Proof of Theorem dchrvmasumlem2
Step | Hyp | Ref
| Expression |
1 | | 1red 10055 |
. 2
⊢ (𝜑 → 1 ∈
ℝ) |
2 | | dchrvmasum.c |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) |
3 | | elrege0 12278 |
. . . . . . 7
⊢ (𝐶 ∈ (0[,)+∞) ↔
(𝐶 ∈ ℝ ∧ 0
≤ 𝐶)) |
4 | 2, 3 | sylib 208 |
. . . . . 6
⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) |
5 | 4 | simpld 475 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ ℝ) |
6 | 5 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈
ℝ) |
7 | | fzfid 12772 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(1...(⌊‘𝑥))
∈ Fin) |
8 | | simpr 477 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈
ℝ+) |
9 | | elfznn 12370 |
. . . . . . . . 9
⊢ (𝑑 ∈
(1...(⌊‘𝑥))
→ 𝑑 ∈
ℕ) |
10 | 9 | nnrpd 11870 |
. . . . . . . 8
⊢ (𝑑 ∈
(1...(⌊‘𝑥))
→ 𝑑 ∈
ℝ+) |
11 | | rpdivcl 11856 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑑 ∈
ℝ+) → (𝑥 / 𝑑) ∈
ℝ+) |
12 | 8, 10, 11 | syl2an 494 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑑) ∈
ℝ+) |
13 | | relogcl 24322 |
. . . . . . 7
⊢ ((𝑥 / 𝑑) ∈ ℝ+ →
(log‘(𝑥 / 𝑑)) ∈
ℝ) |
14 | 12, 13 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (log‘(𝑥 /
𝑑)) ∈
ℝ) |
15 | 8 | adantr 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑥 ∈
ℝ+) |
16 | 14, 15 | rerpdivcld 11903 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((log‘(𝑥 /
𝑑)) / 𝑥) ∈ ℝ) |
17 | 7, 16 | fsumrecl 14465 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ) |
18 | 6, 17 | remulcld 10070 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐶 · Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ ℝ) |
19 | | dchrvmasum.r |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ ℝ) |
20 | | 3nn 11186 |
. . . . . . 7
⊢ 3 ∈
ℕ |
21 | | nnrp 11842 |
. . . . . . 7
⊢ (3 ∈
ℕ → 3 ∈ ℝ+) |
22 | | relogcl 24322 |
. . . . . . 7
⊢ (3 ∈
ℝ+ → (log‘3) ∈ ℝ) |
23 | 20, 21, 22 | mp2b 10 |
. . . . . 6
⊢
(log‘3) ∈ ℝ |
24 | | 1re 10039 |
. . . . . 6
⊢ 1 ∈
ℝ |
25 | 23, 24 | readdcli 10053 |
. . . . 5
⊢
((log‘3) + 1) ∈ ℝ |
26 | | remulcl 10021 |
. . . . 5
⊢ ((𝑅 ∈ ℝ ∧
((log‘3) + 1) ∈ ℝ) → (𝑅 · ((log‘3) + 1)) ∈
ℝ) |
27 | 19, 25, 26 | sylancl 694 |
. . . 4
⊢ (𝜑 → (𝑅 · ((log‘3) + 1)) ∈
ℝ) |
28 | 27 | adantr 481 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑅 · ((log‘3) + 1))
∈ ℝ) |
29 | | rpssre 11843 |
. . . . 5
⊢
ℝ+ ⊆ ℝ |
30 | 5 | recnd 10068 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ ℂ) |
31 | | o1const 14350 |
. . . . 5
⊢
((ℝ+ ⊆ ℝ ∧ 𝐶 ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ 𝐶) ∈
𝑂(1)) |
32 | 29, 30, 31 | sylancr 695 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ 𝐶) ∈
𝑂(1)) |
33 | | logfacrlim2 24951 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
↦ Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ⇝𝑟
1 |
34 | | rlimo1 14347 |
. . . . 5
⊢ ((𝑥 ∈ ℝ+
↦ Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ⇝𝑟 1 →
(𝑥 ∈
ℝ+ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ 𝑂(1)) |
35 | 33, 34 | mp1i 13 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ 𝑂(1)) |
36 | 6, 17, 32, 35 | o1mul2 14355 |
. . 3
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝐶 · Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥))) ∈ 𝑂(1)) |
