Step | Hyp | Ref
| Expression |
1 | | 4nn 11464 |
. . . . . 6
⊢ 4 ∈
ℕ |
2 | | eluznn 12070 |
. . . . . . . 8
⊢ ((4
∈ ℕ ∧ 𝑁
∈ (ℤ≥‘4)) → 𝑁 ∈ ℕ) |
3 | 1, 2 | mpan 680 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘4) → 𝑁 ∈ ℕ) |
4 | 3 | nnnn0d 11707 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘4) → 𝑁 ∈
ℕ0) |
5 | | nnexpcl 13196 |
. . . . . 6
⊢ ((4
∈ ℕ ∧ 𝑁
∈ ℕ0) → (4↑𝑁) ∈ ℕ) |
6 | 1, 4, 5 | sylancr 581 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘4) → (4↑𝑁) ∈ ℕ) |
7 | 6 | nnrpd 12184 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘4) → (4↑𝑁) ∈
ℝ+) |
8 | 3 | nnrpd 12184 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘4) → 𝑁 ∈
ℝ+) |
9 | 7, 8 | rpdivcld 12203 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘4) → ((4↑𝑁) / 𝑁) ∈
ℝ+) |
10 | 9 | relogcld 24817 |
. 2
⊢ (𝑁 ∈
(ℤ≥‘4) → (log‘((4↑𝑁) / 𝑁)) ∈ ℝ) |
11 | | fzctr 12775 |
. . . . . 6
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈ (0...(2
· 𝑁))) |
12 | 4, 11 | syl 17 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘4) → 𝑁 ∈ (0...(2 · 𝑁))) |
13 | | bccl2 13434 |
. . . . 5
⊢ (𝑁 ∈ (0...(2 · 𝑁)) → ((2 · 𝑁)C𝑁) ∈ ℕ) |
14 | 12, 13 | syl 17 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘4) → ((2 · 𝑁)C𝑁) ∈ ℕ) |
15 | 14 | nnrpd 12184 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘4) → ((2 · 𝑁)C𝑁) ∈
ℝ+) |
16 | 15 | relogcld 24817 |
. 2
⊢ (𝑁 ∈
(ℤ≥‘4) → (log‘((2 · 𝑁)C𝑁)) ∈ ℝ) |
17 | | 2z 11766 |
. . . . . . 7
⊢ 2 ∈
ℤ |
18 | | eluzelz 12007 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘4) → 𝑁 ∈ ℤ) |
19 | | zmulcl 11783 |
. . . . . . 7
⊢ ((2
∈ ℤ ∧ 𝑁
∈ ℤ) → (2 · 𝑁) ∈ ℤ) |
20 | 17, 18, 19 | sylancr 581 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘4) → (2 · 𝑁) ∈ ℤ) |
21 | 20 | zred 11839 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘4) → (2 · 𝑁) ∈ ℝ) |
22 | | ppicl 25320 |
. . . . 5
⊢ ((2
· 𝑁) ∈ ℝ
→ (π‘(2 · 𝑁)) ∈
ℕ0) |
23 | 21, 22 | syl 17 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘4) → (π‘(2 · 𝑁)) ∈
ℕ0) |
24 | 23 | nn0red 11708 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘4) → (π‘(2 · 𝑁)) ∈
ℝ) |
25 | | 2nn 11453 |
. . . . . 6
⊢ 2 ∈
ℕ |
26 | | nnmulcl 11404 |
. . . . . 6
⊢ ((2
∈ ℕ ∧ 𝑁
∈ ℕ) → (2 · 𝑁) ∈ ℕ) |
27 | 25, 3, 26 | sylancr 581 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘4) → (2 · 𝑁) ∈ ℕ) |
28 | 27 | nnrpd 12184 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘4) → (2 · 𝑁) ∈
ℝ+) |
29 | 28 | relogcld 24817 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘4) → (log‘(2 · 𝑁)) ∈
ℝ) |
30 | 24, 29 | remulcld 10409 |
. 2
⊢ (𝑁 ∈
(ℤ≥‘4) → ((π‘(2 · 𝑁)) · (log‘(2
· 𝑁))) ∈
ℝ) |
31 | | bclbnd 25468 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘4) → ((4↑𝑁) / 𝑁) < ((2 · 𝑁)C𝑁)) |
32 | | logltb 24794 |
. . . 4
⊢
((((4↑𝑁) /
𝑁) ∈
ℝ+ ∧ ((2 · 𝑁)C𝑁) ∈ ℝ+) →
(((4↑𝑁) / 𝑁) < ((2 · 𝑁)C𝑁) ↔ (log‘((4↑𝑁) / 𝑁)) < (log‘((2 · 𝑁)C𝑁)))) |
33 | 9, 15, 32 | syl2anc 579 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘4) → (((4↑𝑁) / 𝑁) < ((2 · 𝑁)C𝑁) ↔ (log‘((4↑𝑁) / 𝑁)) < (log‘((2 · 𝑁)C𝑁)))) |
34 | 31, 33 | mpbid 224 |
. 2
⊢ (𝑁 ∈
(ℤ≥‘4) → (log‘((4↑𝑁) / 𝑁)) < (log‘((2 · 𝑁)C𝑁))) |
35 | | chebbnd1lem1.1 |
. . . . . . . 8
⊢ 𝐾 = if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) |
36 | 27, 14 | ifcld 4352 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘4) → if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ∈ ℕ) |
37 | 35, 36 | syl5eqel 2863 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘4) → 𝐾 ∈ ℕ) |
