Step | Hyp | Ref
| Expression |
1 | | id 22 |
. 2
⊢ (𝜑 → 𝜑) |
2 | | dvnmul.n |
. . 3
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
3 | | nn0uz 12004 |
. . . . 5
⊢
ℕ0 = (ℤ≥‘0) |
4 | 2, 3 | syl6eleq 2916 |
. . . 4
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘0)) |
5 | | eluzfz2 12642 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘0) → 𝑁 ∈ (0...𝑁)) |
6 | 4, 5 | syl 17 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) |
7 | | eleq1 2894 |
. . . . 5
⊢ (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁))) |
8 | | fveq2 6433 |
. . . . . . 7
⊢ (𝑛 = 𝑁 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁)) |
9 | | oveq2 6913 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁)) |
10 | 9 | sumeq1d 14808 |
. . . . . . . . 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 6912 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘)) |
12 | | fvoveq1 6928 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑁 → (𝐷‘(𝑛 − 𝑘)) = (𝐷‘(𝑁 − 𝑘))) |
13 | 12 | fveq1d 6435 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑁 → ((𝐷‘(𝑛 − 𝑘))‘𝑥) = ((𝐷‘(𝑁 − 𝑘))‘𝑥)) |
14 | 13 | oveq2d 6921 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑁 → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))) |
15 | 11, 14 | oveq12d 6923 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑁 → ((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = ((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))) |
16 | 15 | sumeq2ad 14811 |
. . . . . . . . 9
⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))) |
17 | 10, 16 | eqtrd 2861 |
. . . . . . . 8
⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))) |
18 | 17 | mpteq2dv 4968 |
. . . . . . 7
⊢ (𝑛 = 𝑁 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))) |
19 | 8, 18 | eqeq12d 2840 |
. . . . . 6
⊢ (𝑛 = 𝑁 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))) |
20 | 19 | imbi2d 332 |
. . . . 5
⊢ (𝑛 = 𝑁 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))))) |
21 | 7, 20 | imbi12d 336 |
. . . 4
⊢ (𝑛 = 𝑁 → ((𝑛 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))))) ↔ (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))))) |
22 | | fveq2 6433 |
. . . . . . 7
⊢ (𝑚 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0)) |
23 | | simpl 476 |
. . . . . . . . . 10
⊢ ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 0) |
24 | 23 | oveq2d 6921 |
. . . . . . . . 9
⊢ ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...0)) |
25 | | simpll 785 |
. . . . . . . . . . 11
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → 𝑚 = 0) |
26 | 25 | oveq1d 6920 |
. . . . . . . . . 10
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (𝑚C𝑘) = (0C𝑘)) |
27 | 25 | fvoveq1d 6927 |
. . . . . . . . . . . 12
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(0 − 𝑘))) |
28 | 27 | fveq1d 6435 |
. . . . . . . . . . 11
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(0 − 𝑘))‘𝑥)) |
29 | 28 | oveq2d 6921 |
. . . . . . . . . 10
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) |
30 | 26, 29 | oveq12d 6923 |
. . . . . . . . 9
⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))) |
31 | 24, 30 | sumeq12rdv 14815 |
. . . . . . . 8
⊢ ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))) |
32 | 31 | mpteq2dva 4967 |
. . . . . . 7
⊢ (𝑚 = 0 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))) |
33 | 22, 32 | eqeq12d 2840 |
. . . . . 6
⊢ (𝑚 = 0 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))) |
34 | 33 | imbi2d 332 |
. . . . 5
⊢ (𝑚 = 0 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))))) |
35 | | fveq2 6433 |
. . . . . . 7
⊢ (𝑚 = 𝑖 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)) |
36 | | simpl 476 |
. . . . . . . . . 10
⊢ ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 𝑖) |
37 | 36 | oveq2d 6921 |
. . . . . . . . 9
⊢ ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...𝑖)) |
38 | | simpll 785 |
. . . . . . . . . . 11
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → 𝑚 = 𝑖) |
39 | 38 | oveq1d 6920 |
. . . . . . . . . 10
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑚C𝑘) = (𝑖C𝑘)) |
40 | 38 | fvoveq1d 6927 |
. . . . . . . . . . . 12
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(𝑖 − 𝑘))) |
41 | 40 | fveq1d 6435 |
. . . . . . . . . . 11
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(𝑖 − 𝑘))‘𝑥)) |
42 | 41 | oveq2d 6921 |
. . . . . . . . . 10
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) |
43 | 39, 42 | oveq12d 6923 |
. . . . . . . . 9
⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))) |
44 | 37, 43 | sumeq12rdv 14815 |
. . . . . . . 8
⊢ ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))) |
45 | 44 | mpteq2dva 4967 |
. . . . . . 7
⊢ (𝑚 = 𝑖 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) |
46 | 35, 45 | eqeq12d 2840 |
. . . . . 6
⊢ (𝑚 = 𝑖 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))) |
47 | 46 | imbi2d 332 |
. . . . 5
⊢ (𝑚 = 𝑖 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))))) |
48 | | fveq2 6433 |
. . . . . . 7
⊢ (𝑚 = (𝑖 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1))) |
49 | | simpl 476 |
. . . . . . . . . 10
⊢ ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → 𝑚 = (𝑖 + 1)) |
50 | 49 | oveq2d 6921 |
. . . . . . . . 9
⊢ ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...(𝑖 + 1))) |
51 | | simpll 785 |
. . . . . . . . . . 11
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑚 = (𝑖 + 1)) |
52 | 51 | oveq1d 6920 |
. . . . . . . . . 10
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑚C𝑘) = ((𝑖 + 1)C𝑘)) |
53 | 51 | fvoveq1d 6927 |
. . . . . . . . . . . 12
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘))) |
54 | 53 | fveq1d 6435 |
. . . . . . . . . . 11
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) |
55 | 54 | oveq2d 6921 |
. . . . . . . . . 10
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
56 | 52, 55 | oveq12d 6923 |
. . . . . . . . 9
⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
57 | 50, 56 | sumeq12rdv 14815 |
. . . . . . . 8
⊢ ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) |
58 | 57 | mpteq2dva 4967 |
. . . . . . 7
⊢ (𝑚 = (𝑖 + 1) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) |
59 | 48, 58 | eqeq12d 2840 |
. . . . . 6
⊢ (𝑚 = (𝑖 + 1) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))) |
60 | 59 | imbi2d 332 |
. . . . 5
⊢ (𝑚 = (𝑖 + 1) → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))) |
61 | | fveq2 6433 |
. . . . . . 7
⊢ (𝑚 = 𝑛 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛)) |
62 | | simpl 476 |
. . . . . . . . . 10
⊢ ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 𝑛) |
63 | 62 | oveq2d 6921 |
. . . . . . . . 9
⊢ ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...𝑛)) |
64 | | simpll 785 |
. . . . . . . . . . 11
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → 𝑚 = 𝑛) |
65 | 64 | oveq1d 6920 |
. . . . . . . . . 10
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝑚C𝑘) = (𝑛C𝑘)) |
66 | 64 | fvoveq1d 6927 |
. . . . . . . . . . . 12
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(𝑛 − 𝑘))) |
67 | 66 | fveq1d 6435 |
. . . . . . . . . . 11
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(𝑛 − 𝑘))‘𝑥)) |
68 | 67 | oveq2d 6921 |
. . . . . . . . . 10
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) |
69 | 65, 68 | oveq12d 6923 |
. . . . . . . . 9
⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) |
70 | 63, 69 | sumeq12rdv 14815 |
. . . . . . . 8
⊢ ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) |
71 | 70 | mpteq2dva 4967 |
. . . . . . 7
⊢ (𝑚 = 𝑛 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))) |
72 | 61, 71 | eqeq12d 2840 |
. . . . . 6
⊢ (𝑚 = 𝑛 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))))) |
73 | 72 | imbi2d 332 |
. . . . 5
⊢ (𝑚 = 𝑛 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))))) |
74 | | dvnmul.s |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
75 | | recnprss 24067 |
. . . . . . . . 9
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
76 | 74, 75 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
77 | | dvnmul.a |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) |
78 | | dvnmul.cc |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) |
79 | 77, 78 | mulcld 10377 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 · 𝐵) ∈ ℂ) |
80 | | restsspw 16445 |
. . . . . . . . . . 11
⊢
((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆 |
81 | | dvnmul.x |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
82 | 80, 81 | sseldi 3825 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆) |
83 | | elpwi 4388 |
. . . . . . . . . 10
⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆) |
84 | 82, 83 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
85 | | cnex 10333 |
. . . . . . . . . 10
⊢ ℂ
∈ V |
86 | 85 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ℂ ∈
V) |
87 | 79, 84, 86, 74 | mptelpm 40166 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ
↑pm 𝑆)) |
88 | | dvn0 24086 |
. . . . . . . 8
⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ
↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))) |
89 | 76, 87, 88 | syl2anc 581 |
. . . . . . 7
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))) |
90 | | 0z 11715 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℤ |
91 | | fzsn 12676 |
. . . . . . . . . . . 12
⊢ (0 ∈
ℤ → (0...0) = {0}) |
92 | 90, 91 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (0...0) =
{0} |
93 | 92 | sumeq1i 14805 |
. . . . . . . . . 10
⊢
Σ𝑘 ∈
(0...0)((0C𝑘) ·
(((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) |
94 | 93 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))) |
95 | | nfcvd 2970 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Ⅎ𝑘(𝐴 · 𝐵)) |
96 | | nfv 2015 |
. . . . . . . . . 10
⊢
Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ 𝑋) |
97 | | oveq2 6913 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → (0C𝑘) = (0C0)) |
98 | | 0nn0 11635 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
ℕ0 |
99 | | bcn0 13390 |
. . . . . . . . . . . . . . . 16
⊢ (0 ∈
ℕ0 → (0C0) = 1) |
100 | 98, 99 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ (0C0) =
1 |
101 | 100 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → (0C0) =
1) |
102 | 97, 101 | eqtrd 2861 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 0 → (0C𝑘) = 1) |
103 | 102 | adantl 475 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (0C𝑘) = 1) |
104 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (𝐶‘𝑘) = (𝐶‘0)) |
105 | 104 | adantl 475 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = (𝐶‘0)) |
106 | | dvnmul.c |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) |
107 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐹)‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑛)) |
108 | 107 | cbvmptv 4973 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛)) |
109 | 106, 108 | eqtri 2849 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛)) |
110 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 0 → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘0)) |
111 | | eluzfz1 12641 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ≥‘0) → 0 ∈ (0...𝑁)) |
112 | 4, 111 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 0 ∈ (0...𝑁)) |
113 | | fvexd 6448 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘0) ∈ V) |
114 | 109, 110,
112, 113 | fvmptd3 6550 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0)) |
115 | 114 | adantr 474 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0)) |
116 | 105, 115 | eqtrd 2861 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘0)) |
117 | | dvnmulf |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 𝐴) |
118 | 77, 84, 86, 74 | mptelpm 40166 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm
𝑆)) |
119 | 117, 118 | syl5eqel 2910 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm
𝑆)) |
