd 2648 |
. . . . . . . . . . . . . . 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))))) |
713 | 522, 526,
712 | 3eqtrd 2648 |
. . . . . . . . . . . . . 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))))) |
714 | | nfcv 2751 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(𝑖C0) |
715 | 370, 683 | nffv 6110 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(𝐶‘0) |
716 | 715, 487 | nffv 6110 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝐶‘0)‘𝑥) |
717 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝑖 + 1) − 0) |
718 | 490, 717 | nffv 6110 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(𝐷‘((𝑖 + 1) − 0)) |
719 | 718, 487 | nffv 6110 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝐷‘((𝑖 + 1) − 0))‘𝑥) |
720 | 716, 484,
719 | nfov 6575 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) |
721 | 714, 484,
720 | nfov 6575 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) |
722 | 702 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → (ℤ≥‘0) =
ℕ0) |
723 | 193, 722 | eleqtrrd 2691 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈
(ℤ≥‘0)) |
724 | | eluzfz1 12219 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈
(ℤ≥‘0) → 0 ∈ (0...𝑖)) |
725 | 723, 724 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (0..^𝑁) → 0 ∈ (0...𝑖)) |
726 | 725 | ad2antlr 759 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ (0...𝑖)) |
727 | | oveq2 6557 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 → (𝑖C𝑘) = (𝑖C0)) |
728 | 109 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 0 → ((𝐶‘𝑘)‘𝑥) = ((𝐶‘0)‘𝑥)) |
729 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 0 → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − 0)) |
730 | 729 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 0))) |
731 | 730 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 0 → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) |
732 | 728, 731 | oveq12d 6567 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) |
733 | 727, 732 | oveq12d 6567 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))) |
734 | 501, 721,
467, 469, 726, 733 | fsumsplit1 38639 |
. . . . . . . . . . . . . . 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))))) |
735 | 658 | subid1d 10260 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 0) = (𝑖 + 1)) |
736 | 735 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − 0)) = (𝐷‘(𝑖 + 1))) |
737 | 736 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − 0))‘𝑥) = ((𝐷‘(𝑖 + 1))‘𝑥)) |
738 | 737 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) |
739 | 738 | oveq2d 6565 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))) |
740 | 739 | oveq1d 6564 |
. . . . . . . . . . . . . . . 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))))) |
741 | 740 | ad2antlr 759 |
. . . . . . . . . . . . . . 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))))) |
742 | | bcn0 12959 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ ℕ0
→ (𝑖C0) =
1) |
743 | 193, 742 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C0) = 1) |
744 | 743 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))) |
745 | 744 | ad2antlr 759 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))) |
746 | 715, 373,
374 | nff 5954 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐶‘0):𝑋⟶ℂ |
747 | 682, 746 | nfim 1813 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ) |
748 | 109 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = 0 → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘0):𝑋⟶ℂ)) |
749 | 689, 748 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = 0 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ))) |
750 | 747, 687,
749, 240 | vtoclf 3231 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ) |
751 | 1, 119, 750 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐶‘0):𝑋⟶ℂ) |
752 | 751 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐶‘0):𝑋⟶ℂ) |
753 | 752 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘0)‘𝑥) ∈ ℂ) |
754 | 490, 669 | nffv 6110 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐷‘(𝑖 + 1)) |
755 | 754, 373,
374 | nff 5954 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘(𝐷‘(𝑖 + 1)):𝑋⟶ℂ |
756 | 668, 755 | nfim 1813 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ) |
757 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = (𝑖 + 1) → (𝐷‘𝑘) = (𝐷‘(𝑖 + 1))) |
758 | 757 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = (𝑖 + 1) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)) |
759 | 675, 758 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = (𝑖 + 1) → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ))) |
760 | 756, 673,
759, 620 | vtoclf 3231 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ) |
761 | 665, 667,
760 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ) |
762 | 761 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 + 1))‘𝑥) ∈ ℂ) |
763 | 753, 762 | mulcld 9939 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) ∈ ℂ) |
764 | 763 | mulid2d 9937 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) |
765 | 745, 764 | eqtrd 2644 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) |
766 | | nfv 1830 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑗 𝑖 ∈ (0..^𝑁) |
767 | | 1zzd 11285 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ∈