38 | 37 | nnred 11396 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘4) → 𝐾 ∈ ℝ) |
39 | | ppicl 25320 |
. . . . . 6
⊢ (𝐾 ∈ ℝ →
(π‘𝐾)
∈ ℕ0) |
40 | 38, 39 | syl 17 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘4) → (π‘𝐾) ∈
ℕ0) |
41 | 40 | nn0red 11708 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘4) → (π‘𝐾) ∈ ℝ) |
42 | 41, 29 | remulcld 10409 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘4) → ((π‘𝐾) · (log‘(2 · 𝑁))) ∈
ℝ) |
43 | | fzfid 13096 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘4) → (1...𝐾) ∈ Fin) |
44 | | inss1 4053 |
. . . . . 6
⊢
((1...𝐾) ∩
ℙ) ⊆ (1...𝐾) |
45 | | ssfi 8470 |
. . . . . 6
⊢
(((1...𝐾) ∈ Fin
∧ ((1...𝐾) ∩
ℙ) ⊆ (1...𝐾))
→ ((1...𝐾) ∩
ℙ) ∈ Fin) |
46 | 43, 44, 45 | sylancl 580 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘4) → ((1...𝐾) ∩ ℙ) ∈
Fin) |
47 | 37 | nnzd 11838 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘4) → 𝐾 ∈ ℤ) |
48 | 14 | nnzd 11838 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘4) → ((2 · 𝑁)C𝑁) ∈ ℤ) |
49 | 14 | nnred 11396 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈
(ℤ≥‘4) → ((2 · 𝑁)C𝑁) ∈ ℝ) |
50 | | min2 12338 |
. . . . . . . . . . . 12
⊢ (((2
· 𝑁) ∈ ℝ
∧ ((2 · 𝑁)C𝑁) ∈ ℝ) → if((2
· 𝑁) ≤ ((2
· 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ≤ ((2 · 𝑁)C𝑁)) |
51 | 21, 49, 50 | syl2anc 579 |
. . . . . . . . . . 11
⊢ (𝑁 ∈
(ℤ≥‘4) → if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ≤ ((2 · 𝑁)C𝑁)) |
52 | 35, 51 | syl5eqbr 4923 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘4) → 𝐾 ≤ ((2 · 𝑁)C𝑁)) |
53 | | eluz2 12003 |
. . . . . . . . . 10
⊢ (((2
· 𝑁)C𝑁) ∈
(ℤ≥‘𝐾) ↔ (𝐾 ∈ ℤ ∧ ((2 · 𝑁)C𝑁) ∈ ℤ ∧ 𝐾 ≤ ((2 · 𝑁)C𝑁))) |
54 | 47, 48, 52, 53 | syl3anbrc 1400 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘4) → ((2 · 𝑁)C𝑁) ∈ (ℤ≥‘𝐾)) |
55 | | fzss2 12703 |
. . . . . . . . 9
⊢ (((2
· 𝑁)C𝑁) ∈
(ℤ≥‘𝐾) → (1...𝐾) ⊆ (1...((2 · 𝑁)C𝑁))) |
56 | 54, 55 | syl 17 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘4) → (1...𝐾) ⊆ (1...((2 · 𝑁)C𝑁))) |
57 | 56 | ssrind 4060 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘4) → ((1...𝐾) ∩ ℙ) ⊆ ((1...((2 ·
𝑁)C𝑁)) ∩ ℙ)) |
58 | 57 | sselda 3821 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) |
59 | | inss1 4053 |
. . . . . . . . . . 11
⊢ ((1...((2
· 𝑁)C𝑁)) ∩ ℙ) ⊆
(1...((2 · 𝑁)C𝑁)) |
60 | | simpr 479 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) |
61 | 59, 60 | sseldi 3819 |
. . . . . . . . . 10
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → 𝑘 ∈ (1...((2 · 𝑁)C𝑁))) |
62 | | elfznn 12692 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (1...((2 · 𝑁)C𝑁)) → 𝑘 ∈ ℕ) |
63 | 61, 62 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → 𝑘 ∈ ℕ) |
64 | | inss2 4054 |
. . . . . . . . . . 11
⊢ ((1...((2
· 𝑁)C𝑁)) ∩ ℙ) ⊆
ℙ |
65 | 64, 60 | sseldi 3819 |
. . . . . . . . . 10
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → 𝑘 ∈ ℙ) |
66 | 14 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → ((2 · 𝑁)C𝑁) ∈ ℕ) |
67 | 65, 66 | pccld 15970 |
. . . . . . . . 9
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈
ℕ0) |
68 | 63, 67 | nnexpcld 13357 |
. . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ∈ ℕ) |
69 | 68 | nnrpd 12184 |
. . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ∈
ℝ+) |
70 | 69 | relogcld 24817 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ∈ ℝ) |
71 | 58, 70 | syldan 585 |
. . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ∈ ℝ) |
72 | 29 | adantr 474 |
. . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (log‘(2
· 𝑁)) ∈
ℝ) |
73 | | elin 4019 |
