Step | Hyp | Ref
| Expression |
1 | | bpolydiflem.1 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ ℕ) |
2 | 1 | nnnn0d 12291 |
. . . 4
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
3 | | bpolydiflem.2 |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ ℂ) |
4 | | peano2cn 11145 |
. . . . 5
⊢ (𝑋 ∈ ℂ → (𝑋 + 1) ∈
ℂ) |
5 | 3, 4 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑋 + 1) ∈ ℂ) |
6 | | bpolyval 15755 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ (𝑋 + 1) ∈
ℂ) → (𝑁 BernPoly
(𝑋 + 1)) = (((𝑋 + 1)↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))))) |
7 | 2, 5, 6 | syl2anc 584 |
. . 3
⊢ (𝜑 → (𝑁 BernPoly (𝑋 + 1)) = (((𝑋 + 1)↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))))) |
8 | | bpolyval 15755 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) |
9 | 2, 3, 8 | syl2anc 584 |
. . 3
⊢ (𝜑 → (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) |
10 | 7, 9 | oveq12d 7287 |
. 2
⊢ (𝜑 → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = ((((𝑋 + 1)↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))) − ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))) |
11 | 5, 2 | expcld 13860 |
. . 3
⊢ (𝜑 → ((𝑋 + 1)↑𝑁) ∈ ℂ) |
12 | | fzfid 13689 |
. . . 4
⊢ (𝜑 → (0...(𝑁 − 1)) ∈ Fin) |
13 | | elfzelz 13253 |
. . . . . . 7
⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℤ) |
14 | | bccl 14032 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ (𝑁C𝑘) ∈
ℕ0) |
15 | 2, 13, 14 | syl2an 596 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁C𝑘) ∈
ℕ0) |
16 | 15 | nn0cnd 12293 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁C𝑘) ∈ ℂ) |
17 | | elfznn0 13346 |
. . . . . . 7
⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ0) |
18 | | bpolycl 15758 |
. . . . . . 7
⊢ ((𝑘 ∈ ℕ0
∧ (𝑋 + 1) ∈
ℂ) → (𝑘 BernPoly
(𝑋 + 1)) ∈
ℂ) |
19 | 17, 5, 18 | syl2anr 597 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 BernPoly (𝑋 + 1)) ∈ ℂ) |
20 | | fzssp1 13296 |
. . . . . . . . . . 11
⊢
(0...(𝑁 − 1))
⊆ (0...((𝑁 − 1)
+ 1)) |
21 | 1 | nncnd 11987 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℂ) |
22 | | ax-1cn 10928 |
. . . . . . . . . . . . 13
⊢ 1 ∈
ℂ |
23 | | npcan 11228 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑁 −
1) + 1) = 𝑁) |
24 | 21, 22, 23 | sylancl 586 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
25 | 24 | oveq2d 7285 |
. . . . . . . . . . 11
⊢ (𝜑 → (0...((𝑁 − 1) + 1)) = (0...𝑁)) |
26 | 20, 25 | sseqtrid 3978 |
. . . . . . . . . 10
⊢ (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁)) |
27 | 26 | sselda 3926 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ (0...𝑁)) |
28 | | fznn0sub 13285 |
. . . . . . . . 9
⊢ (𝑘 ∈ (0...𝑁) → (𝑁 − 𝑘) ∈
ℕ0) |
29 | 27, 28 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁 − 𝑘) ∈
ℕ0) |
30 | | nn0p1nn 12270 |
. . . . . . . 8
⊢ ((𝑁 − 𝑘) ∈ ℕ0 → ((𝑁 − 𝑘) + 1) ∈ ℕ) |
31 | 29, 30 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ∈ ℕ) |
32 | 31 | nncnd 11987 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ∈ ℂ) |
33 | 31 | nnne0d 12021 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ≠ 0) |
34 | 19, 32, 33 | divcld 11749 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) ∈ ℂ) |
35 | 16, 34 | mulcld 10994 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) ∈ ℂ) |
36 | 12, 35 | fsumcl 15441 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) ∈ ℂ) |
37 | 3, 2 | expcld 13860 |
. . 3
⊢ (𝜑 → (𝑋↑𝑁) ∈ ℂ) |
38 | | bpolycl 15758 |
. . . . . . 7
⊢ ((𝑘 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (𝑘 BernPoly 𝑋) ∈
ℂ) |
39 | 17, 3, 38 | syl2anr 597 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 BernPoly 𝑋) ∈ ℂ) |
