Proof of Theorem vmalogdivsum
| Step | Hyp | Ref
| Expression |
| 1 | | elioore 13417 |
. . . . . . . 8
⊢ (𝑥 ∈ (1(,)+∞) →
𝑥 ∈
ℝ) |
| 2 | 1 | adantl 481 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 𝑥 ∈ ℝ) |
| 3 | | 1rp 13038 |
. . . . . . . 8
⊢ 1 ∈
ℝ+ |
| 4 | 3 | a1i 11 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 1 ∈ ℝ+) |
| 5 | | 1red 11262 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 1 ∈ ℝ) |
| 6 | | eliooord 13446 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (1(,)+∞) → (1
< 𝑥 ∧ 𝑥 <
+∞)) |
| 7 | 6 | adantl 481 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞)) |
| 8 | 7 | simpld 494 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 1 < 𝑥) |
| 9 | 5, 2, 8 | ltled 11409 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 1 ≤ 𝑥) |
| 10 | 2, 4, 9 | rpgecld 13116 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 𝑥 ∈ ℝ+) |
| 11 | 10 | ex 412 |
. . . . 5
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) → 𝑥
∈ ℝ+)) |
| 12 | 11 | ssrdv 3989 |
. . . 4
⊢ (⊤
→ (1(,)+∞) ⊆ ℝ+) |
| 13 | | vmadivsum 27526 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
↦ (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1) |
| 14 | 13 | a1i 11 |
. . . 4
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)) |
| 15 | 12, 14 | o1res2 15599 |
. . 3
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)) |
| 16 | | fzfid 14014 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin) |
| 17 | | elfznn 13593 |
. . . . . . . . . 10
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℕ) |
| 18 | 17 | adantl 481 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ) |
| 19 | | vmacl 27161 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ →
(Λ‘𝑛) ∈
ℝ) |
| 20 | 18, 19 | syl 17 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈
ℝ) |
| 21 | 20, 18 | nndivred 12320 |
. . . . . . 7
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
((Λ‘𝑛) / 𝑛) ∈
ℝ) |
| 22 | 21 | recnd 11289 |
. . . . . 6
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
((Λ‘𝑛) / 𝑛) ∈
ℂ) |
| 23 | 16, 22 | fsumcl 15769 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ) |
| 24 | 10 | relogcld 26665 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ) |
| 25 | 24 | recnd 11289 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ) |
| 26 | 23, 25 | subcld 11620 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ) |
| 27 | 18 | nnrpd 13075 |
. . . . . . . . . 10
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+) |
| 28 | 27 | relogcld 26665 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈
ℝ) |
| 29 | 21, 28 | remulcld 11291 |
. . . . . . . 8
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛) /
𝑛) ·
(log‘𝑛)) ∈
ℝ) |
| 30 | 16, 29 | fsumrecl 15770 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) ∈ ℝ) |
| 31 | 2, 8 | rplogcld 26671 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (log‘𝑥) ∈
ℝ+) |
| 32 | 30, 31 | rerpdivcld 13108 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) ∈ ℝ) |
| 33 | 24 | rehalfcld 12513 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℝ) |
| 34 | 32, 33 | resubcld 11691 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℝ) |
| 35 | 34 | recnd 11289 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℂ) |
| 36 | 33 | recnd 11289 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ) |
| 37 | 23, 36 | subcld 11620 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) ∈ ℂ) |
| 38 | 32 | recnd 11289 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) ∈ ℂ) |
| 39 | 37, 38, 36 | nnncan2d 11655 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)))) |
| 40 | 23, 36, 36 | subsub4d 11651 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − ((log‘𝑥) / 2)) = (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (((log‘𝑥) / 2) + ((log‘𝑥) / 2)))) |
| 41 | 25 | 2halvesd 12512 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((log‘𝑥) / 2) + ((log‘𝑥) / 2)) = (log‘𝑥)) |
| 42 | 41 | oveq2d 7447 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (((log‘𝑥) / 2) + ((log‘𝑥) / 2))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) |
| 43 | 40, 42 | eqtrd 2777 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − ((log‘𝑥) / 2)) = (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) |
| 44 | 43 | oveq1d 7446 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)))) |
