Proof of Theorem selberg3r
| Step | Hyp | Ref
| Expression |
| 1 | | elioore 13417 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (1(,)+∞) →
𝑥 ∈
ℝ) |
| 2 | 1 | adantl 481 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 𝑥 ∈ ℝ) |
| 3 | | 1rp 13038 |
. . . . . . . . . . . . 13
⊢ 1 ∈
ℝ+ |
| 4 | 3 | a1i 11 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 1 ∈ ℝ+) |
| 5 | | 1red 11262 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 1 ∈ ℝ) |
| 6 | | eliooord 13446 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (1(,)+∞) → (1
< 𝑥 ∧ 𝑥 <
+∞)) |
| 7 | 6 | adantl 481 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞)) |
| 8 | 7 | simpld 494 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 1 < 𝑥) |
| 9 | 5, 2, 8 | ltled 11409 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 1 ≤ 𝑥) |
| 10 | 2, 4, 9 | rpgecld 13116 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 𝑥 ∈ ℝ+) |
| 11 | 10 | relogcld 26665 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ) |
| 12 | 11 | recnd 11289 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ) |
| 13 | 12 | 2timesd 12509 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (2 · (log‘𝑥)) = ((log‘𝑥) + (log‘𝑥))) |
| 14 | 13 | oveq2d 7447 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (2 · (log‘𝑥))) = (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − ((log‘𝑥) + (log‘𝑥)))) |
| 15 | | chpcl 27167 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ →
(ψ‘𝑥) ∈
ℝ) |
| 16 | 2, 15 | syl 17 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℝ) |
| 17 | 16, 11 | remulcld 11291 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ) |
| 18 | | 2re 12340 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℝ |
| 19 | 18 | a1i 11 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 2 ∈ ℝ) |
| 20 | 2, 8 | rplogcld 26671 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (log‘𝑥) ∈
ℝ+) |
| 21 | 19, 20 | rerpdivcld 13108 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ) |
| 22 | | fzfid 14014 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin) |
| 23 | | elfznn 13593 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℕ) |
| 24 | 23 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ) |
| 25 | | vmacl 27161 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ →
(Λ‘𝑛) ∈
ℝ) |
| 26 | 24, 25 | syl 17 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈
ℝ) |
| 27 | 2 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ) |
| 28 | 27, 24 | nndivred 12320 |
. . . . . . . . . . . . . . . 16
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ) |
| 29 | | chpcl 27167 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ) |
| 30 | 28, 29 | syl 17 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ) |
| 31 | 26, 30 | remulcld 11291 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
((Λ‘𝑛)
· (ψ‘(𝑥 /
𝑛))) ∈
ℝ) |
| 32 | 24 | nnrpd 13075 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+) |
| 33 | 32 | relogcld 26665 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈
ℝ) |
| 34 | 31, 33 | remulcld 11291 |
. . . . . . . . . . . . 13
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛)
· (ψ‘(𝑥 /
𝑛))) ·
(log‘𝑛)) ∈
ℝ) |
| 35 | 22, 34 | fsumrecl 15770 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ) |
| 36 | 21, 35 | remulcld 11291 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℝ) |
| 37 | 17, 36 | readdcld 11290 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) ∈ ℝ) |
| 38 | 37, 10 | rerpdivcld 13108 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) ∈ ℝ) |
| 39 | 38 | recnd 11289 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) ∈ ℂ) |
| 40 | 39, 12, 12 | subsub4d 11651 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (log‘𝑥)) − (log‘𝑥)) = (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − ((log‘𝑥) + (log‘𝑥)))) |
| 41 | 14, 40 | eqtr4d 2780 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (2 · (log‘𝑥))) = ((((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (log‘𝑥)) − (log‘𝑥))) |
| 42 | 41 | oveq1d 7446 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − (((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) − (log‘𝑥))) = (((((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (log‘𝑥)) − (log‘𝑥)) − (((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) − (log‘𝑥)))) |