120 | | dvn0 24086 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ
↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹) |
121 | 76, 119, 120 | syl2anc 581 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹) |
122 | 121 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹) |
123 | 116, 122 | eqtrd 2861 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = 𝐹) |
124 | 123 | fveq1d 6435 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = (𝐹‘𝑥)) |
125 | 124 | adantlr 708 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = (𝐹‘𝑥)) |
126 | | simpr 479 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
127 | 117 | fvmpt2 6538 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ℂ) → (𝐹‘𝑥) = 𝐴) |
128 | 126, 77, 127 | syl2anc 581 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) = 𝐴) |
129 | 128 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (𝐹‘𝑥) = 𝐴) |
130 | 125, 129 | eqtrd 2861 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = 𝐴) |
131 | | oveq2 6913 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (0 − 𝑘) = (0 −
0)) |
132 | | 0m0e0 11478 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0
− 0) = 0 |
133 | 132 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (0 − 0) =
0) |
134 | 131, 133 | eqtrd 2861 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 0 → (0 − 𝑘) = 0) |
135 | 134 | fveq2d 6437 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 → (𝐷‘(0 − 𝑘)) = (𝐷‘0)) |
136 | 135 | fveq1d 6435 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥)) |
137 | 136 | adantl 475 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥)) |
138 | 137 | adantlr 708 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥)) |
139 | | dvnmul.d |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) |
140 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑛)) |
141 | 140 | cbvmptv 4973 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) |
142 | 139, 141 | eqtri 2849 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) |
143 | 142 | fveq1i 6434 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0) |
144 | 143 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0)) |
145 | | eqidd 2826 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))) |
146 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 0 → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0)) |
147 | 146 | adantl 475 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0)) |
148 | | dvnmul.f |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ 𝐵) |
149 | 78, 84, 86, 74 | mptelpm 40166 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (ℂ ↑pm
𝑆)) |
150 | 148, 149 | syl5eqel 2910 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝐺 ∈ (ℂ ↑pm
𝑆)) |
151 | | dvn0 24086 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ
↑pm 𝑆)) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺) |
152 | 76, 150, 151 | syl2anc 581 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑆 D𝑛 𝐺)‘0) = 𝐺) |
153 | 152 | adantr 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺) |
154 | 147, 153 | eqtrd 2861 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = 𝐺) |
155 | 148 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝑋 ↦ 𝐵)) |
156 | | mptexg 6740 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑋 ∈ 𝒫 𝑆 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ V) |
157 | 82, 156 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ V) |
158 | 155, 157 | eqeltrd 2906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐺 ∈ V) |
159 | 145, 154,
112, 158 | fvmptd 6535 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0) = 𝐺) |
160 | 144, 159 | eqtrd 2861 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐷‘0) = 𝐺) |
161 | 160 | fveq1d 6435 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐷‘0)‘𝑥) = (𝐺‘𝑥)) |
162 | 161 | ad2antrr 719 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘0)‘𝑥) = (𝐺‘𝑥)) |
163 | 155, 78 | fvmpt2d 6540 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) = 𝐵) |
164 | 163 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (𝐺‘𝑥) = 𝐵) |
165 | 138, 162,
164 | 3eqtrd 2865 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = 𝐵) |
166 | 130, 165 | oveq12d 6923 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)) = (𝐴 · 𝐵)) |
167 | 103, 166 | oveq12d 6923 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (1 · (𝐴 · 𝐵))) |
168 | 79 | mulid2d 10375 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵)) |
169 | 168 | adantr 474 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵)) |
170 | 167, 169 | eqtrd 2861 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵)) |
171 | | 0re 10358 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
172 | 171 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ) |
173 | 95, 96, 170, 172, 79 | sumsnd 40003 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵)) |
174 | 94, 173 | eqtr2d 2862 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))) |
175 | 174 | mpteq2dva 4967 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))) |
176 | 89, 175 | eqtrd 2861 |
. . . . . 6
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))) |
177 | 176 | a1i 11 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘0) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))) |
178 | | simp3 1174 |
. . . . . . 7
⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → 𝜑) |
179 | | simp1 1172 |
. . . . . . 7
⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → 𝑖 ∈ (0..^𝑁)) |
180 | | simp2 1173 |
. . . . . . . 8
⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))) |
181 | | pm3.35 839 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) |
182 | 178, 180,
181 | syl2anc 581 |
. . . . . . 7
⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) |
183 | 76 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆 ⊆ ℂ) |
184 | 87 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ
↑pm 𝑆)) |
185 | | elfzonn0 12808 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0) |
186 | 185 | adantl 475 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0) |
187 | | dvnp1 24087 |
. . . . . . . . . 10
⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ
↑pm 𝑆) ∧ 𝑖 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖))) |
188 | 183, 184,
186, 187 | syl3anc 1496 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖))) |
189 | 188 | adantr 474 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖))) |
190 | | simpr 479 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) |
191 | 190 | oveq2d 6921 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))) |
192 | | eqid 2825 |
. . . . . . . . . . 11
⊢
((TopOpen‘ℂfld) ↾t 𝑆) =
((TopOpen‘ℂfld) ↾t 𝑆) |
193 | | eqid 2825 |
. . . . . . . . . . 11
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
194 | 74 | adantr 474 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆 ∈ {ℝ, ℂ}) |
195 | 81 | adantr 474 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
196 | | fzfid 13067 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (0...𝑖) ∈ Fin) |
197 | 185 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℕ0) |
198 | | elfzelz 12635 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℤ) |
199 | 198 | adantl 475 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℤ) |
200 | 197, 199 | bccld 40328 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈
ℕ0) |
201 | 200 | nn0cnd 11680 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
202 | 201 | adantll 707 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
203 | 202 | 3adant3 1168 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝑖C𝑘) ∈ ℂ) |
204 | | simpll 785 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝜑) |
205 | | 0zd 11716 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℤ) |
206 | | elfzoel2 12764 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℤ) |
207 | 206 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℤ) |
208 | 205, 207,
199 | 3jca 1164 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈
ℤ)) |
209 | | elfzle1 12637 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (0...𝑖) → 0 ≤ 𝑘) |
210 | 209 | adantl 475 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ 𝑘) |
211 | 199 | zred 11810 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℝ) |
212 | 206 | zred 11810 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℝ) |
213 | 212 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℝ) |
214 | 185 | nn0red 11679 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℝ) |
215 | 214 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℝ) |
216 | | elfzle2 12638 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ≤ 𝑖) |
217 | 216 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ 𝑖) |
218 | | elfzolt2 12774 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 < 𝑁) |
219 | 218 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 < 𝑁) |
220 | 211, 215,
213, 217, 219 | lelttrd 10514 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 < 𝑁) |
221 | 211, 213,
220 | ltled 10504 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ 𝑁) |
222 | 208, 210,
221 | jca32 513 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤
𝑘 ∧ 𝑘 ≤ 𝑁))) |
223 | | elfz2 12626 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤
𝑘 ∧ 𝑘 ≤ 𝑁))) |
224 | 222, 223 | sylibr 226 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁)) |
225 | 224 | adantll 707 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁)) |
226 | | dvnmul.dvnf |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ) |
227 | 106 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))) |
228 | | fvexd 6448 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘) ∈ V) |
229 | 227, 228 | fvmpt2d 6540 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘)) |
230 | 229 | feq1d 6263 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐶‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)) |
231 | 226, 230 | mpbird 249 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) |
232 | 204, 225,
231 | syl2anc 581 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘):𝑋⟶ℂ) |
233 | 232 | 3adant3 1168 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝐶‘𝑘):𝑋⟶ℂ) |
234 | | simp3 1174 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
235 | 233, 234 | ffvelrnd 6609 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
236 | 185 | nn0zd 11808 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℤ) |
237 | 236 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℤ) |
238 | 237, 199 | zsubcld 11815 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ ℤ) |
239 | 205, 207,
238 | 3jca 1164 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖 − 𝑘) ∈ ℤ)) |
240 | | elfzel2 12633 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℤ) |
241 | 240 | zred 11810 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℝ) |
242 | 198 | zred 11810 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℝ) |
243 | 241, 242 | subge0d 10942 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ (0...𝑖) → (0 ≤ (𝑖 − 𝑘) ↔ 𝑘 ≤ 𝑖)) |
244 | 216, 243 | mpbird 249 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑖 − 𝑘)) |
245 | 244 | adantl 475 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑖 − 𝑘)) |
246 | 215, 211 | resubcld 10782 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ ℝ) |
247 | 213, 211 | resubcld 10782 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ∈ ℝ) |
248 | 171 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℝ) |
249 | 213, 248 | jca 509 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 ∈ ℝ ∧ 0 ∈
ℝ)) |
250 | | resubcl 10666 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑁 ∈ ℝ ∧ 0 ∈
ℝ) → (𝑁 −
0) ∈ ℝ) |
251 | 249, 250 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) ∈ ℝ) |
252 | 215, 213,
211, 219 | ltsub1dd 10964 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < (𝑁 − 𝑘)) |
253 | 248, 211,
