Step | Hyp | Ref
| Expression |
1 | | 1re 10976 |
. . . . . . . 8
⊢ 1 ∈
ℝ |
2 | | elicopnf 13176 |
. . . . . . . 8
⊢ (1 ∈
ℝ → (𝑦 ∈
(1[,)+∞) ↔ (𝑦
∈ ℝ ∧ 1 ≤ 𝑦))) |
3 | 1, 2 | ax-mp 5 |
. . . . . . 7
⊢ (𝑦 ∈ (1[,)+∞) ↔
(𝑦 ∈ ℝ ∧ 1
≤ 𝑦)) |
4 | 3 | simplbi 498 |
. . . . . 6
⊢ (𝑦 ∈ (1[,)+∞) →
𝑦 ∈
ℝ) |
5 | 4 | ssriv 3930 |
. . . . 5
⊢
(1[,)+∞) ⊆ ℝ |
6 | 5 | a1i 11 |
. . . 4
⊢ (⊤
→ (1[,)+∞) ⊆ ℝ) |
7 | 1 | a1i 11 |
. . . 4
⊢ (⊤
→ 1 ∈ ℝ) |
8 | | fzfid 13691 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑦
∈ (1[,)+∞)) → (1...(⌊‘𝑦)) ∈ Fin) |
9 | | elfznn 13284 |
. . . . . . . . . . 11
⊢ (𝑚 ∈
(1...(⌊‘𝑦))
→ 𝑚 ∈
ℕ) |
10 | 9 | adantl 482 |
. . . . . . . . . 10
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℕ) |
11 | | vmacl 26265 |
. . . . . . . . . 10
⊢ (𝑚 ∈ ℕ →
(Λ‘𝑚) ∈
ℝ) |
12 | 10, 11 | syl 17 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑚) ∈
ℝ) |
13 | 10 | nnrpd 12769 |
. . . . . . . . . 10
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℝ+) |
14 | 13 | relogcld 25776 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → (log‘𝑚) ∈
ℝ) |
15 | 12, 14 | remulcld 11006 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) →
((Λ‘𝑚)
· (log‘𝑚))
∈ ℝ) |
16 | 8, 15 | fsumrecl 15444 |
. . . . . . 7
⊢
((⊤ ∧ 𝑦
∈ (1[,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ∈
ℝ) |
17 | 4 | adantl 482 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑦
∈ (1[,)+∞)) → 𝑦 ∈ ℝ) |
18 | | chpcl 26271 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℝ →
(ψ‘𝑦) ∈
ℝ) |
19 | 17, 18 | syl 17 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑦
∈ (1[,)+∞)) → (ψ‘𝑦) ∈ ℝ) |
20 | | 1rp 12733 |
. . . . . . . . . . 11
⊢ 1 ∈
ℝ+ |
21 | 20 | a1i 11 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑦
∈ (1[,)+∞)) → 1 ∈ ℝ+) |
22 | 3 | simprbi 497 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (1[,)+∞) → 1
≤ 𝑦) |
23 | 22 | adantl 482 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑦
∈ (1[,)+∞)) → 1 ≤ 𝑦) |
24 | 17, 21, 23 | rpgecld 12810 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑦
∈ (1[,)+∞)) → 𝑦 ∈ ℝ+) |
25 | 24 | relogcld 25776 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑦
∈ (1[,)+∞)) → (log‘𝑦) ∈ ℝ) |
26 | 19, 25 | remulcld 11006 |
. . . . . . 7
⊢
((⊤ ∧ 𝑦
∈ (1[,)+∞)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ) |
27 | 16, 26 | resubcld 11403 |
. . . . . 6
⊢
((⊤ ∧ 𝑦
∈ (1[,)+∞)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈
ℝ) |
28 | 27, 24 | rerpdivcld 12802 |
. . . . 5
⊢
((⊤ ∧ 𝑦
∈ (1[,)+∞)) → ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦) ∈ ℝ) |
29 | 28 | recnd 11004 |
. . . 4
⊢
((⊤ ∧ 𝑦
∈ (1[,)+∞)) → ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦) ∈ ℂ) |
30 | 24 | ex 413 |
. . . . . 6
⊢ (⊤
→ (𝑦 ∈
(1[,)+∞) → 𝑦
∈ ℝ+)) |
31 | 30 | ssrdv 3932 |
. . . . 5
⊢ (⊤
→ (1[,)+∞) ⊆ ℝ+) |
32 | | selberg2lem 26696 |
. . . . . 6
⊢ (𝑦 ∈ ℝ+
↦ ((Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ∈ 𝑂(1) |
33 | 32 | a1i 11 |
. . . . 5
⊢ (⊤
→ (𝑦 ∈
ℝ+ ↦ ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ∈ 𝑂(1)) |
34 | 31, 33 | o1res2 15270 |
. . . 4
⊢ (⊤
→ (𝑦 ∈
(1[,)+∞) ↦ ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ∈ 𝑂(1)) |
35 | | fzfid 13691 |
. . . . . 6
⊢
((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin) |
36 | | elfznn 13284 |
. . . . . . . . 9
⊢ (𝑚 ∈
(1...(⌊‘𝑥))
→ 𝑚 ∈
ℕ) |
37 | 36 | adantl 482 |
. . . . . . . 8
⊢
(((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ) |
38 | 37, 11 | syl 17 |
. . . . . . 7
⊢
(((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈
ℝ) |
39 | 37 | nnrpd 12769 |
. . . . . . . 8
⊢
(((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ+) |
40 | 39 | relogcld 25776 |
. . . . . . 7
⊢
(((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈
ℝ) |
41 | 38, 40 | remulcld 11006 |
. . . . . 6
⊢
(((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) →
((Λ‘𝑚)
· (log‘𝑚))
∈ ℝ) |
42 | 35, 41 | fsumrecl 15444 |
. . . . 5
⊢
((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) ∈
ℝ) |
43 | | chpcl 26271 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ →
(ψ‘𝑥) ∈
ℝ) |
44 | 43 | ad2antrl 725 |
. . . . . 6
⊢
((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) → (ψ‘𝑥) ∈ ℝ) |
45 | | simprl 768 |
. . . . . . . 8
⊢
((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ) |
46 | 20 | a1i 11 |
. . . . . . . 8
⊢
((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) → 1 ∈
ℝ+) |
47 | | simprr 770 |
. . . . . . . 8
⊢
((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥) |
48 | 45, 46, 47 | rpgecld 12810 |
. . . . . . 7
⊢
((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+) |
49 | 48 | relogcld 25776 |
. . . . . 6
⊢
((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ) |
50 | 44, 49 | remulcld 11006 |
. . . . 5
⊢
((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ) |
51 | 42, 50 | readdcld 11005 |
. . . 4
⊢
((⊤ ∧ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥))) ∈
ℝ) |
52 | 27 | adantr 481 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈
ℝ) |
53 | 52 | recnd 11004 |
. . . . . . 7
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈
ℂ) |
54 | 24 | adantr 481 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℝ+) |
55 | 54 | rpcnd 12773 |
. . . . . . 7
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℂ) |
56 | 54 | rpne0d 12776 |
. . . . . . 7
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ≠ 0) |
57 | 53, 55, 56 | absdivd 15165 |
. . . . . 6
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘((Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) = ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / (abs‘𝑦))) |
58 | 17 | adantr 481 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℝ) |
59 | 54 | rpge0d 12775 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ 𝑦) |
60 | 58, 59 | absidd 15132 |
. . . . . . 7
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘𝑦) = 𝑦) |
61 | 60 | oveq2d 7287 |
. . . . . 6
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘(Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / (abs‘𝑦)) = ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 𝑦)) |
62 | 57, 61 | eqtrd 2780 |
. . . . 5
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘((Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) = ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 𝑦)) |
63 | 53 | abscld 15146 |
. . . . . . 7
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘(Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) ∈ ℝ) |
64 | 63, 54 | rerpdivcld 12802 |
. . . . . 6
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘(Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 𝑦) ∈ ℝ) |
65 | 42 | ad2ant2r 744 |
. . . . . . 7
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) ∈
ℝ) |
66 | | simprll 776 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑥 ∈ ℝ) |
67 | 66, 43 | syl 17 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (ψ‘𝑥) ∈ ℝ) |
68 | | simprr 770 |
. . . . . . . . . . 11
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 < 𝑥) |
69 | 58, 66, 68 | ltled 11123 |
. . . . . . . . . 10
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ≤ 𝑥) |
70 | 66, 54, 69 | rpgecld 12810 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑥 ∈ ℝ+) |
71 | 70 | relogcld 25776 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑥) ∈ ℝ) |
72 | 67, 71 | remulcld 11006 |
. . . . . . 7
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ) |
73 | 65, 72 | readdcld 11005 |
. . . . . 6
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥))) ∈
ℝ) |
74 | 20 | a1i 11 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 1 ∈
ℝ+) |
75 | 53 | absge0d 15154 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ (abs‘(Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))))) |
76 | 23 | adantr 481 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 1 ≤ 𝑦) |
77 | 74, 54, 63, 75, 76 | lediv2ad 12793 |
. . . . . . 7
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘(Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 𝑦) ≤ ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 1)) |
