Step | Hyp | Ref
| Expression |
1 | | fzfid 12634 |
. . . 4
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (1...(2
· 𝑁)) ∈
Fin) |
2 | | 2nn 11062 |
. . . . . . . . . . 11
⊢ 2 ∈
ℕ |
3 | | nnmulcl 10920 |
. . . . . . . . . . 11
⊢ ((2
∈ ℕ ∧ 𝑁
∈ ℕ) → (2 · 𝑁) ∈ ℕ) |
4 | 2, 3 | mpan 702 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℕ → (2
· 𝑁) ∈
ℕ) |
5 | 4 | ad2antrr 758 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · 𝑁) ∈
ℕ) |
6 | | prmnn 15226 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
7 | 6 | ad2antlr 759 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑃 ∈ ℕ) |
8 | | elfznn 12241 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ (1...(2 · 𝑁)) → 𝑘 ∈ ℕ) |
9 | 8 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑘 ∈ ℕ) |
10 | 9 | nnnn0d 11228 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑘 ∈ ℕ0) |
11 | 7, 10 | nnexpcld 12892 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ∈ ℕ) |
12 | | nnrp 11718 |
. . . . . . . . . 10
⊢ ((2
· 𝑁) ∈ ℕ
→ (2 · 𝑁)
∈ ℝ+) |
13 | | nnrp 11718 |
. . . . . . . . . 10
⊢ ((𝑃↑𝑘) ∈ ℕ → (𝑃↑𝑘) ∈
ℝ+) |
14 | | rpdivcl 11732 |
. . . . . . . . . 10
⊢ (((2
· 𝑁) ∈
ℝ+ ∧ (𝑃↑𝑘) ∈ ℝ+) → ((2
· 𝑁) / (𝑃↑𝑘)) ∈
ℝ+) |
15 | 12, 13, 14 | syl2an 493 |
. . . . . . . . 9
⊢ (((2
· 𝑁) ∈ ℕ
∧ (𝑃↑𝑘) ∈ ℕ) → ((2
· 𝑁) / (𝑃↑𝑘)) ∈
ℝ+) |
16 | 5, 11, 15 | syl2anc 691 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈
ℝ+) |
17 | 16 | rpred 11748 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ) |
18 | 17 | flcld 12461 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘((2
· 𝑁) / (𝑃↑𝑘))) ∈ ℤ) |
19 | | 2z 11286 |
. . . . . . 7
⊢ 2 ∈
ℤ |
20 | | simpll 786 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 ∈ ℕ) |
21 | | nnrp 11718 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℝ+) |
22 | | rpdivcl 11732 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℝ+
∧ (𝑃↑𝑘) ∈ ℝ+)
→ (𝑁 / (𝑃↑𝑘)) ∈
ℝ+) |
23 | 21, 13, 22 | syl2an 493 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ ∧ (𝑃↑𝑘) ∈ ℕ) → (𝑁 / (𝑃↑𝑘)) ∈
ℝ+) |
24 | 20, 11, 23 | syl2anc 691 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) ∈
ℝ+) |
25 | 24 | rpred 11748 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ) |
26 | 25 | flcld 12461 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℤ) |
27 | | zmulcl 11303 |
. . . . . . 7
⊢ ((2
∈ ℤ ∧ (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℤ) → (2 ·
(⌊‘(𝑁 / (𝑃↑𝑘)))) ∈ ℤ) |
28 | 19, 26, 27 | sylancr 694 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 ·
(⌊‘(𝑁 / (𝑃↑𝑘)))) ∈ ℤ) |
29 | 18, 28 | zsubcld 11363 |
. . . . 5
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2
· 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℤ) |
30 | 29 | zred 11358 |
. . . 4
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2
· 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℝ) |
31 | | 1re 9918 |
. . . . . 6
⊢ 1 ∈
ℝ |
32 | | 0re 9919 |
. . . . . 6
⊢ 0 ∈
ℝ |
33 | 31, 32 | keepel 4105 |
. . . . 5
⊢ if(𝑘 ∈
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0) ∈ ℝ |
34 | 33 | a1i 11 |
. . . 4
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → if(𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))), 1, 0)
∈ ℝ) |
35 | 28 | zred 11358 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 ·
(⌊‘(𝑁 / (𝑃↑𝑘)))) ∈ ℝ) |
36 | 17, 35 | resubcld 10337 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℝ) |
37 | | 2re 10967 |
. . . . . . . . . 10
⊢ 2 ∈
ℝ |
38 | 37 | a1i 11 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 2 ∈
ℝ) |
39 | 18 | zred 11358 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘((2
· 𝑁) / (𝑃↑𝑘))) ∈ ℝ) |
40 | | flle 12462 |
. . . . . . . . . . 11
⊢ (((2