ℤ) |
768 | 245 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑖 ∈ ℤ) |
769 | | eldifi 3694 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (0...𝑖)) |
770 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ) |
771 | 769, 770 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℤ) |
772 | 771 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ ℤ) |
773 | 767, 768,
772 | 3jca 1235 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → (1 ∈ ℤ
∧ 𝑖 ∈ ℤ
∧ 𝑗 ∈
ℤ)) |
774 | | elfznn0 12302 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℕ0) |
775 | 769, 774 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ0) |
776 | | eldifsni 4261 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≠ 0) |
777 | 775, 776 | jca 553 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → (𝑗 ∈ ℕ0 ∧ 𝑗 ≠ 0)) |
778 | | elnnne0 11183 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ℕ ↔ (𝑗 ∈ ℕ0
∧ 𝑗 ≠
0)) |
779 | 777, 778 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ) |
780 | | nnge1 10923 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ℕ → 1 ≤
𝑗) |
781 | 779, 780 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 1 ≤ 𝑗) |
782 | 781 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ≤ 𝑗) |
783 | | elfzle2 12216 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ≤ 𝑖) |
784 | 769, 783 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≤ 𝑖) |
785 | 784 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ≤ 𝑖) |
786 | 773, 782,
785 | jca32 556 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → ((1 ∈ ℤ
∧ 𝑖 ∈ ℤ
∧ 𝑗 ∈ ℤ)
∧ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑖))) |
787 | | elfz2 12204 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ (1...𝑖) ↔ ((1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (1 ≤
𝑗 ∧ 𝑗 ≤ 𝑖))) |
788 | 786, 787 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ (1...𝑖)) |
789 | 788 | ex 449 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (1...𝑖))) |
790 | | 0zd 11266 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 0 ∈ ℤ) |
791 | | elfzel2 12211 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 𝑖 ∈ ℤ) |
792 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℤ) |
793 | 790, 791,
792 | 3jca 1235 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...𝑖) → (0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈
ℤ)) |
794 | 179 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 0 ∈ ℝ) |
795 | 792 | zred 11358 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℝ) |
796 | | 1red 9934 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (1...𝑖) → 1 ∈ ℝ) |
797 | 563 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (1...𝑖) → 0 < 1) |
798 | | elfzle1 12215 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (1...𝑖) → 1 ≤ 𝑗) |
799 | 794, 796,
795, 797, 798 | ltletrd 10076 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 0 < 𝑗) |
800 | 794, 795,
799 | ltled 10064 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...𝑖) → 0 ≤ 𝑗) |
801 | | elfzle2 12216 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ≤ 𝑖) |
802 | 793, 800,
801 | jca32 556 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...𝑖) → ((0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤
𝑗 ∧ 𝑗 ≤ 𝑖))) |
803 | | elfz2 12204 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (0...𝑖) ↔ ((0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤
𝑗 ∧ 𝑗 ≤ 𝑖))) |
804 | 802, 803 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (0...𝑖)) |
805 | 794, 799 | gtned 10051 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ≠ 0) |
806 | | nelsn 4159 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ≠ 0 → ¬ 𝑗 ∈ {0}) |
807 | 805, 806 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (1...𝑖) → ¬ 𝑗 ∈ {0}) |
808 | 804, 807 | eldifd 3551 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0})) |
809 | 808 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ((0...𝑖) ∖ {0})) |
810 | 809 | ex 449 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0}))) |
811 | 789, 810 | impbid 201 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖))) |
812 | 766, 811 | alrimi 2069 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖))) |
813 | | dfcleq 2604 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((0...𝑖) ∖
{0}) = (1...𝑖) ↔
∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖))) |
814 | 812, 813 | sylibr 223 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → ((0...𝑖) ∖ {0}) = (1...𝑖)) |
815 | 814 | sumeq1d 14279 |
. . . . . . . . . . . . . . . . 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)))) |
816 | 815 | ad2antlr 759 |
. . . . . . . . . . . . . . . 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)))) |
817 | 765, 816 | oveq12d 6567 |
. . . . . . . . . . . . . . 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))))) |
818 | 734, 741,
817 | 3eqtrd 2648 |
. . . . . . . . . . . . . 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))))) |
819 | 713, 818 | oveq12d 6567 |
. . . . . . . . . . . . 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)))))) |
820 | | fzfid 12634 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...𝑖) ∈ Fin) |
821 | 193 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0) |
822 | 809, 771 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ℤ) |
