Proof of Theorem bcpasc
Step | Hyp | Ref
| Expression |
1 | | peano2nn0 11333 |
. . . . . 6
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℕ0) |
2 | | elfzp12 12419 |
. . . . . . 7
⊢ ((𝑁 + 1) ∈
(ℤ≥‘0) → (𝐾 ∈ (0...(𝑁 + 1)) ↔ (𝐾 = 0 ∨ 𝐾 ∈ ((0 + 1)...(𝑁 + 1))))) |
3 | | nn0uz 11722 |
. . . . . . 7
⊢
ℕ0 = (ℤ≥‘0) |
4 | 2, 3 | eleq2s 2719 |
. . . . . 6
⊢ ((𝑁 + 1) ∈ ℕ0
→ (𝐾 ∈
(0...(𝑁 + 1)) ↔ (𝐾 = 0 ∨ 𝐾 ∈ ((0 + 1)...(𝑁 + 1))))) |
5 | 1, 4 | syl 17 |
. . . . 5
⊢ (𝑁 ∈ ℕ0
→ (𝐾 ∈
(0...(𝑁 + 1)) ↔ (𝐾 = 0 ∨ 𝐾 ∈ ((0 + 1)...(𝑁 + 1))))) |
6 | | 1p0e1 11133 |
. . . . . . . 8
⊢ (1 + 0) =
1 |
7 | | bcn0 13097 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ0
→ (𝑁C0) =
1) |
8 | | 0z 11388 |
. . . . . . . . . . 11
⊢ 0 ∈
ℤ |
9 | | 1z 11407 |
. . . . . . . . . . 11
⊢ 1 ∈
ℤ |
10 | | zsubcl 11419 |
. . . . . . . . . . 11
⊢ ((0
∈ ℤ ∧ 1 ∈ ℤ) → (0 − 1) ∈
ℤ) |
11 | 8, 9, 10 | mp2an 708 |
. . . . . . . . . 10
⊢ (0
− 1) ∈ ℤ |
12 | | 0re 10040 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℝ |
13 | | ltm1 10863 |
. . . . . . . . . . . 12
⊢ (0 ∈
ℝ → (0 − 1) < 0) |
14 | 12, 13 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (0
− 1) < 0 |
15 | 14 | orci 405 |
. . . . . . . . . 10
⊢ ((0
− 1) < 0 ∨ 𝑁
< (0 − 1)) |
16 | | bcval4 13094 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ (0 − 1) ∈ ℤ ∧ ((0 − 1) < 0 ∨ 𝑁 < (0 − 1))) →
(𝑁C(0 − 1)) =
0) |
17 | 11, 15, 16 | mp3an23 1416 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ0
→ (𝑁C(0 − 1)) =
0) |
18 | 7, 17 | oveq12d 6668 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ0
→ ((𝑁C0) + (𝑁C(0 − 1))) = (1 +
0)) |
19 | | bcn0 13097 |
. . . . . . . . 9
⊢ ((𝑁 + 1) ∈ ℕ0
→ ((𝑁 + 1)C0) =
1) |
20 | 1, 19 | syl 17 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ0
→ ((𝑁 + 1)C0) =
1) |
21 | 6, 18, 20 | 3eqtr4a 2682 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ0
→ ((𝑁C0) + (𝑁C(0 − 1))) = ((𝑁 + 1)C0)) |
22 | | oveq2 6658 |
. . . . . . . . 9
⊢ (𝐾 = 0 → (𝑁C𝐾) = (𝑁C0)) |
23 | | oveq1 6657 |
. . . . . . . . . 10
⊢ (𝐾 = 0 → (𝐾 − 1) = (0 −
1)) |
24 | 23 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝐾 = 0 → (𝑁C(𝐾 − 1)) = (𝑁C(0 − 1))) |
25 | 22, 24 | oveq12d 6668 |
. . . . . . . 8
⊢ (𝐾 = 0 → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁C0) + (𝑁C(0 − 1)))) |
26 | | oveq2 6658 |
. . . . . . . 8
⊢ (𝐾 = 0 → ((𝑁 + 1)C𝐾) = ((𝑁 + 1)C0)) |
27 | 25, 26 | eqeq12d 2637 |
. . . . . . 7
⊢ (𝐾 = 0 → (((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾) ↔ ((𝑁C0) + (𝑁C(0 − 1))) = ((𝑁 + 1)C0))) |
28 | 21, 27 | syl5ibrcom 237 |
. . . . . 6
⊢ (𝑁 ∈ ℕ0
→ (𝐾 = 0 →
((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾))) |
29 | | simpr 477 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ((0 +
1)...(𝑁 + 1))) → 𝐾 ∈ ((0 + 1)...(𝑁 + 1))) |
30 | | 0p1e1 11132 |
. . . . . . . . . 10
⊢ (0 + 1) =
1 |
31 | 30 | oveq1i 6660 |
. . . . . . . . 9
⊢ ((0 +
1)...(𝑁 + 1)) = (1...(𝑁 + 1)) |
32 | 29, 31 | syl6eleq 2711 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ((0 +
1)...(𝑁 + 1))) → 𝐾 ∈ (1...(𝑁 + 1))) |
33 | | nn0p1nn 11332 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℕ) |
34 | | nnuz 11723 |
. . . . . . . . . . 11
⊢ ℕ =
(ℤ≥‘1) |
35 | 33, 34 | syl6eleq 2711 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
(ℤ≥‘1)) |
36 | | fzm1 12420 |
. . . . . . . . . . 11
⊢ ((𝑁 + 1) ∈
