Proof of Theorem selberg4lem1
Step | Hyp | Ref
| Expression |
1 | | 2cnd 10970 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈
ℂ) |
2 | | fzfid 12634 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(1...(⌊‘𝑥))
∈ Fin) |
3 | | elfznn 12241 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℕ) |
4 | 3 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℕ) |
5 | | vmacl 24644 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ →
(Λ‘𝑛) ∈
ℝ) |
6 | 4, 5 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Λ‘𝑛)
∈ ℝ) |
7 | 6, 4 | nndivred 10946 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
/ 𝑛) ∈
ℝ) |
8 | | elioore 12076 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (1(,)+∞) →
𝑥 ∈
ℝ) |
9 | 8 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈
ℝ) |
10 | | 1rp 11712 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
ℝ+ |
11 | 10 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈
ℝ+) |
12 | | 1red 9934 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈
ℝ) |
13 | | eliooord 12104 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ (1(,)+∞) → (1
< 𝑥 ∧ 𝑥 <
+∞)) |
14 | 13 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞)) |
15 | 14 | simpld 474 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥) |
16 | 12, 9, 15 | ltled 10064 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥) |
17 | 9, 11, 16 | rpgecld 11787 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈
ℝ+) |
18 | 17 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑥 ∈
ℝ+) |
19 | 4 | nnrpd 11746 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℝ+) |
20 | 18, 19 | rpdivcld 11765 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑛) ∈
ℝ+) |
21 | 20 | relogcld 24173 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (log‘(𝑥 /
𝑛)) ∈
ℝ) |
22 | 7, 21 | remulcld 9949 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((Λ‘𝑛)
/ 𝑛) ·
(log‘(𝑥 / 𝑛))) ∈
ℝ) |
23 | 2, 22 | fsumrecl 14312 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ) |
24 | 9, 15 | rplogcld 24179 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(log‘𝑥) ∈
ℝ+) |
25 | 23, 24 | rerpdivcld 11779 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) ∈ ℝ) |
26 | 25 | recnd 9947 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) ∈ ℂ) |
27 | 17 | relogcld 24173 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(log‘𝑥) ∈
ℝ) |
28 | 27 | rehalfcld 11156 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((log‘𝑥) / 2) ∈
ℝ) |
29 | 28 | recnd 9947 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((log‘𝑥) / 2) ∈
ℂ) |
30 | 1, 26, 29 | subdid 10365 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 ·
((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((2 · (Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (2 · ((log‘𝑥) / 2)))) |
31 | 27 | recnd 9947 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(log‘𝑥) ∈
ℂ) |
32 | | 2ne0 10990 |
. . . . . . . 8
⊢ 2 ≠
0 |
33 | 32 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ≠
0) |
34 | 31, 1, 33 | divcan2d 10682 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 ·
((log‘𝑥) / 2)) =
(log‘𝑥)) |
35 | 34 | oveq2d 6565 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 ·
(Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (2 · ((log‘𝑥) / 2))) = ((2 ·
(Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥))) |
36 | 30, 35 | eqtrd 2644 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 ·
((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((2 · (Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥))) |
37 | 36 | mpteq2dva 4672 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (2 ·
((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))) = (𝑥 ∈ (1(,)+∞) ↦ ((2 ·
(Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥)))) |
38 | | 2re 10967 |
. . . . 5
⊢ 2 ∈
ℝ |
39 | 38 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈
ℝ) |
40 | 25, 28 | resubcld 10337 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℝ) |
41 | | ioossre 12106 |
. . . . . 6
⊢
(1(,)+∞) ⊆ ℝ |
42 | | 2cn 10968 |
. . . . . 6
⊢ 2 ∈
ℂ |
43 | | o1const 14198 |
. . . . . 6
⊢
(((1(,)+∞) ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦
2) ∈ 𝑂(1)) |
44 | 41, 42, 43 | mp2an 704 |
. . . . 5
⊢ (𝑥 ∈ (1(,)+∞) ↦
2) ∈ 𝑂(1) |
45 | 44 | a1i 11 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 2) ∈
𝑂(1)) |
46 | | vmalogdivsum2 25027 |
. . . . 5
⊢ (𝑥 ∈ (1(,)+∞) ↦
((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1) |
47 | 46 | a1i 11 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈
𝑂(1)) |
48 | 39, 40, 45, 47 | o1mul2 14203 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (2 ·
((Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))) ∈
𝑂(1)) |
49 | 37, 48 | eqeltrrd 2689 |
. 2
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((2 ·
(Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1)) |