823 | | 1zzd 11285 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 1 ∈ ℤ) |
824 | 822, 823 | zsubcld 11363 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑗 − 1) ∈ ℤ) |
825 | 821, 824 | bccld 38472 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈
ℕ0) |
826 | 825 | nn0cnd 11230 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
827 | 826 | adantll 746 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
828 | 827 | adantlr 747 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
829 | | simpl 472 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋)) |
830 | | fzelp1 12263 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (1...(𝑖 + 1))) |
831 | 830 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ (1...(𝑖 + 1))) |
832 | 829, 831,
585 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐶‘𝑗)‘𝑥) ∈ ℂ) |
833 | 831, 624 | syldan 486 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ) |
834 | 832, 833 | mulcld 9939 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ) |
835 | 828, 834 | mulcld 9939 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ) |
836 | 820, 835 | fsumcl 14311 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ) |
837 | 193 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0) |
838 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ ℤ) |
839 | 838 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ ℤ) |
840 | 837, 839 | bccld 38472 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈
ℕ0) |
841 | 840 | nn0cnd 11230 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
842 | 841 | adantll 746 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
843 | 842 | adantlr 747 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
844 | | simpll 786 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝜑 ∧ 𝑖 ∈ (0..^𝑁))) |
845 | | simplr 788 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑥 ∈ 𝑋) |
846 | 804 | ssriv 3572 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(1...𝑖) ⊆
(0...𝑖) |
847 | | id 22 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (1...𝑖)) |
848 | 846, 847 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (0...𝑖)) |
849 | 848 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ (0...𝑖)) |
850 | 844, 845,
849, 461 | syl21anc 1317 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
851 | 849, 463 | syldan 486 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) |
852 | 850, 851 | mulcld 9939 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ) |
853 | 843, 852 | mulcld 9939 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
854 | 820, 853 | fsumcl 14311 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
855 | 697, 836,
763, 854 | add4d 10143 |
. . . . . . . . . . . . 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)))))) |
856 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑘 → (𝑗 − 1) = (𝑘 − 1)) |
857 | 856 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑘 → (𝑖C(𝑗 − 1)) = (𝑖C(𝑘 − 1))) |
858 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 𝑘 → (𝐶‘𝑗) = (𝐶‘𝑘)) |
859 | 858 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑘 → ((𝐶‘𝑗)‘𝑥) = ((𝐶‘𝑘)‘𝑥)) |
860 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 = 𝑘 → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − 𝑘)) |
861 | 860 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 𝑘 → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − 𝑘))) |
862 | 861 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑘 → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) |
863 | 859, 862 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑘 → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
864 | 857, 863 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . 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)))) |
865 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(1...𝑖) |
866 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑗(1...𝑖) |
867 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘(𝑖C(𝑗 − 1)) |
868 | 372, 487 | nffv 6110 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘((𝐶‘𝑗)‘𝑥) |
869 | 606, 487 | nffv 6110 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) |
870 | 868, 484,
869 | nfov 6575 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘(((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) |
871 | 867, 484,
870 | nfov 6575 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) |
872 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑗((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
873 | 864, 865,
866, 871, 872 | cbvsum 14273 |
. . . . . . . . . . . . . . . . . 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))) |
874 | 873 | 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)))) |
875 | 874 | oveq1d 6564 |
. . . . . . . . . . . . . . . 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))))) |
876 | | peano2zm 11297 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ ℤ → (𝑘 − 1) ∈
ℤ) |
877 | 839, 876 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑘 − 1) ∈ ℤ) |
878 | 837, 877 | bccld 38472 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈
ℕ0) |
879 | 878 | nn0cnd 11230 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ) |
880 | 879 | adantll 746 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ) |
