Proof of Theorem pntrlog2bndlem6
Step | Hyp | Ref
| Expression |
1 | | elioore 12076 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (1(,)+∞) →
𝑥 ∈
ℝ) |
2 | 1 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈
ℝ) |
3 | | 1rp 11712 |
. . . . . . . . . . . . 13
⊢ 1 ∈
ℝ+ |
4 | 3 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈
ℝ+) |
5 | | 1red 9934 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈
ℝ) |
6 | | eliooord 12104 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (1(,)+∞) → (1
< 𝑥 ∧ 𝑥 <
+∞)) |
7 | 6 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞)) |
8 | 7 | simpld 474 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥) |
9 | 5, 2, 8 | ltled 10064 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥) |
10 | 2, 4, 9 | rpgecld 11787 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈
ℝ+) |
11 | | pntrlog2bnd.r |
. . . . . . . . . . . . 13
⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦
((ψ‘𝑎) −
𝑎)) |
12 | 11 | pntrf 25052 |
. . . . . . . . . . . 12
⊢ 𝑅:ℝ+⟶ℝ |
13 | 12 | ffvelrni 6266 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ+
→ (𝑅‘𝑥) ∈
ℝ) |
14 | 10, 13 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℝ) |
15 | 14 | recnd 9947 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℂ) |
16 | 15 | abscld 14023 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(abs‘(𝑅‘𝑥)) ∈
ℝ) |
17 | 10 | relogcld 24173 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(log‘𝑥) ∈
ℝ) |
18 | 16, 17 | remulcld 9949 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈
ℝ) |
19 | | 2re 10967 |
. . . . . . . . . 10
⊢ 2 ∈
ℝ |
20 | 19 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈
ℝ) |
21 | 2, 8 | rplogcld 24179 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(log‘𝑥) ∈
ℝ+) |
22 | 20, 21 | rerpdivcld 11779 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 /
(log‘𝑥)) ∈
ℝ) |
23 | | fzfid 12634 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(1...(⌊‘𝑥))
∈ Fin) |
24 | 10 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑥 ∈
ℝ+) |
25 | | elfznn 12241 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℕ) |
26 | 25 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℕ) |
27 | 26 | nnrpd 11746 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℝ+) |
28 | 24, 27 | rpdivcld 11765 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑛) ∈
ℝ+) |
29 | 12 | ffvelrni 6266 |
. . . . . . . . . . . . 13
⊢ ((𝑥 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ) |
30 | 28, 29 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑅‘(𝑥 / 𝑛)) ∈ ℝ) |
31 | 30 | recnd 9947 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑅‘(𝑥 / 𝑛)) ∈ ℂ) |
32 | 31 | abscld 14023 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ) |
33 | 27 | relogcld 24173 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (log‘𝑛) ∈
ℝ) |
34 | 32, 33 | remulcld 9949 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ) |
35 | 23, 34 | fsumrecl 14312 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ) |
36 | 22, 35 | remulcld 9949 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℝ) |
37 | 18, 36 | resubcld 10337 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) ∈ ℝ) |
38 | 37 | recnd 9947 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) ∈ ℂ) |
39 | | fzfid 12634 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) ∈ Fin) |
40 | | ssun2 3739 |
. . . . . . . . . . 11
⊢
(((⌊‘(𝑥
/ 𝐴)) +
1)...(⌊‘𝑥))
⊆ ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) |
41 | | pntsval.1 |
. . . . . . . . . . . 12
⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈
(1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) |
42 | | pntrlog2bnd.t |
. . . . . . . . . . . 12
⊢ 𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0)) |
43 | | pntrlog2bndlem5.1 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 ∈
ℝ+) |
44 | | pntrlog2bndlem5.2 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑦 ∈ ℝ+
(abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵) |
45 | | pntrlog2bndlem6.1 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈ ℝ) |
46 | | pntrlog2bndlem6.2 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 ≤ 𝐴) |
47 | 41, 11, 42, 43, 44, 45, 46 | pntrlog2bndlem6a 25071 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(1...(⌊‘𝑥)) =