| 45 | 23, 36, 38 | sub32d 11652 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) − ((log‘𝑥) / 2))) |
| 46 | 10 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+) |
| 47 | 46 | relogcld 26665 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑥) ∈
ℝ) |
| 48 | 21, 47 | remulcld 11291 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛) /
𝑛) ·
(log‘𝑥)) ∈
ℝ) |
| 49 | 48 | recnd 11289 |
. . . . . . . . . . . . 13
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛) /
𝑛) ·
(log‘𝑥)) ∈
ℂ) |
| 50 | 29 | recnd 11289 |
. . . . . . . . . . . . 13
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛) /
𝑛) ·
(log‘𝑛)) ∈
ℂ) |
| 51 | 16, 49, 50 | fsumsub 15824 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − (((Λ‘𝑛) / 𝑛) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) |
| 52 | 46, 27 | relogdivd 26668 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) = ((log‘𝑥) − (log‘𝑛))) |
| 53 | 52 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛) /
𝑛) ·
(log‘(𝑥 / 𝑛))) = (((Λ‘𝑛) / 𝑛) · ((log‘𝑥) − (log‘𝑛)))) |
| 54 | 25 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑥) ∈
ℂ) |
| 55 | 28 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈
ℂ) |
| 56 | 22, 54, 55 | subdid 11719 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛) /
𝑛) ·
((log‘𝑥) −
(log‘𝑛))) =
((((Λ‘𝑛) /
𝑛) ·
(log‘𝑥)) −
(((Λ‘𝑛) /
𝑛) ·
(log‘𝑛)))) |
| 57 | 53, 56 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛) /
𝑛) ·
(log‘(𝑥 / 𝑛))) = ((((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − (((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) |
| 58 | 57 | sumeq2dv 15738 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − (((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) |
| 59 | 20 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈
ℂ) |
| 60 | 18 | nncnd 12282 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ) |
| 61 | 18 | nnne0d 12316 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0) |
| 62 | 59, 60, 61 | divcld 12043 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
((Λ‘𝑛) / 𝑛) ∈
ℂ) |
| 63 | 16, 25, 62 | fsummulc1 15821 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑥))) |
| 64 | 63 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) |
| 65 | 51, 58, 64 | 3eqtr4d 2787 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) |
| 66 | 65 | oveq1d 7446 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) / (log‘𝑥))) |
| 67 | 23, 25 | mulcld 11281 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) ∈ ℂ) |
| 68 | 30 | recnd 11289 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) ∈ ℂ) |
| 69 | 31 | rpne0d 13082 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (log‘𝑥) ≠ 0) |
| 70 | 67, 68, 25, 69 | divsubdird 12082 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) / (log‘𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) / (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)))) |
| 71 | 23, 25, 69 | divcan4d 12049 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) / (log‘𝑥)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛)) |
| 72 | 71 | oveq1d 7446 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) / (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)))) |
| 73 | 66, 70, 72 | 3eqtrd 2781 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)))) |
| 74 | 73 | oveq1d 7446 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) − ((log‘𝑥) / 2))) |
| 75 | 45, 74 | eqtr4d 2780 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) |
| 76 | 39, 44, 75 | 3eqtr3d 2785 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) |
| 77 | 76 | mpteq2dva 5242 |
. . . . 5
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)))) = (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))) |
| 78 | | vmalogdivsum2 27582 |
. . . . 5
⊢ (𝑥 ∈ (1(,)+∞) ↦
((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1) |
| 79 | 77, 78 | eqeltrdi 2849 |
. . . 4
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)))) ∈
𝑂(1)) |
| 80 | 26, 35, 79 | o1dif 15666 |
. . 3
⊢ (⊤
→ ((𝑥 ∈
(1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦
((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈
𝑂(1))) |
| 81 | 15, 80 | mpbid 232 |
. 2
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈
𝑂(1)) |
| 82 | 81 | mptru 1547 |
1
⊢ (𝑥 ∈ (1(,)+∞) ↦
((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1) |