881 | 880 | adantlr 747 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ) |
882 | 881, 852 | mulcld 9939 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
883 | 820, 882,
853 | fsumadd 14317 |
. . . . . . . . . . . . . . . . 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))))) |
884 | 883 | eqcomd 2616 |
. . . . . . . . . . . . . . . 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))))) |
885 | 879, 841 | addcomd 10117 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) = ((𝑖C𝑘) + (𝑖C(𝑘 − 1)))) |
886 | | bcpasc 12970 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘)) |
887 | 837, 839,
886 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘)) |
888 | 885, 887 | eqtr2d 2645 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) = ((𝑖C(𝑘 − 1)) + (𝑖C𝑘))) |
889 | 888 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . 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)))) |
890 | 889 | adantll 746 |
. . . . . . . . . . . . . . . . . . 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)))) |
891 | 890 | adantlr 747 |
. . . . . . . . . . . . . . . . . 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)))) |
892 | 881, 843,
852 | adddird 9944 |
. . . . . . . . . . . . . . . . . 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))))) |
893 | 891, 892 | eqtr2d 2645 |
. . . . . . . . . . . . . . . . 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)))) |
894 | 893 | sumeq2dv 14281 |
. . . . . . . . . . . . . . . 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)))) |
895 | 875, 884,
894 | 3eqtrd 2648 |
. . . . . . . . . . . . . . 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)))) |
896 | 895 | oveq2d 6565 |
. . . . . . . . . . . . . 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))))) |
897 | | peano2nn0 11210 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ ℕ0
→ (𝑖 + 1) ∈
ℕ0) |
898 | 837, 897 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖 + 1) ∈
ℕ0) |
899 | 898, 839 | bccld 38472 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈
ℕ0) |
900 | 899 | nn0cnd 11230 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
901 | 900 | adantll 746 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
902 | 901 | adantlr 747 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
903 | 902, 852 | mulcld 9939 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
904 | 820, 903 | fsumcl 14311 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
905 | 697, 763,
904 | addassd 9941 |
. . . . . . . . . . . . . 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)))))) |
906 | 193, 897 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈
ℕ0) |
907 | | bcn0 12959 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 + 1) ∈ ℕ0
→ ((𝑖 + 1)C0) =
1) |
908 | 906, 907 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C0) = 1) |
909 | 908, 738 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))) |
910 | 909 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . 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)))) |
911 | 910, 764 | eqtr2d 2645 |
. . . . . . . . . . . . . . . . . 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)))) |
912 | 814 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((0...𝑖) ∖ {0}) = (1...𝑖)) |
913 | 912 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...𝑖) = ((0...𝑖) ∖ {0})) |
914 | 913 | sumeq1d 14279 |
. . . . . . . . . . . . . . . . . 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)))) |
915 | 911, 914 | oveq12d 6567 |
. . . . . . . . . . . . . . . . 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))))) |
916 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝑖 + 1)C0) |
917 | 916, 484,
720 | nfov 6575 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) |
918 | 205, 897 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈
ℕ0) |
919 | 918, 207 | bccld 38472 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈
ℕ0) |
920 | 919 | nn0cnd 11230 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
921 | 920 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
922 | 921 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
923 | 922, 464 | mulcld 9939 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
924 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 0 → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C0)) |
925 | 924, 732 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . 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)))) |
926 | 501, 917,
467, 923, 726, 925 | fsumsplit1 38639 |
. . . . . . . . . . . . . . . . . 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))))) |
927 | 926 | eqcomd 2616 |
. . . . . . . . . . . . . . . . 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)))) |
928 | 915, 927 | eqtrd 2644 |
. . . . . . . . . . . . . . . 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)))) |
929 | 928 | oveq2d 6565 |
. . . . . . . . . . . . . . 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))))) |
930 | | bcnn 12961 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 + 1) ∈ ℕ0
→ ((𝑖 + 1)C(𝑖 + 1)) = 1) |
931 | 906, 930 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C(𝑖 + 1)) = 1) |
932 | 931 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖 + 1)C(𝑖 + 1)) = 1) |
