Proof of Theorem mulogsumlem
Step | Hyp | Ref
| Expression |
1 | | fzfid 13432 |
. . . . . 6
⊢ (𝑥 ∈ ℝ+
→ (1...(⌊‘𝑥)) ∈ Fin) |
2 | | elfznn 13027 |
. . . . . . . . . . 11
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℕ) |
3 | 2 | adantl 485 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℕ) |
4 | | mucl 25878 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ →
(μ‘𝑛) ∈
ℤ) |
5 | 3, 4 | syl 17 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑛)
∈ ℤ) |
6 | 5 | zred 12168 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑛)
∈ ℝ) |
7 | 6, 3 | nndivred 11770 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((μ‘𝑛) /
𝑛) ∈
ℝ) |
8 | 7 | recnd 10747 |
. . . . . 6
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((μ‘𝑛) /
𝑛) ∈
ℂ) |
9 | 1, 8 | fsumcl 15183 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ) |
10 | 9 | adantl 485 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ) |
11 | | emre 25743 |
. . . . . 6
⊢ γ
∈ ℝ |
12 | 11 | recni 10733 |
. . . . 5
⊢ γ
∈ ℂ |
13 | 12 | a1i 11 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → γ ∈ ℂ) |
14 | | mudivsum 26266 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
↦ Σ𝑛 ∈
(1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1) |
15 | 14 | a1i 11 |
. . . 4
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1)) |
16 | | rpssre 12479 |
. . . . . 6
⊢
ℝ+ ⊆ ℝ |
17 | | o1const 15067 |
. . . . . 6
⊢
((ℝ+ ⊆ ℝ ∧ γ ∈ ℂ)
→ (𝑥 ∈
ℝ+ ↦ γ) ∈ 𝑂(1)) |
18 | 16, 12, 17 | mp2an 692 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
↦ γ) ∈ 𝑂(1) |
19 | 18 | a1i 11 |
. . . 4
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ γ) ∈ 𝑂(1)) |
20 | 10, 13, 15, 19 | o1mul2 15072 |
. . 3
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ)) ∈
𝑂(1)) |
21 | | fzfid 13432 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin) |
22 | | elfznn 13027 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛))) → 𝑚 ∈
ℕ) |
23 | 22 | adantl 485 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → 𝑚 ∈
ℕ) |
24 | 23 | nnrecred 11767 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → (1 / 𝑚) ∈
ℝ) |
25 | 21, 24 | fsumrecl 15184 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) ∈ ℝ) |
26 | 2 | nnrpd 12512 |
. . . . . . . . . . 11
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℝ+) |
27 | | rpdivcl 12497 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
ℝ+) → (𝑥 / 𝑛) ∈
ℝ+) |
28 | 26, 27 | sylan2 596 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑛) ∈
ℝ+) |
29 | 28 | relogcld 25366 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (log‘(𝑥 /
𝑛)) ∈
ℝ) |
30 | 25, 29 | resubcld 11146 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℝ) |
31 | 7, 30 | remulcld 10749 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ) |
32 | 1, 31 | fsumrecl 15184 |
. . . . . 6
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ) |
33 | 32 | recnd 10747 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ) |
34 | 33 | adantl 485 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ) |
35 | | mulcl 10699 |
. . . . . 6
⊢
((Σ𝑛 ∈
(1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ ∧ γ ∈
ℂ) → (Σ𝑛
∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈
ℂ) |
36 | 9, 12, 35 | sylancl 589 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
→ (Σ𝑛 ∈
(1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈
ℂ) |
37 | 36 | adantl 485 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈
ℂ) |
38 | | nnrecre 11758 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ → (1 /
𝑚) ∈
ℝ) |
39 | 38 | recnd 10747 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ ℕ → (1 /
𝑚) ∈
ℂ) |
40 | 23, 39 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → (1 / 𝑚) ∈
ℂ) |
41 | 21, 40 | fsumcl 15183 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) ∈ ℂ) |
42 | 29 | recnd 10747 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (log‘(𝑥 /
𝑛)) ∈
ℂ) |
43 | 41, 42 | subcld 11075 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℂ) |
44 | 8, 43 | mulcld 10739 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ) |
45 | | mulcl 10699 |