40 | 39, 32, 33 | divcld 11749 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) ∈ ℂ) |
41 | 16, 40 | mulcld 10994 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ) |
42 | 12, 41 | fsumcl 15441 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ) |
43 | 11, 36, 37, 42 | sub4d 11379 |
. 2
⊢ (𝜑 → ((((𝑋 + 1)↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))) − ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) = ((((𝑋 + 1)↑𝑁) − (𝑋↑𝑁)) − (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))) |
44 | 26 | sselda 3926 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑚 ∈ (0...𝑁)) |
45 | | bccl2 14033 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ (0...𝑁) → (𝑁C𝑚) ∈ ℕ) |
46 | 45 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℕ) |
47 | 46 | nncnd 11987 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℂ) |
48 | | elfznn0 13346 |
. . . . . . . . . 10
⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℕ0) |
49 | | expcl 13796 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ ℂ ∧ 𝑚 ∈ ℕ0)
→ (𝑋↑𝑚) ∈
ℂ) |
50 | 3, 48, 49 | syl2an 596 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑋↑𝑚) ∈ ℂ) |
51 | 47, 50 | mulcld 10994 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → ((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ) |
52 | 44, 51 | syldan 591 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ) |
53 | 12, 52 | fsumcl 15441 |
. . . . . 6
⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ) |
54 | | addcom 11159 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ ℂ ∧ 1 ∈
ℂ) → (𝑋 + 1) =
(1 + 𝑋)) |
55 | 3, 22, 54 | sylancl 586 |
. . . . . . . . 9
⊢ (𝜑 → (𝑋 + 1) = (1 + 𝑋)) |
56 | 55 | oveq1d 7284 |
. . . . . . . 8
⊢ (𝜑 → ((𝑋 + 1)↑𝑁) = ((1 + 𝑋)↑𝑁)) |
57 | | binom1p 15539 |
. . . . . . . . 9
⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ ((1 + 𝑋)↑𝑁) = Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (𝑋↑𝑚))) |
58 | 3, 2, 57 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → ((1 + 𝑋)↑𝑁) = Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (𝑋↑𝑚))) |
59 | 56, 58 | eqtrd 2780 |
. . . . . . 7
⊢ (𝜑 → ((𝑋 + 1)↑𝑁) = Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (𝑋↑𝑚))) |
60 | | nn0uz 12617 |
. . . . . . . . 9
⊢
ℕ0 = (ℤ≥‘0) |
61 | 2, 60 | eleqtrdi 2851 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘0)) |
62 | | oveq2 7277 |
. . . . . . . . 9
⊢ (𝑚 = 𝑁 → (𝑁C𝑚) = (𝑁C𝑁)) |
63 | | oveq2 7277 |
. . . . . . . . 9
⊢ (𝑚 = 𝑁 → (𝑋↑𝑚) = (𝑋↑𝑁)) |
64 | 62, 63 | oveq12d 7287 |
. . . . . . . 8
⊢ (𝑚 = 𝑁 → ((𝑁C𝑚) · (𝑋↑𝑚)) = ((𝑁C𝑁) · (𝑋↑𝑁))) |
65 | 61, 51, 64 | fsumm1 15459 |
. . . . . . 7
⊢ (𝜑 → Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (𝑋↑𝑚)) = (Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + ((𝑁C𝑁) · (𝑋↑𝑁)))) |
66 | | bcnn 14022 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ0
→ (𝑁C𝑁) = 1) |
67 | 2, 66 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁C𝑁) = 1) |
68 | 67 | oveq1d 7284 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁C𝑁) · (𝑋↑𝑁)) = (1 · (𝑋↑𝑁))) |
69 | 37 | mulid2d 10992 |
. . . . . . . . 9
⊢ (𝜑 → (1 · (𝑋↑𝑁)) = (𝑋↑𝑁)) |
70 | 68, 69 | eqtrd 2780 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁C𝑁) · (𝑋↑𝑁)) = (𝑋↑𝑁)) |
71 | 70 | oveq2d 7285 |
. . . . . . 7
⊢ (𝜑 → (Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + ((𝑁C𝑁) · (𝑋↑𝑁))) = (Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑋↑𝑁))) |
72 | 59, 65, 71 | 3eqtrd 2784 |
. . . . . 6
⊢ (𝜑 → ((𝑋 + 1)↑𝑁) = (Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑋↑𝑁))) |
73 | 53, 37, 72 | mvrraddd 11385 |
. . . . 5
⊢ (𝜑 → (((𝑋 + 1)↑𝑁) − (𝑋↑𝑁)) = Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚))) |