933 | 932 | oveq1d 6564 |
. . . . . . . . . . . . . . . . 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)))) |
934 | 660 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0)) |
935 | 934 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ)) |
936 | 695, 935 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ) |
937 | 936 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ) |
938 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
939 | 937, 938 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) ∈ ℂ) |
940 | 681, 939 | mulcld 9939 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) ∈ ℂ) |
941 | 940 | mulid2d 9937 |
. . . . . . . . . . . . . . . . 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))) |
942 | 662 | ad2antlr 759 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) |
943 | 933, 941,
942 | 3eqtrrd 2649 |
. . . . . . . . . . . . . . . 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)))) |
944 | | fzdifsuc 12270 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈
(ℤ≥‘0) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})) |
945 | 723, 944 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})) |
946 | 945 | sumeq1d 14279 |
. . . . . . . . . . . . . . . . 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)))) |
947 | 946 | ad2antlr 759 |
. . . . . . . . . . . . . . . 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)))) |
948 | 943, 947 | oveq12d 6567 |
. . . . . . . . . . . . . . 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))))) |
949 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝑖 + 1)C(𝑖 + 1)) |
950 | 670, 487 | nffv 6110 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝐶‘(𝑖 + 1))‘𝑥) |
951 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘((𝑖 + 1) − (𝑖 + 1)) |
952 | 490, 951 | nffv 6110 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘(𝐷‘((𝑖 + 1) − (𝑖 + 1))) |
953 | 952, 487 | nffv 6110 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) |
954 | 950, 484,
953 | nfov 6575 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘(((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) |
955 | 949, 484,
954 | nfov 6575 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) |
956 | | fzfid 12634 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (0...(𝑖 + 1)) ∈ Fin) |
957 | 906 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈
ℕ0) |
958 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ∈ ℤ) |
959 | 958 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℤ) |
960 | 957, 959 | bccld 38472 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈
ℕ0) |
961 | 960 | nn0cnd 11230 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
962 | 961 | adantll 746 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
963 | 962 | adantlr 747 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
964 | 665 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑) |
965 | 95 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈
ℤ) |
966 | 214 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℤ) |
967 | 965, 966,
959 | 3jca 1235 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈
ℤ)) |
968 | | elfzle1 12215 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 ∈ (0...(𝑖 + 1)) → 0 ≤ 𝑘) |
969 | 968 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ 𝑘) |
970 | 959 | zred 11358 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℝ) |
971 | 957 | nn0red 11229 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ) |
972 | 220 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℝ) |
973 | | elfzle2 12216 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ≤ (𝑖 + 1)) |
974 | 973 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ (𝑖 + 1)) |
975 | 320 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁) |
976 | 970, 971,
972, 974, 975 | letrd 10073 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ 𝑁) |
977 | 967, 969,
976 | jca32 556 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤
𝑘 ∧ 𝑘 ≤ 𝑁))) |
978 | 977, 231 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁)) |
979 | 978 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁)) |
980 | 964, 979,
240 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶‘𝑘):𝑋⟶ℂ) |
981 | 980 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶‘𝑘):𝑋⟶ℂ) |
982 | | simplr 788 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑥 ∈ 𝑋) |
983 | 981, 982 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
984 | 964 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑) |
985 | 628 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ) |
986 | 985, 959 | zsubcld 11363 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℤ) |
987 | 965, 966,
986 | 3jca 1235 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ)) |
988 | 971, 970 | subge0d 10496 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1))) |
989 | 974, 988 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑘)) |
990 | 971, 970 | resubcld 10337 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℝ) |
991 | 972, 970 | resubcld 10337 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 𝑘) ∈ ℝ) |
992 | 972, 179,
259 | sylancl 693 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) ∈ ℝ) |
993 | 971, 972,
970, 975 | lesub1dd 10522 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 𝑘)) |
