Proof of Theorem mulogsumlem
| Step | Hyp | Ref
| Expression |
| 1 | | fzfid 14014 |
. . . . . 6
⊢ (𝑥 ∈ ℝ+
→ (1...(⌊‘𝑥)) ∈ Fin) |
| 2 | | elfznn 13593 |
. . . . . . . . . . 11
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℕ) |
| 3 | 2 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℕ) |
| 4 | | mucl 27184 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ →
(μ‘𝑛) ∈
ℤ) |
| 5 | 3, 4 | syl 17 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑛)
∈ ℤ) |
| 6 | 5 | zred 12722 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑛)
∈ ℝ) |
| 7 | 6, 3 | nndivred 12320 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((μ‘𝑛) /
𝑛) ∈
ℝ) |
| 8 | 7 | recnd 11289 |
. . . . . 6
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((μ‘𝑛) /
𝑛) ∈
ℂ) |
| 9 | 1, 8 | fsumcl 15769 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ) |
| 10 | 9 | adantl 481 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ) |
| 11 | | emre 27049 |
. . . . . 6
⊢ γ
∈ ℝ |
| 12 | 11 | recni 11275 |
. . . . 5
⊢ γ
∈ ℂ |
| 13 | 12 | a1i 11 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → γ ∈ ℂ) |
| 14 | | mudivsum 27574 |
. . . . 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 13042 |
. . . . . 6
⊢
ℝ+ ⊆ ℝ |
| 17 | | o1const 15656 |
. . . . . 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 15661 |
. . 3
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ)) ∈
𝑂(1)) |
| 21 | | fzfid 14014 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin) |
| 22 | | elfznn 13593 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛))) → 𝑚 ∈
ℕ) |
| 23 | 22 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → 𝑚 ∈
ℕ) |
| 24 | 23 | nnrecred 12317 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → (1 / 𝑚) ∈
ℝ) |
| 25 | 21, 24 | fsumrecl 15770 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) ∈ ℝ) |
| 26 | 2 | nnrpd 13075 |
. . . . . . . . . . 11
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℝ+) |
| 27 | | rpdivcl 13060 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
ℝ+) → (𝑥 / 𝑛) ∈
ℝ+) |
| 28 | 26, 27 | sylan2 593 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑛) ∈
ℝ+) |
| 29 | 28 | relogcld 26665 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (log‘(𝑥 /
𝑛)) ∈
ℝ) |
| 30 | 25, 29 | resubcld 11691 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℝ) |
| 31 | 7, 30 | remulcld 11291 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ) |
| 32 | 1, 31 | fsumrecl 15770 |
. . . . . 6
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ) |
| 33 | 32 | recnd 11289 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ) |
| 34 | 33 | adantl 481 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ) |
| 35 | | mulcl 11239 |
. . . . . 6
⊢
((Σ𝑛 ∈
(1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ ∧ γ ∈
ℂ) → (Σ𝑛
∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈
ℂ) |
| 36 | 9, 12, 35 | sylancl 586 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
→ (Σ𝑛 ∈
(1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈
ℂ) |
| 37 | 36 | adantl 481 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) ∈
ℂ) |
| 38 | | nnrecre 12308 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ → (1 /
𝑚) ∈
ℝ) |
| 39 | 38 | recnd 11289 |
. . . . . . . . . . . 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 15769 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) ∈ ℂ) |
| 42 | 29 | recnd 11289 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (log‘(𝑥 /
𝑛)) ∈
ℂ) |
| 43 | 41, 42 | subcld 11620 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℂ) |
| 44 | 8, 43 | mulcld 11281 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ) |
| 45 | | mulcl 11239 |
. . . . . . . . 9
⊢
((((μ‘𝑛) /
𝑛) ∈ ℂ ∧
γ ∈ ℂ) → (((μ‘𝑛) / 𝑛) · γ) ∈
ℂ) |
| 46 | 8, 12, 45 | sylancl 586 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · γ) ∈
ℂ) |
| 47 | 1, 44, 46 | fsumsub 15824 |
. . . . . . 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 11651 |
. . . . . . . . . 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 7447 |
. . . . . . . . 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 11719 |
. . . . . . . . 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 2779 |
. . . . . . . 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 15738 |
. . . . . . 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 15821 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ (Σ𝑛 ∈
(1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) · γ) = Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · γ)) |
| 56 | 55 | oveq2d 7447 |