74 | | nnm1nn0 12272 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈
ℕ0) |
75 | 1, 74 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑁 − 1) ∈
ℕ0) |
76 | 75, 60 | eleqtrdi 2851 |
. . . . . 6
⊢ (𝜑 → (𝑁 − 1) ∈
(ℤ≥‘0)) |
77 | | oveq2 7277 |
. . . . . . 7
⊢ (𝑚 = (𝑁 − 1) → (𝑁C𝑚) = (𝑁C(𝑁 − 1))) |
78 | | oveq2 7277 |
. . . . . . 7
⊢ (𝑚 = (𝑁 − 1) → (𝑋↑𝑚) = (𝑋↑(𝑁 − 1))) |
79 | 77, 78 | oveq12d 7287 |
. . . . . 6
⊢ (𝑚 = (𝑁 − 1) → ((𝑁C𝑚) · (𝑋↑𝑚)) = ((𝑁C(𝑁 − 1)) · (𝑋↑(𝑁 − 1)))) |
80 | 76, 52, 79 | fsumm1 15459 |
. . . . 5
⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) = (Σ𝑚 ∈ (0...((𝑁 − 1) − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + ((𝑁C(𝑁 − 1)) · (𝑋↑(𝑁 − 1))))) |
81 | | 1cnd 10969 |
. . . . . . . . . 10
⊢ (𝜑 → 1 ∈
ℂ) |
82 | 21, 81, 81 | subsub4d 11361 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − (1 + 1))) |
83 | | df-2 12034 |
. . . . . . . . . 10
⊢ 2 = (1 +
1) |
84 | 83 | oveq2i 7280 |
. . . . . . . . 9
⊢ (𝑁 − 2) = (𝑁 − (1 + 1)) |
85 | 82, 84 | eqtr4di 2798 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − 2)) |
86 | 85 | oveq2d 7285 |
. . . . . . 7
⊢ (𝜑 → (0...((𝑁 − 1) − 1)) = (0...(𝑁 − 2))) |
87 | 86 | sumeq1d 15409 |
. . . . . 6
⊢ (𝜑 → Σ𝑚 ∈ (0...((𝑁 − 1) − 1))((𝑁C𝑚) · (𝑋↑𝑚)) = Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚))) |
88 | | bcnm1 14037 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ0
→ (𝑁C(𝑁 − 1)) = 𝑁) |
89 | 2, 88 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑁C(𝑁 − 1)) = 𝑁) |
90 | 89 | oveq1d 7284 |
. . . . . 6
⊢ (𝜑 → ((𝑁C(𝑁 − 1)) · (𝑋↑(𝑁 − 1))) = (𝑁 · (𝑋↑(𝑁 − 1)))) |
91 | 87, 90 | oveq12d 7287 |
. . . . 5
⊢ (𝜑 → (Σ𝑚 ∈ (0...((𝑁 − 1) − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + ((𝑁C(𝑁 − 1)) · (𝑋↑(𝑁 − 1)))) = (Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑁 · (𝑋↑(𝑁 − 1))))) |
92 | 73, 80, 91 | 3eqtrd 2784 |
. . . 4
⊢ (𝜑 → (((𝑋 + 1)↑𝑁) − (𝑋↑𝑁)) = (Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑁 · (𝑋↑(𝑁 − 1))))) |
93 | | oveq2 7277 |
. . . . . . . . 9
⊢ (𝑘 = 0 → (𝑁C𝑘) = (𝑁C0)) |
94 | | oveq1 7276 |
. . . . . . . . . 10
⊢ (𝑘 = 0 → (𝑘 BernPoly (𝑋 + 1)) = (0 BernPoly (𝑋 + 1))) |
95 | | oveq2 7277 |
. . . . . . . . . . 11
⊢ (𝑘 = 0 → (𝑁 − 𝑘) = (𝑁 − 0)) |
96 | 95 | oveq1d 7284 |
. . . . . . . . . 10
⊢ (𝑘 = 0 → ((𝑁 − 𝑘) + 1) = ((𝑁 − 0) + 1)) |
97 | 94, 96 | oveq12d 7287 |
. . . . . . . . 9
⊢ (𝑘 = 0 → ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) = ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1))) |
98 | 93, 97 | oveq12d 7287 |
. . . . . . . 8
⊢ (𝑘 = 0 → ((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) = ((𝑁C0) · ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1)))) |
99 | 76, 35, 98 | fsum1p 15461 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) = (((𝑁C0) · ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))))) |
100 | | bpoly0 15756 |
. . . . . . . . . . 11
⊢ ((𝑋 + 1) ∈ ℂ → (0
BernPoly (𝑋 + 1)) =
1) |
101 | 5, 100 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (0 BernPoly (𝑋 + 1)) = 1) |
102 | 101 | oveq1d 7284 |
. . . . . . . . 9
⊢ (𝜑 → ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1)) = (1 / ((𝑁 − 0) + 1))) |
103 | 102 | oveq2d 7285 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁C0) · ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1))) = ((𝑁C0) · (1 / ((𝑁 − 0) + 1)))) |
104 | 103 | oveq1d 7284 |
. . . . . . 7
⊢ (𝜑 → (((𝑁C0) · ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))) = (((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))))) |
105 | 99, 104 | eqtrd 2780 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) = (((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))))) |
106 | | oveq1 7276 |
. . . . . . . . . 10
⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋)) |
107 | 106, 96 | oveq12d 7287 |
. . . . . . . . 9
⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1))) |
108 | 93, 107 | oveq12d 7287 |
. . . . . . . 8
⊢ (𝑘 = 0 → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = ((𝑁C0) · ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1)))) |
109 | 76, 41, 108 | fsum1p 15461 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = (((𝑁C0) · ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) |
110 | | bpoly0 15756 |
. . . . . . . . . . 11
⊢ (𝑋 ∈ ℂ → (0
BernPoly 𝑋) =
1) |
111 | 3, 110 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (0 BernPoly 𝑋) = 1) |
112 | 111 | oveq1d 7284 |
. . . . . . . . 9
⊢ (𝜑 → ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1)) = (1 / ((𝑁 − 0) + 1))) |
113 | 112 | oveq2d 7285 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁C0) · ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1))) = ((𝑁C0) · (1 / ((𝑁 − 0) + 1)))) |
114 | 113 | oveq1d 7284 |
. . . . . . 7
⊢ (𝜑 → (((𝑁C0) · ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = (((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) |
115 | 109, 114 | eqtrd 2780 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = (((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) |
116 | 105, 115 | oveq12d 7287 |
. . . . 5
⊢ (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = ((((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))) − (((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))) |
117 | | 0z 12328 |
. . . . . . . . 9
⊢ 0 ∈
ℤ |
118 | | bccl 14032 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ0
∧ 0 ∈ ℤ) → (𝑁C0) ∈
ℕ0) |
119 | 2, 117, 118 | sylancl 586 |
. . . . . . . 8
⊢ (𝜑 → (𝑁C0) ∈
ℕ0) |
120 | 119 | nn0cnd 12293 |
. . . . . . 7
⊢ (𝜑 → (𝑁C0) ∈ ℂ) |
121 | 21 | subid1d 11319 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁 − 0) = 𝑁) |
122 | 121, 1 | eqeltrd 2841 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 − 0) ∈ ℕ) |
123 | 122 | peano2nnd 11988 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁 − 0) + 1) ∈
ℕ) |
124 | 123 | nnrecred 12022 |
. . . . . . . 8
⊢ (𝜑 → (1 / ((𝑁 − 0) + 1)) ∈
ℝ) |
125 | 124 | recnd 11002 |
. . . . . . 7
⊢ (𝜑 → (1 / ((𝑁 − 0) + 1)) ∈
ℂ) |
126 | 120, 125 | mulcld 10994 |
. . . . . 6
⊢ (𝜑 → ((𝑁C0) · (1 / ((𝑁 − 0) + 1))) ∈
ℂ) |
127 | | fzfid 13689 |
. . . . . . 7
⊢ (𝜑 → ((0 + 1)...(𝑁 − 1)) ∈
Fin) |
128 | | fzp1ss 13304 |
. . . . . . . . . 10
⊢ (0 ∈
ℤ → ((0 + 1)...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))) |
129 | 117, 128 | ax-mp 5 |
. . . . . . . . 9
⊢ ((0 +
1)...(𝑁 − 1)) ⊆
(0...(𝑁 −
1)) |
130 | 129 | sseli 3922 |
. . . . . . . 8
⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → 𝑘 ∈ (0...(𝑁 − 1))) |
131 | 130, 35 | sylan2 593 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) ∈ ℂ) |
132 | 127, 131 | fsumcl 15441 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) ∈ ℂ) |
133 | 130, 41 | sylan2 593 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ) |
134 | 127, 133 | fsumcl 15441 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ) |
135 | 126, 132,
134 | pnpcand 11367 |
. . . . 5
⊢ (𝜑 → ((((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))) − (((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) = (Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) |
136 | | 1zzd 12349 |
. . . . . . . 8
⊢ (𝜑 → 1 ∈
ℤ) |
137 | | 0zd 12329 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℤ) |
138 | 1 | nnzd 12422 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℤ) |
139 | | 2z 12350 |
. . . . . . . . 9
⊢ 2 ∈
ℤ |
140 | | zsubcl 12360 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℤ ∧ 2 ∈
