Proof of Theorem binom1dif
Step | Hyp | Ref
| Expression |
1 | | fzfid 13546 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ (0...(𝑁 − 1))
∈ Fin) |
2 | | fzssp1 13155 |
. . . . . 6
⊢
(0...(𝑁 − 1))
⊆ (0...((𝑁 − 1)
+ 1)) |
3 | | nn0cn 12100 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℂ) |
4 | 3 | adantl 485 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ 𝑁 ∈
ℂ) |
5 | | ax-1cn 10787 |
. . . . . . . 8
⊢ 1 ∈
ℂ |
6 | | npcan 11087 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑁 −
1) + 1) = 𝑁) |
7 | 4, 5, 6 | sylancl 589 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ ((𝑁 − 1) + 1)
= 𝑁) |
8 | 7 | oveq2d 7229 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ (0...((𝑁 − 1)
+ 1)) = (0...𝑁)) |
9 | 2, 8 | sseqtrid 3953 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ (0...(𝑁 − 1))
⊆ (0...𝑁)) |
10 | 9 | sselda 3901 |
. . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ (0...𝑁)) |
11 | | bccl2 13889 |
. . . . . . 7
⊢ (𝑘 ∈ (0...𝑁) → (𝑁C𝑘) ∈ ℕ) |
12 | 11 | adantl 485 |
. . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℕ) |
13 | 12 | nncnd 11846 |
. . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℂ) |
14 | | simpl 486 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ 𝐴 ∈
ℂ) |
15 | | elfznn0 13205 |
. . . . . 6
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) |
16 | | expcl 13653 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ (𝐴↑𝑘) ∈
ℂ) |
17 | 14, 15, 16 | syl2an 599 |
. . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑𝑘) ∈ ℂ) |
18 | 13, 17 | mulcld 10853 |
. . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · (𝐴↑𝑘)) ∈ ℂ) |
19 | 10, 18 | syldan 594 |
. . 3
⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁C𝑘) · (𝐴↑𝑘)) ∈ ℂ) |
20 | 1, 19 | fsumcl 15297 |
. 2
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ Σ𝑘 ∈
(0...(𝑁 − 1))((𝑁C𝑘) · (𝐴↑𝑘)) ∈ ℂ) |
21 | | expcl 13653 |
. 2
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ (𝐴↑𝑁) ∈
ℂ) |
22 | | addcom 11018 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 1 ∈
ℂ) → (𝐴 + 1) =
(1 + 𝐴)) |
23 | 14, 5, 22 | sylancl 589 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ (𝐴 + 1) = (1 + 𝐴)) |
24 | 23 | oveq1d 7228 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ ((𝐴 + 1)↑𝑁) = ((1 + 𝐴)↑𝑁)) |
25 | | binom1p 15395 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ ((1 + 𝐴)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (𝐴↑𝑘))) |
26 | | simpr 488 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ 𝑁 ∈
ℕ0) |
27 | | nn0uz 12476 |
. . . . . 6
⊢
ℕ0 = (ℤ≥‘0) |
28 | 26, 27 | eleqtrdi 2848 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ 𝑁 ∈
(ℤ≥‘0)) |
29 | | oveq2 7221 |
. . . . . 6
⊢ (𝑘 = 𝑁 → (𝑁C𝑘) = (𝑁C𝑁)) |
30 | | oveq2 7221 |
. . . . . 6
⊢ (𝑘 = 𝑁 → (𝐴↑𝑘) = (𝐴↑𝑁)) |
31 | 29, 30 | oveq12d 7231 |
. . . . 5
⊢ (𝑘 = 𝑁 → ((𝑁C𝑘) · (𝐴↑𝑘)) = ((𝑁C𝑁) · (𝐴↑𝑁))) |
32 | 28, 18, 31 | fsumm1 15315 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ Σ𝑘 ∈
(0...𝑁)((𝑁C𝑘) · (𝐴↑𝑘)) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · (𝐴↑𝑘)) + ((𝑁C𝑁) · (𝐴↑𝑁)))) |
33 | | bcnn 13878 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ0
→ (𝑁C𝑁) = 1) |
34 | 33 | adantl 485 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ (𝑁C𝑁) = 1) |
35 | 34 | oveq1d 7228 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ ((𝑁C𝑁) · (𝐴↑𝑁)) = (1 · (𝐴↑𝑁))) |
36 | 21 | mulid2d 10851 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ (1 · (𝐴↑𝑁)) = (𝐴↑𝑁)) |
37 | 35, 36 | eqtrd 2777 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ ((𝑁C𝑁) · (𝐴↑𝑁)) = (𝐴↑𝑁)) |
38 | 37 | oveq2d 7229 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ (Σ𝑘 ∈
(0...(𝑁 − 1))((𝑁C𝑘) · (𝐴↑𝑘)) + ((𝑁C𝑁) · (𝐴↑𝑁))) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · (𝐴↑𝑘)) + (𝐴↑𝑁))) |
39 | 32, 38 | eqtrd 2777 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ Σ𝑘 ∈
(0...𝑁)((𝑁C𝑘) · (𝐴↑𝑘)) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · (𝐴↑𝑘)) + (𝐴↑𝑁))) |
40 | 24, 25, 39 | 3eqtrd 2781 |
. 2
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ ((𝐴 + 1)↑𝑁) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · (𝐴↑𝑘)) + (𝐴↑𝑁))) |
41 | 20, 21, 40 | mvrraddd 11244 |
1
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0)
→ (((𝐴 + 1)↑𝑁) − (𝐴↑𝑁)) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · (𝐴↑𝑘))) |