((1...(⌊‘(𝑥 /
𝐴))) ∪
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)))) |
48 | 40, 47 | syl5sseqr 3617 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) ⊆
(1...(⌊‘𝑥))) |
49 | 48 | sselda 3568 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ (1...(⌊‘𝑥))) |
50 | 49, 34 | syldan 486 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ) |
51 | 39, 50 | fsumrecl 14312 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ) |
52 | 22, 51 | remulcld 9949 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℝ) |
53 | 52 | recnd 9947 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℂ) |
54 | 2 | recnd 9947 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈
ℂ) |
55 | 10 | rpne0d 11753 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0) |
56 | 38, 53, 54, 55 | divdird 10718 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) = (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥))) |
57 | 18 | recnd 9947 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈
ℂ) |
58 | 36 | recnd 9947 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℂ) |
59 | 57, 58, 53 | subsubd 10299 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))) = ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))) |
60 | 22 | recnd 9947 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 /
(log‘𝑥)) ∈
ℂ) |
61 | 35 | recnd 9947 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ) |
62 | 51 | recnd 9947 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ) |
63 | 60, 61, 62 | subdid 10365 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 /
(log‘𝑥)) ·
(Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))) |
64 | 3 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 1 ∈
ℝ+) |
65 | 45, 64, 46 | rpgecld 11787 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐴 ∈
ℝ+) |
66 | 65 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈
ℝ+) |
67 | 2, 66 | rerpdivcld 11779 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ∈ ℝ) |
68 | | reflcl 12459 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 / 𝐴) ∈ ℝ →
(⌊‘(𝑥 / 𝐴)) ∈
ℝ) |
69 | 67, 68 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(⌊‘(𝑥 / 𝐴)) ∈
ℝ) |
70 | 69 | ltp1d 10833 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(⌊‘(𝑥 / 𝐴)) < ((⌊‘(𝑥 / 𝐴)) + 1)) |
71 | | fzdisj 12239 |
. . . . . . . . . . . . 13
⊢
((⌊‘(𝑥 /
𝐴)) <
((⌊‘(𝑥 / 𝐴)) + 1) →
((1...(⌊‘(𝑥 /
𝐴))) ∩
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) = ∅) |
72 | 70, 71 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((1...(⌊‘(𝑥 /
𝐴))) ∩
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) = ∅) |
73 | 34 | recnd 9947 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ) |
74 | 72, 47, 23, 73 | fsumsplit 14318 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) = (Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) |
75 | 74 | oveq1d 6564 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) = ((Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) |
76 | | fzfid 12634 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(1...(⌊‘(𝑥 /
𝐴))) ∈
Fin) |
77 | | ssun1 3738 |
. . . . . . . . . . . . . . . 16
⊢
(1...(⌊‘(𝑥 / 𝐴))) ⊆ ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) |
78 | 77, 47 | syl5sseqr 3617 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(1...(⌊‘(𝑥 /
𝐴))) ⊆
(1...(⌊‘𝑥))) |
79 | 78 | sselda 3568 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))) → 𝑛 ∈
(1...(⌊‘𝑥))) |
80 | 79, 34 | syldan 486 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))) →
((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ) |
81 | 76, 80 | fsumrecl 14312 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ) |
82 | 81 | recnd 9947 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ) |
83 | 82, 62 | pncand 10272 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) |
84 | 75, 83 | eqtrd 2644 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) |
85 | 84 | oveq2d 6565 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 /
(log‘𝑥)) ·
(Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) |
86 | 63, 85 | eqtr3d 2646 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) |