78 | 63 | recnd 11004 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘(Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) ∈ ℂ) |
79 | 78 | div1d 11743 |
. . . . . . 7
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘(Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 1) = (abs‘(Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))))) |
80 | 77, 79 | breqtrd 5105 |
. . . . . 6
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘(Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 𝑦) ≤ (abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))))) |
81 | 16 | adantr 481 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ∈
ℝ) |
82 | 58, 18 | syl 17 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (ψ‘𝑦) ∈ ℝ) |
83 | 54 | relogcld 25776 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑦) ∈ ℝ) |
84 | 82, 83 | remulcld 11006 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ) |
85 | 81, 84 | readdcld 11005 |
. . . . . . 7
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑦) · (log‘𝑦))) ∈
ℝ) |
86 | 81 | recnd 11004 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ∈
ℂ) |
87 | 26 | adantr 481 |
. . . . . . . . . 10
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ) |
88 | 87 | recnd 11004 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℂ) |
89 | 86, 88 | abs2dif2d 15168 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘(Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) ≤ ((abs‘Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚))) + (abs‘((ψ‘𝑦) · (log‘𝑦))))) |
90 | | vmage0 26268 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℕ → 0 ≤
(Λ‘𝑚)) |
91 | 10, 90 | syl 17 |
. . . . . . . . . . . . 13
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤
(Λ‘𝑚)) |
92 | 10 | nnred 11988 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℝ) |
93 | 10 | nnge1d 12021 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 1 ≤ 𝑚) |
94 | 92, 93 | logge0d 25783 |
. . . . . . . . . . . . 13
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤
(log‘𝑚)) |
95 | 12, 14, 91, 94 | mulge0d 11552 |
. . . . . . . . . . . 12
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤
((Λ‘𝑚)
· (log‘𝑚))) |
96 | 8, 15, 95 | fsumge0 15505 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑦
∈ (1[,)+∞)) → 0 ≤ Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚))) |
97 | 96 | adantr 481 |
. . . . . . . . . 10
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚))) |
98 | 81, 97 | absidd 15132 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚))) = Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚))) |
99 | | chpge0 26273 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℝ → 0 ≤
(ψ‘𝑦)) |
100 | 58, 99 | syl 17 |
. . . . . . . . . . 11
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ (ψ‘𝑦)) |
101 | 58, 76 | logge0d 25783 |
. . . . . . . . . . 11
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ (log‘𝑦)) |
102 | 82, 83, 100, 101 | mulge0d 11552 |
. . . . . . . . . 10
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ ((ψ‘𝑦) · (log‘𝑦))) |
103 | 87, 102 | absidd 15132 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘((ψ‘𝑦) · (log‘𝑦))) = ((ψ‘𝑦) · (log‘𝑦))) |
104 | 98, 103 | oveq12d 7289 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚))) + (abs‘((ψ‘𝑦) · (log‘𝑦)))) = (Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑦) · (log‘𝑦)))) |
105 | 89, 104 | breqtrd 5105 |
. . . . . . 7
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘(Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) ≤ (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑦) · (log‘𝑦)))) |
106 | | fzfid 13691 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin) |
107 | 36 | adantl 482 |
. . . . . . . . . . 11
⊢
((((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ) |
108 | 107, 11 | syl 17 |
. . . . . . . . . 10
⊢
((((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈
ℝ) |
109 | 107 | nnrpd 12769 |
. . . . . . . . . . 11
⊢