. . . . . . 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 2787 |
. . . . . 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 5245 |
. . . . 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 11280 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((log‘(𝑥 /
𝑛)) + γ) ∈
ℂ) |
| 61 | 41, 60 | subcld 11620 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)) ∈
ℂ) |
| 62 | 8, 61 | mulcld 11281 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈
ℂ) |
| 63 | 1, 62 | fsumcl 15769 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈
ℂ) |
| 64 | 63 | adantl 481 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ))) ∈
ℂ) |
| 65 | | 1red 11262 |
. . . . . 6
⊢ (⊤
→ 1 ∈ ℝ) |
| 66 | 63 | abscld 15475 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ (abs‘Σ𝑛
∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈
ℝ) |
| 67 | 62 | abscld 15475 |
. . . . . . . . 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 15770 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ∈
ℝ) |
| 69 | | 1red 11262 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ+
→ 1 ∈ ℝ) |
| 70 | 1, 62 | fsumabs 15837 |
. . . . . . . 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 13050 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
∧ 0 ≤ 𝑥)) |
| 72 | | flge0nn0 13860 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ 0 ≤
𝑥) →
(⌊‘𝑥) ∈
ℕ0) |
| 73 | 71, 72 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (⌊‘𝑥)
∈ ℕ0) |
| 74 | 73 | nn0red 12588 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ (⌊‘𝑥)
∈ ℝ) |
| 75 | | rerpdivcl 13065 |
. . . . . . . . . 10
⊢
(((⌊‘𝑥)
∈ ℝ ∧ 𝑥
∈ ℝ+) → ((⌊‘𝑥) / 𝑥) ∈ ℝ) |
| 76 | 74, 75 | mpancom 688 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ ((⌊‘𝑥) /
𝑥) ∈
ℝ) |
| 77 | | rpreccl 13061 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ+
→ (1 / 𝑥) ∈
ℝ+) |
| 78 | 77 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑥) ∈
ℝ+) |
| 79 | 78 | rpred 13077 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑥) ∈
ℝ) |
| 80 | 8 | abscld 15475 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑛) / 𝑛)) ∈ ℝ) |
| 81 | 3 | nnrecred 12317 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑛) ∈
ℝ) |
| 82 | 61 | abscld 15475 |
. . . . . . . . . . . . 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 13060 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ ℝ+
∧ 𝑥 ∈
ℝ+) → (𝑛 / 𝑥) ∈
ℝ+) |
| 85 | 26, 83, 84 | syl2anr 597 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑛 / 𝑥) ∈
ℝ+) |
| 86 | 85 | rpred 13077 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑛 / 𝑥) ∈
ℝ) |
| 87 | 8 | absge0d 15483 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘((μ‘𝑛) / 𝑛))) |
| 88 | 61 | absge0d 15483 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) |
| 89 | 6 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑛)
∈ ℂ) |
| 90 | 3 | nncnd 12282 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℂ) |
| 91 | 3 | nnne0d 12316 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ≠
0) |
| 92 | 89, 90, 91 | absdivd 15494 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑛) / 𝑛)) = ((abs‘(μ‘𝑛)) / (abs‘𝑛))) |
| 93 | 3 | nnrpd 13075 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℝ+) |
| 94 | | rprege0 13050 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℝ+
→ (𝑛 ∈ ℝ
∧ 0 ≤ 𝑛)) |
| 95 | 93, 94 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑛 ∈ ℝ
∧ 0 ≤ 𝑛)) |
| 96 | | absid 15335 |
. . . . . . . . . . . . . . . . 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 7447 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(μ‘𝑛)) / (abs‘𝑛)) = ((abs‘(μ‘𝑛)) / 𝑛)) |
| 99 | 92, 98 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑛) / 𝑛)) = ((abs‘(μ‘𝑛)) / 𝑛)) |
| 100 | 89 | abscld 15475 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘(μ‘𝑛)) ∈ ℝ) |
| 101 | | 1red 11262 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 1 ∈ ℝ) |
| 102 | | mule1 27191 |
. . . . . . . . . . . . . . . 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 13135 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(μ‘𝑛)) / 𝑛) ≤ (1 / 𝑛)) |
| 105 | 99, 104 | eqbrtrd 5165 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑛) / 𝑛)) ≤ (1 / 𝑛)) |
| 106 | | harmonicbnd4 27054 |
. . . . . . . . . . . . . . 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 13053 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) |
| 109 | 108 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) |
| 110 | | rpcnne0 13053 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℝ+
→ (𝑛 ∈ ℂ
∧ 𝑛 ≠
0)) |