37 | 27 | recnd 10068 |
. . . 4
⊢ (𝜑 → (𝑅 · ((log‘3) + 1)) ∈
ℂ) |
38 | | o1const 14350 |
. . . 4
⊢
((ℝ+ ⊆ ℝ ∧ (𝑅 · ((log‘3) + 1)) ∈
ℂ) → (𝑥 ∈
ℝ+ ↦ (𝑅 · ((log‘3) + 1))) ∈
𝑂(1)) |
39 | 29, 37, 38 | sylancr 695 |
. . 3
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝑅 · ((log‘3) + 1)))
∈ 𝑂(1)) |
40 | 18, 28, 36, 39 | o1add2 14354 |
. 2
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ ((𝐶 · Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))) ∈
𝑂(1)) |
41 | 18, 28 | readdcld 10069 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((𝐶 · Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))) ∈
ℝ) |
42 | | dchrvmasum.f |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ℝ+) → 𝐹 ∈
ℂ) |
43 | 42 | ralrimiva 2966 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑚 ∈ ℝ+ 𝐹 ∈ ℂ) |
44 | 43 | ad2antrr 762 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ∀𝑚 ∈
ℝ+ 𝐹
∈ ℂ) |
45 | | dchrvmasum.g |
. . . . . . . . . 10
⊢ (𝑚 = (𝑥 / 𝑑) → 𝐹 = 𝐾) |
46 | 45 | eleq1d 2686 |
. . . . . . . . 9
⊢ (𝑚 = (𝑥 / 𝑑) → (𝐹 ∈ ℂ ↔ 𝐾 ∈ ℂ)) |
47 | 46 | rspcv 3305 |
. . . . . . . 8
⊢ ((𝑥 / 𝑑) ∈ ℝ+ →
(∀𝑚 ∈
ℝ+ 𝐹
∈ ℂ → 𝐾
∈ ℂ)) |
48 | 12, 44, 47 | sylc 65 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝐾 ∈
ℂ) |
49 | | dchrvmasum.t |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ∈ ℂ) |
50 | 49 | ad2antrr 762 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑇 ∈
ℂ) |
51 | 48, 50 | subcld 10392 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐾 − 𝑇) ∈
ℂ) |
52 | 51 | abscld 14175 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(𝐾
− 𝑇)) ∈
ℝ) |
53 | 9 | adantl 482 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℕ) |
54 | 52, 53 | nndivred 11069 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(𝐾
− 𝑇)) / 𝑑) ∈
ℝ) |
55 | 7, 54 | fsumrecl 14465 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
Σ𝑑 ∈
(1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℝ) |
56 | 55 | recnd 10068 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
Σ𝑑 ∈
(1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑) ∈ ℂ) |
57 | 53 | nnrpd 11870 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℝ+) |
58 | 51 | absge0d 14183 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘(𝐾
− 𝑇))) |
59 | 52, 57, 58 | divge0d 11912 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ ((abs‘(𝐾 − 𝑇)) / 𝑑)) |
60 | 7, 54, 59 | fsumge0 14527 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 0 ≤
Σ𝑑 ∈
(1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) |
61 | 55, 60 | absidd 14161 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(abs‘Σ𝑑 ∈
(1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) |
62 | 61, 55 | eqeltrd 2701 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(abs‘Σ𝑑 ∈
(1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ ℝ) |
63 | 41 | recnd 10068 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((𝐶 · Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))) ∈
ℂ) |
64 | 63 | abscld 14175 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(abs‘((𝐶 ·
Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1)))) ∈
ℝ) |
65 | | 3re 11094 |