((((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ+) |
110 | 109 | relogcld 25776 |
. . . . . . . . . 10
⊢
((((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈
ℝ) |
111 | 108, 110 | remulcld 11006 |
. . . . . . . . 9
⊢
((((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) →
((Λ‘𝑚)
· (log‘𝑚))
∈ ℝ) |
112 | 107, 90 | syl 17 |
. . . . . . . . . 10
⊢
((((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤
(Λ‘𝑚)) |
113 | 107 | nnred 11988 |
. . . . . . . . . . 11
⊢
((((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ) |
114 | 107 | nnge1d 12021 |
. . . . . . . . . . 11
⊢
((((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑚) |
115 | 113, 114 | logge0d 25783 |
. . . . . . . . . 10
⊢
((((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤
(log‘𝑚)) |
116 | 108, 110,
112, 115 | mulge0d 11552 |
. . . . . . . . 9
⊢
((((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤
((Λ‘𝑚)
· (log‘𝑚))) |
117 | | flword2 13531 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑦 ≤ 𝑥) → (⌊‘𝑥) ∈
(ℤ≥‘(⌊‘𝑦))) |
118 | 58, 66, 69, 117 | syl3anc 1370 |
. . . . . . . . . 10
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (⌊‘𝑥) ∈
(ℤ≥‘(⌊‘𝑦))) |
119 | | fzss2 13295 |
. . . . . . . . . 10
⊢
((⌊‘𝑥)
∈ (ℤ≥‘(⌊‘𝑦)) → (1...(⌊‘𝑦)) ⊆
(1...(⌊‘𝑥))) |
120 | 118, 119 | syl 17 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (1...(⌊‘𝑦)) ⊆
(1...(⌊‘𝑥))) |
121 | 106, 111,
116, 120 | fsumless 15506 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ≤ Σ𝑚 ∈
(1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚))) |
122 | | chpwordi 26304 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑦 ≤ 𝑥) → (ψ‘𝑦) ≤ (ψ‘𝑥)) |
123 | 58, 66, 69, 122 | syl3anc 1370 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (ψ‘𝑦) ≤ (ψ‘𝑥)) |
124 | 54, 70 | logled 25780 |
. . . . . . . . . 10
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (𝑦 ≤ 𝑥 ↔ (log‘𝑦) ≤ (log‘𝑥))) |
125 | 69, 124 | mpbid 231 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑦) ≤ (log‘𝑥)) |
126 | 82, 67, 83, 71, 100, 101, 123, 125 | lemul12ad 11917 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ≤ ((ψ‘𝑥) · (log‘𝑥))) |
127 | 81, 84, 65, 72, 121, 126 | le2addd 11594 |
. . . . . . 7
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑦) · (log‘𝑦))) ≤ (Σ𝑚 ∈
(1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥)))) |
128 | 63, 85, 73, 105, 127 | letrd 11132 |
. . . . . 6
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘(Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) ≤ (Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥)))) |
129 | 64, 63, 73, 80, 128 | letrd 11132 |
. . . . 5
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘(Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 𝑦) ≤ (Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥)))) |
130 | 62, 129 | eqbrtrd 5101 |
. . . 4
⊢
(((⊤ ∧ 𝑦
∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘((Σ𝑚 ∈
(1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥)))) |
131 | 6, 7, 29, 34, 51, 130 | o1bddrp 15249 |
. . 3
⊢ (⊤
→ ∃𝑐 ∈
ℝ+ ∀𝑦 ∈
(1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐) |
132 | 131 | mptru 1549 |
. 2
⊢
∃𝑐 ∈
ℝ+ ∀𝑦 ∈
(1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐 |
133 | | simpl 483 |
. . . 4
⊢ ((𝑐 ∈ ℝ+
∧ ∀𝑦 ∈
(1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐) → 𝑐 ∈ ℝ+) |
134 | | simpr 485 |
. . . 4
⊢ ((𝑐 ∈ ℝ+
∧ ∀𝑦 ∈
(1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐) → ∀𝑦 ∈
(1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐) |
135 | 133, 134 | selberg3lem1 26703 |
. . 3
⊢ ((𝑐 ∈ ℝ+
∧ ∀𝑦 ∈
(1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐) → (𝑥 ∈ (1(,)+∞) ↦ ((((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1)) |
136 | 135 | rexlimiva 3212 |
. 2
⊢
(∃𝑐 ∈
ℝ+ ∀𝑦 ∈
(1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐 → (𝑥 ∈ (1(,)+∞) ↦ ((((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1)) |
137 | 132, 136 | ax-mp 5 |
1
⊢ (𝑥 ∈ (1(,)+∞) ↦
((((2 / (log‘𝑥))
· Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1) |