(ℤ≥‘1) → (𝐾 ∈ (1...(𝑁 + 1)) ↔ (𝐾 ∈ (1...((𝑁 + 1) − 1)) ∨ 𝐾 = (𝑁 + 1)))) |
37 | 36 | biimpa 501 |
. . . . . . . . . 10
⊢ (((𝑁 + 1) ∈
(ℤ≥‘1) ∧ 𝐾 ∈ (1...(𝑁 + 1))) → (𝐾 ∈ (1...((𝑁 + 1) − 1)) ∨ 𝐾 = (𝑁 + 1))) |
38 | 35, 37 | sylan 488 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ (1...(𝑁 + 1))) → (𝐾 ∈ (1...((𝑁 + 1) − 1)) ∨ 𝐾 = (𝑁 + 1))) |
39 | | nn0cn 11302 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℂ) |
40 | | ax-1cn 9994 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
ℂ |
41 | | pncan 10287 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑁 + 1)
− 1) = 𝑁) |
42 | 39, 40, 41 | sylancl 694 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℕ0
→ ((𝑁 + 1) − 1)
= 𝑁) |
43 | 42 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ0
→ (1...((𝑁 + 1)
− 1)) = (1...𝑁)) |
44 | 43 | eleq2d 2687 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ0
→ (𝐾 ∈
(1...((𝑁 + 1) − 1))
↔ 𝐾 ∈ (1...𝑁))) |
45 | 44 | biimpa 501 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ (1...((𝑁 + 1) − 1))) → 𝐾 ∈ (1...𝑁)) |
46 | | 1eluzge0 11732 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
(ℤ≥‘0) |
47 | | fzss1 12380 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
(ℤ≥‘0) → (1...𝑁) ⊆ (0...𝑁)) |
48 | 46, 47 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢
(1...𝑁) ⊆
(0...𝑁) |
49 | 48 | sseli 3599 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ (0...𝑁)) |
50 | | bcp1n 13103 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∈ (0...𝑁) → ((𝑁 + 1)C𝐾) = ((𝑁C𝐾) · ((𝑁 + 1) / ((𝑁 + 1) − 𝐾)))) |
51 | 49, 50 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁 + 1)C𝐾) = ((𝑁C𝐾) · ((𝑁 + 1) / ((𝑁 + 1) − 𝐾)))) |
52 | | bcrpcl 13095 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈
ℝ+) |
53 | 49, 52 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (1...𝑁) → (𝑁C𝐾) ∈
ℝ+) |
54 | 53 | rpcnd 11874 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ (1...𝑁) → (𝑁C𝐾) ∈ ℂ) |
55 | | elfzuz2 12346 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈ (1...𝑁) → 𝑁 ∈
(ℤ≥‘1)) |
56 | 55, 34 | syl6eleqr 2712 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ (1...𝑁) → 𝑁 ∈ ℕ) |
57 | 56 | peano2nnd 11037 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (1...𝑁) → (𝑁 + 1) ∈ ℕ) |
58 | 57 | nncnd 11036 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ (1...𝑁) → (𝑁 + 1) ∈ ℂ) |
59 | 56 | nncnd 11036 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈ (1...𝑁) → 𝑁 ∈ ℂ) |
60 | | 1cnd 10056 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈ (1...𝑁) → 1 ∈ ℂ) |
61 | | elfzelz 12342 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℤ) |
62 | 61 | zcnd 11483 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℂ) |
63 | 59, 60, 62 | addsubd 10413 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁 + 1) − 𝐾) = ((𝑁 − 𝐾) + 1)) |
64 | | fznn0sub 12373 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈ (1...𝑁) → (𝑁 − 𝐾) ∈
ℕ0) |
65 | | nn0p1nn 11332 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 − 𝐾) ∈ ℕ0 → ((𝑁 − 𝐾) + 1) ∈ ℕ) |
66 | 64, 65 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁 − 𝐾) + 1) ∈ ℕ) |
67 | 63, 66 | eqeltrd 2701 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁 + 1) − 𝐾) ∈ ℕ) |
68 | 67 | nncnd 11036 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁 + 1) − 𝐾) ∈ ℂ) |