· 𝑁) / (𝑃↑𝑘)) ∈ ℝ → (⌊‘((2
· 𝑁) / (𝑃↑𝑘))) ≤ ((2 · 𝑁) / (𝑃↑𝑘))) |
41 | 17, 40 | syl 17 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘((2
· 𝑁) / (𝑃↑𝑘))) ≤ ((2 · 𝑁) / (𝑃↑𝑘))) |
42 | 39, 17, 35, 41 | lesub1dd 10522 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2
· 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))))) |
43 | | resubcl 10224 |
. . . . . . . . . . . . 13
⊢ (((𝑁 / (𝑃↑𝑘)) ∈ ℝ ∧ 1 ∈ ℝ)
→ ((𝑁 / (𝑃↑𝑘)) − 1) ∈
ℝ) |
44 | 25, 31, 43 | sylancl 693 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 / (𝑃↑𝑘)) − 1) ∈
ℝ) |
45 | | remulcl 9900 |
. . . . . . . . . . . 12
⊢ ((2
∈ ℝ ∧ ((𝑁 /
(𝑃↑𝑘)) − 1) ∈ ℝ) → (2
· ((𝑁 / (𝑃↑𝑘)) − 1)) ∈
ℝ) |
46 | 37, 44, 45 | sylancr 694 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) ∈
ℝ) |
47 | | flltp1 12463 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 / (𝑃↑𝑘)) ∈ ℝ → (𝑁 / (𝑃↑𝑘)) < ((⌊‘(𝑁 / (𝑃↑𝑘))) + 1)) |
48 | 25, 47 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) < ((⌊‘(𝑁 / (𝑃↑𝑘))) + 1)) |
49 | | 1red 9934 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 1 ∈
ℝ) |
50 | 26 | zred 11358 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℝ) |
51 | 25, 49, 50 | ltsubaddd 10502 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((𝑁 / (𝑃↑𝑘)) − 1) < (⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (𝑁 / (𝑃↑𝑘)) < ((⌊‘(𝑁 / (𝑃↑𝑘))) + 1))) |
52 | 48, 51 | mpbird 246 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 / (𝑃↑𝑘)) − 1) < (⌊‘(𝑁 / (𝑃↑𝑘)))) |
53 | | 2pos 10989 |
. . . . . . . . . . . . . . 15
⊢ 0 <
2 |
54 | 37, 53 | pm3.2i 470 |
. . . . . . . . . . . . . 14
⊢ (2 ∈
ℝ ∧ 0 < 2) |
55 | | ltmul2 10753 |
. . . . . . . . . . . . . 14
⊢ ((((𝑁 / (𝑃↑𝑘)) − 1) ∈ ℝ ∧
(⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℝ ∧ (2 ∈ ℝ
∧ 0 < 2)) → (((𝑁 / (𝑃↑𝑘)) − 1) < (⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) < (2 ·
(⌊‘(𝑁 / (𝑃↑𝑘)))))) |
56 | 54, 55 | mp3an3 1405 |
. . . . . . . . . . . . 13
⊢ ((((𝑁 / (𝑃↑𝑘)) − 1) ∈ ℝ ∧
(⌊‘(𝑁 / (𝑃↑𝑘))) ∈ ℝ) → (((𝑁 / (𝑃↑𝑘)) − 1) < (⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) < (2 ·
(⌊‘(𝑁 / (𝑃↑𝑘)))))) |
57 | 44, 50, 56 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((𝑁 / (𝑃↑𝑘)) − 1) < (⌊‘(𝑁 / (𝑃↑𝑘))) ↔ (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) < (2 ·
(⌊‘(𝑁 / (𝑃↑𝑘)))))) |
58 | 52, 57 | mpbid 221 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) < (2 ·
(⌊‘(𝑁 / (𝑃↑𝑘))))) |
59 | 46, 35, 17, 58 | ltsub2dd 10519 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · ((𝑁 / (𝑃↑𝑘)) − 1)))) |
60 | | 2cnd 10970 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 2 ∈
ℂ) |
61 | | nncn 10905 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℂ) |
62 | 61 | ad2antrr 758 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 ∈ ℂ) |
63 | 11 | nncnd 10913 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ∈ ℂ) |
64 | 11 | nnne0d 10942 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ≠ 0) |
65 | 60, 62, 63, 64 | divassd 10715 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) / (𝑃↑𝑘)) = (2 · (𝑁 / (𝑃↑𝑘)))) |
66 | 25 | recnd 9947 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 / (𝑃↑𝑘)) ∈ ℂ) |
67 | | 1cnd 9935 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 1 ∈
ℂ) |
68 | 60, 66, 67 | subdid 10365 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) = ((2 · (𝑁 / (𝑃↑𝑘))) − (2 · 1))) |
69 | | 2t1e2 11053 |
. . . . . . . . . . . . . 14
⊢ (2
· 1) = 2 |
70 | 69 | oveq2i 6560 |
. . . . . . . . . . . . 13
⊢ ((2
· (𝑁 / (𝑃↑𝑘))) − (2 · 1)) = ((2 ·
(𝑁 / (𝑃↑𝑘))) − 2) |
71 | 68, 70 | syl6eq 2660 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · ((𝑁 / (𝑃↑𝑘)) − 1)) = ((2 · (𝑁 / (𝑃↑𝑘))) − 2)) |