| 43 | 39, 12 | subcld 11620 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (log‘𝑥)) ∈ ℂ) |
| 44 | | 2cn 12341 |
. . . . . . . . 9
⊢ 2 ∈
ℂ |
| 45 | 44 | a1i 11 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 2 ∈ ℂ) |
| 46 | 20 | rpne0d 13082 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (log‘𝑥) ≠ 0) |
| 47 | 45, 12, 46 | divcld 12043 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ) |
| 48 | 26, 24 | nndivred 12320 |
. . . . . . . . . 10
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
((Λ‘𝑛) / 𝑛) ∈
ℝ) |
| 49 | 48, 33 | remulcld 11291 |
. . . . . . . . 9
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛) /
𝑛) ·
(log‘𝑛)) ∈
ℝ) |
| 50 | 22, 49 | fsumrecl 15770 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) ∈ ℝ) |
| 51 | 50 | recnd 11289 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) ∈ ℂ) |
| 52 | 47, 51 | mulcld 11281 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) ∈ ℂ) |
| 53 | 43, 52, 12 | nnncan2d 11655 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (log‘𝑥)) − (log‘𝑥)) − (((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) − (log‘𝑥))) = ((((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) |
| 54 | | pntrval.r |
. . . . . . . . . . . . 13
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦
((ψ‘𝑎) −
𝑎)) |
| 55 | 54 | pntrf 27607 |
. . . . . . . . . . . 12
⊢ 𝑅:ℝ+⟶ℝ |
| 56 | 55 | ffvelcdmi 7103 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (𝑅‘𝑥) ∈
ℝ) |
| 57 | 10, 56 | syl 17 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℝ) |
| 58 | 57 | recnd 11289 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℂ) |
| 59 | 58, 12 | mulcld 11281 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((𝑅‘𝑥) · (log‘𝑥)) ∈ ℂ) |
| 60 | 36 | recnd 11289 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℂ) |
| 61 | 59, 60 | addcld 11280 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) ∈ ℂ) |
| 62 | 2 | recnd 11289 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 𝑥 ∈ ℂ) |
| 63 | 62, 52 | mulcld 11281 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (𝑥 · ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) ∈ ℂ) |
| 64 | 10 | rpne0d 13082 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 𝑥 ≠ 0) |
| 65 | 61, 63, 62, 64 | divsubdird 12082 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) − (𝑥 · ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) / 𝑥) = (((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − ((𝑥 · ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) / 𝑥))) |
| 66 | 59, 60, 63 | addsubassd 11640 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) − (𝑥 · ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) = (((𝑅‘𝑥) · (log‘𝑥)) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (𝑥 · ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))))) |
| 67 | 35 | recnd 11289 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ) |
| 68 | 62, 51 | mulcld 11281 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) ∈ ℂ) |
| 69 | 47, 67, 68 | subdid 11719 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) − (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))))) |
| 70 | 49 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛) /
𝑛) ·
(log‘𝑛)) ∈
ℂ) |
| 71 | 22, 62, 70 | fsummulc2 15820 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑥 · (((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) |
| 72 | 71 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) − (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑥 · (((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) |
| 73 | 34 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛)
· (ψ‘(𝑥 /
𝑛))) ·
(log‘𝑛)) ∈
ℂ) |
| 74 | 62 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ) |
| 75 | 74, 70 | mulcld 11281 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · (((Λ‘𝑛) / 𝑛) · (log‘𝑛))) ∈ ℂ) |
| 76 | 22, 73, 75 | fsumsub 15824 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) − (𝑥 · (((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑥 · (((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) |
| 77 | 72, 76 | eqtr4d 2780 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) − (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) − (𝑥 · (((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) |