69 | 67 | nnne0d 11065 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁 + 1) − 𝐾) ≠ 0) |
70 | 54, 58, 68, 69 | div12d 10837 |
. . . . . . . . . . . . . 14
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁C𝐾) · ((𝑁 + 1) / ((𝑁 + 1) − 𝐾))) = ((𝑁 + 1) · ((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)))) |
71 | 67 | nnrpd 11870 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁 + 1) − 𝐾) ∈
ℝ+) |
72 | 53, 71 | rpdivcld 11889 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) ∈
ℝ+) |
73 | 72 | rpcnd 11874 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) ∈ ℂ) |
74 | 58, 73 | mulcomd 10061 |
. . . . . . . . . . . . . 14
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁 + 1) · ((𝑁C𝐾) / ((𝑁 + 1) − 𝐾))) = (((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · (𝑁 + 1))) |
75 | 70, 74 | eqtrd 2656 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁C𝐾) · ((𝑁 + 1) / ((𝑁 + 1) − 𝐾))) = (((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · (𝑁 + 1))) |
76 | 58, 62 | npcand 10396 |
. . . . . . . . . . . . . 14
⊢ (𝐾 ∈ (1...𝑁) → (((𝑁 + 1) − 𝐾) + 𝐾) = (𝑁 + 1)) |
77 | 76 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∈ (1...𝑁) → (((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · (((𝑁 + 1) − 𝐾) + 𝐾)) = (((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · (𝑁 + 1))) |
78 | 73, 68, 62 | adddid 10064 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∈ (1...𝑁) → (((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · (((𝑁 + 1) − 𝐾) + 𝐾)) = ((((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · ((𝑁 + 1) − 𝐾)) + (((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · 𝐾))) |
79 | 75, 77, 78 | 3eqtr2d 2662 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁C𝐾) · ((𝑁 + 1) / ((𝑁 + 1) − 𝐾))) = ((((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · ((𝑁 + 1) − 𝐾)) + (((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · 𝐾))) |
80 | 54, 68, 69 | divcan1d 10802 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∈ (1...𝑁) → (((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · ((𝑁 + 1) − 𝐾)) = (𝑁C𝐾)) |
81 | | elfznn 12370 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℕ) |
82 | 81 | nnne0d 11065 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ≠ 0) |
83 | 54, 68, 62, 69, 82 | divdiv2d 10833 |
. . . . . . . . . . . . . 14
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁C𝐾) / (((𝑁 + 1) − 𝐾) / 𝐾)) = (((𝑁C𝐾) · 𝐾) / ((𝑁 + 1) − 𝐾))) |
84 | | bcm1k 13102 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (1...𝑁) → (𝑁C𝐾) = ((𝑁C(𝐾 − 1)) · ((𝑁 − (𝐾 − 1)) / 𝐾))) |
85 | 59, 62, 60 | subsub3d 10422 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈ (1...𝑁) → (𝑁 − (𝐾 − 1)) = ((𝑁 + 1) − 𝐾)) |
86 | 85 | oveq1d 6665 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁 − (𝐾 − 1)) / 𝐾) = (((𝑁 + 1) − 𝐾) / 𝐾)) |
87 | 86 | oveq2d 6666 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁C(𝐾 − 1)) · ((𝑁 − (𝐾 − 1)) / 𝐾)) = ((𝑁C(𝐾 − 1)) · (((𝑁 + 1) − 𝐾) / 𝐾))) |
88 | 84, 87 | eqtrd 2656 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ (1...𝑁) → (𝑁C𝐾) = ((𝑁C(𝐾 − 1)) · (((𝑁 + 1) − 𝐾) / 𝐾))) |
89 | | fzelp1 12393 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ (1...(𝑁 + 1))) |
90 | 57 | nnzd 11481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐾 ∈ (1...𝑁) → (𝑁 + 1) ∈ ℤ) |
