Proof of Theorem selberg2
Step | Hyp | Ref
| Expression |
1 | | reex 10962 |
. . . . . . 7
⊢ ℝ
∈ V |
2 | | rpssre 12737 |
. . . . . . 7
⊢
ℝ+ ⊆ ℝ |
3 | 1, 2 | ssexi 5246 |
. . . . . 6
⊢
ℝ+ ∈ V |
4 | 3 | a1i 11 |
. . . . 5
⊢ (⊤
→ ℝ+ ∈ V) |
5 | | ovexd 7310 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ V) |
6 | | ovexd 7310 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) ∈ V) |
7 | | eqidd 2739 |
. . . . 5
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦
((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))))) |
8 | | eqidd 2739 |
. . . . 5
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) = (𝑥 ∈ ℝ+ ↦
((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) |
9 | 4, 5, 6, 7, 8 | offval2 7553 |
. . . 4
⊢ (⊤
→ ((𝑥 ∈
ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘f −
(𝑥 ∈
ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ+ ↦
(((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)))) |
10 | 9 | mptru 1546 |
. . 3
⊢ ((𝑥 ∈ ℝ+
↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘f −
(𝑥 ∈
ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ+ ↦
(((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) |
11 | | fzfid 13693 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ (1...(⌊‘𝑥)) ∈ Fin) |
12 | | elfznn 13285 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℕ) |
13 | 12 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℕ) |
14 | | vmacl 26267 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ →
(Λ‘𝑛) ∈
ℝ) |
15 | 13, 14 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Λ‘𝑛)
∈ ℝ) |
16 | 15 | recnd 11003 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Λ‘𝑛)
∈ ℂ) |
17 | 13 | nnrpd 12770 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℝ+) |
18 | | relogcl 25731 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℝ+
→ (log‘𝑛) ∈
ℝ) |
19 | 17, 18 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (log‘𝑛) ∈
ℝ) |
20 | 19 | recnd 11003 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (log‘𝑛) ∈
ℂ) |
21 | | rpre 12738 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ) |
22 | | nndivre 12014 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑥 / 𝑛) ∈ ℝ) |
23 | 21, 12, 22 | syl2an 596 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑛) ∈
ℝ) |
24 | | chpcl 26273 |
. . . . . . . . . . . 12
⊢ ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ) |
25 | 23, 24 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (ψ‘(𝑥 /
𝑛)) ∈
ℝ) |
26 | 25 | recnd 11003 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (ψ‘(𝑥 /
𝑛)) ∈
ℂ) |
27 | 20, 26 | addcld 10994 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((log‘𝑛) +
(ψ‘(𝑥 / 𝑛))) ∈
ℂ) |
28 | 16, 27 | mulcld 10995 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
· ((log‘𝑛) +
(ψ‘(𝑥 / 𝑛)))) ∈
ℂ) |
29 | 11, 28 | fsumcl 15445 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ∈ ℂ) |
30 | | rpcn 12740 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℂ) |
31 | | rpne0 12746 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ≠
0) |
32 | 29, 30, 31 | divcld 11751 |
. . . . . 6
⊢ (𝑥 ∈ ℝ+
→ (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℂ) |
33 | | 2cn 12048 |
. . . . . . 7
⊢ 2 ∈
ℂ |
34 | | relogcl 25731 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ (log‘𝑥) ∈
ℝ) |
35 | 34 | recnd 11003 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ (log‘𝑥) ∈
ℂ) |
36 | | mulcl 10955 |
. . . . . . 7
⊢ ((2
∈ ℂ ∧ (log‘𝑥) ∈ ℂ) → (2 ·
(log‘𝑥)) ∈
ℂ) |
37 | 33, 35, 36 | sylancr 587 |
. . . . . 6
⊢ (𝑥 ∈ ℝ+
→ (2 · (log‘𝑥)) ∈ ℂ) |
38 | 16, 20 | mulcld 10995 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
· (log‘𝑛))
∈ ℂ) |
39 | 11, 38 | fsumcl 15445 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) ∈ ℂ) |
40 | | chpcl 26273 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ →
(ψ‘𝑥) ∈
ℝ) |
41 | 21, 40 | syl 17 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ (ψ‘𝑥)
∈ ℝ) |
42 | 41 | recnd 11003 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ (ψ‘𝑥)
∈ ℂ) |