213, 210 | lesub2dd 10969 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ≤ (𝑁 − 0)) |
254 | 246, 247,
251, 252, 253 | ltletrd 10516 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < (𝑁 − 0)) |
255 | 212 | recnd 10385 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℂ) |
256 | 255 | subid1d 10702 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → (𝑁 − 0) = 𝑁) |
257 | 256 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) = 𝑁) |
258 | 254, 257 | breqtrd 4899 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < 𝑁) |
259 | 246, 213,
258 | ltled 10504 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ≤ 𝑁) |
260 | 239, 245,
259 | jca32 513 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝑖 − 𝑘) ∧ (𝑖 − 𝑘) ≤ 𝑁))) |
261 | | elfz2 12626 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 − 𝑘) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝑖 − 𝑘) ∧ (𝑖 − 𝑘) ≤ 𝑁))) |
262 | 260, 261 | sylibr 226 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ (0...𝑁)) |
263 | 262 | adantll 707 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ (0...𝑁)) |
264 | | ovex 6937 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 − 𝑘) ∈ V |
265 | | eleq1 2894 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑖 − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ (𝑖 − 𝑘) ∈ (0...𝑁))) |
266 | 265 | anbi2d 624 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 − 𝑘) ∈ (0...𝑁)))) |
267 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑖 − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
268 | 267 | feq1d 6263 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ)) |
269 | 266, 268 | imbi12d 336 |
. . . . . . . . . . . . . . . . . 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))) |
270 | | nfv 2015 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) |
271 | | eleq1 2894 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑗 → (𝑘 ∈ (0...𝑁) ↔ 𝑗 ∈ (0...𝑁))) |
272 | 271 | anbi2d 624 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑗 ∈ (0...𝑁)))) |
273 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑗 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑗)) |
274 | 273 | feq1d 6263 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑗 → (((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)) |
275 | 272, 274 | imbi12d 336 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ))) |
276 | | dvnmul.dvng |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ) |
277 | 270, 275,
276 | chvar 2416 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) |
278 | 264, 269,
277 | vtocl 3475 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑖 − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ) |
279 | 204, 263,
278 | syl2anc 581 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ) |
280 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = (𝑖 − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
281 | | fvexd 6448 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)) ∈ V) |
282 | 142, 280,
262, 281 | fvmptd3 6550 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
283 | 282 | adantll 707 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
284 | 283 | feq1d 6263 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ)) |
285 | 279, 284 | mpbird 249 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ) |
286 | 285 | 3adant3 1168 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ) |
287 | 286, 234 | ffvelrnd 6609 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 − 𝑘))‘𝑥) ∈ ℂ) |
288 | 235, 287 | mulcld 10377 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ) |
289 | 203, 288 | mulcld 10377 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ) |
290 | 203 | 3expa 1153 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (𝑖C𝑘) ∈ ℂ) |
291 | 237 | peano2zd 11813 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℤ) |
292 | 291, 199 | zsubcld 11815 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℤ) |
293 | 205, 207,
292 | 3jca 1164 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ)) |
294 | | peano2re 10528 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑖 ∈ ℝ → (𝑖 + 1) ∈
ℝ) |
295 | 241, 294 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ (0...𝑖) → (𝑖 + 1) ∈ ℝ) |
296 | | peano2re 10528 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈
ℝ) |
297 | 242, 296 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ∈ ℝ) |
298 | 242 | ltp1d 11284 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑘 + 1)) |
299 | | 1red 10357 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑘 ∈ (0...𝑖) → 1 ∈ ℝ) |
300 | 242, 241,
299, 216 | leadd1dd 10966 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ≤ (𝑖 + 1)) |
301 | 242, 297,
295, 298, 300 | ltletrd 10516 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑖 + 1)) |
302 | 242, 295,
301 | ltled 10504 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ≤ (𝑖 + 1)) |
303 | 302 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ (𝑖 + 1)) |
304 | 215, 294 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℝ) |
305 | 304, 211 | subge0d 10942 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1))) |
306 | 303, 305 | mpbird 249 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ ((𝑖 + 1) − 𝑘)) |
307 | 304, 211 | resubcld 10782 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℝ) |
308 | | elfzop1le2 40301 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ≤ 𝑁) |
309 | 308 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ≤ 𝑁) |
310 | 304, 213,
211, 309 | lesub1dd 10968 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 𝑘)) |
311 | 253, 257 | breqtrd 4899 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ≤ 𝑁) |
312 | 307, 247,
213, 310, 311 | letrd 10513 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ 𝑁) |
313 | 293, 306,
312 | jca32 513 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑘) ∧ ((𝑖 + 1) − 𝑘) ≤ 𝑁))) |
314 | | elfz2 12626 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑖 + 1) − 𝑘) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑘) ∧ ((𝑖 + 1) − 𝑘) ≤ 𝑁))) |
315 | 313, 314 | sylibr 226 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
316 | 315 | adantll 707 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
317 | | ovex 6937 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 + 1) − 𝑘) ∈ V |
318 | | eleq1 2894 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))) |
319 | 318 | anbi2d 624 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)))) |
320 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
321 | 320 | feq1d 6263 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)) |
322 | 319, 321 | imbi12d 336 |
. . . . . . . . . . . . . . . . . . . 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))) |
323 | 317, 322,
277 | vtocl 3475 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ) |
324 | 204, 316,
323 | syl2anc 581 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ) |
325 | 142 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))) |
326 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → 𝑛 = ((𝑖 + 1) − 𝑘)) |
327 | 326 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
328 | | fvexd 6448 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) ∈ V) |
329 | 325, 327,
316, 328 | fvmptd 6535 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
330 | 329 | feq1d 6263 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)) |
331 | 324, 330 | mpbird 249 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ) |
332 | 331 | ffvelrnda 6608 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) |
333 | 235 | 3expa 1153 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
334 | 332, 333 | mulcomd 10378 |
. . . . . . . . . . . . . . 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))) |
335 | 334 | oveq2d 6921 |
. . . . . . . . . . . . . 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)))) |
336 | 199 | peano2zd 11813 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℤ) |
337 | 205, 207,
336 | 3jca 1164 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ∈
ℤ)) |
338 | 171 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...𝑖) → 0 ∈ ℝ) |
339 | 338, 242,
297, 209, 298 | lelttrd 10514 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...𝑖) → 0 < (𝑘 + 1)) |
340 | 338, 297,
339 | ltled 10504 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑘 + 1)) |
341 | 340 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑘 + 1)) |
342 | 211, 296 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℝ) |
343 | 300 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ (𝑖 + 1)) |
344 | 342, 304,
213, 343, 309 | letrd 10513 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ 𝑁) |
345 | 337, 341,
344 | jca32 513 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) ∧ (0
≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑁))) |
346 | | elfz2 12626 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 + 1) ∈ (0...𝑁) ↔ ((0 ∈ ℤ
∧ 𝑁 ∈ ℤ
∧ (𝑘 + 1) ∈
ℤ) ∧ (0 ≤ (𝑘 +
1) ∧ (𝑘 + 1) ≤ 𝑁))) |
347 | 345, 346 | sylibr 226 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁)) |
348 | 347 | adantll 707 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁)) |
349 | | ovex 6937 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 + 1) ∈ V |
350 | | eleq1 2894 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = (𝑘 + 1) → (𝑗 ∈ (0...𝑁) ↔ (𝑘 + 1) ∈ (0...𝑁))) |
351 | 350 | anbi2d 624 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑘 + 1) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)))) |
352 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = (𝑘 + 1) → (𝐶‘𝑗) = (𝐶‘(𝑘 + 1))) |
353 | 352 | feq1d 6263 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑘 + 1) → ((𝐶‘𝑗):𝑋⟶ℂ ↔ (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)) |
354 | 351, 353 | imbi12d 336 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑘 + 1) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ))) |
355 | | nfv 2015 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (0...𝑁)) |
356 | | nfmpt1 4970 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) |
357 | 106, 356 | nfcxfr 2967 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘𝐶 |
358 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘𝑗 |
359 | 357, 358 | nffv 6443 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘(𝐶‘𝑗) |
360 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘𝑋 |
361 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘ℂ |
362 | 359, 360,
361 | nff 6274 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘(𝐶‘𝑗):𝑋⟶ℂ |
363 | 355, 362 | nfim 2001 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ) |
364 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑗 → (𝐶‘𝑘) = (𝐶‘𝑗)) |
365 | 364 | feq1d 6263 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑗 → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘𝑗):𝑋⟶ℂ)) |
366 | 272, 365 | imbi12d 336 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ))) |
367 | 363, 366,
231 | chvar 2416 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ) |
368 | 349, 354,
367 | vtocl 3475 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ) |
369 | 204, 348,
368 | syl2anc 581 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ) |
370 | 369 | ffvelrnda 6608 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘(𝑘 + 1))‘𝑥) ∈ ℂ) |
371 | 287 | 3expa 1153 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 − 𝑘))‘𝑥) ∈ ℂ) |
372 | 370, 371 | mulcld 10377 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ) |
373 | 332, 333 | mulcld 10377 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥)) ∈ ℂ) |
374 | 372, 373 | addcld 10376 |
. . . . . . . . . . . . . 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) |
375 | 335, 374 | eqeltrrd 2907 |
. . . . . . . . . . . . 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) |
376 | 290, 375 | mulcld 10377 |
. . . . . . . . . . . 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) |