91 | | elfzm1b 12418 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐾 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) →
(𝐾 ∈ (1...(𝑁 + 1)) ↔ (𝐾 − 1) ∈ (0...((𝑁 + 1) − 1)))) |
92 | 61, 90, 91 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐾 ∈ (1...𝑁) → (𝐾 ∈ (1...(𝑁 + 1)) ↔ (𝐾 − 1) ∈ (0...((𝑁 + 1) − 1)))) |
93 | 89, 92 | mpbid 222 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐾 ∈ (1...𝑁) → (𝐾 − 1) ∈ (0...((𝑁 + 1) − 1))) |
94 | 59, 40, 41 | sylancl 694 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁 + 1) − 1) = 𝑁) |
95 | 94 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐾 ∈ (1...𝑁) → (0...((𝑁 + 1) − 1)) = (0...𝑁)) |
96 | 93, 95 | eleqtrd 2703 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈ (1...𝑁) → (𝐾 − 1) ∈ (0...𝑁)) |
97 | | bcrpcl 13095 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐾 − 1) ∈ (0...𝑁) → (𝑁C(𝐾 − 1)) ∈
ℝ+) |
98 | 96, 97 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ (1...𝑁) → (𝑁C(𝐾 − 1)) ∈
ℝ+) |
99 | 98 | rpcnd 11874 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (1...𝑁) → (𝑁C(𝐾 − 1)) ∈
ℂ) |
100 | 81 | nnrpd 11870 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈
ℝ+) |
101 | 71, 100 | rpdivcld 11889 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ (1...𝑁) → (((𝑁 + 1) − 𝐾) / 𝐾) ∈
ℝ+) |
102 | 101 | rpcnd 11874 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (1...𝑁) → (((𝑁 + 1) − 𝐾) / 𝐾) ∈ ℂ) |
103 | 101 | rpne0d 11877 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ (1...𝑁) → (((𝑁 + 1) − 𝐾) / 𝐾) ≠ 0) |
104 | 54, 99, 102, 103 | divmul3d 10835 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ (1...𝑁) → (((𝑁C𝐾) / (((𝑁 + 1) − 𝐾) / 𝐾)) = (𝑁C(𝐾 − 1)) ↔ (𝑁C𝐾) = ((𝑁C(𝐾 − 1)) · (((𝑁 + 1) − 𝐾) / 𝐾)))) |
105 | 88, 104 | mpbird 247 |
. . . . . . . . . . . . . 14
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁C𝐾) / (((𝑁 + 1) − 𝐾) / 𝐾)) = (𝑁C(𝐾 − 1))) |
106 | 54, 62, 68, 69 | div23d 10838 |
. . . . . . . . . . . . . 14
⊢ (𝐾 ∈ (1...𝑁) → (((𝑁C𝐾) · 𝐾) / ((𝑁 + 1) − 𝐾)) = (((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · 𝐾)) |
107 | 83, 105, 106 | 3eqtr3rd 2665 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∈ (1...𝑁) → (((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · 𝐾) = (𝑁C(𝐾 − 1))) |
108 | 80, 107 | oveq12d 6668 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ (1...𝑁) → ((((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · ((𝑁 + 1) − 𝐾)) + (((𝑁C𝐾) / ((𝑁 + 1) − 𝐾)) · 𝐾)) = ((𝑁C𝐾) + (𝑁C(𝐾 − 1)))) |
109 | 51, 79, 108 | 3eqtrrd 2661 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ (1...𝑁) → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾)) |
110 | 45, 109 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ (1...((𝑁 + 1) − 1))) →
((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾)) |
111 | | oveq2 6658 |
. . . . . . . . . . . . 13
⊢ (𝐾 = (𝑁 + 1) → (𝑁C𝐾) = (𝑁C(𝑁 + 1))) |
112 | 33 | nnzd 11481 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℤ) |
113 | | nn0re 11301 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℝ) |
114 | 113 | ltp1d 10954 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℕ0
→ 𝑁 < (𝑁 + 1)) |
115 | 114 | olcd 408 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℕ0
→ ((𝑁 + 1) < 0 ∨
𝑁 < (𝑁 + 1))) |
116 | | bcval4 13094 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ0