50 | | fzfid 12634 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin) |
51 | | elfznn 12241 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛))) → 𝑚 ∈
ℕ) |
52 | 51 | adantl 481 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → 𝑚 ∈
ℕ) |
53 | | vmacl 24644 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ ℕ →
(Λ‘𝑚) ∈
ℝ) |
54 | 52, 53 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) →
(Λ‘𝑚) ∈
ℝ) |
55 | 52 | nnrpd 11746 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → 𝑚 ∈
ℝ+) |
56 | 55 | relogcld 24173 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) →
(log‘𝑚) ∈
ℝ) |
57 | 9 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑥 ∈
ℝ) |
58 | 57, 4 | nndivred 10946 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑛) ∈
ℝ) |
59 | 58 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → (𝑥 / 𝑛) ∈ ℝ) |
60 | 59, 52 | nndivred 10946 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → ((𝑥 / 𝑛) / 𝑚) ∈ ℝ) |
61 | | chpcl 24650 |
. . . . . . . . . . . 12
⊢ (((𝑥 / 𝑛) / 𝑚) ∈ ℝ → (ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ) |
62 | 60, 61 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) →
(ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ) |
63 | 56, 62 | readdcld 9948 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) →
((log‘𝑚) +
(ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℝ) |
64 | 54, 63 | remulcld 9949 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) →
((Λ‘𝑚)
· ((log‘𝑚) +
(ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ) |
65 | 50, 64 | fsumrecl 14312 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ) |
66 | 6, 65 | remulcld 9949 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
· Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℝ) |
67 | 2, 66 | fsumrecl 14312 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℝ) |
68 | 17, 24 | rpmulcld 11764 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈
ℝ+) |
69 | 67, 68 | rerpdivcld 11779 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) ∈ ℝ) |
70 | 69, 27 | resubcld 10337 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)) ∈ ℝ) |
71 | 70 | recnd 9947 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)) ∈ ℂ) |
72 | 23 | recnd 9947 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ) |
73 | 24 | rpne0d 11753 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(log‘𝑥) ≠
0) |
74 | 72, 31, 73 | divcld 10680 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) ∈ ℂ) |
75 | 1, 74 | mulcld 9939 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 ·
(Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) ∈ ℂ) |
76 | 75, 31 | subcld 10271 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 ·
(Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥)) ∈ ℂ) |
77 | 69 | recnd 9947 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) ∈ ℂ) |
78 | 77, 75, 31 | nnncan2d 10306 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)) − ((2 · (Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (2 · (Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))))) |
79 | 67 | recnd 9947 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℂ) |
80 | 9 | recnd 9947 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈
ℂ) |
81 | 17 | rpne0d 11753 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0) |
82 | 79, 80, 31, 81, 73 | divdiv1d 10711 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) / (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥)))) |
83 | 1, 72, 31, 73 | divassd 10715 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 ·
Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) = (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)))) |
84 | 82, 83 | oveq12d 6567 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) / (log‘𝑥)) − ((2 · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (2 · (Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))))) |
85 | 67, 17 | rerpdivcld 11779 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) ∈ ℝ) |
86 | 85 | recnd 9947 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) ∈ ℂ) |
87 | 1, 72 | mulcld 9939 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 ·
Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ ℂ) |
88 | 86, 87, 31, 73 | divsubdird 10719 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) / (log‘𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) / (log‘𝑥)) − ((2 · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥)))) |
89 | 81 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑥 ≠
0) |
90 | 66, 57, 89 | redivcld 10732 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((Λ‘𝑛)
· Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) ∈ ℝ) |
91 | 90 | recnd 9947 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((Λ‘𝑛)
· Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) ∈ ℂ) |
92 | 38 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 2 ∈ ℝ) |