377 | 376 | 3impa 1142 |
. . . . . . . . . . 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) |
378 | 204, 74 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ∈ {ℝ, ℂ}) |
379 | 171 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ) |
380 | 204, 81 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
381 | 378, 380,
202 | dvmptconst 40924 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝑖C𝑘))) = (𝑥 ∈ 𝑋 ↦ 0)) |
382 | 288 | 3expa 1153 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ) |
383 | 204, 225,
229 | syl2anc 581 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘)) |
384 | 383 | eqcomd 2831 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘𝑘) = (𝐶‘𝑘)) |
385 | 232 | feqmptd 6496 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥))) |
386 | 384, 385 | eqtr2d 2862 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥)) = ((𝑆 D𝑛 𝐹)‘𝑘)) |
387 | 386 | oveq2d 6921 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘))) |
388 | 378, 75 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ⊆ ℂ) |
389 | 204, 119 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐹 ∈ (ℂ ↑pm
𝑆)) |
390 | | elfznn0 12727 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ0) |
391 | 390 | adantl 475 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ0) |
392 | | dvnp1 24087 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ
↑pm 𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘))) |
393 | 388, 389,
391, 392 | syl3anc 1496 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘))) |
394 | 393 | eqcomd 2831 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1))) |
395 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = (𝑘 + 1) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1))) |
396 | | fvexd 6448 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) ∈ V) |
397 | 109, 395,
348, 396 | fvmptd3 6550 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1))) |
398 | 397 | eqcomd 2831 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝐶‘(𝑘 + 1))) |
399 | 369 | feqmptd 6496 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥))) |
400 | 398, 399 | eqtrd 2861 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥))) |
401 | 387, 394,
400 | 3eqtrd 2865 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥))) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥))) |
402 | 283 | eqcomd 2831 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)) = (𝐷‘(𝑖 − 𝑘))) |
403 | 285 | feqmptd 6496 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥))) |
404 | 402, 403 | eqtr2d 2862 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) |
405 | 404 | oveq2d 6921 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))) |
406 | 204, 150 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐺 ∈ (ℂ ↑pm
𝑆)) |
407 | | fznn0sub 12666 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...𝑖) → (𝑖 − 𝑘) ∈
ℕ0) |
408 | 407 | adantl 475 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈
ℕ0) |
409 | | dvnp1 24087 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ
↑pm 𝑆) ∧ (𝑖 − 𝑘) ∈ ℕ0) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))) |
410 | 388, 406,
408, 409 | syl3anc 1496 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))) |
411 | 410 | eqcomd 2831 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) = ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1))) |
412 | 215 | recnd 10385 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℂ) |
413 | | 1cnd 10351 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 1 ∈ ℂ) |
414 | 211 | recnd 10385 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℂ) |
415 | 412, 413,
414 | addsubd 10734 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) = ((𝑖 − 𝑘) + 1)) |
416 | 415 | eqcomd 2831 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 − 𝑘) + 1) = ((𝑖 + 1) − 𝑘)) |
417 | 416 | fveq2d 6437 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
418 | 417 | adantll 707 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘))) |
419 | 329 | eqcomd 2831 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘))) |
420 | 331 | feqmptd 6496 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
421 | 418, 419,
420 | 3eqtrd 2865 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
422 | 405, 411,
421 | 3eqtrd 2865 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
423 | 378, 333,
370, 401, 371, 332, 422 | dvmptmul 24123 |
. . . . . . . . . . . . 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))))) |
424 | 378, 290,
379, 381, 382, 374, 423 | dvmptmul 24123 |
. . . . . . . . . . . 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))))) |
425 | 382 | mul02d 10553 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = 0) |
426 | 335 | oveq1d 6920 |
. . . . . . . . . . . . . . . 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))) |
427 | 375, 290 | mulcomd 10378 |
. . . . . . . . . . . . . . . 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))))) |
428 | 426, 427 | eqtrd 2861 |
. . . . . . . . . . . . . . 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))))) |
429 | 425, 428 | oveq12d 6923 |
. . . . . . . . . . . . . 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)))))) |
430 | 376 | addid2d 10556 |
. . . . . . . . . . . . . 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))))) |
431 | 429, 430 | eqtrd 2861 |
. . . . . . . . . . . . 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))))) |
432 | 431 | mpteq2dva 4967 |
. . . . . . . . . . . 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)))))) |
433 | 424, 432 | eqtrd 2861 |
. . . . . . . . . . 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)))))) |
434 | 192, 193,
194, 195, 196, 289, 377, 433 | dvmptfsum 24137 |
. . . . . . . . . 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)))))) |
435 | 202 | adantlr 708 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
436 | 372 | an32s 644 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ) |
437 | | anass 462 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋))) |
438 | | ancom 454 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖))) |
439 | 438 | anbi2i 618 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋)) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖)))) |
440 | | anass 462 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖)))) |
441 | 440 | bicomi 216 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖))) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖))) |
442 | 439, 441 | bitri 267 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋)) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖))) |
443 | 437, 442 | bitri 267 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖))) |
444 | 443 | imbi1i 341 |
. . . . . . . . . . . . . . . 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)) |
445 | 333, 444 | mpbi 222 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
446 | 443 | imbi1i 341 |
. . . . . . . . . . . . . . . 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)) |
447 | 332, 446 | mpbi 222 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) |
448 | 445, 447 | mulcld 10377 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ) |
449 | 435, 436,
448 | adddid 10381 |
. . . . . . . . . . . . 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))))) |
450 | 449 | sumeq2dv 14810 |
. . . . . . . . . . . 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))))) |
451 | 196 | adantr 474 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (0...𝑖) ∈ Fin) |
452 | 435, 436 | mulcld 10377 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ) |
453 | 435, 448 | mulcld 10377 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
454 | 451, 452,
453 | fsumadd 14847 |
. . . . . . . . . . . 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))))) |
455 | | oveq2 6913 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = ℎ → (𝑖C𝑘) = (𝑖Cℎ)) |
456 | | fvoveq1 6928 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = ℎ → (𝐶‘(𝑘 + 1)) = (𝐶‘(ℎ + 1))) |
457 | 456 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = ℎ → ((𝐶‘(𝑘 + 1))‘𝑥) = ((𝐶‘(ℎ + 1))‘𝑥)) |
458 | | oveq2 6913 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = ℎ → (𝑖 − 𝑘) = (𝑖 − ℎ)) |
459 | 458 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = ℎ → (𝐷‘(𝑖 − 𝑘)) = (𝐷‘(𝑖 − ℎ))) |
460 | 459 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = ℎ → ((𝐷‘(𝑖 − 𝑘))‘𝑥) = ((𝐷‘(𝑖 − ℎ))‘𝑥)) |
461 | 457, 460 | oveq12d 6923 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = ℎ → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) = (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) |
462 | 455, 461 | oveq12d 6923 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = ℎ → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)))) |
463 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎℎ(0...𝑖) |
464 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘(0...𝑖) |
465 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎℎ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) |
466 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(𝑖Cℎ) |
467 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘
· |
468 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘(ℎ + 1) |
469 | 357, 468 | nffv 6443 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘(𝐶‘(ℎ + 1)) |
470 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘𝑥 |
471 | 469, 470 | nffv 6443 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝐶‘(ℎ + 1))‘𝑥) |
472 | | nfmpt1 4970 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) |
473 | 139, 472 | nfcxfr 2967 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘𝐷 |
474 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘(𝑖 − ℎ) |
475 | 473, 474 | nffv 6443 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘(𝐷‘(𝑖 − ℎ)) |
476 | 475, 470 | nffv 6443 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝐷‘(𝑖 − ℎ))‘𝑥) |
477 | 471, 467,
476 | nfov 6935 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)) |
478 | 466, 467,
477 | nfov 6935 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) |
479 | 462, 463,
464, 465, 478 | cbvsum 14802 |
. . . . . . . . . . . . . . . . 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))) |
480 | 479 | 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)))) |
481 | | 1zzd 11736 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℤ) |
482 | 90 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℤ) |
483 | 236 | ad2antlr 720 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 𝑖 ∈ ℤ) |
484 | | nfv 2015 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) |
485 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘ℎ |
486 | 485, 464 | nfel 2982 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘 ℎ ∈ (0...𝑖) |
487 | 484, 486 | nfan 2004 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) |
488 | 478, 361 | nfel 2982 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ |
489 | 487, 488 | nfim 2001 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ) |
490 | | eleq1 2894 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = ℎ → (𝑘 ∈ (0...𝑖) ↔ ℎ ∈ (0...𝑖))) |
491 | 490 | anbi2d 624 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = ℎ → ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)))) |
492 | 462 | eleq1d 2891 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = ℎ → (((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ ↔ ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ)) |
493 | 491, 492 | imbi12d 336 |
. . . . . . . . . . . . . . . . . 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))) |
494 | 489, 493,
452 | chvar 2416 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ) |
495 | | oveq2 6913 |
. . . . . . . . . . . . . . . . . 18
⊢ (ℎ = (𝑗 − 1) → (𝑖Cℎ) = (𝑖C(𝑗 − 1))) |