∧ (𝑁 + 1) ∈
ℤ ∧ ((𝑁 + 1) <
0 ∨ 𝑁 < (𝑁 + 1))) → (𝑁C(𝑁 + 1)) = 0) |
117 | 112, 115,
116 | mpd3an23 1426 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ0
→ (𝑁C(𝑁 + 1)) = 0) |
118 | 111, 117 | sylan9eqr 2678 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 = (𝑁 + 1)) → (𝑁C𝐾) = 0) |
119 | | oveq1 6657 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 = (𝑁 + 1) → (𝐾 − 1) = ((𝑁 + 1) − 1)) |
120 | 119, 42 | sylan9eqr 2678 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 = (𝑁 + 1)) → (𝐾 − 1) = 𝑁) |
121 | 120 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 = (𝑁 + 1)) → (𝑁C(𝐾 − 1)) = (𝑁C𝑁)) |
122 | | bcnn 13099 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℕ0
→ (𝑁C𝑁) = 1) |
123 | 122 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 = (𝑁 + 1)) → (𝑁C𝑁) = 1) |
124 | 121, 123 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 = (𝑁 + 1)) → (𝑁C(𝐾 − 1)) = 1) |
125 | 118, 124 | oveq12d 6668 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 = (𝑁 + 1)) → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = (0 + 1)) |
126 | | oveq2 6658 |
. . . . . . . . . . . 12
⊢ (𝐾 = (𝑁 + 1) → ((𝑁 + 1)C𝐾) = ((𝑁 + 1)C(𝑁 + 1))) |
127 | | bcnn 13099 |
. . . . . . . . . . . . 13
⊢ ((𝑁 + 1) ∈ ℕ0
→ ((𝑁 + 1)C(𝑁 + 1)) = 1) |
128 | 1, 127 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ0
→ ((𝑁 + 1)C(𝑁 + 1)) = 1) |
129 | 126, 128 | sylan9eqr 2678 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 = (𝑁 + 1)) → ((𝑁 + 1)C𝐾) = 1) |
130 | 30, 125, 129 | 3eqtr4a 2682 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 = (𝑁 + 1)) → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾)) |
131 | 110, 130 | jaodan 826 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ0
∧ (𝐾 ∈
(1...((𝑁 + 1) − 1))
∨ 𝐾 = (𝑁 + 1))) → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾)) |
132 | 38, 131 | syldan 487 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ (1...(𝑁 + 1))) → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾)) |
133 | 32, 132 | syldan 487 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ((0 +
1)...(𝑁 + 1))) →
((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾)) |
134 | 133 | ex 450 |
. . . . . 6
⊢ (𝑁 ∈ ℕ0
→ (𝐾 ∈ ((0 +
1)...(𝑁 + 1)) →
((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾))) |
135 | 28, 134 | jaod 395 |
. . . . 5
⊢ (𝑁 ∈ ℕ0
→ ((𝐾 = 0 ∨ 𝐾 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾))) |
136 | 5, 135 | sylbid 230 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ (𝐾 ∈
(0...(𝑁 + 1)) →
((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾))) |
137 | 136 | imp 445 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ (0...(𝑁 + 1))) → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾)) |
138 | 137 | adantlr 751 |
. 2
⊢ (((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
∧ 𝐾 ∈ (0...(𝑁 + 1))) → ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾)) |
139 | | 00id 10211 |
. . 3
⊢ (0 + 0) =
0 |
140 | | fzelp1 12393 |
. . . . . 6
⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈ (0...(𝑁 + 1))) |
141 | 140 | con3i 150 |
. . . . 5
⊢ (¬
𝐾 ∈ (0...(𝑁 + 1)) → ¬ 𝐾 ∈ (0...𝑁)) |
142 | | bcval3 13093 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ
∧ ¬ 𝐾 ∈
(0...𝑁)) → (𝑁C𝐾) = 0) |
143 | 142 | 3expa 1265 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
∧ ¬ 𝐾 ∈