. . . . . . . . 9
⊢
((((μ‘𝑛) /
𝑛) ∈ ℂ ∧
γ ∈ ℂ) → (((μ‘𝑛) / 𝑛) · γ) ∈
ℂ) |
46 | 8, 12, 45 | sylancl 589 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · γ) ∈
ℂ) |
47 | 1, 44, 46 | fsumsub 15236 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))((((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (((μ‘𝑛) / 𝑛) · γ)) = (Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · γ))) |
48 | 12 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ γ ∈ ℂ) |
49 | 41, 42, 48 | subsub4d 11106 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) − γ) = (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) |
50 | 49 | oveq2d 7186 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) − γ)) = (((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) |
51 | 8, 43, 48 | subdid 11174 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) − γ)) = ((((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (((μ‘𝑛) / 𝑛) · γ))) |
52 | 50, 51 | eqtr3d 2775 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) = ((((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (((μ‘𝑛) / 𝑛) · γ))) |
53 | 52 | sumeq2dv 15153 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (((μ‘𝑛) / 𝑛) · γ))) |
54 | 12 | a1i 11 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ γ ∈ ℂ) |
55 | 1, 54, 8 | fsummulc1 15233 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ (Σ𝑛 ∈
(1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) = Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · γ)) |
56 | 55 | oveq2d 7186 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ (Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ)) = (Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · γ))) |
57 | 47, 53, 56 | 3eqtr4d 2783 |
. . . . . 6
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ))) |
58 | 57 | mpteq2ia 5121 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
↦ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) = (𝑥 ∈ ℝ+ ↦
(Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ))) |
59 | 16 | a1i 11 |
. . . . . 6
⊢ (⊤
→ ℝ+ ⊆ ℝ) |
60 | 42, 48 | addcld 10738 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((log‘(𝑥 /
𝑛)) + γ) ∈
ℂ) |
61 | 41, 60 | subcld 11075 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)) ∈
ℂ) |
62 | 8, 61 | mulcld 10739 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈
ℂ) |
63 | 1, 62 | fsumcl 15183 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈
ℂ) |
64 | 63 | adantl 485 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈
ℂ) |
65 | | 1red 10720 |
. . . . . 6
⊢ (⊤
→ 1 ∈ ℝ) |
66 | 63 | abscld 14886 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ (abs‘Σ𝑛
∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈
ℝ) |
67 | 62 | abscld 14886 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈
ℝ) |
68 | 1, 67 | fsumrecl 15184 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈
ℝ) |
69 | | 1red 10720 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ 1 ∈ ℝ) |
70 | 1, 62 | fsumabs 15249 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ (abs‘Σ𝑛
∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ Σ𝑛 ∈
(1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))))) |
71 | | rprege0 12487 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
∧ 0 ≤ 𝑥)) |
72 | | flge0nn0 13281 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ 0 ≤
𝑥) →
(⌊‘𝑥) ∈
ℕ0) |
73 | 71, 72 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (⌊‘𝑥)
∈ ℕ0) |
74 | 73 | nn0red 12037 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ (⌊‘𝑥)
∈ ℝ) |
75 | | rerpdivcl 12502 |
. . . . . . . . . 10
⊢
(((⌊‘𝑥)
∈ ℝ ∧ 𝑥
∈ ℝ+) → ((⌊‘𝑥) / 𝑥) ∈ ℝ) |
76 | 74, 75 | mpancom 688 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ ((⌊‘𝑥) /
𝑥) ∈
ℝ) |
77 | | rpreccl 12498 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ+
→ (1 / 𝑥) ∈
ℝ+) |
78 | 77 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑥) ∈
ℝ+) |
79 | 78 | rpred 12514 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑥) ∈
ℝ) |
80 | 8 | abscld 14886 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑛) / 𝑛)) ∈ ℝ) |