72 | 65, 71 | oveq12d 6567 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · ((𝑁 / (𝑃↑𝑘)) − 1))) = ((2 · (𝑁 / (𝑃↑𝑘))) − ((2 · (𝑁 / (𝑃↑𝑘))) − 2))) |
73 | | remulcl 9900 |
. . . . . . . . . . . . . 14
⊢ ((2
∈ ℝ ∧ (𝑁 /
(𝑃↑𝑘)) ∈ ℝ) → (2 · (𝑁 / (𝑃↑𝑘))) ∈ ℝ) |
74 | 37, 25, 73 | sylancr 694 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · (𝑁 / (𝑃↑𝑘))) ∈ ℝ) |
75 | 74 | recnd 9947 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · (𝑁 / (𝑃↑𝑘))) ∈ ℂ) |
76 | | 2cn 10968 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℂ |
77 | | nncan 10189 |
. . . . . . . . . . . 12
⊢ (((2
· (𝑁 / (𝑃↑𝑘))) ∈ ℂ ∧ 2 ∈ ℂ)
→ ((2 · (𝑁 /
(𝑃↑𝑘))) − ((2 · (𝑁 / (𝑃↑𝑘))) − 2)) = 2) |
78 | 75, 76, 77 | sylancl 693 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · (𝑁 / (𝑃↑𝑘))) − ((2 · (𝑁 / (𝑃↑𝑘))) − 2)) = 2) |
79 | 72, 78 | eqtrd 2644 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · ((𝑁 / (𝑃↑𝑘)) − 1))) = 2) |
80 | 59, 79 | breqtrd 4609 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < 2) |
81 | 30, 36, 38, 42, 80 | lelttrd 10074 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2
· 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < 2) |
82 | | df-2 10956 |
. . . . . . . 8
⊢ 2 = (1 +
1) |
83 | 81, 82 | syl6breq 4624 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2
· 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < (1 + 1)) |
84 | | 1z 11284 |
. . . . . . . 8
⊢ 1 ∈
ℤ |
85 | | zleltp1 11305 |
. . . . . . . 8
⊢
((((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ∈ ℤ ∧ 1 ∈ ℤ)
→ (((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 1 ↔ ((⌊‘((2
· 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < (1 + 1))) |
86 | 29, 84, 85 | sylancl 693 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((⌊‘((2
· 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 1 ↔ ((⌊‘((2
· 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) < (1 + 1))) |
87 | 83, 86 | mpbird 246 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2
· 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 1) |
88 | | iftrue 4042 |
. . . . . . 7
⊢ (𝑘 ∈
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → if(𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))), 1, 0) =
1) |
89 | 88 | breq2d 4595 |
. . . . . 6
⊢ (𝑘 ∈
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → (((⌊‘((2 ·
𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ if(𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))), 1, 0)
↔ ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 1)) |
90 | 87, 89 | syl5ibrcom 236 |
. . . . 5
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))) →
((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ if(𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))), 1,
0))) |
91 | 9 | nnge1d 10940 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 1 ≤ 𝑘) |
92 | 91 | biantrurd 528 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑘 ≤ (⌊‘((log‘(2 ·
𝑁)) / (log‘𝑃))) ↔ (1 ≤ 𝑘 ∧ 𝑘 ≤ (⌊‘((log‘(2 ·
𝑁)) / (log‘𝑃)))))) |
93 | 6 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈
ℕ) |
94 | 93 | nnred 10912 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈
ℝ) |
95 | | prmuz2 15246 |
. . . . . . . . . . . . . . 15
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
(ℤ≥‘2)) |
96 | 95 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈
(ℤ≥‘2)) |
97 | | eluz2b1 11635 |
. . . . . . . . . . . . . . 15
⊢ (𝑃 ∈
(ℤ≥‘2) ↔ (𝑃 ∈ ℤ ∧ 1 < 𝑃)) |
98 | 97 | simprbi 479 |
. . . . . . . . . . . . . 14
⊢ (𝑃 ∈
(ℤ≥‘2) → 1 < 𝑃) |
99 | 96, 98 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 <
𝑃) |
100 | 94, 99 | jca 553 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 ∈ ℝ ∧ 1 <
𝑃)) |