(0...𝑁)) → (𝑁C𝐾) = 0) |
144 | 141, 143 | sylan2 491 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
∧ ¬ 𝐾 ∈
(0...(𝑁 + 1))) →
(𝑁C𝐾) = 0) |
145 | | simpll 790 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
∧ ¬ 𝐾 ∈
(0...(𝑁 + 1))) → 𝑁 ∈
ℕ0) |
146 | | simplr 792 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
∧ ¬ 𝐾 ∈
(0...(𝑁 + 1))) → 𝐾 ∈
ℤ) |
147 | | peano2zm 11420 |
. . . . . 6
⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈
ℤ) |
148 | 146, 147 | syl 17 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
∧ ¬ 𝐾 ∈
(0...(𝑁 + 1))) →
(𝐾 − 1) ∈
ℤ) |
149 | 39 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
→ 𝑁 ∈
ℂ) |
150 | 149, 40, 41 | sylancl 694 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
→ ((𝑁 + 1) − 1)
= 𝑁) |
151 | 150 | oveq2d 6666 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
→ (0...((𝑁 + 1)
− 1)) = (0...𝑁)) |
152 | 151 | eleq2d 2687 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
→ ((𝐾 − 1)
∈ (0...((𝑁 + 1)
− 1)) ↔ (𝐾
− 1) ∈ (0...𝑁))) |
153 | | id 22 |
. . . . . . . . 9
⊢ (𝐾 ∈ ℤ → 𝐾 ∈
ℤ) |
154 | 1 | nn0zd 11480 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℤ) |
155 | 153, 154,
91 | syl2anr 495 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
→ (𝐾 ∈
(1...(𝑁 + 1)) ↔ (𝐾 − 1) ∈ (0...((𝑁 + 1) −
1)))) |
156 | | fzp1ss 12392 |
. . . . . . . . . . 11
⊢ (0 ∈
ℤ → ((0 + 1)...(𝑁 + 1)) ⊆ (0...(𝑁 + 1))) |
157 | 8, 156 | ax-mp 5 |
. . . . . . . . . 10
⊢ ((0 +
1)...(𝑁 + 1)) ⊆
(0...(𝑁 +
1)) |
158 | 31, 157 | eqsstr3i 3636 |
. . . . . . . . 9
⊢
(1...(𝑁 + 1))
⊆ (0...(𝑁 +
1)) |
159 | 158 | sseli 3599 |
. . . . . . . 8
⊢ (𝐾 ∈ (1...(𝑁 + 1)) → 𝐾 ∈ (0...(𝑁 + 1))) |
160 | 155, 159 | syl6bir 244 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
→ ((𝐾 − 1)
∈ (0...((𝑁 + 1)
− 1)) → 𝐾 ∈
(0...(𝑁 +
1)))) |
161 | 152, 160 | sylbird 250 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
→ ((𝐾 − 1)
∈ (0...𝑁) → 𝐾 ∈ (0...(𝑁 + 1)))) |
162 | 161 | con3dimp 457 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
∧ ¬ 𝐾 ∈
(0...(𝑁 + 1))) → ¬
(𝐾 − 1) ∈
(0...𝑁)) |
163 | | bcval3 13093 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ (𝐾 − 1) ∈
ℤ ∧ ¬ (𝐾
− 1) ∈ (0...𝑁))
→ (𝑁C(𝐾 − 1)) =
0) |
164 | 145, 148,
162, 163 | syl3anc 1326 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
∧ ¬ 𝐾 ∈
(0...(𝑁 + 1))) →
(𝑁C(𝐾 − 1)) = 0) |
165 | 144, 164 | oveq12d 6668 |
. . 3
⊢ (((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
∧ ¬ 𝐾 ∈
(0...(𝑁 + 1))) →
((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = (0 + 0)) |
166 | 145, 1 | syl 17 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
∧ ¬ 𝐾 ∈
(0...(𝑁 + 1))) →
(𝑁 + 1) ∈
ℕ0) |
167 | | simpr 477 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
∧ ¬ 𝐾 ∈
(0...(𝑁 + 1))) → ¬
𝐾 ∈ (0...(𝑁 + 1))) |
168 | | bcval3 13093 |
. . . 4
⊢ (((𝑁 + 1) ∈ ℕ0
∧ 𝐾 ∈ ℤ
∧ ¬ 𝐾 ∈
(0...(𝑁 + 1))) →
((𝑁 + 1)C𝐾) = 0) |
169 | 166, 146,
167, 168 | syl3anc 1326 |
. . 3
⊢ (((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
∧ ¬ 𝐾 ∈
(0...(𝑁 + 1))) →
((𝑁 + 1)C𝐾) = 0) |
170 | 139, 165,
169 | 3eqtr4a 2682 |
. 2
⊢ (((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
∧ ¬ 𝐾 ∈
(0...(𝑁 + 1))) →
((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾)) |
171 | 138, 170 | pm2.61dan 832 |
1
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈ ℤ)
→ ((𝑁C𝐾) + (𝑁C(𝐾 − 1))) = ((𝑁 + 1)C𝐾)) |