. . . . . . . 8
⊢ 3 ∈
ℝ |
66 | 65 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 3 ∈
ℝ) |
67 | | 1le3 11244 |
. . . . . . 7
⊢ 1 ≤
3 |
68 | 66, 67 | jctir 561 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (3 ∈
ℝ ∧ 1 ≤ 3)) |
69 | 19 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑅 ∈
ℝ) |
70 | 24 | rexri 10097 |
. . . . . . . . . 10
⊢ 1 ∈
ℝ* |
71 | 65 | rexri 10097 |
. . . . . . . . . 10
⊢ 3 ∈
ℝ* |
72 | | 1lt3 11196 |
. . . . . . . . . 10
⊢ 1 <
3 |
73 | | lbico1 12228 |
. . . . . . . . . 10
⊢ ((1
∈ ℝ* ∧ 3 ∈ ℝ* ∧ 1 < 3)
→ 1 ∈ (1[,)3)) |
74 | 70, 71, 72, 73 | mp3an 1424 |
. . . . . . . . 9
⊢ 1 ∈
(1[,)3) |
75 | | 0red 10041 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ∈
ℝ) |
76 | | elico2 12237 |
. . . . . . . . . . . . . . 15
⊢ ((1
∈ ℝ ∧ 3 ∈ ℝ*) → (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚 ∧ 𝑚 < 3))) |
77 | 24, 71, 76 | mp2an 708 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤
𝑚 ∧ 𝑚 < 3)) |
78 | 77 | simp1bi 1076 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ (1[,)3) → 𝑚 ∈
ℝ) |
79 | | 0red 10041 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ (1[,)3) → 0 ∈
ℝ) |
80 | | 1red 10055 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ (1[,)3) → 1 ∈
ℝ) |
81 | | 0lt1 10550 |
. . . . . . . . . . . . . . 15
⊢ 0 <
1 |
82 | 81 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ (1[,)3) → 0 <
1) |
83 | 77 | simp2bi 1077 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ (1[,)3) → 1 ≤
𝑚) |
84 | 79, 80, 78, 82, 83 | ltletrd 10197 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ (1[,)3) → 0 <
𝑚) |
85 | 78, 84 | elrpd 11869 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ (1[,)3) → 𝑚 ∈
ℝ+) |
86 | 49 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℝ+) → 𝑇 ∈
ℂ) |
87 | 42, 86 | subcld 10392 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℝ+) → (𝐹 − 𝑇) ∈ ℂ) |
88 | 87 | abscld 14175 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℝ+) →
(abs‘(𝐹 − 𝑇)) ∈
ℝ) |
89 | 85, 88 | sylan2 491 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘(𝐹 − 𝑇)) ∈ ℝ) |
90 | 19 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 𝑅 ∈ ℝ) |
91 | 87 | absge0d 14183 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℝ+) → 0 ≤
(abs‘(𝐹 − 𝑇))) |
92 | 85, 91 | sylan2 491 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ≤
(abs‘(𝐹 − 𝑇))) |
93 | | dchrvmasum.2 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹 − 𝑇)) ≤ 𝑅) |
94 | 93 | r19.21bi 2932 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘(𝐹 − 𝑇)) ≤ 𝑅) |
95 | 75, 89, 90, 92, 94 | letrd 10194 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ≤ 𝑅) |
96 | 95 | ralrimiva 2966 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑚 ∈ (1[,)3)0 ≤ 𝑅) |
97 | | biidd 252 |
. . . . . . . . . 10
⊢ (𝑚 = 1 → (0 ≤ 𝑅 ↔ 0 ≤ 𝑅)) |
98 | 97 | rspcv 3305 |
. . . . . . . . 9
⊢ (1 ∈
(1[,)3) → (∀𝑚
∈ (1[,)3)0 ≤ 𝑅
→ 0 ≤ 𝑅)) |
99 | 74, 96, 98 | mpsyl 68 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ 𝑅) |
100 | 99 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 0 ≤
𝑅) |
101 | 69, 100 | jca 554 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑅 ∈ ℝ ∧ 0 ≤
𝑅)) |