101 | 100 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃 ∈ ℝ ∧ 1 < 𝑃)) |
102 | | elfzelz 12213 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ (1...(2 · 𝑁)) → 𝑘 ∈ ℤ) |
103 | 102 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑘 ∈ ℤ) |
104 | 4 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2
· 𝑁) ∈
ℕ) |
105 | 104 | nnrpd 11746 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2
· 𝑁) ∈
ℝ+) |
106 | 105 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · 𝑁) ∈
ℝ+) |
107 | | efexple 24806 |
. . . . . . . . . . 11
⊢ (((𝑃 ∈ ℝ ∧ 1 <
𝑃) ∧ 𝑘 ∈ ℤ ∧ (2 · 𝑁) ∈ ℝ+)
→ ((𝑃↑𝑘) ≤ (2 · 𝑁) ↔ 𝑘 ≤ (⌊‘((log‘(2 ·
𝑁)) / (log‘𝑃))))) |
108 | 101, 103,
106, 107 | syl3anc 1318 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑃↑𝑘) ≤ (2 · 𝑁) ↔ 𝑘 ≤ (⌊‘((log‘(2 ·
𝑁)) / (log‘𝑃))))) |
109 | 9 | nnzd 11357 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑘 ∈ ℤ) |
110 | 84 | a1i 11 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 1 ∈
ℤ) |
111 | 104 | nnred 10912 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2
· 𝑁) ∈
ℝ) |
112 | | 1red 9934 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 ∈
ℝ) |
113 | 37 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 2 ∈
ℝ) |
114 | | 1lt2 11071 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 <
2 |
115 | 114 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 <
2) |
116 | | nnre 10904 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℝ) |
117 | 116 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑁 ∈
ℝ) |
118 | | 0le2 10988 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 0 ≤
2 |
119 | 37, 118 | pm3.2i 470 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (2 ∈
ℝ ∧ 0 ≤ 2) |
120 | 119 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2
∈ ℝ ∧ 0 ≤ 2)) |
121 | | nnge1 10923 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ → 1 ≤
𝑁) |
122 | 121 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 ≤
𝑁) |
123 | | lemul2a 10757 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((1
∈ ℝ ∧ 𝑁
∈ ℝ ∧ (2 ∈ ℝ ∧ 0 ≤ 2)) ∧ 1 ≤ 𝑁) → (2 · 1) ≤ (2
· 𝑁)) |
124 | 112, 117,
120, 122, 123 | syl31anc 1321 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2
· 1) ≤ (2 · 𝑁)) |
125 | 69, 124 | syl5eqbrr 4619 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 2 ≤
(2 · 𝑁)) |
126 | 112, 113,
111, 115, 125 | ltletrd 10076 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 <
(2 · 𝑁)) |
127 | 111, 126 | rplogcld 24179 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(log‘(2 · 𝑁))
∈ ℝ+) |
128 | 94, 99 | rplogcld 24179 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(log‘𝑃) ∈
ℝ+) |
129 | 127, 128 | rpdivcld 11765 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
((log‘(2 · 𝑁))
/ (log‘𝑃)) ∈
ℝ+) |
130 | 129 | rpred 11748 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
((log‘(2 · 𝑁))
/ (log‘𝑃)) ∈
ℝ) |
131 | 130 | flcld 12461 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℤ) |
132 | 131 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) →
(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℤ) |
133 | | elfz 12203 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℤ ∧ 1 ∈
ℤ ∧ (⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℤ) → (𝑘 ∈
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ (1 ≤ 𝑘 ∧ 𝑘 ≤ (⌊‘((log‘(2 ·
𝑁)) / (log‘𝑃)))))) |
134 | 109, 110,
132, 133 | syl3anc 1318 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))) ↔ (1
≤ 𝑘 ∧ 𝑘 ≤
(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))))) |
135 | 92, 108, 134 | 3bitr4rd 300 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))) ↔
(𝑃↑𝑘) ≤ (2 · 𝑁))) |
136 | 135 | notbid 307 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (¬ 𝑘 ∈