| 111 | 93, 110 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑛 ∈ ℂ
∧ 𝑛 ≠
0)) |
| 112 | | recdiv 11973 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (1 / (𝑥 / 𝑛)) = (𝑛 / 𝑥)) |
| 113 | 109, 111,
112 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / (𝑥 / 𝑛)) = (𝑛 / 𝑥)) |
| 114 | 107, 113 | breqtrd 5169 |
. . . . . . . . . . . . 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 12210 |
. . . . . . . . . . . 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 15493 |
. . . . . . . . . . . 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 11256 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 1 ∈ ℂ) |
| 118 | | dmdcan 11977 |
. . . . . . . . . . . . . 14
⊢ (((𝑛 ∈ ℂ ∧ 𝑛 ≠ 0) ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ 1 ∈ ℂ)
→ ((𝑛 / 𝑥) · (1 / 𝑛)) = (1 / 𝑥)) |
| 119 | 111, 109,
117, 118 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((𝑛 / 𝑥) · (1 / 𝑛)) = (1 / 𝑥)) |
| 120 | 85 | rpcnd 13079 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑛 / 𝑥) ∈
ℂ) |
| 121 | 81 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑛) ∈
ℂ) |
| 122 | 120, 121 | mulcomd 11282 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((𝑛 / 𝑥) · (1 / 𝑛)) = ((1 / 𝑛) · (𝑛 / 𝑥))) |
| 123 | 119, 122 | eqtr3d 2779 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑥) = ((1 /
𝑛) · (𝑛 / 𝑥))) |
| 124 | 115, 116,
123 | 3brtr4d 5175 |
. . . . . . . . . . 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 15835 |
. . . . . . . . . 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 14385 |
. . . . . . . . . . . . 13
⊢
((⌊‘𝑥)
∈ ℕ0 → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) |
| 127 | 73, 126 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) |
| 128 | 127 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)) = ((⌊‘𝑥) · (1 / 𝑥))) |
| 129 | 77 | rpcnd 13079 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ (1 / 𝑥) ∈
ℂ) |
| 130 | | fsumconst 15826 |
. . . . . . . . . . . 12
⊢
(((1...(⌊‘𝑥)) ∈ Fin ∧ (1 / 𝑥) ∈ ℂ) → Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑥) =
((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥))) |
| 131 | 1, 129, 130 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑥) =
((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥))) |
| 132 | 73 | nn0cnd 12589 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ (⌊‘𝑥)
∈ ℂ) |
| 133 | | rpcn 13045 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℂ) |
| 134 | | rpne0 13051 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ≠
0) |
| 135 | 132, 133,
134 | divrecd 12046 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ ((⌊‘𝑥) /
𝑥) = ((⌊‘𝑥) · (1 / 𝑥))) |
| 136 | 128, 131,
135 | 3eqtr4d 2787 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑥) = ((⌊‘𝑥) / 𝑥)) |
| 137 | 125, 136 | breqtrd 5169 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(abs‘(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − ((log‘(𝑥 / 𝑛)) + γ)))) ≤ ((⌊‘𝑥) / 𝑥)) |
| 138 | | rpre 13043 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ) |
| 139 | | flle 13839 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ≤
𝑥) |
| 140 | 138, 139 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (⌊‘𝑥)
≤ 𝑥) |
| 141 | 133 | mulridd 11278 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (𝑥 · 1) =
𝑥) |
| 142 | 140, 141 | breqtrrd 5171 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ (⌊‘𝑥)
≤ (𝑥 ·
1)) |
| 143 | | reflcl 13836 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ∈
ℝ) |
| 144 | 138, 143 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (⌊‘𝑥)
∈ ℝ) |
| 145 | | rpregt0 13049 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
∧ 0 < 𝑥)) |
| 146 | | ledivmul 12144 |
. . . . . . . . . . 11
⊢
(((⌊‘𝑥)
∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1))) |
| 147 | 144, 69, 145, 146 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ (((⌊‘𝑥)
/ 𝑥) ≤ 1 ↔
(⌊‘𝑥) ≤
(𝑥 ·
1))) |
| 148 | 142, 147 | mpbird 257 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ ((⌊‘𝑥) /
𝑥) ≤ 1) |
| 149 | 68, 76, 69, 137, 148 | letrd 11418 |
. . . . . . . 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 11418 |
. . . . . . 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 15572 |
. . . . 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 2846 |
. . . 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 15666 |
. . 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 257 |
. 2
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1)) |
| 156 | 155 | mptru 1547 |
1
⊢ (𝑥 ∈ ℝ+
↦ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1) |