ℤ) → (𝑁 −
2) ∈ ℤ) |
141 | 138, 139,
140 | sylancl 586 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 − 2) ∈ ℤ) |
142 | | fzssp1 13296 |
. . . . . . . . . . 11
⊢
(0...(𝑁 − 2))
⊆ (0...((𝑁 − 2)
+ 1)) |
143 | | 2cnd 12049 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 2 ∈
ℂ) |
144 | 21, 143, 81 | subsubd 11358 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 − (2 − 1)) = ((𝑁 − 2) +
1)) |
145 | | 2m1e1 12097 |
. . . . . . . . . . . . . 14
⊢ (2
− 1) = 1 |
146 | 145 | oveq2i 7280 |
. . . . . . . . . . . . 13
⊢ (𝑁 − (2 − 1)) = (𝑁 − 1) |
147 | 144, 146 | eqtr3di 2795 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑁 − 2) + 1) = (𝑁 − 1)) |
148 | 147 | oveq2d 7285 |
. . . . . . . . . . 11
⊢ (𝜑 → (0...((𝑁 − 2) + 1)) = (0...(𝑁 − 1))) |
149 | 142, 148 | sseqtrid 3978 |
. . . . . . . . . 10
⊢ (𝜑 → (0...(𝑁 − 2)) ⊆ (0...(𝑁 − 1))) |
150 | 149 | sselda 3926 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑁 − 2))) → 𝑚 ∈ (0...(𝑁 − 1))) |
151 | 150, 52 | syldan 591 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑁 − 2))) → ((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ) |
152 | | oveq2 7277 |
. . . . . . . . 9
⊢ (𝑚 = (𝑘 − 1) → (𝑁C𝑚) = (𝑁C(𝑘 − 1))) |
153 | | oveq2 7277 |
. . . . . . . . 9
⊢ (𝑚 = (𝑘 − 1) → (𝑋↑𝑚) = (𝑋↑(𝑘 − 1))) |
154 | 152, 153 | oveq12d 7287 |
. . . . . . . 8
⊢ (𝑚 = (𝑘 − 1) → ((𝑁C𝑚) · (𝑋↑𝑚)) = ((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1)))) |
155 | 136, 137,
141, 151, 154 | fsumshft 15488 |
. . . . . . 7
⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) = Σ𝑘 ∈ ((0 + 1)...((𝑁 − 2) + 1))((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1)))) |
156 | 147 | oveq2d 7285 |
. . . . . . . 8
⊢ (𝜑 → ((0 + 1)...((𝑁 − 2) + 1)) = ((0 +
1)...(𝑁 −
1))) |
157 | 156 | sumeq1d 15409 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...((𝑁 − 2) + 1))((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))) = Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1)))) |
158 | 155, 157 | eqtrd 2780 |
. . . . . 6
⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) = Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1)))) |
159 | | 0p1e1 12093 |
. . . . . . . . . 10
⊢ (0 + 1) =
1 |
160 | 159 | oveq1i 7279 |
. . . . . . . . 9
⊢ ((0 +
1)...(𝑁 − 1)) =
(1...(𝑁 −
1)) |
161 | 160 | eleq2i 2832 |
. . . . . . . 8
⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) ↔ 𝑘 ∈ (1...(𝑁 − 1))) |
162 | | fzssp1 13296 |
. . . . . . . . . . . . . 14
⊢
(1...(𝑁 − 1))
⊆ (1...((𝑁 − 1)
+ 1)) |
163 | 24 | oveq2d 7285 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁)) |
164 | 162, 163 | sseqtrid 3978 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
165 | 164 | sselda 3926 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ (1...𝑁)) |
166 | | bcm1k 14025 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ (1...𝑁) → (𝑁C𝑘) = ((𝑁C(𝑘 − 1)) · ((𝑁 − (𝑘 − 1)) / 𝑘))) |
167 | 165, 166 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁C𝑘) = ((𝑁C(𝑘 − 1)) · ((𝑁 − (𝑘 − 1)) / 𝑘))) |
168 | 1 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ) |
169 | 168 | nncnd 11987 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℂ) |
170 | | elfznn 13282 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (1...(𝑁 − 1)) → 𝑘 ∈ ℕ) |
171 | 170 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℕ) |
172 | 171 | nncnd 11987 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℂ) |
173 | | 1cnd 10969 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 1 ∈
ℂ) |
174 | 169, 172,
173 | subsubd 11358 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁 − (𝑘 − 1)) = ((𝑁 − 𝑘) + 1)) |