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ ¬ (𝑃↑𝑘) ≤ (2 · 𝑁))) |
137 | 111 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (2 · 𝑁) ∈
ℝ) |
138 | 11 | nnred 10912 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑃↑𝑘) ∈ ℝ) |
139 | 137, 138 | ltnled 10063 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) ↔ ¬ (𝑃↑𝑘) ≤ (2 · 𝑁))) |
140 | 136, 139 | bitr4d 270 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (¬ 𝑘 ∈
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔ (2 · 𝑁) < (𝑃↑𝑘))) |
141 | 16 | rpge0d 11752 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 0 ≤ ((2 ·
𝑁) / (𝑃↑𝑘))) |
142 | 141 | adantrr 749 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → 0 ≤ ((2 · 𝑁) / (𝑃↑𝑘))) |
143 | 11 | nngt0d 10941 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 0 < (𝑃↑𝑘)) |
144 | | ltdivmul 10777 |
. . . . . . . . . . . . . . . . 17
⊢ (((2
· 𝑁) ∈ ℝ
∧ 1 ∈ ℝ ∧ ((𝑃↑𝑘) ∈ ℝ ∧ 0 < (𝑃↑𝑘))) → (((2 · 𝑁) / (𝑃↑𝑘)) < 1 ↔ (2 · 𝑁) < ((𝑃↑𝑘) · 1))) |
145 | 137, 49, 138, 143, 144 | syl112anc 1322 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) < 1 ↔ (2 · 𝑁) < ((𝑃↑𝑘) · 1))) |
146 | 63 | mulid1d 9936 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑃↑𝑘) · 1) = (𝑃↑𝑘)) |
147 | 146 | breq2d 4595 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < ((𝑃↑𝑘) · 1) ↔ (2 · 𝑁) < (𝑃↑𝑘))) |
148 | 145, 147 | bitrd 267 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (((2 · 𝑁) / (𝑃↑𝑘)) < 1 ↔ (2 · 𝑁) < (𝑃↑𝑘))) |
149 | 148 | biimprd 237 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) → ((2 · 𝑁) / (𝑃↑𝑘)) < 1)) |
150 | 149 | impr 647 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((2 · 𝑁) / (𝑃↑𝑘)) < 1) |
151 | | 0p1e1 11009 |
. . . . . . . . . . . . 13
⊢ (0 + 1) =
1 |
152 | 150, 151 | syl6breqr 4625 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((2 · 𝑁) / (𝑃↑𝑘)) < (0 + 1)) |
153 | 17 | adantrr 749 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((2 · 𝑁) / (𝑃↑𝑘)) ∈ ℝ) |
154 | | 0z 11265 |
. . . . . . . . . . . . 13
⊢ 0 ∈
ℤ |
155 | | flbi 12479 |
. . . . . . . . . . . . 13
⊢ ((((2
· 𝑁) / (𝑃↑𝑘)) ∈ ℝ ∧ 0 ∈ ℤ)
→ ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) = 0 ↔ (0 ≤ ((2 · 𝑁) / (𝑃↑𝑘)) ∧ ((2 · 𝑁) / (𝑃↑𝑘)) < (0 + 1)))) |
156 | 153, 154,
155 | sylancl 693 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) = 0 ↔ (0 ≤ ((2 · 𝑁) / (𝑃↑𝑘)) ∧ ((2 · 𝑁) / (𝑃↑𝑘)) < (0 + 1)))) |
157 | 142, 152,
156 | mpbir2and 959 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (⌊‘((2 · 𝑁) / (𝑃↑𝑘))) = 0) |
158 | 24 | rpge0d 11752 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 0 ≤ (𝑁 / (𝑃↑𝑘))) |
159 | 158 | adantrr 749 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → 0 ≤ (𝑁 / (𝑃↑𝑘))) |
160 | 116, 21 | ltaddrp2d 11782 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ → 𝑁 < (𝑁 + 𝑁)) |
161 | 61 | 2timesd 11152 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ → (2
· 𝑁) = (𝑁 + 𝑁)) |
162 | 160, 161 | breqtrrd 4611 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ → 𝑁 < (2 · 𝑁)) |
163 | 162 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 < (2 · 𝑁)) |
164 | 116 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → 𝑁 ∈ ℝ) |
165 | | lttr 9993 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ ℝ ∧ (2
· 𝑁) ∈ ℝ
∧ (𝑃↑𝑘) ∈ ℝ) → ((𝑁 < (2 · 𝑁) ∧ (2 · 𝑁) < (𝑃↑𝑘)) → 𝑁 < (𝑃↑𝑘))) |
166 | 164, 137,
138, 165 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 < (2 · 𝑁) ∧ (2 · 𝑁) < (𝑃↑𝑘)) → 𝑁 < (𝑃↑𝑘))) |
167 | 163, 166 | mpand 707 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) → 𝑁 < (𝑃↑𝑘))) |
168 | | ltdivmul 10777 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ ℝ ∧ 1 ∈