| 78 | 26 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈
ℂ) |
| 79 | 30 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ) |
| 80 | 33 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈
ℂ) |
| 81 | 78, 79, 80 | mul32d 11471 |
. . . . . . . . . . . . . . . 16
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛)
· (ψ‘(𝑥 /
𝑛))) ·
(log‘𝑛)) =
(((Λ‘𝑛)
· (log‘𝑛))
· (ψ‘(𝑥 /
𝑛)))) |
| 82 | 24 | nncnd 12282 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ) |
| 83 | 24 | nnne0d 12316 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0) |
| 84 | 78, 80, 82, 83 | div23d 12080 |
. . . . . . . . . . . . . . . . . 18
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛)
· (log‘𝑛)) /
𝑛) =
(((Λ‘𝑛) /
𝑛) ·
(log‘𝑛))) |
| 85 | 84 | oveq2d 7447 |
. . . . . . . . . . . . . . . . 17
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · (((Λ‘𝑛) · (log‘𝑛)) / 𝑛)) = (𝑥 · (((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) |
| 86 | 78, 80 | mulcld 11281 |
. . . . . . . . . . . . . . . . . 18
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
((Λ‘𝑛)
· (log‘𝑛))
∈ ℂ) |
| 87 | 74, 86, 82, 83 | div12d 12079 |
. . . . . . . . . . . . . . . . 17
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · (((Λ‘𝑛) · (log‘𝑛)) / 𝑛)) = (((Λ‘𝑛) · (log‘𝑛)) · (𝑥 / 𝑛))) |
| 88 | 85, 87 | eqtr3d 2779 |
. . . . . . . . . . . . . . . 16
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · (((Λ‘𝑛) / 𝑛) · (log‘𝑛))) = (((Λ‘𝑛) · (log‘𝑛)) · (𝑥 / 𝑛))) |
| 89 | 81, 88 | oveq12d 7449 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
((((Λ‘𝑛)
· (ψ‘(𝑥 /
𝑛))) ·
(log‘𝑛)) −
(𝑥 ·
(((Λ‘𝑛) /
𝑛) ·
(log‘𝑛)))) =
((((Λ‘𝑛)
· (log‘𝑛))
· (ψ‘(𝑥 /
𝑛))) −
(((Λ‘𝑛)
· (log‘𝑛))
· (𝑥 / 𝑛)))) |
| 90 | 10 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+) |
| 91 | 90, 32 | rpdivcld 13094 |
. . . . . . . . . . . . . . . . . 18
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈
ℝ+) |
| 92 | 54 | pntrval 27606 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑥 / 𝑛)) = ((ψ‘(𝑥 / 𝑛)) − (𝑥 / 𝑛))) |
| 93 | 91, 92 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) = ((ψ‘(𝑥 / 𝑛)) − (𝑥 / 𝑛))) |
| 94 | 93 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛)
· (log‘𝑛))
· (𝑅‘(𝑥 / 𝑛))) = (((Λ‘𝑛) · (log‘𝑛)) · ((ψ‘(𝑥 / 𝑛)) − (𝑥 / 𝑛)))) |
| 95 | 28 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ) |
| 96 | 86, 79, 95 | subdid 11719 |
. . . . . . . . . . . . . . . 16
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛)
· (log‘𝑛))
· ((ψ‘(𝑥 /
𝑛)) − (𝑥 / 𝑛))) = ((((Λ‘𝑛) · (log‘𝑛)) · (ψ‘(𝑥 / 𝑛))) − (((Λ‘𝑛) · (log‘𝑛)) · (𝑥 / 𝑛)))) |
| 97 | 94, 96 | eqtrd 2777 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛)
· (log‘𝑛))
· (𝑅‘(𝑥 / 𝑛))) = ((((Λ‘𝑛) · (log‘𝑛)) · (ψ‘(𝑥 / 𝑛))) − (((Λ‘𝑛) · (log‘𝑛)) · (𝑥 / 𝑛)))) |
| 98 | 89, 97 | eqtr4d 2780 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
((((Λ‘𝑛)
· (ψ‘(𝑥 /
𝑛))) ·
(log‘𝑛)) −
(𝑥 ·
(((Λ‘𝑛) /
𝑛) ·
(log‘𝑛)))) =
(((Λ‘𝑛)
· (log‘𝑛))
· (𝑅‘(𝑥 / 𝑛)))) |
| 99 | 55 | ffvelcdmi 7103 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ) |
| 100 | 91, 99 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ) |
| 101 | 100 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ) |
| 102 | 78, 101, 80 | mul32d 11471 |
. . . . . . . . . . . . . 14
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
(((Λ‘𝑛)
· (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) = (((Λ‘𝑛) · (log‘𝑛)) · (𝑅‘(𝑥 / 𝑛)))) |
| 103 | 98, 102 | eqtr4d 2780 |
. . . . . . . . . . . . 13
⊢
(((⊤ ∧ 𝑥
∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) →
((((Λ‘𝑛)
· (ψ‘(𝑥 /
𝑛))) ·
(log‘𝑛)) −
(𝑥 ·
(((Λ‘𝑛) /
𝑛) ·
(log‘𝑛)))) =
(((Λ‘𝑛)
· (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) |
| 104 | 103 | sumeq2dv 15738 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) − (𝑥 · (((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) |
| 105 | 77, 104 | eqtrd 2777 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) − (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) |
| 106 | 105 | oveq2d 7447 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) − (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) |
| 107 | 47, 62, 51 | mul12d 11470 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) = (𝑥 · ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) |