175 | 174 | oveq1d 7284 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁 − (𝑘 − 1)) / 𝑘) = (((𝑁 − 𝑘) + 1) / 𝑘)) |
176 | 175 | oveq2d 7285 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁C(𝑘 − 1)) · ((𝑁 − (𝑘 − 1)) / 𝑘)) = ((𝑁C(𝑘 − 1)) · (((𝑁 − 𝑘) + 1) / 𝑘))) |
177 | 167, 176 | eqtrd 2780 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁C𝑘) = ((𝑁C(𝑘 − 1)) · (((𝑁 − 𝑘) + 1) / 𝑘))) |
178 | | bpolydiflem.3 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1)))) |
179 | 178 | oveq1d 7284 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) / ((𝑁 − 𝑘) + 1)) = ((𝑘 · (𝑋↑(𝑘 − 1))) / ((𝑁 − 𝑘) + 1))) |
180 | 161, 130 | sylbir 234 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (1...(𝑁 − 1)) → 𝑘 ∈ (0...(𝑁 − 1))) |
181 | 180, 19 | sylan2 593 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 BernPoly (𝑋 + 1)) ∈ ℂ) |
182 | 180, 39 | sylan2 593 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 BernPoly 𝑋) ∈ ℂ) |
183 | 180, 32 | sylan2 593 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ∈ ℂ) |
184 | 180, 33 | sylan2 593 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ≠ 0) |
185 | 181, 182,
183, 184 | divsubdird 11788 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) / ((𝑁 − 𝑘) + 1)) = (((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) − ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) |
186 | 3 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℂ) |
187 | | nnm1nn0 12272 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈
ℕ0) |
188 | 171, 187 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 − 1) ∈
ℕ0) |
189 | 186, 188 | expcld 13860 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑋↑(𝑘 − 1)) ∈ ℂ) |
190 | 172, 189,
183, 184 | div23d 11786 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 · (𝑋↑(𝑘 − 1))) / ((𝑁 − 𝑘) + 1)) = ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1)))) |
191 | 179, 185,
190 | 3eqtr3d 2788 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) − ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1)))) |
192 | 177, 191 | oveq12d 7287 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁C𝑘) · (((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) − ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = (((𝑁C(𝑘 − 1)) · (((𝑁 − 𝑘) + 1) / 𝑘)) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1))))) |
193 | 180, 16 | sylan2 593 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁C𝑘) ∈ ℂ) |
194 | 181, 183,
184 | divcld 11749 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) ∈ ℂ) |
195 | 182, 183,
184 | divcld 11749 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) ∈ ℂ) |
196 | 193, 194,
195 | subdid 11429 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁C𝑘) · (((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) − ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = (((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) |
197 | 168 | nnnn0d 12291 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑁 ∈
ℕ0) |
198 | 188 | nn0zd 12421 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 − 1) ∈ ℤ) |
199 | | bccl 14032 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ0
∧ (𝑘 − 1) ∈
ℤ) → (𝑁C(𝑘 − 1)) ∈
ℕ0) |
200 | 197, 198,
199 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁C(𝑘 − 1)) ∈
ℕ0) |
201 | 200 | nn0cnd 12293 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁C(𝑘 − 1)) ∈ ℂ) |
202 | 171 | nnne0d 12021 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ≠ 0) |
203 | 183, 172,
202 | divcld 11749 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑁 − 𝑘) + 1) / 𝑘) ∈ ℂ) |
204 | 172, 183,
184 | divcld 11749 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 / ((𝑁 − 𝑘) + 1)) ∈ ℂ) |