87 | 86 | oveq2d 6565 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))) = (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))) |
88 | 59, 87 | eqtr3d 2646 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) = (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))) |
89 | 88 | oveq1d 6564 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) = ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) |
90 | 56, 89 | eqtr3d 2646 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥)) = ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) |
91 | 90 | mpteq2dva 4672 |
. 2
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦
(((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥))) = (𝑥 ∈ (1(,)+∞) ↦
((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥))) |
92 | 37, 10 | rerpdivcld 11779 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) ∈ ℝ) |
93 | 52, 10 | rerpdivcld 11779 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥) ∈ ℝ) |
94 | 41, 11, 42, 43, 44 | pntrlog2bndlem5 25070 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦
((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1)) |
95 | | ioossre 12106 |
. . . . 5
⊢
(1(,)+∞) ⊆ ℝ |
96 | 95 | a1i 11 |
. . . 4
⊢ (𝜑 → (1(,)+∞) ⊆
ℝ) |
97 | | 1red 9934 |
. . . 4
⊢ (𝜑 → 1 ∈
ℝ) |
98 | 19 | a1i 11 |
. . . . 5
⊢ (𝜑 → 2 ∈
ℝ) |
99 | 43 | rpred 11748 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℝ) |
100 | 65 | relogcld 24173 |
. . . . . . 7
⊢ (𝜑 → (log‘𝐴) ∈
ℝ) |
101 | 100, 97 | readdcld 9948 |
. . . . . 6
⊢ (𝜑 → ((log‘𝐴) + 1) ∈
ℝ) |
102 | 99, 101 | remulcld 9949 |
. . . . 5
⊢ (𝜑 → (𝐵 · ((log‘𝐴) + 1)) ∈ ℝ) |
103 | 98, 102 | remulcld 9949 |
. . . 4
⊢ (𝜑 → (2 · (𝐵 · ((log‘𝐴) + 1))) ∈
ℝ) |
104 | 51, 21 | rerpdivcld 11779 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ∈ ℝ) |
105 | 99 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈
ℝ) |
106 | 66 | relogcld 24173 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(log‘𝐴) ∈
ℝ) |
107 | 106, 5 | readdcld 9948 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((log‘𝐴) + 1) ∈
ℝ) |
108 | 105, 107 | remulcld 9949 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · ((log‘𝐴) + 1)) ∈
ℝ) |
109 | 2, 108 | remulcld 9949 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (𝐵 · ((log‘𝐴) + 1))) ∈ ℝ) |
110 | | 2rp 11713 |
. . . . . . . . . 10
⊢ 2 ∈
ℝ+ |
111 | 110 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈
ℝ+) |
112 | 111 | rpge0d 11752 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤
2) |
113 | 105, 2 | remulcld 9949 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · 𝑥) ∈ ℝ) |
114 | 49, 25 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℕ) |
115 | 114 | nnrecred 10943 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ) |
116 | 39, 115 | fsumrecl 14312 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ) |
117 | 113, 116 | remulcld 9949 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) ∈ ℝ) |
118 | 21 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑥) ∈
ℝ+) |
119 | 50, 118 | rerpdivcld 11779 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ∈ ℝ) |
120 | 105 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝐵 ∈ ℝ) |
121 | 2 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑥 ∈ ℝ) |
122 | 120, 121 | remulcld 9949 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · 𝑥) ∈ ℝ) |
123 | 122, 115 | remulcld 9949 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) · (1 / 𝑛)) ∈ ℝ) |
124 | 49, 32 | syldan 486 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ) |
125 | 121, 114 | nndivred 10946 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ) |
126 | 120, 125 | remulcld 9949 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · (𝑥 / 𝑛)) ∈ ℝ) |
127 | 49, 27 | syldan 486 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℝ+) |
128 | 127 | relogcld 24173 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ) |
129 | 10 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑥 ∈ ℝ+) |
130 | 129 | relogcld 24173 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑥) ∈ ℝ) |
131 | 49, 31 | syldan 486 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ) |
132 | 131 | absge0d 14031 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑅‘(𝑥 / 𝑛)))) |