43 | 42, 35 | mulcld 10995 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ ((ψ‘𝑥)
· (log‘𝑥))
∈ ℂ) |
44 | 39, 43 | subcld 11332 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) ∈ ℂ) |
45 | 44, 30, 31 | divcld 11751 |
. . . . . 6
⊢ (𝑥 ∈ ℝ+
→ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) ∈ ℂ) |
46 | 32, 37, 45 | sub32d 11364 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
→ (((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) − (2 · (log‘𝑥)))) |
47 | | rpcnne0 12748 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) |
48 | | divsubdir 11669 |
. . . . . . . 8
⊢
((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ∈ ℂ ∧ (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) |
49 | 29, 44, 47, 48 | syl3anc 1370 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) |
50 | 16, 20, 26 | adddid 10999 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
· ((log‘𝑛) +
(ψ‘(𝑥 / 𝑛)))) = (((Λ‘𝑛) · (log‘𝑛)) + ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))) |
51 | 50 | sumeq2dv 15415 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (log‘𝑛)) + ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))) |
52 | 16, 26 | mulcld 10995 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
· (ψ‘(𝑥 /
𝑛))) ∈
ℂ) |
53 | 11, 38, 52 | fsumadd 15452 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) · (log‘𝑛)) + ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))) |
54 | 51, 53 | eqtrd 2778 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))) |
55 | 54 | oveq1d 7290 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) = ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))))) |
56 | 11, 52 | fsumcl 15445 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ) |
57 | 39, 56, 43 | pnncand 11371 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) = (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((ψ‘𝑥) · (log‘𝑥)))) |
58 | 56, 43 | addcomd 11177 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((ψ‘𝑥) · (log‘𝑥))) = (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))) |
59 | 55, 57, 58 | 3eqtrd 2782 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) = (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))) |
60 | 59 | oveq1d 7290 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) / 𝑥) = ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) |
61 | 49, 60 | eqtr3d 2780 |
. . . . . 6
⊢ (𝑥 ∈ ℝ+
→ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) = ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) |
62 | 61 | oveq1d 7290 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
→ (((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) − (2 · (log‘𝑥))) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) |
63 | 46, 62 | eqtrd 2778 |
. . . 4
⊢ (𝑥 ∈ ℝ+
→ (((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) |
64 | 63 | mpteq2ia 5177 |
. . 3
⊢ (𝑥 ∈ ℝ+
↦ (((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ+ ↦
(((((ψ‘𝑥)
· (log‘𝑥)) +
Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) |
65 | 10, 64 | eqtri 2766 |
. 2
⊢ ((𝑥 ∈ ℝ+
↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘f −
(𝑥 ∈
ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ+ ↦
(((((ψ‘𝑥)
· (log‘𝑥)) +
Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) |
66 | | selberg 26696 |
. . 3
⊢ (𝑥 ∈ ℝ+
↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈
𝑂(1) |
67 | | selberg2lem 26698 |
. . 3
⊢ (𝑥 ∈ ℝ+
↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1) |
68 | | o1sub 15325 |
. . 3
⊢ (((𝑥 ∈ ℝ+
↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1) ∧
(𝑥 ∈
ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ+
↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘f −
(𝑥 ∈
ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) ∈ 𝑂(1)) |
69 | 66, 67, 68 | mp2an 689 |
. 2
⊢ ((𝑥 ∈ ℝ+
↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘f −
(𝑥 ∈
ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) ∈ 𝑂(1) |
70 | 65, 69 | eqeltrri 2836 |
1
⊢ (𝑥 ∈ ℝ+
↦ (((((ψ‘𝑥)
· (log‘𝑥)) +
Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈
𝑂(1) |