102 | 52 | recnd 10068 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(𝐾
− 𝑇)) ∈
ℂ) |
103 | 5 | ad2antrr 762 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝐶 ∈
ℝ) |
104 | 103, 16 | remulcld 10070 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 ·
((log‘(𝑥 / 𝑑)) / 𝑥)) ∈ ℝ) |
105 | 4 | ad2antrr 762 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 ∈ ℝ
∧ 0 ≤ 𝐶)) |
106 | | log1 24332 |
. . . . . . . . 9
⊢
(log‘1) = 0 |
107 | 53 | nncnd 11036 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℂ) |
108 | 107 | mulid2d 10058 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 · 𝑑) =
𝑑) |
109 | | rpre 11839 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ) |
110 | 109 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈
ℝ) |
111 | | fznnfl 12661 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ → (𝑑 ∈
(1...(⌊‘𝑥))
↔ (𝑑 ∈ ℕ
∧ 𝑑 ≤ 𝑥))) |
112 | 110, 111 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑑 ∈
(1...(⌊‘𝑥))
↔ (𝑑 ∈ ℕ
∧ 𝑑 ≤ 𝑥))) |
113 | 112 | simplbda 654 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ≤ 𝑥) |
114 | 108, 113 | eqbrtrd 4675 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 · 𝑑) ≤
𝑥) |
115 | | 1red 10055 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 1 ∈ ℝ) |
116 | 109 | ad2antlr 763 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑥 ∈
ℝ) |
117 | 115, 116,
57 | lemuldivd 11921 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((1 · 𝑑) ≤
𝑥 ↔ 1 ≤ (𝑥 / 𝑑))) |
118 | 114, 117 | mpbid 222 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 1 ≤ (𝑥 / 𝑑)) |
119 | | 1rp 11836 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℝ+ |
120 | 119 | a1i 11 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 1 ∈ ℝ+) |
121 | 120, 12 | logled 24373 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 ≤ (𝑥 / 𝑑) ↔ (log‘1) ≤
(log‘(𝑥 / 𝑑)))) |
122 | 118, 121 | mpbid 222 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (log‘1) ≤ (log‘(𝑥 / 𝑑))) |
123 | 106, 122 | syl5eqbrr 4689 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (log‘(𝑥
/ 𝑑))) |
124 | | rpregt0 11846 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
∧ 0 < 𝑥)) |
125 | 124 | ad2antlr 763 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑥 ∈ ℝ
∧ 0 < 𝑥)) |
126 | | divge0 10892 |
. . . . . . . 8
⊢
((((log‘(𝑥 /
𝑑)) ∈ ℝ ∧ 0
≤ (log‘(𝑥 / 𝑑))) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤
((log‘(𝑥 / 𝑑)) / 𝑥)) |
127 | 14, 123, 125, 126 | syl21anc 1325 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ ((log‘(𝑥 / 𝑑)) / 𝑥)) |
128 | | mulge0 10546 |
. . . . . . 7
⊢ (((𝐶 ∈ ℝ ∧ 0 ≤
𝐶) ∧
(((log‘(𝑥 / 𝑑)) / 𝑥) ∈ ℝ ∧ 0 ≤
((log‘(𝑥 / 𝑑)) / 𝑥))) → 0 ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥))) |
129 | 105, 16, 127, 128 | syl12anc 1324 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (𝐶 ·
((log‘(𝑥 / 𝑑)) / 𝑥))) |
130 | | absidm 14063 |
. . . . . . . . 9
⊢ ((𝐾 − 𝑇) ∈ ℂ →
(abs‘(abs‘(𝐾
− 𝑇))) =
(abs‘(𝐾 − 𝑇))) |
131 | 51, 130 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(abs‘(𝐾 − 𝑇))) = (abs‘(𝐾 − 𝑇))) |
132 | 131 | adantr 481 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 3 ≤ (𝑥 / 𝑑)) →
(abs‘(abs‘(𝐾
− 𝑇))) =
(abs‘(𝐾 − 𝑇))) |