133 | | elfzle2 12216 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) → 𝑛 ≤ (⌊‘𝑥)) |
134 | 133 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ≤ (⌊‘𝑥)) |
135 | 114 | nnzd 11357 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℤ) |
136 | | flge 12468 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝑥 ↔ 𝑛 ≤ (⌊‘𝑥))) |
137 | 121, 135,
136 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑛 ≤ 𝑥 ↔ 𝑛 ≤ (⌊‘𝑥))) |
138 | 134, 137 | mpbird 246 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ≤ 𝑥) |
139 | 127, 129 | logled 24177 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑛 ≤ 𝑥 ↔ (log‘𝑛) ≤ (log‘𝑥))) |
140 | 138, 139 | mpbid 221 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑛) ≤ (log‘𝑥)) |
141 | 128, 130,
124, 132, 140 | lemul2ad 10843 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ≤ ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑥))) |
142 | 50, 124, 118 | ledivmul2d 11802 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (abs‘(𝑅‘(𝑥 / 𝑛))) ↔ ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ≤ ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑥)))) |
143 | 141, 142 | mpbird 246 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (abs‘(𝑅‘(𝑥 / 𝑛)))) |
144 | 125 | recnd 9947 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ) |
145 | 49, 28 | syldan 486 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈
ℝ+) |
146 | 145 | rpne0d 11753 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ≠ 0) |
147 | 131, 144,
146 | absdivd 14042 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛)))) |
148 | 10 | rpge0d 11752 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝑥) |
149 | 148 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ 𝑥) |
150 | 121, 127,
149 | divge0d 11788 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ (𝑥 / 𝑛)) |
151 | 125, 150 | absidd 14009 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘(𝑥 / 𝑛)) = (𝑥 / 𝑛)) |
152 | 151 | oveq2d 6565 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛))) |
153 | 147, 152 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛))) |
154 | 44 | ad2antrr 758 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ∀𝑦 ∈ ℝ+
(abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵) |
155 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = (𝑥 / 𝑛) → (𝑅‘𝑦) = (𝑅‘(𝑥 / 𝑛))) |
156 | | id 22 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛)) |
157 | 155, 156 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = (𝑥 / 𝑛) → ((𝑅‘𝑦) / 𝑦) = ((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) |
158 | 157 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = (𝑥 / 𝑛) → (abs‘((𝑅‘𝑦) / 𝑦)) = (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛)))) |
159 | 158 | breq1d 4593 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = (𝑥 / 𝑛) → ((abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵 ↔ (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵)) |
160 | 159 | rspcv 3278 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 / 𝑛) ∈ ℝ+ →
(∀𝑦 ∈
ℝ+ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵 → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵)) |
161 | 145, 154,
160 | sylc 63 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵) |
162 | 153, 161 | eqbrtrrd 4607 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)) ≤ 𝐵) |
163 | 124, 120,
145 | ledivmul2d 11802 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)) ≤ 𝐵 ↔ (abs‘(𝑅‘(𝑥 / 𝑛))) ≤ (𝐵 · (𝑥 / 𝑛)))) |
164 | 162, 163 | mpbid 221 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ≤ (𝐵 · (𝑥 / 𝑛))) |
165 | 119, 124,
126, 143, 164 | letrd 10073 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (𝐵 · (𝑥 / 𝑛))) |
166 | 120 | recnd 9947 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝐵 ∈ ℂ) |
167 | 54 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑥 ∈ ℂ) |
168 | 114 | nncnd 10913 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℂ) |
169 | 114 | nnne0d 10942 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ≠ 0) |
170 | 166, 167,
168, 169 | divassd 10715 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) / 𝑛) = (𝐵 · (𝑥 / 𝑛))) |
171 | 166, 167 | mulcld 9939 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · 𝑥) ∈ ℂ) |