496 | | fvoveq1 6928 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℎ = (𝑗 − 1) → (𝐶‘(ℎ + 1)) = (𝐶‘((𝑗 − 1) + 1))) |
497 | 496 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℎ = (𝑗 − 1) → ((𝐶‘(ℎ + 1))‘𝑥) = ((𝐶‘((𝑗 − 1) + 1))‘𝑥)) |
498 | | oveq2 6913 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (ℎ = (𝑗 − 1) → (𝑖 − ℎ) = (𝑖 − (𝑗 − 1))) |
499 | 498 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℎ = (𝑗 − 1) → (𝐷‘(𝑖 − ℎ)) = (𝐷‘(𝑖 − (𝑗 − 1)))) |
500 | 499 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℎ = (𝑗 − 1) → ((𝐷‘(𝑖 − ℎ))‘𝑥) = ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)) |
501 | 497, 500 | oveq12d 6923 |
. . . . . . . . . . . . . . . . . 18
⊢ (ℎ = (𝑗 − 1) → (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)) = (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) |
502 | 495, 501 | oveq12d 6923 |
. . . . . . . . . . . . . . . . 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)))) |
503 | 481, 482,
483, 494, 502 | fsumshft 14886 |
. . . . . . . . . . . . . . . 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)))) |
504 | 480, 503 | eqtrd 2861 |
. . . . . . . . . . . . . . 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)))) |
505 | | 0p1e1 11480 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 + 1) =
1 |
506 | 505 | oveq1i 6915 |
. . . . . . . . . . . . . . . . 17
⊢ ((0 +
1)...(𝑖 + 1)) = (1...(𝑖 + 1)) |
507 | 506 | sumeq1i 14805 |
. . . . . . . . . . . . . . . 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))) |
508 | 507 | 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)))) |
509 | | elfzelz 12635 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℤ) |
510 | 509 | zcnd 11811 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℂ) |
511 | | 1cnd 10351 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈
ℂ) |
512 | 510, 511 | npcand 10717 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → ((𝑗 − 1) + 1) = 𝑗) |
513 | 512 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → (𝐶‘((𝑗 − 1) + 1)) = (𝐶‘𝑗)) |
514 | 513 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶‘𝑗)‘𝑥)) |
515 | 514 | adantl 475 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶‘𝑗)‘𝑥)) |
516 | 214 | recnd 10385 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℂ) |
517 | 516 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℂ) |
518 | 510 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℂ) |
519 | 511 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈
ℂ) |
520 | 517, 518,
519 | subsub3d 10743 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 − (𝑗 − 1)) = ((𝑖 + 1) − 𝑗)) |
521 | 520 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘(𝑖 − (𝑗 − 1))) = (𝐷‘((𝑖 + 1) − 𝑗))) |
522 | 521 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) |
523 | 515, 522 | oveq12d 6923 |
. . . . . . . . . . . . . . . . . . 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))) |
524 | 523 | oveq2d 6921 |
. . . . . . . . . . . . . . . . . 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)))) |
525 | 524 | sumeq2dv 14810 |
. . . . . . . . . . . . . . . . 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)))) |
526 | 525 | ad2antlr 720 |
. . . . . . . . . . . . . . . 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)))) |
527 | | nfv 2015 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑗((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) |
528 | | nfcv 2969 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑗((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) |
529 | | fzfid 13067 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...(𝑖 + 1)) ∈ Fin) |
530 | 185 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℕ0) |
531 | 509 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℤ) |
532 | | 1zzd 11736 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈
ℤ) |
533 | 531, 532 | zsubcld 11815 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑗 − 1) ∈ ℤ) |
534 | 530, 533 | bccld 40328 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈
ℕ0) |
535 | 534 | nn0cnd 11680 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
536 | 535 | adantll 707 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
537 | 536 | adantlr 708 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
538 | 1 | ad2antrr 719 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝜑) |
539 | | 0zd 11716 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ∈
ℤ) |
540 | 206 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℤ) |
541 | 539, 540,
531 | 3jca 1164 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈
ℤ)) |
542 | 171 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 ∈
ℝ) |
543 | 509 | zred 11810 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ) |
544 | | 1red 10357 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈
ℝ) |
545 | | 0lt1 10874 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ 0 <
1 |
546 | 545 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 1) |
547 | | elfzle1 12637 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 1 ≤ 𝑗) |
548 | 542, 544,
543, 546, 547 | ltletrd 10516 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 𝑗) |
549 | 542, 543,
548 | ltled 10504 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 ≤ 𝑗) |
550 | 549 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ 𝑗) |
551 | 543 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ) |
552 | 214 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℝ) |
553 | | 1red 10357 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈
ℝ) |
554 | 552, 553 | readdcld 10386 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ) |
555 | 212 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℝ) |
556 | | elfzle2 12638 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ≤ (𝑖 + 1)) |
557 | 556 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ (𝑖 + 1)) |
558 | 308 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁) |
559 | 551, 554,
555, 557, 558 | letrd 10513 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ 𝑁) |
560 | 541, 550,
559 | jca32 513 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤
𝑗 ∧ 𝑗 ≤ 𝑁))) |
561 | | elfz2 12626 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤
𝑗 ∧ 𝑗 ≤ 𝑁))) |
562 | 560, 561 | sylibr 226 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁)) |
563 | 562 | adantll 707 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁)) |
564 | 538, 563,
367 | syl2anc 581 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶‘𝑗):𝑋⟶ℂ) |
565 | 564 | adantlr 708 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶‘𝑗):𝑋⟶ℂ) |
566 | | simplr 787 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑥 ∈ 𝑋) |
567 | 565, 566 | ffvelrnd 6609 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘𝑗)‘𝑥) ∈ ℂ) |
568 | 236 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℤ) |
569 | 568 | peano2zd 11813 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ) |
570 | 569, 531 | zsubcld 11815 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℤ) |
571 | 539, 540,
570 | 3jca 1164 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ)) |
572 | 554, 551 | subge0d 10942 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑗) ↔ 𝑗 ≤ (𝑖 + 1))) |
573 | 557, 572 | mpbird 249 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑗)) |
574 | 554, 551 | resubcld 10782 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℝ) |
575 | 574 | leidd 10918 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ ((𝑖 + 1) − 𝑗)) |
576 | 543, 548 | elrpd 12153 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ+) |
577 | 576 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ+) |
578 | 554, 577 | ltsubrpd 12188 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < (𝑖 + 1)) |
579 | 574, 554,
555, 578, 558 | ltletrd 10516 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁) |
580 | 574, 574,
555, 575, 579 | lelttrd 10514 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁) |
581 | 574, 555,
580 | ltled 10504 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ 𝑁) |
582 | 571, 573,
581 | jca32 513 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑗) ∧ ((𝑖 + 1) − 𝑗) ≤ 𝑁))) |
583 | | elfz2 12626 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑖 + 1) − 𝑗) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑗) ∧ ((𝑖 + 1) − 𝑗) ≤ 𝑁))) |
584 | 582, 583 | sylibr 226 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) |
585 | 584 | adantll 707 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) |
586 | | nfv 2015 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) |
587 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑘((𝑖 + 1) − 𝑗) |
588 | 473, 587 | nffv 6443 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝐷‘((𝑖 + 1) − 𝑗)) |
589 | 588, 360,
361 | nff 6274 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ |
590 | 586, 589 | nfim 2001 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ) |
591 | | ovex 6937 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 + 1) − 𝑗) ∈ V |
592 | | eleq1 2894 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → (𝑘 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))) |
593 | 592 | anbi2d 624 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)))) |
594 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → (𝐷‘𝑘) = (𝐷‘((𝑖 + 1) − 𝑗))) |
595 | 594 | feq1d 6263 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)) |
596 | 593, 595 | imbi12d 336 |
. . . . . . . . . . . . . . . . . . . . . . 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))) |
597 | 139 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))) |
598 | | fvexd 6448 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘) ∈ V) |
599 | 597, 598 | fvmpt2d 6540 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑘)) |
600 | 599 | feq1d 6263 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ)) |
601 | 276, 600 | mpbird 249 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) |
602 | 590, 591,
596, 601 | vtoclf 3474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ) |
603 | 538, 585,
602 | syl2anc 581 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ) |
604 | 603 | adantlr 708 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ) |
605 | 604, 566 | ffvelrnd 6609 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ) |
606 | 567, 605 | mulcld 10377 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ) |
607 | 537, 606 | mulcld 10377 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ) |
608 | | 1zzd 11736 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 1 ∈ ℤ) |
609 | 236 | peano2zd 11813 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℤ) |
610 | 505 | eqcomi 2834 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 1 = (0 +
1) |
611 | 610 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → 1 = (0 + 1)) |
612 | 171 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 0 ∈ ℝ) |
613 | | 1red 10357 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 1 ∈ ℝ) |
614 | 185 | nn0ge0d 11681 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 0 ≤ 𝑖) |
615 | 612, 214,
613, 614 | leadd1dd 10966 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → (0 + 1) ≤ (𝑖 + 1)) |
616 | 611, 615 | eqbrtrd 4895 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 1 ≤ (𝑖 + 1)) |
617 | 608, 609,
616 | 3jca 1164 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1
≤ (𝑖 +
1))) |
618 | | eluz2 11974 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 + 1) ∈
(ℤ≥‘1) ↔ (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1
≤ (𝑖 +
1))) |
619 | 617, 618 | sylibr 226 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈
(ℤ≥‘1)) |
620 | | eluzfz2 12642 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 + 1) ∈
(ℤ≥‘1) → (𝑖 + 1) ∈ (1...(𝑖 + 1))) |
621 | 619, 620 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (1...(𝑖 + 1))) |
622 | 621 | ad2antlr 720 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝑖 + 1) ∈ (1...(𝑖 + 1))) |