205 | 204, 189 | mulcld 10994 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1))) ∈
ℂ) |
206 | 201, 203,
205 | mulassd 10997 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑁C(𝑘 − 1)) · (((𝑁 − 𝑘) + 1) / 𝑘)) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1)))) = ((𝑁C(𝑘 − 1)) · ((((𝑁 − 𝑘) + 1) / 𝑘) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1)))))) |
207 | 183, 172,
184, 202 | divcan6d 11768 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((((𝑁 − 𝑘) + 1) / 𝑘) · (𝑘 / ((𝑁 − 𝑘) + 1))) = 1) |
208 | 207 | oveq1d 7284 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((((𝑁 − 𝑘) + 1) / 𝑘) · (𝑘 / ((𝑁 − 𝑘) + 1))) · (𝑋↑(𝑘 − 1))) = (1 · (𝑋↑(𝑘 − 1)))) |
209 | 203, 204,
189 | mulassd 10997 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((((𝑁 − 𝑘) + 1) / 𝑘) · (𝑘 / ((𝑁 − 𝑘) + 1))) · (𝑋↑(𝑘 − 1))) = ((((𝑁 − 𝑘) + 1) / 𝑘) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1))))) |
210 | 189 | mulid2d 10992 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (1 · (𝑋↑(𝑘 − 1))) = (𝑋↑(𝑘 − 1))) |
211 | 208, 209,
210 | 3eqtr3d 2788 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((((𝑁 − 𝑘) + 1) / 𝑘) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1)))) = (𝑋↑(𝑘 − 1))) |
212 | 211 | oveq2d 7285 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁C(𝑘 − 1)) · ((((𝑁 − 𝑘) + 1) / 𝑘) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1))))) = ((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1)))) |
213 | 206, 212 | eqtrd 2780 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑁C(𝑘 − 1)) · (((𝑁 − 𝑘) + 1) / 𝑘)) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1)))) = ((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1)))) |
214 | 192, 196,
213 | 3eqtr3d 2788 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = ((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1)))) |
215 | 161, 214 | sylan2b 594 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 − 1))) → (((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = ((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1)))) |
216 | 215 | sumeq2dv 15411 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))(((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1)))) |
217 | 127, 131,
133 | fsumsub 15496 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))(((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = (Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) |
218 | 158, 216,
217 | 3eqtr2rd 2787 |
. . . . 5
⊢ (𝜑 → (Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚))) |
219 | 116, 135,
218 | 3eqtrd 2784 |
. . . 4
⊢ (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚))) |
220 | 92, 219 | oveq12d 7287 |
. . 3
⊢ (𝜑 → ((((𝑋 + 1)↑𝑁) − (𝑋↑𝑁)) − (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) = ((Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑁 · (𝑋↑(𝑁 − 1)))) − Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)))) |
221 | | fzfid 13689 |
. . . . 5
⊢ (𝜑 → (0...(𝑁 − 2)) ∈ Fin) |
222 | 221, 151 | fsumcl 15441 |
. . . 4
⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ) |
223 | 3, 75 | expcld 13860 |
. . . . 5
⊢ (𝜑 → (𝑋↑(𝑁 − 1)) ∈
ℂ) |
224 | 21, 223 | mulcld 10994 |
. . . 4
⊢ (𝜑 → (𝑁 · (𝑋↑(𝑁 − 1))) ∈
ℂ) |
225 | 222, 224 | pncan2d 11332 |
. . 3
⊢ (𝜑 → ((Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑁 · (𝑋↑(𝑁 − 1)))) − Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚))) = (𝑁 · (𝑋↑(𝑁 − 1)))) |
226 | 220, 225 | eqtrd 2780 |
. 2
⊢ (𝜑 → ((((𝑋 + 1)↑𝑁) − (𝑋↑𝑁)) − (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) = (𝑁 · (𝑋↑(𝑁 − 1)))) |
227 | 10, 43, 226 | 3eqtrd 2784 |
1
⊢ (𝜑 → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1)))) |