93 | 92, 22 | remulcld 9949 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ ℝ) |
94 | 93 | recnd 9947 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ ℂ) |
95 | 2, 91, 94 | fsumsub 14362 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))((((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − Σ𝑛 ∈ (1...(⌊‘𝑥))(2 ·
(((Λ‘𝑛) /
𝑛) ·
(log‘(𝑥 / 𝑛)))))) |
96 | 6 | recnd 9947 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Λ‘𝑛)
∈ ℂ) |
97 | 65, 57, 89 | redivcld 10732 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) ∈ ℝ) |
98 | 97 | recnd 9947 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) ∈ ℂ) |
99 | | 2cnd 10970 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 2 ∈ ℂ) |
100 | 21 | recnd 9947 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (log‘(𝑥 /
𝑛)) ∈
ℂ) |
101 | 4 | nncnd 10913 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℂ) |
102 | 4 | nnne0d 10942 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ≠
0) |
103 | 100, 101,
102 | divcld 10680 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((log‘(𝑥 /
𝑛)) / 𝑛) ∈ ℂ) |
104 | 99, 103 | mulcld 9939 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)) ∈ ℂ) |
105 | 96, 98, 104 | subdid 10365 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
· ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) = (((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥)) − ((Λ‘𝑛) · (2 ·
((log‘(𝑥 / 𝑛)) / 𝑛))))) |
106 | 65 | recnd 9947 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℂ) |
107 | 80 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑥 ∈
ℂ) |
108 | 96, 106, 107, 89 | divassd 10715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((Λ‘𝑛)
· Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) = ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥))) |
109 | 96, 101, 100, 102 | div32d 10703 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((Λ‘𝑛)
/ 𝑛) ·
(log‘(𝑥 / 𝑛))) = ((Λ‘𝑛) · ((log‘(𝑥 / 𝑛)) / 𝑛))) |
110 | 109 | oveq2d 6565 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (2 · ((Λ‘𝑛) · ((log‘(𝑥 / 𝑛)) / 𝑛)))) |
111 | 99, 96, 103 | mul12d 10124 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (2 · ((Λ‘𝑛) · ((log‘(𝑥 / 𝑛)) / 𝑛))) = ((Λ‘𝑛) · (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) |
112 | 110, 111 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = ((Λ‘𝑛) · (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) |
113 | 108, 112 | oveq12d 6567 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = (((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥)) − ((Λ‘𝑛) · (2 ·
((log‘(𝑥 / 𝑛)) / 𝑛))))) |
114 | 105, 113 | eqtr4d 2647 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
· ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) = ((((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))) |
115 | 114 | sumeq2dv 14281 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) · Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))) |
116 | 66 | recnd 9947 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
· Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℂ) |
117 | 2, 80, 116, 81 | fsumdivc 14360 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥)) |
118 | 22 | recnd 9947 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((Λ‘𝑛)
/ 𝑛) ·
(log‘(𝑥 / 𝑛))) ∈
ℂ) |
119 | 2, 1, 118 | fsummulc2 14358 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 ·
Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(2 ·
(((Λ‘𝑛) /
𝑛) ·
(log‘(𝑥 / 𝑛))))) |
120 | 117, 119 | oveq12d 6567 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − Σ𝑛 ∈ (1...(⌊‘𝑥))(2 ·
(((Λ‘𝑛) /
𝑛) ·
(log‘(𝑥 / 𝑛)))))) |
121 | 95, 115, 120 | 3eqtr4rd 2655 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) |
122 | 121 | oveq1d 6564 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) / (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))) |
123 | 88, 122 | eqtr3d 2646 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) / (log‘𝑥)) − ((2 · Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))) |
124 | 78, 84, 123 | 3eqtr2d 2650 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)) − ((2 · (Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))) |
125 | 124 | mpteq2dva 4672 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦
(((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)) − ((2 · (Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥)))) = (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥)))) |
126 | | 1red 9934 |
. . . . 5
⊢ (𝜑 → 1 ∈
ℝ) |
127 | | selberg4lem1.1 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
ℝ+) |
128 | 127 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈
ℝ+) |
129 | 128 | rpred 11748 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈
ℝ) |
130 | 2, 7 | fsumrecl 14312 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℝ) |
131 | 130, 24 | rerpdivcld 11779 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℝ) |
