Proof of Theorem pcbc
Step | Hyp | Ref
| Expression |
1 | | simp3 1063 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℙ) |
2 | | nnnn0 11299 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℕ0) |
3 | 2 | 3ad2ant1 1082 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈
ℕ0) |
4 | | faccl 13070 |
. . . . 5
⊢ (𝑁 ∈ ℕ0
→ (!‘𝑁) ∈
ℕ) |
5 | 3, 4 | syl 17 |
. . . 4
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝑁) ∈
ℕ) |
6 | 5 | nnzd 11481 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝑁) ∈
ℤ) |
7 | 5 | nnne0d 11065 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝑁) ≠ 0) |
8 | | fznn0sub 12373 |
. . . . . 6
⊢ (𝐾 ∈ (0...𝑁) → (𝑁 − 𝐾) ∈
ℕ0) |
9 | 8 | 3ad2ant2 1083 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ∈
ℕ0) |
10 | | faccl 13070 |
. . . . 5
⊢ ((𝑁 − 𝐾) ∈ ℕ0 →
(!‘(𝑁 − 𝐾)) ∈
ℕ) |
11 | 9, 10 | syl 17 |
. . . 4
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘(𝑁 − 𝐾)) ∈ ℕ) |
12 | | elfznn0 12433 |
. . . . . 6
⊢ (𝐾 ∈ (0...𝑁) → 𝐾 ∈
ℕ0) |
13 | 12 | 3ad2ant2 1083 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝐾 ∈
ℕ0) |
14 | | faccl 13070 |
. . . . 5
⊢ (𝐾 ∈ ℕ0
→ (!‘𝐾) ∈
ℕ) |
15 | 13, 14 | syl 17 |
. . . 4
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝐾) ∈
ℕ) |
16 | 11, 15 | nnmulcld 11068 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → ((!‘(𝑁 − 𝐾)) · (!‘𝐾)) ∈ ℕ) |
17 | | pcdiv 15557 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧
((!‘𝑁) ∈ ℤ
∧ (!‘𝑁) ≠ 0)
∧ ((!‘(𝑁 −
𝐾)) · (!‘𝐾)) ∈ ℕ) → (𝑃 pCnt ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) = ((𝑃 pCnt (!‘𝑁)) − (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))) |
18 | 1, 6, 7, 16, 17 | syl121anc 1331 |
. 2
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) = ((𝑃 pCnt (!‘𝑁)) − (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))) |
19 | | bcval2 13092 |
. . . 4
⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) |
20 | 19 | 3ad2ant2 1083 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) |
21 | 20 | oveq2d 6666 |
. 2
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (𝑁C𝐾)) = (𝑃 pCnt ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))) |
22 | | fzfid 12772 |
. . . 4
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (1...𝑁) ∈ Fin) |
23 | | nnre 11027 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℝ) |
24 | 23 | 3ad2ant1 1082 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℝ) |
25 | 24 | adantr 481 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑁 ∈ ℝ) |
26 | | simpl3 1066 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑃 ∈ ℙ) |
27 | | prmnn 15388 |
. . . . . . . . 9
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
28 | 26, 27 | syl 17 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑃 ∈ ℕ) |
29 | | elfznn 12370 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ) |
30 | 29 | nnnn0d 11351 |
. . . . . . . . 9
⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ0) |
31 | 30 | adantl 482 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ0) |
32 | 28, 31 | nnexpcld 13030 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑃↑𝑘) ∈ ℕ) |
33 | 25, 32 | nndivred 11069 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ) |
34 | 33 | flcld 12599 |
. . . . 5
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℤ) |
35 | 34 | zcnd 11483 |
. . . 4
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℂ) |
36 | 13 | nn0red 11352 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝐾 ∈ ℝ) |
37 | 24, 36 | resubcld 10458 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ∈ ℝ) |
38 | 37 | adantr 481 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑁 − 𝐾) ∈ ℝ) |
39 | 38, 32 | nndivred 11069 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝑁 − 𝐾) / (𝑃↑𝑘)) ∈ ℝ) |
40 | 39 | flcld 12599 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) ∈ ℤ) |
41 | 40 | zcnd 11483 |
. . . . 5
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) ∈ ℂ) |
42 | 36 | adantr 481 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → 𝐾 ∈ ℝ) |
43 | 42, 32 | nndivred 11069 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (𝐾 / (𝑃↑𝑘)) ∈ ℝ) |
44 | 43 | flcld 12599 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝐾 / (𝑃↑𝑘))) ∈ ℤ) |
45 | 44 | zcnd 11483 |
. . . . 5
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → (⌊‘(𝐾 / (𝑃↑𝑘))) ∈ ℂ) |
46 | 41, 45 | addcld 10059 |
. . . 4
⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...𝑁)) → ((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘)))) ∈ ℂ) |