623 | | oveq1 6912 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 + 1) → (𝑗 − 1) = ((𝑖 + 1) − 1)) |
624 | 623 | oveq2d 6921 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑖 + 1) → (𝑖C(𝑗 − 1)) = (𝑖C((𝑖 + 1) − 1))) |
625 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑖 + 1) → (𝐶‘𝑗) = (𝐶‘(𝑖 + 1))) |
626 | 625 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 + 1) → ((𝐶‘𝑗)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥)) |
627 | | oveq2 6913 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = (𝑖 + 1) → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − (𝑖 + 1))) |
628 | 627 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1)))) |
629 | 628 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) |
630 | 626, 629 | oveq12d 6923 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑖 + 1) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) |
631 | 624, 630 | oveq12d 6923 |
. . . . . . . . . . . . . . . . 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)))) |
632 | 527, 528,
529, 607, 622, 631 | fsumsplit1 40599 |
. . . . . . . . . . . . . . . 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))))) |
633 | | 1cnd 10351 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → 1 ∈ ℂ) |
634 | 516, 633 | pncand 10714 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 1) = 𝑖) |
635 | 634 | oveq2d 6921 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = (𝑖C𝑖)) |
636 | | bcnn 13392 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ ℕ0
→ (𝑖C𝑖) = 1) |
637 | 185, 636 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C𝑖) = 1) |
638 | 635, 637 | eqtrd 2861 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = 1) |
639 | 516, 633 | addcld 10376 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℂ) |
640 | 639 | subidd 10701 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − (𝑖 + 1)) = 0) |
641 | 640 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0)) |
642 | 641 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) = ((𝐷‘0)‘𝑥)) |
643 | 642 | oveq2d 6921 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) |
644 | 638, 643 | oveq12d 6923 |
. . . . . . . . . . . . . . . . . . 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)))) |
645 | 644 | ad2antlr 720 |
. . . . . . . . . . . . . . . . . 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)))) |
646 | | simpl 476 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝜑) |
647 | | fzofzp1 12860 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁)) |
648 | 647 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁)) |
649 | | nfv 2015 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) |
650 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑘(𝑖 + 1) |
651 | 357, 650 | nffv 6443 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝐶‘(𝑖 + 1)) |
652 | 651, 360,
361 | nff 6274 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐶‘(𝑖 + 1)):𝑋⟶ℂ |
653 | 649, 652 | nfim 2001 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ) |
654 | | ovex 6937 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 + 1) ∈ V |
655 | | eleq1 2894 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = (𝑖 + 1) → (𝑘 ∈ (0...𝑁) ↔ (𝑖 + 1) ∈ (0...𝑁))) |
656 | 655 | anbi2d 624 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = (𝑖 + 1) → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)))) |
657 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = (𝑖 + 1) → (𝐶‘𝑘) = (𝐶‘(𝑖 + 1))) |
658 | 657 | feq1d 6263 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = (𝑖 + 1) → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)) |
659 | 656, 658 | imbi12d 336 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = (𝑖 + 1) → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ))) |
660 | 653, 654,
659, 231 | vtoclf 3474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ) |
661 | 646, 648,
660 | syl2anc 581 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ) |
662 | 661 | ffvelrnda 6608 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘(𝑖 + 1))‘𝑥) ∈ ℂ) |
663 | | nfv 2015 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝜑 ∧ 0 ∈ (0...𝑁)) |
664 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
Ⅎ𝑘0 |
665 | 473, 664 | nffv 6443 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑘(𝐷‘0) |
666 | 665, 360,
361 | nff 6274 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝐷‘0):𝑋⟶ℂ |
667 | 663, 666 | nfim 2001 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ) |
668 | | c0ex 10350 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 0 ∈
V |
669 | | eleq1 2894 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 = 0 → (𝑘 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁))) |
670 | 669 | anbi2d 624 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = 0 → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁)))) |
671 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 = 0 → (𝐷‘𝑘) = (𝐷‘0)) |
672 | 671 | feq1d 6263 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = 0 → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ)) |
673 | 670, 672 | imbi12d 336 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = 0 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ))) |
674 | 667, 668,
673, 601 | vtoclf 3474 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ) |
675 | 1, 112, 674 | syl2anc 581 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝐷‘0):𝑋⟶ℂ) |
676 | 675 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘0):𝑋⟶ℂ) |
677 | 676 | ffvelrnda 6608 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘0)‘𝑥) ∈ ℂ) |
678 | 662, 677 | mulcld 10377 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) ∈ ℂ) |
679 | 678 | mulid2d 10375 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) |
680 | 645, 679 | eqtrd 2861 |
. . . . . . . . . . . . . . . . 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))) |
681 | | 1m1e0 11423 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (1
− 1) = 0 |
682 | 681 | fveq2i 6436 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(ℤ≥‘(1 − 1)) =
(ℤ≥‘0) |
683 | 3 | eqcomi 2834 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(ℤ≥‘0) = ℕ0 |
684 | 682, 683 | eqtr2i 2850 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
ℕ0 = (ℤ≥‘(1 −
1)) |
685 | 684 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → ℕ0 =
(ℤ≥‘(1 − 1))) |
686 | 185, 685 | eleqtrd 2908 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ≥‘(1
− 1))) |
687 | | fzdifsuc2 40322 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈
(ℤ≥‘(1 − 1)) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})) |
688 | 686, 687 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})) |
689 | 688 | eqcomd 2831 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}) = (1...𝑖)) |
690 | 689 | sumeq1d 14808 |
. . . . . . . . . . . . . . . . . 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)))) |
691 | 690 | ad2antlr 720 |
. . . . . . . . . . . . . . . . 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)))) |
692 | 680, 691 | oveq12d 6923 |
. . . . . . . . . . . . . . . 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))))) |
693 | 526, 632,
692 | 3eqtrd 2865 |
. . . . . . . . . . . . . . 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))))) |
694 | 504, 508,
693 | 3eqtrd 2865 |
. . . . . . . . . . . . . 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))))) |
695 | | nfcv 2969 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(𝑖C0) |
696 | 357, 664 | nffv 6443 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(𝐶‘0) |
697 | 696, 470 | nffv 6443 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝐶‘0)‘𝑥) |
698 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝑖 + 1) − 0) |
699 | 473, 698 | nffv 6443 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(𝐷‘((𝑖 + 1) − 0)) |
700 | 699, 470 | nffv 6443 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝐷‘((𝑖 + 1) − 0))‘𝑥) |
701 | 697, 467,
700 | nfov 6935 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) |
702 | 695, 467,
701 | nfov 6935 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) |
703 | 683 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → (ℤ≥‘0) =
ℕ0) |
704 | 185, 703 | eleqtrrd 2909 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈
(ℤ≥‘0)) |
705 | | eluzfz1 12641 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈
(ℤ≥‘0) → 0 ∈ (0...𝑖)) |
706 | 704, 705 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (0..^𝑁) → 0 ∈ (0...𝑖)) |
707 | 706 | ad2antlr 720 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ (0...𝑖)) |
708 | | oveq2 6913 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 → (𝑖C𝑘) = (𝑖C0)) |
709 | 104 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 0 → ((𝐶‘𝑘)‘𝑥) = ((𝐶‘0)‘𝑥)) |
710 | | oveq2 6913 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 0 → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − 0)) |
711 | 710 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 0))) |
712 | 711 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 0 → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) |
713 | 709, 712 | oveq12d 6923 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) |
714 | 708, 713 | oveq12d 6923 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))) |
715 | 484, 702,
451, 453, 707, 714 | fsumsplit1 40599 |
. . . . . . . . . . . . . . 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))))) |
716 | 639 | subid1d 10702 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 0) = (𝑖 + 1)) |
717 | 716 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − 0)) = (𝐷‘(𝑖 + 1))) |
718 | 717 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − 0))‘𝑥) = ((𝐷‘(𝑖 + 1))‘𝑥)) |
719 | 718 | oveq2d 6921 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) |
720 | 719 | oveq2d 6921 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))) |
721 | 720 | oveq1d 6920 |
. . . . . . . . . . . . . . . 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))))) |
722 | 721 | ad2antlr 720 |
. . . . . . . . . . . . . . 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))))) |
723 | | bcn0 13390 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ ℕ0
→ (𝑖C0) =
1) |
724 | 185, 723 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C0) = 1) |
725 | 724 | oveq1d 6920 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))) |
726 | 725 | ad2antlr 720 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))) |
727 | 696, 360,
361 | nff 6274 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐶‘0):𝑋⟶ℂ |
728 | 663, 727 | nfim 2001 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ) |
729 | 104 | feq1d 6263 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = 0 → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘0):𝑋⟶ℂ)) |
730 | 670, 729 | imbi12d 336 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = 0 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ))) |
731 | 728, 668,
730, 231 | vtoclf 3474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ) |
732 | 1, 112, 731 | syl2anc 581 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐶‘0):𝑋⟶ℂ) |
733 | 732 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐶‘0):𝑋⟶ℂ) |
734 | 733 | ffvelrnda 6608 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘0)‘𝑥) ∈ ℂ) |
735 | 473, 650 | nffv 6443 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝐷‘(𝑖 + 1)) |
736 | 735, 360,
361 | nff 6274 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘(𝐷‘(𝑖 + 1)):𝑋⟶ℂ |
737 | 649, 736 | nfim 2001 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ) |
738 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = (𝑖 + 1) → (𝐷‘𝑘) = (𝐷‘(𝑖 + 1))) |
739 | 738 | feq1d 6263 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = (𝑖 + 1) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)) |
740 | 656, 739 | imbi12d 336 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = (𝑖 + 1) → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ))) |