. . . . . . . . 9
⊢ (𝑘 ∈ ((1...𝐾) ∩ ℙ) ↔ (𝑘 ∈ (1...𝐾) ∧ 𝑘 ∈ ℙ)) |
74 | 73 | simprbi 492 |
. . . . . . . 8
⊢ (𝑘 ∈ ((1...𝐾) ∩ ℙ) → 𝑘 ∈ ℙ) |
75 | | bposlem1 25472 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ ℙ) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁)) |
76 | 3, 74, 75 | syl2an 589 |
. . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁)) |
77 | 58, 69 | syldan 585 |
. . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ∈
ℝ+) |
78 | 77 | reeflogd 24818 |
. . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) →
(exp‘(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))))) = (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) |
79 | 28 | adantr 474 |
. . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (2 · 𝑁) ∈
ℝ+) |
80 | 79 | reeflogd 24818 |
. . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) →
(exp‘(log‘(2 · 𝑁))) = (2 · 𝑁)) |
81 | 76, 78, 80 | 3brtr4d 4920 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) →
(exp‘(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))))) ≤ (exp‘(log‘(2 ·
𝑁)))) |
82 | | efle 15259 |
. . . . . . 7
⊢
(((log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ∈ ℝ ∧ (log‘(2
· 𝑁)) ∈
ℝ) → ((log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ≤ (log‘(2 · 𝑁)) ↔
(exp‘(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))))) ≤ (exp‘(log‘(2 ·
𝑁))))) |
83 | 71, 72, 82 | syl2anc 579 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → ((log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ≤ (log‘(2 · 𝑁)) ↔
(exp‘(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))))) ≤ (exp‘(log‘(2 ·
𝑁))))) |
84 | 81, 83 | mpbird 249 |
. . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ≤ (log‘(2 · 𝑁))) |
85 | 46, 71, 72, 84 | fsumle 14944 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘4) → Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ≤ Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(2 · 𝑁))) |
86 | 70 | recnd 10407 |
. . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ∈ ℂ) |
87 | 58, 86 | syldan 585 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ∈ ℂ) |
88 | | eldifn 3956 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (((1...((2 ·
𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) → ¬
𝑘 ∈ ((1...𝐾) ∩
ℙ)) |
89 | 88 | adantl 475 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → ¬
𝑘 ∈ ((1...𝐾) ∩
ℙ)) |
90 | | simpr 479 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ∈ (((1...((2 ·
𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩
ℙ))) |
91 | 90 | eldifad 3804 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) |
92 | 59, 91 | sseldi 3819 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ∈ (1...((2 · 𝑁)C𝑁))) |
93 | 92, 62 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ∈
ℕ) |
94 | 93 | adantrr 707 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ∈
ℕ) |
95 | 94 | nnred 11396 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ∈
ℝ) |
96 | 91, 68 | syldan 585 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ∈ ℕ) |
97 | 96 | nnred 11396 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ∈ ℝ) |
98 | 97 | adantrr 707 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ∈ ℝ) |
99 | 21 | adantr 474 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (2 · 𝑁) ∈
ℝ) |
100 | 94 | nncnd 11397 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ∈
ℂ) |
101 | 100 | exp1d 13327 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘↑1) = 𝑘) |