132 | 127 | rpcnd 11750 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℂ) |
133 | | o1const 14198 |
. . . . . . 7
⊢
(((1(,)+∞) ⊆ ℝ ∧ 𝐴 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 𝐴) ∈
𝑂(1)) |
134 | 41, 132, 133 | sylancr 694 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 𝐴) ∈
𝑂(1)) |
135 | | 1cnd 9935 |
. . . . . . . 8
⊢ (𝜑 → 1 ∈
ℂ) |
136 | | o1const 14198 |
. . . . . . . 8
⊢
(((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦
1) ∈ 𝑂(1)) |
137 | 41, 135, 136 | sylancr 694 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈
𝑂(1)) |
138 | 131 | recnd 9947 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℂ) |
139 | | 1cnd 9935 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈
ℂ) |
140 | 130 | recnd 9947 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ) |
141 | 140, 31, 31, 73 | divsubdird 10719 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) / (log‘𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − ((log‘𝑥) / (log‘𝑥)))) |
142 | 140, 31 | subcld 10271 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ) |
143 | 142, 31, 73 | divrecd 10683 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) / (log‘𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥)))) |
144 | 31, 73 | dividd 10678 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((log‘𝑥) /
(log‘𝑥)) =
1) |
145 | 144 | oveq2d 6565 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − ((log‘𝑥) / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1)) |
146 | 141, 143,
145 | 3eqtr3d 2652 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥))) = ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1)) |
147 | 146 | mpteq2dva 4672 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥)))) = (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1))) |
148 | 130, 27 | resubcld 10337 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℝ) |
149 | 12, 24 | rerpdivcld 11779 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 /
(log‘𝑥)) ∈
ℝ) |
150 | 17 | ex 449 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈
ℝ+)) |
151 | 150 | ssrdv 3574 |
. . . . . . . . . . 11
⊢ (𝜑 → (1(,)+∞) ⊆
ℝ+) |
152 | | vmadivsum 24971 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
↦ (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1) |
153 | 152 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
(Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)) |
154 | 151, 153 | o1res2 14142 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)) |
155 | | divlogrlim 24181 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (1(,)+∞) ↦
(1 / (log‘𝑥)))
⇝𝑟 0 |
156 | | rlimo1 14195 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (1(,)+∞) ↦
(1 / (log‘𝑥)))
⇝𝑟 0 → (𝑥 ∈ (1(,)+∞) ↦ (1 /
(log‘𝑥))) ∈
𝑂(1)) |
157 | 155, 156 | mp1i 13 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 /
(log‘𝑥))) ∈
𝑂(1)) |
158 | 148, 149,
154, 157 | o1mul2 14203 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥)))) ∈
𝑂(1)) |
159 | 147, 158 | eqeltrrd 2689 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1)) ∈
𝑂(1)) |
160 | 138, 139,
159 | o1dif 14208 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦
1) ∈ 𝑂(1))) |
161 | 137, 160 | mpbird 246 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ 𝑂(1)) |
162 | 129, 131,
134, 161 | o1mul2 14203 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (𝐴 · (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))) ∈ 𝑂(1)) |
163 | 129, 131 | remulcld 9949 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 · (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ ℝ) |
164 | 21, 4 | nndivred 10946 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((log‘(𝑥 /
𝑛)) / 𝑛) ∈ ℝ) |
165 | 92, 164 | remulcld 9949 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)) ∈ ℝ) |
166 | 97, 165 | resubcld 10337 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))) ∈ ℝ) |
167 | 6, 166 | remulcld 9949 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
· ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ∈ ℝ) |
168 | 2, 167 | fsumrecl 14312 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ∈ ℝ) |
169 | 168, 24 | rerpdivcld 11779 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥)) ∈ ℝ) |
170 | 169 | recnd 9947 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥)) ∈ ℂ) |
171 | 168 | recnd 9947 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ∈ ℂ) |
172 | 171 | abscld 14023 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(abs‘Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ∈ ℝ) |
173 | 129, 130 | remulcld 9949 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 · Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛)) ∈ ℝ) |
174 | 98, 104 | subcld 10271 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))) ∈ ℂ) |
175 | 96, 174 | mulcld 9939 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
· ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ∈ ℂ) |
176 | 175 | abscld 14023 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ∈ ℝ) |
177 | 2, 176 | fsumrecl 14312 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ∈ ℝ) |
178 | 167 | recnd 9947 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
· ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ∈ ℂ) |
179 | 2, 178 | fsumabs 14374 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(abs‘Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))))) |
180 | 129 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝐴 ∈
ℝ) |
181 | 180, 7 | remulcld 9949 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝐴 ·
((Λ‘𝑛) / 𝑛)) ∈
ℝ) |
182 | 174 | abscld 14023 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ∈ ℝ) |
183 | 180, 4 | nndivred 10946 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝐴 / 𝑛) ∈
ℝ) |
184 | | vmage0 24647 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 0 ≤
(Λ‘𝑛)) |
185 | 4, 184 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (Λ‘𝑛)) |
186 | 106, 107,
101, 89, 102 | divdiv2d 10712 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) = ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) · 𝑛) / 𝑥)) |
187 | 106, 101,
107, 89 | div23d 10717 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) · 𝑛) / 𝑥) = ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) · 𝑛)) |
188 | 186, 187 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) = ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) · 𝑛)) |
189 | 99, 103, 101 | mulassd 9942 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((2 · ((log‘(𝑥 / 𝑛)) / 𝑛)) · 𝑛) = (2 · (((log‘(𝑥 / 𝑛)) / 𝑛) · 𝑛))) |
190 | 100, 101,
102 | divcan1d 10681 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((log‘(𝑥 /
𝑛)) / 𝑛) · 𝑛) = (log‘(𝑥 / 𝑛))) |
191 | 190 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (2 · (((log‘(𝑥 / 𝑛)) / 𝑛) · 𝑛)) = (2 · (log‘(𝑥 / 𝑛)))) |
192 | 189, 191 | eqtr2d 2645 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (2 · (log‘(𝑥 / 𝑛))) = ((2 · ((log‘(𝑥 / 𝑛)) / 𝑛)) · 𝑛)) |
193 | 188, 192 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛)))) = (((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) · 𝑛) − ((2 · ((log‘(𝑥 / 𝑛)) / 𝑛)) · 𝑛))) |
194 | 98, 104, 101 | subdird 10366 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))) · 𝑛) = (((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) · 𝑛) − ((2 · ((log‘(𝑥 / 𝑛)) / 𝑛)) · 𝑛))) |
195 | 193, 194 | eqtr4d 2647 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛)))) = (((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))) · 𝑛)) |
196 | 195 | fveq2d 6107 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛))))) = (abs‘(((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))) · 𝑛))) |
197 | 174, 101 | absmuld 14041 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘(((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))) · 𝑛)) = ((abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) · (abs‘𝑛))) |
198 | 4 | nnred 10912 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℝ) |
199 | 19 | rpge0d 11752 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ 𝑛) |
200 | 198, 199 | absidd 14009 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘𝑛) =
𝑛) |
201 | 200 | oveq2d 6565 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) · (abs‘𝑛)) = ((abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) · 𝑛)) |
202 | 196, 197,
201 | 3eqtrd 2648 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛))))) = ((abs‘((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) · 𝑛)) |
203 | 101 | mulid2d 9937 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 · 𝑛) =
𝑛) |
204 | | fznnfl 12523 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ ℝ → (𝑛 ∈
(1...(⌊‘𝑥))
↔ (𝑛 ∈ ℕ
∧ 𝑛 ≤ 𝑥))) |
205 | 9, 204 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑛 ∈
(1...(⌊‘𝑥))
↔ (𝑛 ∈ ℕ
∧ 𝑛 ≤ 𝑥))) |
206 | 205 | simplbda 652 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ≤ 𝑥) |
207 | 203, 206 | eqbrtrd 4605 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 · 𝑛) ≤
𝑥) |
208 | | 1red 9934 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 1 ∈ ℝ) |
209 | 208, 57, 19 | lemuldivd 11797 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((1 · 𝑛) ≤
𝑥 ↔ 1 ≤ (𝑥 / 𝑛))) |
210 | 207, 209 | mpbid 221 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 1 ≤ (𝑥 / 𝑛)) |
211 | | 1re 9918 |
. . . . . . . . . . . . . . . . . . 19
⊢ 1 ∈
ℝ |
212 | | elicopnf 12140 |
. . . . . . . . . . . . . . . . . . 19
⊢ (1 ∈
ℝ → ((𝑥 / 𝑛) ∈ (1[,)+∞) ↔
((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛)))) |
213 | 211, 212 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 / 𝑛) ∈ (1[,)+∞) ↔ ((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛))) |
214 | 58, 210, 213 | sylanbrc 695 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑛) ∈
(1[,)+∞)) |
215 | | selberg4lem1.2 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ∀𝑦 ∈
(1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝐴) |
216 | 215 | ad2antrr 758 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ∀𝑦 ∈
(1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝐴) |
217 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑖 = 𝑚 → (Λ‘𝑖) = (Λ‘𝑚)) |
218 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑖 = 𝑚 → (log‘𝑖) = (log‘𝑚)) |
219 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑖 = 𝑚 → (𝑦 / 𝑖) = (𝑦 / 𝑚)) |
220 | 219 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑖 = 𝑚 → (ψ‘(𝑦 / 𝑖)) = (ψ‘(𝑦 / 𝑚))) |
221 | 218, 220 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑖 = 𝑚 → ((log‘𝑖) + (ψ‘(𝑦 / 𝑖))) = ((log‘𝑚) + (ψ‘(𝑦 / 𝑚)))) |
222 | 217, 221 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑖 = 𝑚 → ((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) = ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘(𝑦 / 𝑚))))) |
223 | 222 | cbvsumv 14274 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Σ𝑖 ∈
(1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) = Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘(𝑦 / 𝑚)))) |
224 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 = (𝑥 / 𝑛) → (⌊‘𝑦) = (⌊‘(𝑥 / 𝑛))) |
225 | 224 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = (𝑥 / 𝑛) → (1...(⌊‘𝑦)) = (1...(⌊‘(𝑥 / 𝑛)))) |
226 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑦 = (𝑥 / 𝑛) → (𝑦 / 𝑚) = ((𝑥 / 𝑛) / 𝑚)) |
227 | 226 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑦 = (𝑥 / 𝑛) → (ψ‘(𝑦 / 𝑚)) = (ψ‘((𝑥 / 𝑛) / 𝑚))) |
228 | 227 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = (𝑥 / 𝑛) → ((log‘𝑚) + (ψ‘(𝑦 / 𝑚))) = ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) |
229 | 228 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 = (𝑥 / 𝑛) → ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘(𝑦 / 𝑚)))) = ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) |
230 | 229 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑦 = (𝑥 / 𝑛) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) →
((Λ‘𝑚)
· ((log‘𝑚) +
(ψ‘(𝑦 / 𝑚)))) = ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) |
231 | 225, 230 | sumeq12dv 14284 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = (𝑥 / 𝑛) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘(𝑦 / 𝑚)))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) |
232 | 223, 231 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = (𝑥 / 𝑛) → Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) |
233 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛)) |
234 | 232, 233 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = (𝑥 / 𝑛) → (Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛))) |
235 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = (𝑥 / 𝑛) → (log‘𝑦) = (log‘(𝑥 / 𝑛))) |
236 | 235 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = (𝑥 / 𝑛) → (2 · (log‘𝑦)) = (2 ·
(log‘(𝑥 / 𝑛)))) |
237 | 234, 236 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = (𝑥 / 𝑛) → ((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦))) = ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛))))) |
238 | 237 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = (𝑥 / 𝑛) → (abs‘((Σ𝑖 ∈
(1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) = (abs‘((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛)))))) |
239 | 238 | breq1d 4593 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = (𝑥 / 𝑛) → ((abs‘((Σ𝑖 ∈
(1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝐴 ↔ (abs‘((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛))))) ≤ 𝐴)) |
240 | 239 | rspcv 3278 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 / 𝑛) ∈ (1[,)+∞) → (∀𝑦 ∈
(1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝐴 → (abs‘((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛))))) ≤ 𝐴)) |
241 | 214, 216,
240 | sylc 63 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛))))) ≤ 𝐴) |
242 | 202, 241 | eqbrtrrd 4607 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) · 𝑛) ≤ 𝐴) |
243 | 182, 180,
19 | lemuldivd 11797 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) · 𝑛) ≤ 𝐴 ↔ (abs‘((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ≤ (𝐴 / 𝑛))) |
244 | 242, 243 | mpbid 221 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ≤ (𝐴 / 𝑛)) |
245 | 182, 183,
6, 185, 244 | lemul2ad 10843 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
· (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ≤ ((Λ‘𝑛) · (𝐴 / 𝑛))) |
246 | 96, 174 | absmuld 14041 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) = ((abs‘(Λ‘𝑛)) ·
(abs‘((Σ𝑚
∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))))) |
247 | 6, 185 | absidd 14009 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘(Λ‘𝑛)) = (Λ‘𝑛)) |