| 108 | 107 | oveq2d 7447 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (𝑥 · ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))))) |
| 109 | 69, 106, 108 | 3eqtr3rd 2786 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (𝑥 · ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) |
| 110 | 109 | oveq2d 7447 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((𝑅‘𝑥) · (log‘𝑥)) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (𝑥 · ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))))) = (((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))) |
| 111 | 66, 110 | eqtrd 2777 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) − (𝑥 · ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) = (((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))) |
| 112 | 111 | oveq1d 7446 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) − (𝑥 · ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) / 𝑥) = ((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) |
| 113 | 54 | pntrval 27606 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ+
→ (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥)) |
| 114 | 10, 113 | syl 17 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥)) |
| 115 | 114 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((𝑅‘𝑥) · (log‘𝑥)) = (((ψ‘𝑥) − 𝑥) · (log‘𝑥))) |
| 116 | 16 | recnd 11289 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℂ) |
| 117 | 116, 62, 12 | subdird 11720 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((ψ‘𝑥) − 𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥)))) |
| 118 | 115, 117 | eqtrd 2777 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((𝑅‘𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥)))) |
| 119 | 118 | oveq1d 7446 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) = ((((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))))) |
| 120 | 17 | recnd 11289 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ) |
| 121 | 62, 12 | mulcld 11281 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℂ) |
| 122 | 120, 60, 121 | addsubd 11641 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) − (𝑥 · (log‘𝑥))) = ((((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))))) |
| 123 | 119, 122 | eqtr4d 2780 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) = ((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) − (𝑥 · (log‘𝑥)))) |
| 124 | 123 | oveq1d 7446 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) = (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) − (𝑥 · (log‘𝑥))) / 𝑥)) |
| 125 | 37 | recnd 11289 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) ∈ ℂ) |
| 126 | 125, 121,
62, 64 | divsubdird 12082 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) − (𝑥 · (log‘𝑥))) / 𝑥) = (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − ((𝑥 · (log‘𝑥)) / 𝑥))) |
| 127 | 12, 62, 64 | divcan3d 12048 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((𝑥 · (log‘𝑥)) / 𝑥) = (log‘𝑥)) |
| 128 | 127 | oveq2d 7447 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − ((𝑥 · (log‘𝑥)) / 𝑥)) = (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (log‘𝑥))) |
| 129 | 126, 128 | eqtrd 2777 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) − (𝑥 · (log‘𝑥))) / 𝑥) = (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (log‘𝑥))) |
| 130 | 124, 129 | eqtrd 2777 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) = (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (log‘𝑥))) |
| 131 | 52, 62, 64 | divcan3d 12048 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((𝑥 · ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) / 𝑥) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) |
| 132 | 130, 131 | oveq12d 7449 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − ((𝑥 · ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) / 𝑥)) = ((((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))) |
| 133 | 65, 112, 132 | 3eqtr3rd 2786 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) = ((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) |
| 134 | 42, 53, 133 | 3eqtrrd 2782 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) = ((((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − (((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) − (log‘𝑥)))) |
| 135 | 134 | mpteq2dva 5242 |