994 | 179 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈
ℝ) |
995 | 994, 970,
972, 969 | lesub2dd 10523 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 𝑘) ≤ (𝑁 − 0)) |
996 | 990, 991,
992, 993, 995 | letrd 10073 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 0)) |
997 | 265 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) = 𝑁) |
998 | 996, 997 | breqtrd 4609 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ 𝑁) |
999 | 987, 989,
998 | jca32 556 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑘) ∧ ((𝑖 + 1) − 𝑘) ≤ 𝑁))) |
1000 | 999, 326 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
1001 | 1000 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
1002 | 1001 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
1003 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (𝐷‘𝑗) = (𝐷‘((𝑖 + 1) − 𝑘))) |
1004 | 1003 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝐷‘𝑗):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)) |
1005 | 331, 1004 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . 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))) |
1006 | 490, 371 | nffv 6110 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝐷‘𝑗) |
1007 | 1006, 373, 374 | nff 5954 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐷‘𝑗):𝑋⟶ℂ |
1008 | 368, 1007 | nfim 1813 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ) |
1009 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = 𝑗 → (𝐷‘𝑘) = (𝐷‘𝑗)) |
1010 | 1009 | feq1d 5943 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = 𝑗 → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘𝑗):𝑋⟶ℂ)) |
1011 | 281, 1010 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ))) |
1012 | 1008, 1011, 620 | chvar 2250 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ) |
1013 | 329, 1005, 1012 | vtocl 3232 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ) |
1014 | 984, 1002, 1013 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ) |
1015 | 1014, 982 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) |
1016 | 983, 1015 | mulcld 9939 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ) |
1017 | 963, 1016 | mulcld 9939 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
1018 | 906, 722 | eleqtrrd 2691 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈
(ℤ≥‘0)) |
1019 | | eluzfz2 12220 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 + 1) ∈
(ℤ≥‘0) → (𝑖 + 1) ∈ (0...(𝑖 + 1))) |
1020 | 1018, 1019 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...(𝑖 + 1))) |
1021 | 1020 | ad2antlr 759 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝑖 + 1) ∈ (0...(𝑖 + 1))) |
1022 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = (𝑖 + 1) → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C(𝑖 + 1))) |
1023 | 676 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = (𝑖 + 1) → ((𝐶‘𝑘)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥)) |
1024 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = (𝑖 + 1) → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − (𝑖 + 1))) |
1025 | 1024 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1)))) |
1026 | 1025 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) |
1027 | 1023, 1026 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = (𝑖 + 1) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) |
1028 | 1022, 1027 | oveq12d 6567 |
. . . . . . . . . . . . . . . . 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)))) |
1029 | 501, 955,
956, 1017, 1021, 1028 | fsumsplit1 38639 |
. . . . . . . . . . . . . . . 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))))) |
1030 | 1029 | eqcomd 2616 |
. . . . . . . . . . . . . . 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)))) |
1031 | 929, 948,
1030 | 3eqtrd 2648 |
. . . . . . . . . . . . . 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)))) |
1032 | 896, 905,
1031 | 3eqtrd 2648 |
. . . . . . . . . . . . 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)))) |
1033 | 819, 855,
1032 | 3eqtrd 2648 |
. . . . . . . . . . . 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)))) |
1034 | 466, 470,
1033 | 3eqtrd 2648 |
. . . . . . . . . . 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)))) |
1035 | 1034 | mpteq2dva 4672 |
. . . . . . . . . 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))))) |
1036 | 450, 1035 | eqtrd 2644 |
. . . . . . . . 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))))) |
1037 | 1036 | adantr 480 |
. . . . . . . 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))))) |
1038 | 197, 199,
1037 | 3eqtrd 2648 |
. . . . . . 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))))) |
1039 | 186, 187,
190, 1038 | syl21anc 1317 |
. . . . . 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))))) |
1040 | 1039 | 3exp 1256 |
. . . . 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))))))) |
1041 | 36, 50, 64, 78, 185, 1040 | fzind2 12448 |
. . . 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)))))) |
1042 | 22, 1041 | vtoclg 3239 |
. . 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))))))) |
1043 | 2, 6, 1042 | sylc 63 |
. 2
⊢ (𝜑 → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))) |
1044 | 1, 1043 | mpd 15 |
1
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))) |