81 | 3 | nnrecred 11767 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑛) ∈
ℝ) |
82 | 61 | abscld 14886 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘(Σ𝑚
∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈
ℝ) |
83 | | id 22 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ+) |
84 | | rpdivcl 12497 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ ℝ+
∧ 𝑥 ∈
ℝ+) → (𝑛 / 𝑥) ∈
ℝ+) |
85 | 26, 83, 84 | syl2anr 600 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑛 / 𝑥) ∈
ℝ+) |
86 | 85 | rpred 12514 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑛 / 𝑥) ∈
ℝ) |
87 | 8 | absge0d 14894 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘((μ‘𝑛) / 𝑛))) |
88 | 61 | absge0d 14894 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) |
89 | 6 | recnd 10747 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑛)
∈ ℂ) |
90 | 3 | nncnd 11732 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℂ) |
91 | 3 | nnne0d 11766 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ≠
0) |
92 | 89, 90, 91 | absdivd 14905 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑛) / 𝑛)) = ((abs‘(μ‘𝑛)) / (abs‘𝑛))) |
93 | 3 | nnrpd 12512 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℝ+) |
94 | | rprege0 12487 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℝ+
→ (𝑛 ∈ ℝ
∧ 0 ≤ 𝑛)) |
95 | 93, 94 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑛 ∈ ℝ
∧ 0 ≤ 𝑛)) |
96 | | absid 14746 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ ℝ ∧ 0 ≤
𝑛) → (abs‘𝑛) = 𝑛) |
97 | 95, 96 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘𝑛) =
𝑛) |
98 | 97 | oveq2d 7186 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(μ‘𝑛)) / (abs‘𝑛)) = ((abs‘(μ‘𝑛)) / 𝑛)) |
99 | 92, 98 | eqtrd 2773 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑛) / 𝑛)) = ((abs‘(μ‘𝑛)) / 𝑛)) |
100 | 89 | abscld 14886 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘(μ‘𝑛)) ∈ ℝ) |
101 | | 1red 10720 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 1 ∈ ℝ) |
102 | | mule1 25885 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ →
(abs‘(μ‘𝑛))
≤ 1) |
103 | 3, 102 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘(μ‘𝑛)) ≤ 1) |
104 | 100, 101,
93, 103 | lediv1dd 12572 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(μ‘𝑛)) / 𝑛) ≤ (1 / 𝑛)) |
105 | 99, 104 | eqbrtrd 5052 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑛) / 𝑛)) ≤ (1 / 𝑛)) |
106 | | harmonicbnd4 25748 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 / 𝑛) ∈ ℝ+ →
(abs‘(Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ≤ (1 / (𝑥 / 𝑛))) |
107 | 28, 106 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘(Σ𝑚
∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ≤ (1 / (𝑥 / 𝑛))) |
108 | | rpcnne0 12490 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) |
109 | 108 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) |
110 | | rpcnne0 12490 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℝ+
→ (𝑛 ∈ ℂ
∧ 𝑛 ≠
0)) |
111 | 93, 110 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑛 ∈ ℂ
∧ 𝑛 ≠
0)) |
112 | | recdiv 11424 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (1 / (𝑥 / 𝑛)) = (𝑛 / 𝑥)) |
113 | 109, 111,
112 | syl2anc 587 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / (𝑥 / 𝑛)) = (𝑛 / 𝑥)) |
114 | 107, 113 | breqtrd 5056 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘(Σ𝑚
∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ≤ (𝑛 / 𝑥)) |
115 | 80, 81, 82, 86, 87, 88, 105, 114 | lemul12ad 11660 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((abs‘((μ‘𝑛) / 𝑛)) · (abs‘(Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ ((1 / 𝑛) · (𝑛 / 𝑥))) |
116 | 8, 61 | absmuld 14904 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) =
((abs‘((μ‘𝑛)
/ 𝑛)) ·
(abs‘(Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))))) |
117 | | 1cnd 10714 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 1 ∈ ℂ) |
118 | | dmdcan 11428 |
. . . . . . . . . . . . . 14
⊢ (((𝑛 ∈ ℂ ∧ 𝑛 ≠ 0) ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ 1 ∈ ℂ)
→ ((𝑛 / 𝑥) · (1 / 𝑛)) = (1 / 𝑥)) |
119 | 111, 109,