133 | | nndivre 11056 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ) |
134 | 110, 9, 133 | syl2an 494 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑑) ∈
ℝ) |
135 | 134 | adantr 481 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 3 ≤ (𝑥 / 𝑑)) → (𝑥 / 𝑑) ∈ ℝ) |
136 | | simpr 477 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 3 ≤ (𝑥 / 𝑑)) → 3 ≤ (𝑥 / 𝑑)) |
137 | | elicopnf 12269 |
. . . . . . . . . . 11
⊢ (3 ∈
ℝ → ((𝑥 / 𝑑) ∈ (3[,)+∞) ↔
((𝑥 / 𝑑) ∈ ℝ ∧ 3 ≤ (𝑥 / 𝑑)))) |
138 | 65, 137 | ax-mp 5 |
. . . . . . . . . 10
⊢ ((𝑥 / 𝑑) ∈ (3[,)+∞) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 3 ≤ (𝑥 / 𝑑))) |
139 | 135, 136,
138 | sylanbrc 698 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 3 ≤ (𝑥 / 𝑑)) → (𝑥 / 𝑑) ∈ (3[,)+∞)) |
140 | | dchrvmasum.1 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) →
(abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚))) |
141 | 140 | ralrimiva 2966 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑚 ∈ (3[,)+∞)(abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚))) |
142 | 141 | ad3antrrr 766 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 3 ≤ (𝑥 / 𝑑)) → ∀𝑚 ∈
(3[,)+∞)(abs‘(𝐹
− 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚))) |
143 | 45 | oveq1d 6665 |
. . . . . . . . . . . 12
⊢ (𝑚 = (𝑥 / 𝑑) → (𝐹 − 𝑇) = (𝐾 − 𝑇)) |
144 | 143 | fveq2d 6195 |
. . . . . . . . . . 11
⊢ (𝑚 = (𝑥 / 𝑑) → (abs‘(𝐹 − 𝑇)) = (abs‘(𝐾 − 𝑇))) |
145 | | fveq2 6191 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝑥 / 𝑑) → (log‘𝑚) = (log‘(𝑥 / 𝑑))) |
146 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝑥 / 𝑑) → 𝑚 = (𝑥 / 𝑑)) |
147 | 145, 146 | oveq12d 6668 |
. . . . . . . . . . . 12
⊢ (𝑚 = (𝑥 / 𝑑) → ((log‘𝑚) / 𝑚) = ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) |
148 | 147 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ (𝑚 = (𝑥 / 𝑑) → (𝐶 · ((log‘𝑚) / 𝑚)) = (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)))) |
149 | 144, 148 | breq12d 4666 |
. . . . . . . . . 10
⊢ (𝑚 = (𝑥 / 𝑑) → ((abs‘(𝐹 − 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)) ↔ (abs‘(𝐾 − 𝑇)) ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))))) |
150 | 149 | rspcv 3305 |
. . . . . . . . 9
⊢ ((𝑥 / 𝑑) ∈ (3[,)+∞) → (∀𝑚 ∈
(3[,)+∞)(abs‘(𝐹
− 𝑇)) ≤ (𝐶 · ((log‘𝑚) / 𝑚)) → (abs‘(𝐾 − 𝑇)) ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))))) |
151 | 139, 142,
150 | sylc 65 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(𝐾 − 𝑇)) ≤ (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)))) |
152 | 14 | recnd 10068 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (log‘(𝑥 /
𝑑)) ∈
ℂ) |
153 | | rpcnne0 11850 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) |
154 | 153 | ad2antlr 763 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) |
155 | 57 | rpcnne0d 11881 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑑 ∈ ℂ
∧ 𝑑 ≠
0)) |
156 | | divdiv2 10737 |
. . . . . . . . . . . . 13
⊢
(((log‘(𝑥 /
𝑑)) ∈ ℂ ∧
(𝑥 ∈ ℂ ∧
𝑥 ≠ 0) ∧ (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0)) →
((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑)) = (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥)) |