172 | 171, 168,
169 | divrecd 10683 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) / 𝑛) = ((𝐵 · 𝑥) · (1 / 𝑛))) |
173 | 170, 172 | eqtr3d 2646 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · (𝑥 / 𝑛)) = ((𝐵 · 𝑥) · (1 / 𝑛))) |
174 | 165, 173 | breqtrd 4609 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ ((𝐵 · 𝑥) · (1 / 𝑛))) |
175 | 39, 119, 123, 174 | fsumle 14372 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((𝐵 · 𝑥) · (1 / 𝑛))) |
176 | 17 | recnd 9947 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(log‘𝑥) ∈
ℂ) |
177 | 49, 73 | syldan 486 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ) |
178 | 21 | rpne0d 11753 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(log‘𝑥) ≠
0) |
179 | 39, 176, 177, 178 | fsumdivc 14360 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) = Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥))) |
180 | 105 | recnd 9947 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈
ℂ) |
181 | 180, 54 | mulcld 9939 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · 𝑥) ∈ ℂ) |
182 | 115 | recnd 9947 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ) |
183 | 39, 181, 182 | fsummulc2 14358 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((𝐵 · 𝑥) · (1 / 𝑛))) |
184 | 175, 179,
183 | 3brtr4d 4615 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛))) |
185 | 43 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈
ℝ+) |
186 | 185 | rpge0d 11752 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐵) |
187 | 105, 2, 186, 148 | mulge0d 10483 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (𝐵 · 𝑥)) |
188 | 26 | nnrecred 10943 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑛) ∈
ℝ) |
189 | 23, 188 | fsumrecl 14312 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑛) ∈
ℝ) |
190 | 17, 106 | resubcld 10337 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((log‘𝑥) −
(log‘𝐴)) ∈
ℝ) |
191 | 17, 5 | readdcld 9948 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((log‘𝑥) + 1) ∈
ℝ) |
192 | 79, 188 | syldan 486 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))) → (1 / 𝑛) ∈
ℝ) |
193 | 76, 192 | fsumrecl 14312 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))(1 / 𝑛) ∈ ℝ) |
194 | | harmonicubnd 24536 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 1 ≤
𝑥) → Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑛) ≤ ((log‘𝑥) + 1)) |
195 | 2, 9, 194 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑛) ≤ ((log‘𝑥) + 1)) |
196 | 10, 66 | relogdivd 24176 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(log‘(𝑥 / 𝐴)) = ((log‘𝑥) − (log‘𝐴))) |
197 | 10, 66 | rpdivcld 11765 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ∈
ℝ+) |
198 | | harmoniclbnd 24535 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 / 𝐴) ∈ ℝ+ →
(log‘(𝑥 / 𝐴)) ≤ Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))(1 / 𝑛)) |
199 | 197, 198 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(log‘(𝑥 / 𝐴)) ≤ Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))(1 / 𝑛)) |
200 | 196, 199 | eqbrtrrd 4607 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((log‘𝑥) −
(log‘𝐴)) ≤
Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))(1 / 𝑛)) |
201 | 189, 190,
191, 193, 195, 200 | le2subd 10526 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑛) − Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))(1 / 𝑛)) ≤ (((log‘𝑥) + 1) − ((log‘𝑥) − (log‘𝐴)))) |
202 | 26 | nncnd 10913 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℂ) |
203 | 26 | nnne0d 10942 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ≠
0) |
204 | 202, 203 | reccld 10673 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑛) ∈
ℂ) |
205 | 72, 47, 23, 204 | fsumsplit 14318 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑛) = (Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛))) |
206 | 205 | oveq1d 6564 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑛) − Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))(1 / 𝑛)) = ((Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) − Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛))) |
207 | 79, 25 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))) → 𝑛 ∈
ℕ) |