117, 118 | syl3anc 1372 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((𝑛 / 𝑥) · (1 / 𝑛)) = (1 / 𝑥)) |
120 | 85 | rpcnd 12516 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑛 / 𝑥) ∈
ℂ) |
121 | 81 | recnd 10747 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑛) ∈
ℂ) |
122 | 120, 121 | mulcomd 10740 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((𝑛 / 𝑥) · (1 / 𝑛)) = ((1 / 𝑛) · (𝑛 / 𝑥))) |
123 | 119, 122 | eqtr3d 2775 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑥) = ((1 /
𝑛) · (𝑛 / 𝑥))) |
124 | 115, 116,
123 | 3brtr4d 5062 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ (1 / 𝑥)) |
125 | 1, 67, 79, 124 | fsumle 15247 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑥)) |
126 | | hashfz1 13798 |
. . . . . . . . . . . . 13
⊢
((⌊‘𝑥)
∈ ℕ0 → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) |
127 | 73, 126 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) |
128 | 127 | oveq1d 7185 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)) = ((⌊‘𝑥) · (1 / 𝑥))) |
129 | 77 | rpcnd 12516 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ (1 / 𝑥) ∈
ℂ) |
130 | | fsumconst 15238 |
. . . . . . . . . . . 12
⊢
(((1...(⌊‘𝑥)) ∈ Fin ∧ (1 / 𝑥) ∈ ℂ) → Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑥) =
((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥))) |
131 | 1, 129, 130 | syl2anc 587 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑥) =
((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥))) |
132 | 73 | nn0cnd 12038 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ (⌊‘𝑥)
∈ ℂ) |
133 | | rpcn 12482 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℂ) |
134 | | rpne0 12488 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ≠
0) |
135 | 132, 133,
134 | divrecd 11497 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ ((⌊‘𝑥) /
𝑥) = ((⌊‘𝑥) · (1 / 𝑥))) |
136 | 128, 131,
135 | 3eqtr4d 2783 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑥) = ((⌊‘𝑥) / 𝑥)) |
137 | 125, 136 | breqtrd 5056 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ ((⌊‘𝑥) / 𝑥)) |
138 | | rpre 12480 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ) |
139 | | flle 13260 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ≤
𝑥) |
140 | 138, 139 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (⌊‘𝑥)
≤ 𝑥) |
141 | 133 | mulid1d 10736 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (𝑥 · 1) =
𝑥) |
142 | 140, 141 | breqtrrd 5058 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ (⌊‘𝑥)
≤ (𝑥 ·
1)) |
143 | | reflcl 13257 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ∈
ℝ) |
144 | 138, 143 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (⌊‘𝑥)
∈ ℝ) |
145 | | rpregt0 12486 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
∧ 0 < 𝑥)) |
146 | | ledivmul 11594 |
. . . . . . . . . . 11
⊢
(((⌊‘𝑥)
∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1))) |
147 | 144, 69, 145, 146 | syl3anc 1372 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ (((⌊‘𝑥)
/ 𝑥) ≤ 1 ↔
(⌊‘𝑥) ≤
(𝑥 ·
1))) |
148 | 142, 147 | mpbird 260 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ ((⌊‘𝑥) /
𝑥) ≤ 1) |
149 | 68, 76, 69, 137, 148 | letrd 10875 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ 1) |
150 | 66, 68, 69, 70, 149 | letrd 10875 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ (abs‘Σ𝑛
∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ 1) |
151 | 150 | ad2antrl 728 |
. . . . . 6
⊢
((⊤ ∧ (𝑥
∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ 1) |
152 | 59, 64, 65, 65, 151 | elo1d 14983 |
. . . . 5
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈
𝑂(1)) |
153 | 58, 152 | eqeltrrid 2838 |
. . . 4
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ))) ∈
𝑂(1)) |
154 | 34, 37, 153 | o1dif 15077 |
. . 3
⊢ (⊤
→ ((𝑥 ∈
ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1) ↔ (𝑥 ∈ ℝ+
↦ (Σ𝑛 ∈
(1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ)) ∈
𝑂(1))) |
155 | 20, 154 | mpbird 260 |
. 2
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1)) |
156 | 155 | mptru 1549 |
1
⊢ (𝑥 ∈ ℝ+
↦ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1) |