ℝ ∧ ((𝑃↑𝑘) ∈ ℝ ∧ 0 < (𝑃↑𝑘))) → ((𝑁 / (𝑃↑𝑘)) < 1 ↔ 𝑁 < ((𝑃↑𝑘) · 1))) |
169 | 164, 49, 138, 143, 168 | syl112anc 1322 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 / (𝑃↑𝑘)) < 1 ↔ 𝑁 < ((𝑃↑𝑘) · 1))) |
170 | 146 | breq2d 4595 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (𝑁 < ((𝑃↑𝑘) · 1) ↔ 𝑁 < (𝑃↑𝑘))) |
171 | 169, 170 | bitrd 267 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((𝑁 / (𝑃↑𝑘)) < 1 ↔ 𝑁 < (𝑃↑𝑘))) |
172 | 167, 171 | sylibrd 248 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) → (𝑁 / (𝑃↑𝑘)) < 1)) |
173 | 172 | impr 647 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (𝑁 / (𝑃↑𝑘)) < 1) |
174 | 173, 151 | syl6breqr 4625 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (𝑁 / (𝑃↑𝑘)) < (0 + 1)) |
175 | 25 | adantrr 749 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (𝑁 / (𝑃↑𝑘)) ∈ ℝ) |
176 | | flbi 12479 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 / (𝑃↑𝑘)) ∈ ℝ ∧ 0 ∈ ℤ)
→ ((⌊‘(𝑁 /
(𝑃↑𝑘))) = 0 ↔ (0 ≤ (𝑁 / (𝑃↑𝑘)) ∧ (𝑁 / (𝑃↑𝑘)) < (0 + 1)))) |
177 | 175, 154,
176 | sylancl 693 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((⌊‘(𝑁 / (𝑃↑𝑘))) = 0 ↔ (0 ≤ (𝑁 / (𝑃↑𝑘)) ∧ (𝑁 / (𝑃↑𝑘)) < (0 + 1)))) |
178 | 159, 174,
177 | mpbir2and 959 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (⌊‘(𝑁 / (𝑃↑𝑘))) = 0) |
179 | 178 | oveq2d 6565 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) = (2 · 0)) |
180 | | 2t0e0 11060 |
. . . . . . . . . . . 12
⊢ (2
· 0) = 0 |
181 | 179, 180 | syl6eq 2660 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))) = 0) |
182 | 157, 181 | oveq12d 6567 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) = (0 − 0)) |
183 | | 0m0e0 11007 |
. . . . . . . . . 10
⊢ (0
− 0) = 0 |
184 | 182, 183 | syl6eq 2660 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) = 0) |
185 | | 0le0 10987 |
. . . . . . . . 9
⊢ 0 ≤
0 |
186 | 184, 185 | syl6eqbr 4622 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ (𝑘 ∈ (1...(2 · 𝑁)) ∧ (2 · 𝑁) < (𝑃↑𝑘))) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 0) |
187 | 186 | expr 641 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((2 · 𝑁) < (𝑃↑𝑘) → ((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 0)) |
188 | 140, 187 | sylbid 229 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (¬ 𝑘 ∈
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → ((⌊‘((2 ·
𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 0)) |
189 | | iffalse 4045 |
. . . . . . . 8
⊢ (¬
𝑘 ∈
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → if(𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))), 1, 0) =
0) |
190 | 189 | eqcomd 2616 |
. . . . . . 7
⊢ (¬
𝑘 ∈
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → 0 = if(𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))), 1,
0)) |
191 | 190 | breq2d 4595 |
. . . . . 6
⊢ (¬
𝑘 ∈
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → (((⌊‘((2 ·
𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ 0 ↔ ((⌊‘((2
· 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ if(𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))), 1,
0))) |
192 | 188, 191 | mpbidi 230 |
. . . . 5
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → (¬ 𝑘 ∈
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) → ((⌊‘((2 ·
𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ if(𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))), 1,
0))) |
193 | 90, 192 | pm2.61d 169 |
. . . 4
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ (1...(2 · 𝑁))) → ((⌊‘((2
· 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ if(𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))), 1,
0)) |
194 | 1, 30, 34, 193 | fsumle 14372 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
