Proof of Theorem bpolysum
Step | Hyp | Ref
| Expression |
1 | | simpl 486 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ 𝑁 ∈
ℕ0) |
2 | | nn0uz 12501 |
. . . 4
⊢
ℕ0 = (ℤ≥‘0) |
3 | 1, 2 | eleqtrdi 2849 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ 𝑁 ∈
(ℤ≥‘0)) |
4 | | elfzelz 13137 |
. . . . . 6
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ) |
5 | | bccl 13916 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ (𝑁C𝑘) ∈
ℕ0) |
6 | 1, 4, 5 | syl2an 599 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈
ℕ0) |
7 | 6 | nn0cnd 12177 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℂ) |
8 | | elfznn0 13230 |
. . . . . 6
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) |
9 | | simpr 488 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ 𝑋 ∈
ℂ) |
10 | | bpolycl 15642 |
. . . . . 6
⊢ ((𝑘 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (𝑘 BernPoly 𝑋) ∈
ℂ) |
11 | 8, 9, 10 | syl2anr 600 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
∧ 𝑘 ∈ (0...𝑁)) → (𝑘 BernPoly 𝑋) ∈ ℂ) |
12 | | fznn0sub 13169 |
. . . . . . . 8
⊢ (𝑘 ∈ (0...𝑁) → (𝑁 − 𝑘) ∈
ℕ0) |
13 | 12 | adantl 485 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
∧ 𝑘 ∈ (0...𝑁)) → (𝑁 − 𝑘) ∈
ℕ0) |
14 | | nn0p1nn 12154 |
. . . . . . 7
⊢ ((𝑁 − 𝑘) ∈ ℕ0 → ((𝑁 − 𝑘) + 1) ∈ ℕ) |
15 | 13, 14 | syl 17 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
∧ 𝑘 ∈ (0...𝑁)) → ((𝑁 − 𝑘) + 1) ∈ ℕ) |
16 | 15 | nncnd 11871 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
∧ 𝑘 ∈ (0...𝑁)) → ((𝑁 − 𝑘) + 1) ∈ ℂ) |
17 | 15 | nnne0d 11905 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
∧ 𝑘 ∈ (0...𝑁)) → ((𝑁 − 𝑘) + 1) ≠ 0) |
18 | 11, 16, 17 | divcld 11633 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
∧ 𝑘 ∈ (0...𝑁)) → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) ∈ ℂ) |
19 | 7, 18 | mulcld 10878 |
. . 3
⊢ (((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ) |
20 | | oveq2 7240 |
. . . 4
⊢ (𝑘 = 𝑁 → (𝑁C𝑘) = (𝑁C𝑁)) |
21 | | oveq1 7239 |
. . . . 5
⊢ (𝑘 = 𝑁 → (𝑘 BernPoly 𝑋) = (𝑁 BernPoly 𝑋)) |
22 | | oveq2 7240 |
. . . . . 6
⊢ (𝑘 = 𝑁 → (𝑁 − 𝑘) = (𝑁 − 𝑁)) |
23 | 22 | oveq1d 7247 |
. . . . 5
⊢ (𝑘 = 𝑁 → ((𝑁 − 𝑘) + 1) = ((𝑁 − 𝑁) + 1)) |
24 | 21, 23 | oveq12d 7250 |
. . . 4
⊢ (𝑘 = 𝑁 → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) = ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1))) |
25 | 20, 24 | oveq12d 7250 |
. . 3
⊢ (𝑘 = 𝑁 → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = ((𝑁C𝑁) · ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1)))) |
26 | 3, 19, 25 | fsumm1 15343 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ Σ𝑘 ∈
(0...𝑁)((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) + ((𝑁C𝑁) · ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1))))) |
27 | | bcnn 13906 |
. . . . . 6
⊢ (𝑁 ∈ ℕ0
→ (𝑁C𝑁) = 1) |
28 | 27 | adantr 484 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (𝑁C𝑁) = 1) |
29 | | nn0cn 12125 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℂ) |
30 | 29 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ 𝑁 ∈
ℂ) |
31 | 30 | subidd 11202 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (𝑁 − 𝑁) = 0) |
32 | 31 | oveq1d 7247 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ ((𝑁 − 𝑁) + 1) = (0 +
1)) |
33 | | 0p1e1 11977 |
. . . . . . . 8
⊢ (0 + 1) =
1 |
34 | 32, 33 | eqtrdi 2795 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ ((𝑁 − 𝑁) + 1) = 1) |
35 | 34 | oveq2d 7248 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1)) = ((𝑁 BernPoly 𝑋) / 1)) |
36 | | bpolycl 15642 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (𝑁 BernPoly 𝑋) ∈
ℂ) |
37 | 36 | div1d 11625 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ ((𝑁 BernPoly 𝑋) / 1) = (𝑁 BernPoly 𝑋)) |
38 | 35, 37 | eqtrd 2778 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1)) = (𝑁 BernPoly 𝑋)) |
39 | 28, 38 | oveq12d 7250 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ ((𝑁C𝑁) · ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1))) = (1 · (𝑁 BernPoly 𝑋))) |
40 | 36 | mulid2d 10876 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (1 · (𝑁
BernPoly 𝑋)) = (𝑁 BernPoly 𝑋)) |
41 | 39, 40 | eqtrd 2778 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ ((𝑁C𝑁) · ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1))) = (𝑁 BernPoly 𝑋)) |
42 | 41 | oveq2d 7248 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (Σ𝑘 ∈
(0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) + ((𝑁C𝑁) · ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1)))) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) + (𝑁 BernPoly 𝑋))) |
43 | | bpolyval 15639 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) |
44 | 43 | eqcomd 2744 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = (𝑁 BernPoly 𝑋)) |
45 | | expcl 13680 |
. . . . 5
⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ (𝑋↑𝑁) ∈
ℂ) |
46 | 45 | ancoms 462 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (𝑋↑𝑁) ∈
ℂ) |
47 | | fzfid 13573 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (0...(𝑁 − 1))
∈ Fin) |
48 | | fzssp1 13180 |
. . . . . . . 8
⊢
(0...(𝑁 − 1))
⊆ (0...((𝑁 − 1)
+ 1)) |
49 | | ax-1cn 10812 |
. . . . . . . . . 10
⊢ 1 ∈
ℂ |
50 | | npcan 11112 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑁 −
1) + 1) = 𝑁) |
51 | 30, 49, 50 | sylancl 589 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ ((𝑁 − 1) + 1)
= 𝑁) |
52 | 51 | oveq2d 7248 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (0...((𝑁 − 1)
+ 1)) = (0...𝑁)) |
53 | 48, 52 | sseqtrid 3968 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (0...(𝑁 − 1))
⊆ (0...𝑁)) |
54 | 53 | sselda 3916 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ (0...𝑁)) |
55 | 54, 19 | syldan 594 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ) |
56 | 47, 55 | fsumcl 15325 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ Σ𝑘 ∈
(0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ) |
57 | 46, 56, 36 | subaddd 11232 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = (𝑁 BernPoly 𝑋) ↔ (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) + (𝑁 BernPoly 𝑋)) = (𝑋↑𝑁))) |
58 | 44, 57 | mpbid 235 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ (Σ𝑘 ∈
(0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) + (𝑁 BernPoly 𝑋)) = (𝑋↑𝑁)) |
59 | 26, 42, 58 | 3eqtrd 2782 |
1
⊢ ((𝑁 ∈ ℕ0
∧ 𝑋 ∈ ℂ)
→ Σ𝑘 ∈
(0...𝑁)((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = (𝑋↑𝑁)) |