102 | 94 | nnge1d 11428 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 1 ≤ 𝑘) |
103 | | simprr 763 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ) |
104 | | nnuz 12034 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ℕ =
(ℤ≥‘1) |
105 | 103, 104 | syl6eleq 2869 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈
(ℤ≥‘1)) |
106 | 95, 102, 105 | leexp2ad 13368 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘↑1) ≤ (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) |
107 | 101, 106 | eqbrtrrd 4912 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ≤ (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) |
108 | 3 | adantr 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑁 ∈ ℕ) |
109 | 64, 91 | sseldi 3819 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ∈
ℙ) |
110 | 109 | adantrr 707 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ∈
ℙ) |
111 | 108, 110,
75 | syl2anc 579 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁)) |
112 | 95, 98, 99, 107, 111 | letrd 10535 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ≤ (2 · 𝑁)) |
113 | | elfzle2 12667 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (1...((2 · 𝑁)C𝑁)) → 𝑘 ≤ ((2 · 𝑁)C𝑁)) |
114 | 92, 113 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ≤ ((2 · 𝑁)C𝑁)) |
115 | 114 | adantrr 707 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ≤ ((2 · 𝑁)C𝑁)) |
116 | 49 | adantr 474 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → ((2 ·
𝑁)C𝑁) ∈ ℝ) |
117 | | lemin 12340 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∈ ℝ ∧ (2
· 𝑁) ∈ ℝ
∧ ((2 · 𝑁)C𝑁) ∈ ℝ) → (𝑘 ≤ if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ↔ (𝑘 ≤ (2 · 𝑁) ∧ 𝑘 ≤ ((2 · 𝑁)C𝑁)))) |
118 | 95, 99, 116, 117 | syl3anc 1439 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘 ≤ if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ↔ (𝑘 ≤ (2 · 𝑁) ∧ 𝑘 ≤ ((2 · 𝑁)C𝑁)))) |
119 | 112, 115,
118 | mpbir2and 703 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ≤ if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁))) |
120 | 119, 35 | syl6breqr 4930 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ≤ 𝐾) |
121 | 37 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝐾 ∈ ℕ) |
122 | 121 | nnzd 11838 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝐾 ∈ ℤ) |
123 | | fznn 12731 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ ℤ → (𝑘 ∈ (1...𝐾) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≤ 𝐾))) |
124 | 122, 123 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘 ∈ (1...𝐾) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≤ 𝐾))) |
125 | 94, 120, 124 | mpbir2and 703 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ∈ (1...𝐾)) |
126 | 125, 110 | elind 4021 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ∈ ((1...𝐾) ∩ ℙ)) |
127 | 126 | expr 450 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) →
((𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ → 𝑘 ∈ ((1...𝐾) ∩ ℙ))) |
128 | 89, 127 | mtod 190 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → ¬
(𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ) |
129 | 91, 67 | syldan 585 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈
ℕ0) |
130 | | elnn0 11649 |
. . . . . . . . . . . . 13
⊢ ((𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ0 ↔ ((𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ ∨ (𝑘 pCnt ((2 · 𝑁)C𝑁)) = 0)) |
131 | 129, 130 | sylib 210 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) →
((𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ ∨ (𝑘 pCnt ((2 · 𝑁)C𝑁)) = 0)) |