Σ𝑘 ∈ (1...(2
· 𝑁))((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘))))) ≤ Σ𝑘 ∈ (1...(2 · 𝑁))if(𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))), 1,
0)) |
195 | | pcbcctr 24801 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) = Σ𝑘 ∈ (1...(2 · 𝑁))((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))))) |
196 | 131 | zred 11358 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℝ) |
197 | | flle 12462 |
. . . . . . . . 9
⊢
(((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ →
(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ ((log‘(2 · 𝑁)) / (log‘𝑃))) |
198 | 130, 197 | syl 17 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ ((log‘(2 · 𝑁)) / (log‘𝑃))) |
199 | 104 | nnnn0d 11228 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2
· 𝑁) ∈
ℕ0) |
200 | 93, 199 | nnexpcld 12892 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(2 · 𝑁)) ∈
ℕ) |
201 | 200 | nnred 10912 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(2 · 𝑁)) ∈
ℝ) |
202 | | bernneq3 12854 |
. . . . . . . . . . . . 13
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (2 · 𝑁) ∈ ℕ0) → (2
· 𝑁) < (𝑃↑(2 · 𝑁))) |
203 | 96, 199, 202 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2
· 𝑁) < (𝑃↑(2 · 𝑁))) |
204 | 111, 201,
203 | ltled 10064 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2
· 𝑁) ≤ (𝑃↑(2 · 𝑁))) |
205 | 105 | reeflogd 24174 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(exp‘(log‘(2 · 𝑁))) = (2 · 𝑁)) |
206 | 93 | nnrpd 11746 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈
ℝ+) |
207 | 104 | nnzd 11357 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2
· 𝑁) ∈
ℤ) |
208 | | reexplog 24145 |
. . . . . . . . . . . . 13
⊢ ((𝑃 ∈ ℝ+
∧ (2 · 𝑁) ∈
ℤ) → (𝑃↑(2
· 𝑁)) =
(exp‘((2 · 𝑁)
· (log‘𝑃)))) |
209 | 206, 207,
208 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(2 · 𝑁)) = (exp‘((2 ·
𝑁) ·
(log‘𝑃)))) |
210 | 209 | eqcomd 2616 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(exp‘((2 · 𝑁)
· (log‘𝑃))) =
(𝑃↑(2 · 𝑁))) |
211 | 204, 205,
210 | 3brtr4d 4615 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(exp‘(log‘(2 · 𝑁))) ≤ (exp‘((2 · 𝑁) · (log‘𝑃)))) |
212 | 105 | relogcld 24173 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(log‘(2 · 𝑁))
∈ ℝ) |
213 | 128 | rpred 11748 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(log‘𝑃) ∈
ℝ) |
214 | 111, 213 | remulcld 9949 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((2
· 𝑁) ·
(log‘𝑃)) ∈
ℝ) |
215 | | efle 14687 |
. . . . . . . . . . 11
⊢
(((log‘(2 · 𝑁)) ∈ ℝ ∧ ((2 · 𝑁) · (log‘𝑃)) ∈ ℝ) →
((log‘(2 · 𝑁))
≤ ((2 · 𝑁)
· (log‘𝑃))
↔ (exp‘(log‘(2 · 𝑁))) ≤ (exp‘((2 · 𝑁) · (log‘𝑃))))) |
216 | 212, 214,
215 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
((log‘(2 · 𝑁))
≤ ((2 · 𝑁)
· (log‘𝑃))
↔ (exp‘(log‘(2 · 𝑁))) ≤ (exp‘((2 · 𝑁) · (log‘𝑃))))) |
217 | 211, 216 | mpbird 246 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(log‘(2 · 𝑁))
≤ ((2 · 𝑁)
· (log‘𝑃))) |
218 | 212, 111,
128 | ledivmul2d 11802 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(((log‘(2 · 𝑁)) / (log‘𝑃)) ≤ (2 · 𝑁) ↔ (log‘(2 · 𝑁)) ≤ ((2 · 𝑁) · (log‘𝑃)))) |
219 | 217, 218 | mpbird 246 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
((log‘(2 · 𝑁))
/ (log‘𝑃)) ≤ (2
· 𝑁)) |
220 | 196, 130,
111, 198, 219 | letrd 10073 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ (2 · 𝑁)) |
221 | | eluz 11577 |
. . . . . . . 8
⊢
(((⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℤ ∧ (2 · 𝑁) ∈ ℤ) → ((2
· 𝑁) ∈
(ℤ≥‘(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔
(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ (2 · 𝑁))) |
222 | 131, 207,