47 | 22, 35, 46 | fsumsub 14520 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → Σ𝑘 ∈ (1...𝑁)((⌊‘(𝑁 / (𝑃↑𝑘))) − ((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘))))) = (Σ𝑘 ∈ (1...𝑁)(⌊‘(𝑁 / (𝑃↑𝑘))) − Σ𝑘 ∈ (1...𝑁)((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘)))))) |
48 | 3 | nn0zd 11480 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ ℤ) |
49 | | uzid 11702 |
. . . . . 6
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
(ℤ≥‘𝑁)) |
50 | 48, 49 | syl 17 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ (ℤ≥‘𝑁)) |
51 | | pcfac 15603 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑁 ∈
(ℤ≥‘𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝑁 / (𝑃↑𝑘)))) |
52 | 3, 50, 1, 51 | syl3anc 1326 |
. . . 4
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝑁 / (𝑃↑𝑘)))) |
53 | 13 | nn0ge0d 11354 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 0 ≤ 𝐾) |
54 | 24, 36 | subge02d 10619 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (0 ≤ 𝐾 ↔ (𝑁 − 𝐾) ≤ 𝑁)) |
55 | 53, 54 | mpbid 222 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ≤ 𝑁) |
56 | 13 | nn0zd 11480 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝐾 ∈ ℤ) |
57 | 48, 56 | zsubcld 11487 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 − 𝐾) ∈ ℤ) |
58 | | eluz 11701 |
. . . . . . . . 9
⊢ (((𝑁 − 𝐾) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘(𝑁 − 𝐾)) ↔ (𝑁 − 𝐾) ≤ 𝑁)) |
59 | 57, 48, 58 | syl2anc 693 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑁 ∈ (ℤ≥‘(𝑁 − 𝐾)) ↔ (𝑁 − 𝐾) ≤ 𝑁)) |
60 | 55, 59 | mpbird 247 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ (ℤ≥‘(𝑁 − 𝐾))) |
61 | | pcfac 15603 |
. . . . . . 7
⊢ (((𝑁 − 𝐾) ∈ ℕ0 ∧ 𝑁 ∈
(ℤ≥‘(𝑁 − 𝐾)) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘(𝑁 − 𝐾))) = Σ𝑘 ∈ (1...𝑁)(⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘)))) |
62 | 9, 60, 1, 61 | syl3anc 1326 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘(𝑁 − 𝐾))) = Σ𝑘 ∈ (1...𝑁)(⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘)))) |
63 | | elfzuz3 12339 |
. . . . . . . 8
⊢ (𝐾 ∈ (0...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) |
64 | 63 | 3ad2ant2 1083 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → 𝑁 ∈ (ℤ≥‘𝐾)) |
65 | | pcfac 15603 |
. . . . . . 7
⊢ ((𝐾 ∈ ℕ0
∧ 𝑁 ∈
(ℤ≥‘𝐾) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝐾)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝐾 / (𝑃↑𝑘)))) |
66 | 13, 64, 1, 65 | syl3anc 1326 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝐾)) = Σ𝑘 ∈ (1...𝑁)(⌊‘(𝐾 / (𝑃↑𝑘)))) |
67 | 62, 66 | oveq12d 6668 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → ((𝑃 pCnt (!‘(𝑁 − 𝐾))) + (𝑃 pCnt (!‘𝐾))) = (Σ𝑘 ∈ (1...𝑁)(⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + Σ𝑘 ∈ (1...𝑁)(⌊‘(𝐾 / (𝑃↑𝑘))))) |
68 | 11 | nnzd 11481 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘(𝑁 − 𝐾)) ∈ ℤ) |
69 | 11 | nnne0d 11065 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘(𝑁 − 𝐾)) ≠ 0) |
70 | 15 | nnzd 11481 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝐾) ∈
ℤ) |
71 | 15 | nnne0d 11065 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (!‘𝐾) ≠ 0) |
72 | | pcmul 15556 |
. . . . . 6
⊢ ((𝑃 ∈ ℙ ∧
((!‘(𝑁 − 𝐾)) ∈ ℤ ∧
(!‘(𝑁 − 𝐾)) ≠ 0) ∧ ((!‘𝐾) ∈ ℤ ∧
(!‘𝐾) ≠ 0)) →
(𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) = ((𝑃 pCnt (!‘(𝑁 − 𝐾))) + (𝑃 pCnt (!‘𝐾)))) |
73 | 1, 68, 69, 70, 71, 72 | syl122anc 1335 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) = ((𝑃 pCnt (!‘(𝑁 − 𝐾))) + (𝑃 pCnt (!‘𝐾)))) |
74 | 22, 41, 45 | fsumadd 14470 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → Σ𝑘 ∈ (1...𝑁)((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘)))) = (Σ𝑘 ∈ (1...𝑁)(⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + Σ𝑘 ∈ (1...𝑁)(⌊‘(𝐾 / (𝑃↑𝑘))))) |
75 | 67, 73, 74 | 3eqtr4d 2666 |
. . . 4
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾))) = Σ𝑘 ∈ (1...𝑁)((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘))))) |
76 | 52, 75 | oveq12d 6668 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → ((𝑃 pCnt (!‘𝑁)) − (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) = (Σ𝑘 ∈ (1...𝑁)(⌊‘(𝑁 / (𝑃↑𝑘))) − Σ𝑘 ∈ (1...𝑁)((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘)))))) |
77 | 47, 76 | eqtr4d 2659 |
. 2
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → Σ𝑘 ∈ (1...𝑁)((⌊‘(𝑁 / (𝑃↑𝑘))) − ((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘))))) = ((𝑃 pCnt (!‘𝑁)) − (𝑃 pCnt ((!‘(𝑁 − 𝐾)) · (!‘𝐾))))) |
78 | 18, 21, 77 | 3eqtr4d 2666 |
1
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (𝑁C𝐾)) = Σ𝑘 ∈ (1...𝑁)((⌊‘(𝑁 / (𝑃↑𝑘))) − ((⌊‘((𝑁 − 𝐾) / (𝑃↑𝑘))) + (⌊‘(𝐾 / (𝑃↑𝑘)))))) |