132 | 131 | ord 853 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (¬
(𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ → (𝑘 pCnt ((2 · 𝑁)C𝑁)) = 0)) |
133 | 128, 132 | mpd 15 |
. . . . . . . . . 10
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘 pCnt ((2 · 𝑁)C𝑁)) = 0) |
134 | 133 | oveq2d 6940 |
. . . . . . . . 9
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) = (𝑘↑0)) |
135 | 93 | nncnd 11397 |
. . . . . . . . . 10
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ∈
ℂ) |
136 | 135 | exp0d 13326 |
. . . . . . . . 9
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘↑0) = 1) |
137 | 134, 136 | eqtrd 2814 |
. . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) = 1) |
138 | 137 | fveq2d 6452 |
. . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) →
(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = (log‘1)) |
139 | | log1 24780 |
. . . . . . 7
⊢
(log‘1) = 0 |
140 | 138, 139 | syl6eq 2830 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) →
(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = 0) |
141 | | fzfid 13096 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘4) → (1...((2 · 𝑁)C𝑁)) ∈ Fin) |
142 | | ssfi 8470 |
. . . . . . 7
⊢
(((1...((2 · 𝑁)C𝑁)) ∈ Fin ∧ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ⊆ (1...((2 ·
𝑁)C𝑁))) → ((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∈
Fin) |
143 | 141, 59, 142 | sylancl 580 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘4) → ((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∈
Fin) |
144 | 57, 87, 140, 143 | fsumss 14872 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘4) → Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = Σ𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))))) |
145 | 63 | nnrpd 12184 |
. . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → 𝑘 ∈ ℝ+) |
146 | 67 | nn0zd 11837 |
. . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℤ) |
147 | | relogexp 24790 |
. . . . . . 7
⊢ ((𝑘 ∈ ℝ+
∧ (𝑘 pCnt ((2 ·
𝑁)C𝑁)) ∈ ℤ) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = ((𝑘 pCnt ((2 · 𝑁)C𝑁)) · (log‘𝑘))) |
148 | 145, 146,
147 | syl2anc 579 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = ((𝑘 pCnt ((2 · 𝑁)C𝑁)) · (log‘𝑘))) |
149 | 148 | sumeq2dv 14850 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘4) → Σ𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = Σ𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)((𝑘 pCnt ((2 · 𝑁)C𝑁)) · (log‘𝑘))) |
150 | | pclogsum 25403 |
. . . . . 6
⊢ (((2
· 𝑁)C𝑁) ∈ ℕ →
Σ𝑘 ∈ ((1...((2
· 𝑁)C𝑁)) ∩ ℙ)((𝑘 pCnt ((2 · 𝑁)C𝑁)) · (log‘𝑘)) = (log‘((2 · 𝑁)C𝑁))) |
151 | 14, 150 | syl 17 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘4) → Σ𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)((𝑘 pCnt ((2 · 𝑁)C𝑁)) · (log‘𝑘)) = (log‘((2 · 𝑁)C𝑁))) |
152 | 144, 149,
151 | 3eqtrd 2818 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘4) → Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = (log‘((2 · 𝑁)C𝑁))) |
153 | 29 | recnd 10407 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘4) → (log‘(2 · 𝑁)) ∈
ℂ) |
154 | | fsumconst 14935 |
. . . . . 6
⊢
((((1...𝐾) ∩
ℙ) ∈ Fin ∧ (log‘(2 · 𝑁)) ∈ ℂ) → Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(2 · 𝑁)) = ((♯‘((1...𝐾) ∩ ℙ)) ·
(log‘(2 · 𝑁)))) |
155 | 46, 153, 154 | syl2anc 579 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘4) → Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(2 · 𝑁)) = ((♯‘((1...𝐾) ∩ ℙ)) ·