248 | 247 | oveq1d 6564 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(Λ‘𝑛)) · (abs‘((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) = ((Λ‘𝑛) · (abs‘((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))))) |
249 | 246, 248 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) = ((Λ‘𝑛) · (abs‘((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))))) |
250 | 132 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝐴 ∈
ℂ) |
251 | 250, 96, 101, 102 | div12d 10716 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝐴 ·
((Λ‘𝑛) / 𝑛)) = ((Λ‘𝑛) · (𝐴 / 𝑛))) |
252 | 245, 249,
251 | 3brtr4d 4615 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ≤ (𝐴 · ((Λ‘𝑛) / 𝑛))) |
253 | 2, 176, 181, 252 | fsumle 14372 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(𝐴 · ((Λ‘𝑛) / 𝑛))) |
254 | 132 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈
ℂ) |
255 | 7 | recnd 9947 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((Λ‘𝑛)
/ 𝑛) ∈
ℂ) |
256 | 2, 254, 255 | fsummulc2 14358 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 · Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(𝐴 · ((Λ‘𝑛) / 𝑛))) |
257 | 253, 256 | breqtrrd 4611 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ≤ (𝐴 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛))) |
258 | 172, 177,
173, 179, 257 | letrd 10073 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(abs‘Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ≤ (𝐴 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛))) |
259 | 172, 173,
24, 258 | lediv1dd 11806 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((abs‘Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) / (log‘𝑥)) ≤ ((𝐴 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛)) / (log‘𝑥))) |
260 | 254, 140,
31, 73 | divassd 10715 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 · Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛)) / (log‘𝑥)) = (𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))) |
261 | 259, 260 | breqtrd 4609 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((abs‘Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) / (log‘𝑥)) ≤ (𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))) |
262 | 171, 31, 73 | absdivd 14042 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(abs‘(Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) / (abs‘(log‘𝑥)))) |
263 | 24 | rpge0d 11752 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤
(log‘𝑥)) |
264 | 27, 263 | absidd 14009 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(abs‘(log‘𝑥)) =
(log‘𝑥)) |
265 | 264 | oveq2d 6565 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((abs‘Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) / (abs‘(log‘𝑥))) = ((abs‘Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) / (log‘𝑥))) |
266 | 262, 265 | eqtrd 2644 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(abs‘(Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) / (log‘𝑥))) |
267 | 128 | rpge0d 11752 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐴) |
268 | 6, 19, 185 | divge0d 11788 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ ((Λ‘𝑛) / 𝑛)) |
269 | 2, 7, 268 | fsumge0 14368 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤
Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛)) |
270 | 130, 24, 269 | divge0d 11788 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤
(Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) |
271 | 129, 131,
267, 270 | mulge0d 10483 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (𝐴 · (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))) |
272 | 163, 271 | absidd 14009 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(abs‘(𝐴 ·
(Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))) = (𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))) |
273 | 261, 266,
272 | 3brtr4d 4615 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(abs‘(Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))) ≤ (abs‘(𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))))) |
274 | 273 | adantrr 749 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) →
(abs‘(Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))) ≤ (abs‘(𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))))) |
275 | 126, 162,
163, 170, 274 | o1le 14231 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))) ∈ 𝑂(1)) |
276 | 125, 275 | eqeltrd 2688 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦
(((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)) − ((2 · (Σ𝑛 ∈
(1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥)))) ∈ 𝑂(1)) |
277 | 71, 76, 276 | o1dif 14208 |
. 2
⊢ (𝜑 → ((𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦
((2 · (Σ𝑛
∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1))) |
278 | 49, 277 | mpbird 246 |
1
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈
(1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1)) |