. . 3
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) ↦ ((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) = (𝑥 ∈ (1(,)+∞) ↦
((((((ψ‘𝑥)
· (log‘𝑥)) +
((2 / (log‘𝑥))
· Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − (((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) − (log‘𝑥))))) |
| 136 | 19, 11 | remulcld 11291 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (2 · (log‘𝑥)) ∈ ℝ) |
| 137 | 38, 136 | resubcld 11691 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈
ℝ) |
| 138 | 21, 50 | remulcld 11291 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) ∈ ℝ) |
| 139 | 138, 11 | resubcld 11691 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) − (log‘𝑥)) ∈ ℝ) |
| 140 | | selberg3 27603 |
. . . . 5
⊢ (𝑥 ∈ (1(,)+∞) ↦
(((((ψ‘𝑥)
· (log‘𝑥)) +
((2 / (log‘𝑥))
· Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈
𝑂(1) |
| 141 | 140 | a1i 11 |
. . . 4
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) ↦ (((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈
𝑂(1)) |
| 142 | 19 | recnd 11289 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 2 ∈ ℂ) |
| 143 | 50, 20 | rerpdivcld 13108 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) ∈ ℝ) |
| 144 | 143 | recnd 11289 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) ∈ ℂ) |
| 145 | 11 | rehalfcld 12513 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℝ) |
| 146 | 145 | recnd 11289 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ) |
| 147 | 142, 144,
146 | subdid 11719 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((2 · (Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) − (2 · ((log‘𝑥) / 2)))) |
| 148 | 142, 12, 51, 46 | div32d 12066 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) = (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)))) |
| 149 | 148 | eqcomd 2743 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)))) |
| 150 | | 2ne0 12370 |
. . . . . . . . . 10
⊢ 2 ≠
0 |
| 151 | 150 | a1i 11 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → 2 ≠ 0) |
| 152 | 12, 142, 151 | divcan2d 12045 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (2 · ((log‘𝑥) / 2)) = (log‘𝑥)) |
| 153 | 149, 152 | oveq12d 7449 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) − (2 · ((log‘𝑥) / 2))) = (((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) − (log‘𝑥))) |
| 154 | 147, 153 | eqtrd 2777 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) − (log‘𝑥))) |
| 155 | 154 | mpteq2dva 5242 |
. . . . 5
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) ↦ (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)))) = (𝑥 ∈ (1(,)+∞) ↦ (((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) − (log‘𝑥)))) |
| 156 | 143, 145 | resubcld 11691 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℝ) |
| 157 | | ioossre 13448 |
. . . . . . . 8
⊢
(1(,)+∞) ⊆ ℝ |
| 158 | | o1const 15656 |
. . . . . . . 8
⊢
(((1(,)+∞) ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦
2) ∈ 𝑂(1)) |
| 159 | 157, 44, 158 | mp2an 692 |
. . . . . . 7
⊢ (𝑥 ∈ (1(,)+∞) ↦
2) ∈ 𝑂(1) |
| 160 | 159 | a1i 11 |
. . . . . 6
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) ↦ 2) ∈ 𝑂(1)) |
| 161 | | vmalogdivsum 27583 |
. . . . . . 7
⊢ (𝑥 ∈ (1(,)+∞) ↦
((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1) |
| 162 | 161 | a1i 11 |
. . . . . 6
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈
𝑂(1)) |
| 163 | 19, 156, 160, 162 | o1mul2 15661 |
. . . . 5
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) ↦ (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)))) ∈
𝑂(1)) |
| 164 | 155, 163 | eqeltrrd 2842 |
. . . 4
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) ↦ (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) − (log‘𝑥))) ∈ 𝑂(1)) |
| 165 | 137, 139,
141, 164 | o1sub2 15662 |
. . 3
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) ↦ ((((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − (((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) − (log‘𝑥)))) ∈ 𝑂(1)) |
| 166 | 135, 165 | eqeltrd 2841 |
. 2
⊢ (⊤
→ (𝑥 ∈
(1(,)+∞) ↦ ((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ 𝑂(1)) |
| 167 | 166 | mptru 1547 |
1
⊢ (𝑥 ∈ (1(,)+∞) ↦
((((𝑅‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) · (𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ 𝑂(1) |