741 | 737, 654,
740, 601 | vtoclf 3474 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ) |
742 | 646, 648,
741 | syl2anc 581 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ) |
743 | 742 | ffvelrnda 6608 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 + 1))‘𝑥) ∈ ℂ) |
744 | 734, 743 | mulcld 10377 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) ∈ ℂ) |
745 | 744 | mulid2d 10375 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) |
746 | 726, 745 | eqtrd 2861 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) |
747 | | nfv 2015 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑗 𝑖 ∈ (0..^𝑁) |
748 | | 1zzd 11736 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ∈
ℤ) |
749 | 236 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑖 ∈ ℤ) |
750 | | eldifi 3959 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (0...𝑖)) |
751 | | elfzelz 12635 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ) |
752 | 750, 751 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℤ) |
753 | 752 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ ℤ) |
754 | 748, 749,
753 | 3jca 1164 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → (1 ∈ ℤ
∧ 𝑖 ∈ ℤ
∧ 𝑗 ∈
ℤ)) |
755 | | elfznn0 12727 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℕ0) |
756 | 750, 755 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ0) |
757 | | eldifsni 4540 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≠ 0) |
758 | 756, 757 | jca 509 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → (𝑗 ∈ ℕ0 ∧ 𝑗 ≠ 0)) |
759 | | elnnne0 11634 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ℕ ↔ (𝑗 ∈ ℕ0
∧ 𝑗 ≠
0)) |
760 | 758, 759 | sylibr 226 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ) |
761 | | nnge1 11380 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ℕ → 1 ≤
𝑗) |
762 | 760, 761 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 1 ≤ 𝑗) |
763 | 762 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ≤ 𝑗) |
764 | | elfzle2 12638 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ≤ 𝑖) |
765 | 750, 764 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≤ 𝑖) |
766 | 765 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ≤ 𝑖) |
767 | 754, 763,
766 | jca32 513 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → ((1 ∈ ℤ
∧ 𝑖 ∈ ℤ
∧ 𝑗 ∈ ℤ)
∧ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑖))) |
768 | | elfz2 12626 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ (1...𝑖) ↔ ((1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (1 ≤
𝑗 ∧ 𝑗 ≤ 𝑖))) |
769 | 767, 768 | sylibr 226 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ (1...𝑖)) |
770 | 769 | ex 403 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (1...𝑖))) |
771 | | 0zd 11716 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 0 ∈ ℤ) |
772 | | elfzel2 12633 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 𝑖 ∈ ℤ) |
773 | | elfzelz 12635 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℤ) |
774 | 771, 772,
773 | 3jca 1164 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...𝑖) → (0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈
ℤ)) |
775 | 171 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 0 ∈ ℝ) |
776 | 773 | zred 11810 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℝ) |
777 | | 1red 10357 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (1...𝑖) → 1 ∈ ℝ) |
778 | 545 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (1...𝑖) → 0 < 1) |
779 | | elfzle1 12637 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (1...𝑖) → 1 ≤ 𝑗) |
780 | 775, 777,
776, 778, 779 | ltletrd 10516 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (1...𝑖) → 0 < 𝑗) |
781 | 775, 776,
780 | ltled 10504 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...𝑖) → 0 ≤ 𝑗) |
782 | | elfzle2 12638 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ≤ 𝑖) |
783 | 774, 781,
782 | jca32 513 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...𝑖) → ((0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤
𝑗 ∧ 𝑗 ≤ 𝑖))) |
784 | | elfz2 12626 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (0...𝑖) ↔ ((0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤
𝑗 ∧ 𝑗 ≤ 𝑖))) |
785 | 783, 784 | sylibr 226 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (0...𝑖)) |
786 | 775, 780 | gtned 10491 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ≠ 0) |
787 | | nelsn 4433 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ≠ 0 → ¬ 𝑗 ∈ {0}) |
788 | 786, 787 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (1...𝑖) → ¬ 𝑗 ∈ {0}) |
789 | 785, 788 | eldifd 3809 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0})) |
790 | 789 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ((0...𝑖) ∖ {0})) |
791 | 790 | ex 403 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0}))) |
792 | 770, 791 | impbid 204 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖))) |
793 | 747, 792 | alrimi 2258 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖))) |
794 | | dfcleq 2819 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((0...𝑖) ∖
{0}) = (1...𝑖) ↔
∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖))) |
795 | 793, 794 | sylibr 226 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → ((0...𝑖) ∖ {0}) = (1...𝑖)) |
796 | 795 | sumeq1d 14808 |
. . . . . . . . . . . . . . . . 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)))) |
797 | 796 | ad2antlr 720 |
. . . . . . . . . . . . . . . 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)))) |
798 | 746, 797 | oveq12d 6923 |
. . . . . . . . . . . . . . 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))))) |
799 | 715, 722,
798 | 3eqtrd 2865 |
. . . . . . . . . . . . . 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))))) |
800 | 694, 799 | oveq12d 6923 |
. . . . . . . . . . . . 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)))))) |
801 | | fzfid 13067 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...𝑖) ∈ Fin) |
802 | 185 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0) |
803 | 790, 752 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ℤ) |
804 | | 1zzd 11736 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 1 ∈ ℤ) |
805 | 803, 804 | zsubcld 11815 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑗 − 1) ∈ ℤ) |
806 | 802, 805 | bccld 40328 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈
ℕ0) |
807 | 806 | nn0cnd 11680 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
808 | 807 | adantll 707 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
809 | 808 | adantlr 708 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ) |
810 | | simpl 476 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋)) |
811 | | fzelp1 12686 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (1...(𝑖 + 1))) |
812 | 811 | adantl 475 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ (1...(𝑖 + 1))) |
813 | 810, 812,
567 | syl2anc 581 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐶‘𝑗)‘𝑥) ∈ ℂ) |
814 | 812, 605 | syldan 587 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ) |
815 | 813, 814 | mulcld 10377 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ) |
816 | 809, 815 | mulcld 10377 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ) |
817 | 801, 816 | fsumcl 14841 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ) |
818 | 185 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0) |
819 | | elfzelz 12635 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ ℤ) |
820 | 819 | adantl 475 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ ℤ) |
821 | 818, 820 | bccld 40328 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈
ℕ0) |
822 | 821 | nn0cnd 11680 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
823 | 822 | adantll 707 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
824 | 823 | adantlr 708 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ) |
825 | | simpll 785 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝜑 ∧ 𝑖 ∈ (0..^𝑁))) |
826 | | simplr 787 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑥 ∈ 𝑋) |
827 | 785 | ssriv 3831 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(1...𝑖) ⊆
(0...𝑖) |
828 | | id 22 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (1...𝑖)) |
829 | 827, 828 | sseldi 3825 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (0...𝑖)) |
830 | 829 | adantl 475 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ (0...𝑖)) |
831 | 825, 826,
830, 445 | syl21anc 873 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
832 | 830, 447 | syldan 587 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) |
833 | 831, 832 | mulcld 10377 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ) |
834 | 824, 833 | mulcld 10377 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
835 | 801, 834 | fsumcl 14841 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
836 | 678, 817,
744, 835 | add4d 10583 |
. . . . . . . . . . . . 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)))))) |
837 | | oveq1 6912 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑘 → (𝑗 − 1) = (𝑘 − 1)) |
838 | 837 | oveq2d 6921 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑘 → (𝑖C(𝑗 − 1)) = (𝑖C(𝑘 − 1))) |
839 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 𝑘 → (𝐶‘𝑗) = (𝐶‘𝑘)) |
840 | 839 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑘 → ((𝐶‘𝑗)‘𝑥) = ((𝐶‘𝑘)‘𝑥)) |
841 | | oveq2 6913 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 = 𝑘 → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − 𝑘)) |
842 | 841 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 𝑘 → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − 𝑘))) |
843 | 842 | fveq1d 6435 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑘 → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) |
844 | 840, 843 | oveq12d 6923 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑘 → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
845 | 838, 844 | oveq12d 6923 |
. . . . . . . . . . . . . . . . . . 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)))) |
846 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(1...𝑖) |
847 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑗(1...𝑖) |
848 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘(𝑖C(𝑗 − 1)) |
849 | 359, 470 | nffv 6443 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘((𝐶‘𝑗)‘𝑥) |
850 | 588, 470 | nffv 6443 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) |
851 | 849, 467,
850 | nfov 6935 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘(((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) |
852 | 848, 467,
851 | nfov 6935 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) |
853 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑗((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) |
854 | 845, 846,
847, 852, 853 | cbvsum 14802 |
. . . . . . . . . . . . . . . . . 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))) |
855 | 854 | 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)))) |
856 | 855 | oveq1d 6920 |
. . . . . . . . . . . . . . . 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))))) |
857 | | peano2zm 11748 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ ℤ → (𝑘 − 1) ∈
ℤ) |
858 | 820, 857 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑘 − 1) ∈ ℤ) |
859 | 818, 858 | bccld 40328 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈
ℕ0) |
860 | 859 | nn0cnd 11680 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ) |
861 | 860 | adantll 707 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ) |
862 | 861 | adantlr 708 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ) |
863 | 862, 833 | mulcld 10377 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
864 | 801, 863,
834 | fsumadd 14847 |
. . . . . . . . . . . . . . . . 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))))) |