208 | 207 | nnrecred 10943 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))) → (1 / 𝑛) ∈
ℝ) |
209 | 76, 208 | fsumrecl 14312 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))(1 / 𝑛) ∈ ℝ) |
210 | 209 | recnd 9947 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))(1 / 𝑛) ∈ ℂ) |
211 | 116 | recnd 9947 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ∈ ℂ) |
212 | 210, 211 | pncan2d 10273 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) − Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) |
213 | 206, 212 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈
(1...(⌊‘𝑥))(1 /
𝑛) − Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) |
214 | | 1cnd 9935 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈
ℂ) |
215 | 106 | recnd 9947 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(log‘𝐴) ∈
ℂ) |
216 | 176, 214,
215 | pnncand 10310 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((log‘𝑥) + 1)
− ((log‘𝑥)
− (log‘𝐴))) =
(1 + (log‘𝐴))) |
217 | 214, 215 | addcomd 10117 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 +
(log‘𝐴)) =
((log‘𝐴) +
1)) |
218 | 216, 217 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
(((log‘𝑥) + 1)
− ((log‘𝑥)
− (log‘𝐴))) =
((log‘𝐴) +
1)) |
219 | 201, 213,
218 | 3brtr3d 4614 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝐴) + 1)) |
220 | 116, 107,
113, 187, 219 | lemul2ad 10843 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) ≤ ((𝐵 · 𝑥) · ((log‘𝐴) + 1))) |
221 | 107 | recnd 9947 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((log‘𝐴) + 1) ∈
ℂ) |
222 | 180, 54, 221 | mulassd 9942 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · ((log‘𝐴) + 1)) = (𝐵 · (𝑥 · ((log‘𝐴) + 1)))) |
223 | 180, 54, 221 | mul12d 10124 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · (𝑥 · ((log‘𝐴) + 1))) = (𝑥 · (𝐵 · ((log‘𝐴) + 1)))) |
224 | 222, 223 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · ((log‘𝐴) + 1)) = (𝑥 · (𝐵 · ((log‘𝐴) + 1)))) |
225 | 220, 224 | breqtrd 4609 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) ≤ (𝑥 · (𝐵 · ((log‘𝐴) + 1)))) |
226 | 104, 117,
109, 184, 225 | letrd 10073 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (𝑥 · (𝐵 · ((log‘𝐴) + 1)))) |
227 | 104, 109,
20, 112, 226 | lemul2ad 10843 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 ·
(Σ𝑛 ∈
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥))) ≤ (2 · (𝑥 · (𝐵 · ((log‘𝐴) + 1))))) |
228 | | 2cnd 10970 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈
ℂ) |
229 | 228, 176,
62, 178 | div32d 10703 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) = (2 · (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)))) |
230 | 215, 214 | addcld 9938 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) →
((log‘𝐴) + 1) ∈
ℂ) |
231 | 180, 230 | mulcld 9939 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · ((log‘𝐴) + 1)) ∈
ℂ) |
232 | 54, 228, 231 | mul12d 10124 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (2 · (𝐵 · ((log‘𝐴) + 1)))) = (2 · (𝑥 · (𝐵 · ((log‘𝐴) + 1))))) |
233 | 227, 229,
232 | 3brtr4d 4615 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ≤ (𝑥 · (2 · (𝐵 · ((log‘𝐴) + 1))))) |
234 | 103 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 ·
(𝐵 ·
((log‘𝐴) + 1)))
∈ ℝ) |
235 | 52, 234, 10 | ledivmuld 11801 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥) ≤ (2 · (𝐵 · ((log‘𝐴) + 1))) ↔ ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ≤ (𝑥 · (2 · (𝐵 · ((log‘𝐴) + 1)))))) |
236 | 233, 235 | mpbird 246 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥) ≤ (2 · (𝐵 · ((log‘𝐴) + 1)))) |
237 | 236 | adantrr 749 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥) ≤ (2 · (𝐵 · ((log‘𝐴) + 1)))) |
238 | 96, 93, 97, 103, 237 | ello1d 14102 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((2 /
(log‘𝑥)) ·
Σ𝑛 ∈
(((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥)) ∈ ≤𝑂(1)) |
239 | 92, 93, 94, 238 | lo1add 14205 |
. 2
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦
(((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥))) ∈ ≤𝑂(1)) |
240 | 91, 239 | eqeltrrd 2689 |
1
⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦
((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈
(1...(⌊‘(𝑥 /
𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1)) |