221 | syl2anc 691 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((2
· 𝑁) ∈
(ℤ≥‘(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ↔
(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ≤ (2 · 𝑁))) |
223 | 220, 222 | mpbird 246 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2
· 𝑁) ∈
(ℤ≥‘(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))) |
224 | | fzss2 12252 |
. . . . . 6
⊢ ((2
· 𝑁) ∈
(ℤ≥‘(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) →
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ⊆ (1...(2 · 𝑁))) |
225 | 223, 224 | syl 17 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ⊆ (1...(2 · 𝑁))) |
226 | | sumhash 15438 |
. . . . 5
⊢ (((1...(2
· 𝑁)) ∈ Fin
∧ (1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))) ⊆ (1...(2 · 𝑁))) → Σ𝑘 ∈ (1...(2 · 𝑁))if(𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))), 1, 0) =
(#‘(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))))) |
227 | 1, 225, 226 | syl2anc 691 |
. . . 4
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
Σ𝑘 ∈ (1...(2
· 𝑁))if(𝑘 ∈
(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))), 1, 0) =
(#‘(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃)))))) |
228 | 129 | rprege0d 11755 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ ∧ 0 ≤
((log‘(2 · 𝑁))
/ (log‘𝑃)))) |
229 | | flge0nn0 12483 |
. . . . 5
⊢
((((log‘(2 · 𝑁)) / (log‘𝑃)) ∈ ℝ ∧ 0 ≤
((log‘(2 · 𝑁))
/ (log‘𝑃))) →
(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈
ℕ0) |
230 | | hashfz1 12996 |
. . . . 5
⊢
((⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) ∈ ℕ0 →
(#‘(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))) = (⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))) |
231 | 228, 229,
230 | 3syl 18 |
. . . 4
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(#‘(1...(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))))) = (⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))) |
232 | 227, 231 | eqtr2d 2645 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(⌊‘((log‘(2 · 𝑁)) / (log‘𝑃))) = Σ𝑘 ∈ (1...(2 · 𝑁))if(𝑘 ∈ (1...(⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))), 1,
0)) |
233 | 194, 195,
232 | 3brtr4d 4615 |
. 2
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) ≤ (⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃)))) |
234 | | simpr 476 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈
ℙ) |
235 | | nnnn0 11176 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℕ0) |
236 | | fzctr 12320 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈ (0...(2
· 𝑁))) |
237 | | bccl2 12972 |
. . . . . . 7
⊢ (𝑁 ∈ (0...(2 · 𝑁)) → ((2 · 𝑁)C𝑁) ∈ ℕ) |
238 | 235, 236,
237 | 3syl 18 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → ((2
· 𝑁)C𝑁) ∈
ℕ) |
239 | 238 | adantr 480 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((2
· 𝑁)C𝑁) ∈
ℕ) |
240 | 234, 239 | pccld 15393 |
. . . 4
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) ∈
ℕ0) |
241 | 240 | nn0zd 11356 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) ∈ ℤ) |
242 | | efexple 24806 |
. . 3
⊢ (((𝑃 ∈ ℝ ∧ 1 <
𝑃) ∧ (𝑃 pCnt ((2 · 𝑁)C𝑁)) ∈ ℤ ∧ (2 · 𝑁) ∈ ℝ+)
→ ((𝑃↑(𝑃 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁) ↔ (𝑃 pCnt ((2 · 𝑁)C𝑁)) ≤ (⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃))))) |
243 | 94, 99, 241, 105, 242 | syl211anc 1324 |
. 2
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((𝑃↑(𝑃 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁) ↔ (𝑃 pCnt ((2 · 𝑁)C𝑁)) ≤ (⌊‘((log‘(2
· 𝑁)) /
(log‘𝑃))))) |
244 | 233, 243 | mpbird 246 |
1
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(𝑃 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁)) |