865 | 864 | eqcomd 2831 |
. . . . . . . . . . . . . . . 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))))) |
866 | 860, 822 | addcomd 10557 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) = ((𝑖C𝑘) + (𝑖C(𝑘 − 1)))) |
867 | | bcpasc 13401 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘)) |
868 | 818, 820,
867 | syl2anc 581 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘)) |
869 | 866, 868 | eqtr2d 2862 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) = ((𝑖C(𝑘 − 1)) + (𝑖C𝑘))) |
870 | 869 | oveq1d 6920 |
. . . . . . . . . . . . . . . . . . . 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)))) |
871 | 870 | adantll 707 |
. . . . . . . . . . . . . . . . . . 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)))) |
872 | 871 | adantlr 708 |
. . . . . . . . . . . . . . . . . 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)))) |
873 | 862, 824,
833 | adddird 10382 |
. . . . . . . . . . . . . . . . . 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))))) |
874 | 872, 873 | eqtr2d 2862 |
. . . . . . . . . . . . . . . . 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)))) |
875 | 874 | sumeq2dv 14810 |
. . . . . . . . . . . . . . . 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)))) |
876 | 856, 865,
875 | 3eqtrd 2865 |
. . . . . . . . . . . . . . 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)))) |
877 | 876 | oveq2d 6921 |
. . . . . . . . . . . . . 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))))) |
878 | | peano2nn0 11660 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ ℕ0
→ (𝑖 + 1) ∈
ℕ0) |
879 | 818, 878 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖 + 1) ∈
ℕ0) |
880 | 879, 820 | bccld 40328 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈
ℕ0) |
881 | 880 | nn0cnd 11680 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
882 | 881 | adantll 707 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
883 | 882 | adantlr 708 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
884 | 883, 833 | mulcld 10377 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
885 | 801, 884 | fsumcl 14841 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
886 | 678, 744,
885 | addassd 10379 |
. . . . . . . . . . . . . 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)))))) |
887 | 185, 878 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈
ℕ0) |
888 | | bcn0 13390 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 + 1) ∈ ℕ0
→ ((𝑖 + 1)C0) =
1) |
889 | 887, 888 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C0) = 1) |
890 | 889, 719 | oveq12d 6923 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (0..^𝑁) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))) |
891 | 890 | ad2antlr 720 |
. . . . . . . . . . . . . . . . . . 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)))) |
892 | 891, 745 | eqtr2d 2862 |
. . . . . . . . . . . . . . . . . 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)))) |
893 | 795 | ad2antlr 720 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((0...𝑖) ∖ {0}) = (1...𝑖)) |
894 | 893 | eqcomd 2831 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...𝑖) = ((0...𝑖) ∖ {0})) |
895 | 894 | sumeq1d 14808 |
. . . . . . . . . . . . . . . . . 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)))) |
896 | 892, 895 | oveq12d 6923 |
. . . . . . . . . . . . . . . . 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))))) |
897 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘((𝑖 + 1)C0) |
898 | 897, 467,
701 | nfov 6935 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘(((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) |
899 | 197, 878 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈
ℕ0) |
900 | 899, 199 | bccld 40328 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈
ℕ0) |
901 | 900 | nn0cnd 11680 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
902 | 901 | adantll 707 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
903 | 902 | adantlr 708 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
904 | 903, 448 | mulcld 10377 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ) |
905 | | oveq2 6913 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 0 → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C0)) |
906 | 905, 713 | oveq12d 6923 |
. . . . . . . . . . . . . . . . . . 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)))) |
907 | 484, 898,
451, 904, 707, 906 | fsumsplit1 40599 |
. . . . . . . . . . . . . . . . . 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))))) |
908 | 907 | eqcomd 2831 |
. . . . . . . . . . . . . . . . 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)))) |
909 | 896, 908 | eqtrd 2861 |
. . . . . . . . . . . . . . . 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)))) |
910 | 909 | oveq2d 6921 |
. . . . . . . . . . . . . . 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))))) |
911 | | bcnn 13392 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 + 1) ∈ ℕ0
→ ((𝑖 + 1)C(𝑖 + 1)) = 1) |
912 | 887, 911 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C(𝑖 + 1)) = 1) |
913 | 912 | ad2antlr 720 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖 + 1)C(𝑖 + 1)) = 1) |
914 | 913 | oveq1d 6920 |
. . . . . . . . . . . . . . . . 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)))) |
915 | 641 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0)) |
916 | 915 | feq1d 6263 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ)) |
917 | 676, 916 | mpbird 249 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ) |
918 | 917 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ) |
919 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
920 | 918, 919 | ffvelrnd 6609 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) ∈ ℂ) |
921 | 662, 920 | mulcld 10377 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) ∈ ℂ) |
922 | 921 | mulid2d 10375 |
. . . . . . . . . . . . . . . . 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))) |
923 | 643 | ad2antlr 720 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) |
924 | 914, 922,
923 | 3eqtrrd 2866 |
. . . . . . . . . . . . . . . 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)))) |
925 | | fzdifsuc 12694 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈
(ℤ≥‘0) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})) |
926 | 704, 925 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ (0..^𝑁) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})) |
927 | 926 | sumeq1d 14808 |
. . . . . . . . . . . . . . . . 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)))) |
928 | 927 | ad2antlr 720 |
. . . . . . . . . . . . . . . 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)))) |
929 | 924, 928 | oveq12d 6923 |
. . . . . . . . . . . . . . 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))))) |
930 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝑖 + 1)C(𝑖 + 1)) |
931 | 651, 470 | nffv 6443 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝐶‘(𝑖 + 1))‘𝑥) |
932 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘((𝑖 + 1) − (𝑖 + 1)) |
933 | 473, 932 | nffv 6443 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘(𝐷‘((𝑖 + 1) − (𝑖 + 1))) |
934 | 933, 470 | nffv 6443 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) |
935 | 931, 467,
934 | nfov 6935 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘(((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) |
936 | 930, 467,
935 | nfov 6935 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) |
937 | | fzfid 13067 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (0...(𝑖 + 1)) ∈ Fin) |
938 | 887 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈
ℕ0) |
939 | | elfzelz 12635 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ∈ ℤ) |
940 | 939 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℤ) |
941 | 938, 940 | bccld 40328 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈
ℕ0) |
942 | 941 | nn0cnd 11680 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
943 | 942 | adantll 707 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
944 | 943 | adantlr 708 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ) |
945 | 646 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑) |
946 | 90 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈
ℤ) |
947 | 206 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℤ) |
948 | 946, 947,
940 | 3jca 1164 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈
ℤ)) |
949 | | elfzle1 12637 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 ∈ (0...(𝑖 + 1)) → 0 ≤ 𝑘) |
950 | 949 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ 𝑘) |
951 | 940 | zred 11810 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℝ) |
952 | 938 | nn0red 11679 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ) |
953 | 212 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℝ) |
954 | | elfzle2 12638 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ≤ (𝑖 + 1)) |
955 | 954 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ (𝑖 + 1)) |
956 | 308 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁) |
957 | 951, 952,
953, 955, 956 | letrd 10513 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ 𝑁) |
958 | 948, 950,
957 | jca32 513 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤
𝑘 ∧ 𝑘 ≤ 𝑁))) |
959 | 958, 223 | sylibr 226 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁)) |
960 | 959 | adantll 707 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁)) |
961 | 945, 960,
231 | syl2anc 581 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶‘𝑘):𝑋⟶ℂ) |
962 | 961 | adantlr 708 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶‘𝑘):𝑋⟶ℂ) |
963 | | simplr 787 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑥 ∈ 𝑋) |
964 | 962, 963 | ffvelrnd 6609 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) |
965 | 945 | adantlr 708 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑) |
966 | 609 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ) |
967 | 966, 940 | zsubcld 11815 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℤ) |
968 | 946, 947,
967 | 3jca 1164 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ)) |
969 | 952, 951 | subge0d 10942 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1))) |
970 | 955, 969 | mpbird 249 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑘)) |
971 | 952, 951 | resubcld 10782 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℝ) |
972 | 953, 951 | resubcld 10782 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 𝑘) ∈ ℝ) |
973 | 953, 171,
250 | sylancl 582 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) ∈ ℝ) |
974 | 952, 953,
951, 956 | lesub1dd 10968 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 𝑘)) |
975 | 171 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈
ℝ) |
976 | 975, 951,
953, 950 | lesub2dd 10969 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 𝑘) ≤ (𝑁 − 0)) |
977 | 971, 972,
973, 974, 976 | letrd 10513 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 0)) |
978 | 256 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) = 𝑁) |
979 | 977, 978 | breqtrd 4899 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ 𝑁) |
980 | 968, 970,
979 | jca32 513 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑘) ∧ ((𝑖 + 1) − 𝑘) ≤ 𝑁))) |
981 | 980, 314 | sylibr 226 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
982 | 981 | adantll 707 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
983 | 982 | adantlr 708 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) |
984 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (𝐷‘𝑗) = (𝐷‘((𝑖 + 1) − 𝑘))) |
985 | 984 | feq1d 6263 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝐷‘𝑗):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)) |
986 | 319, 985 | imbi12d 336 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (( |