(log‘(2 · 𝑁)))) |
156 | | 2eluzge1 12045 |
. . . . . . 7
⊢ 2 ∈
(ℤ≥‘1) |
157 | | ppival2g 25318 |
. . . . . . 7
⊢ ((𝐾 ∈ ℤ ∧ 2 ∈
(ℤ≥‘1)) → (π‘𝐾) = (♯‘((1...𝐾) ∩ ℙ))) |
158 | 47, 156, 157 | sylancl 580 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘4) → (π‘𝐾) = (♯‘((1...𝐾) ∩ ℙ))) |
159 | 158 | oveq1d 6939 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘4) → ((π‘𝐾) · (log‘(2 · 𝑁))) =
((♯‘((1...𝐾)
∩ ℙ)) · (log‘(2 · 𝑁)))) |
160 | 155, 159 | eqtr4d 2817 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘4) → Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(2 · 𝑁)) = ((π‘𝐾) · (log‘(2
· 𝑁)))) |
161 | 85, 152, 160 | 3brtr3d 4919 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘4) → (log‘((2 · 𝑁)C𝑁)) ≤ ((π‘𝐾) · (log‘(2
· 𝑁)))) |
162 | | min1 12337 |
. . . . . . 7
⊢ (((2
· 𝑁) ∈ ℝ
∧ ((2 · 𝑁)C𝑁) ∈ ℝ) → if((2
· 𝑁) ≤ ((2
· 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ≤ (2 · 𝑁)) |
163 | 21, 49, 162 | syl2anc 579 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘4) → if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ≤ (2 · 𝑁)) |
164 | 35, 163 | syl5eqbr 4923 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘4) → 𝐾 ≤ (2 · 𝑁)) |
165 | | ppiwordi 25351 |
. . . . 5
⊢ ((𝐾 ∈ ℝ ∧ (2
· 𝑁) ∈ ℝ
∧ 𝐾 ≤ (2 ·
𝑁)) →
(π‘𝐾) ≤
(π‘(2 · 𝑁))) |
166 | 38, 21, 164, 165 | syl3anc 1439 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘4) → (π‘𝐾) ≤ (π‘(2 · 𝑁))) |
167 | | 1red 10379 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘4) → 1 ∈ ℝ) |
168 | | 2re 11454 |
. . . . . . . 8
⊢ 2 ∈
ℝ |
169 | 168 | a1i 11 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘4) → 2 ∈ ℝ) |
170 | | 1lt2 11558 |
. . . . . . . 8
⊢ 1 <
2 |
171 | 170 | a1i 11 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘4) → 1 < 2) |
172 | | 2t1e2 11550 |
. . . . . . . 8
⊢ (2
· 1) = 2 |
173 | 3 | nnge1d 11428 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘4) → 1 ≤ 𝑁) |
174 | | eluzelre 12008 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘4) → 𝑁 ∈ ℝ) |
175 | | 2pos 11490 |
. . . . . . . . . . . 12
⊢ 0 <
2 |
176 | 168, 175 | pm3.2i 464 |
. . . . . . . . . . 11
⊢ (2 ∈
ℝ ∧ 0 < 2) |
177 | 176 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘4) → (2 ∈ ℝ ∧ 0 <
2)) |
178 | | lemul2 11233 |
. . . . . . . . . 10
⊢ ((1
∈ ℝ ∧ 𝑁
∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (1 ≤ 𝑁 ↔ (2 · 1) ≤ (2
· 𝑁))) |
179 | 167, 174,
177, 178 | syl3anc 1439 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘4) → (1 ≤ 𝑁 ↔ (2 · 1) ≤ (2 ·
𝑁))) |
180 | 173, 179 | mpbid 224 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘4) → (2 · 1) ≤ (2 · 𝑁)) |
181 | 172, 180 | syl5eqbrr 4924 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘4) → 2 ≤ (2 · 𝑁)) |
182 | 167, 169,
21, 171, 181 | ltletrd 10538 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘4) → 1 < (2 · 𝑁)) |
183 | 21, 182 | rplogcld 24823 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘4) → (log‘(2 · 𝑁)) ∈
ℝ+) |
184 | 41, 24, 183 | lemul1d 12229 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘4) → ((π‘𝐾) ≤ (π‘(2 · 𝑁)) ↔
((π‘𝐾)
· (log‘(2 · 𝑁))) ≤ ((π‘(2 ·
𝑁)) · (log‘(2
· 𝑁))))) |
185 | 166, 184 | mpbid 224 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘4) → ((π‘𝐾) · (log‘(2 · 𝑁))) ≤ ((π‘(2
· 𝑁)) ·
(log‘(2 · 𝑁)))) |
186 | 16, 42, 30, 161, 185 | letrd 10535 |
. 2
⊢ (𝑁 ∈
(ℤ≥‘4) → (log‘((2 · 𝑁)C𝑁)) ≤ ((π‘(2 ·
𝑁)) · (log‘(2
· 𝑁)))) |
187 | 10, 16, 30, 34, 186 | ltletrd 10538 |
1
⊢ (𝑁 ∈
(ℤ≥‘4) → (log‘((4↑𝑁) / 𝑁)) < ((π‘(2 ·
𝑁)) · (log‘(2
· 𝑁)))) |