157 | 152, 154,
155, 156 | syl3anc 1326 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((log‘(𝑥 /
𝑑)) / (𝑥 / 𝑑)) = (((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥)) |
158 | | div23 10704 |
. . . . . . . . . . . . 13
⊢
(((log‘(𝑥 /
𝑑)) ∈ ℂ ∧
𝑑 ∈ ℂ ∧
(𝑥 ∈ ℂ ∧
𝑥 ≠ 0)) →
(((log‘(𝑥 / 𝑑)) · 𝑑) / 𝑥) = (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑)) |
159 | 152, 107,
154, 158 | syl3anc 1326 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((log‘(𝑥 /
𝑑)) · 𝑑) / 𝑥) = (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑)) |
160 | 157, 159 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((log‘(𝑥 /
𝑑)) / (𝑥 / 𝑑)) = (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑)) |
161 | 160 | oveq2d 6666 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 ·
((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) = (𝐶 · (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑))) |
162 | 30 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝐶 ∈
ℂ) |
163 | 16 | recnd 10068 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((log‘(𝑥 /
𝑑)) / 𝑥) ∈ ℂ) |
164 | 162, 163,
107 | mulassd 10063 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝐶 ·
((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑) = (𝐶 · (((log‘(𝑥 / 𝑑)) / 𝑥) · 𝑑))) |
165 | 161, 164 | eqtr4d 2659 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 ·
((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) = ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑)) |
166 | 165 | adantr 481 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 3 ≤ (𝑥 / 𝑑)) → (𝐶 · ((log‘(𝑥 / 𝑑)) / (𝑥 / 𝑑))) = ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑)) |
167 | 151, 166 | breqtrd 4679 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 3 ≤ (𝑥 / 𝑑)) → (abs‘(𝐾 − 𝑇)) ≤ ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑)) |
168 | 132, 167 | eqbrtrd 4675 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 3 ≤ (𝑥 / 𝑑)) →
(abs‘(abs‘(𝐾
− 𝑇))) ≤ ((𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) · 𝑑)) |
169 | 131 | adantr 481 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ (𝑥 / 𝑑) < 3) →
(abs‘(abs‘(𝐾
− 𝑇))) =
(abs‘(𝐾 − 𝑇))) |
170 | 134 | adantr 481 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) ∈ ℝ) |
171 | 118 | adantr 481 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ (𝑥 / 𝑑) < 3) → 1 ≤ (𝑥 / 𝑑)) |
172 | | simpr 477 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) < 3) |
173 | | elico2 12237 |
. . . . . . . . . 10
⊢ ((1
∈ ℝ ∧ 3 ∈ ℝ*) → ((𝑥 / 𝑑) ∈ (1[,)3) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑) ∧ (𝑥 / 𝑑) < 3))) |
174 | 24, 71, 173 | mp2an 708 |
. . . . . . . . 9
⊢ ((𝑥 / 𝑑) ∈ (1[,)3) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑) ∧ (𝑥 / 𝑑) < 3)) |
175 | 170, 171,
172, 174 | syl3anbrc 1246 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ (𝑥 / 𝑑) < 3) → (𝑥 / 𝑑) ∈ (1[,)3)) |
176 | 93 | ad3antrrr 766 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ (𝑥 / 𝑑) < 3) → ∀𝑚 ∈ (1[,)3)(abs‘(𝐹 − 𝑇)) ≤ 𝑅) |
177 | 144 | breq1d 4663 |
. . . . . . . . 9
⊢ (𝑚 = (𝑥 / 𝑑) → ((abs‘(𝐹 − 𝑇)) ≤ 𝑅 ↔ (abs‘(𝐾 − 𝑇)) ≤ 𝑅)) |
178 | 177 | rspcv 3305 |
. . . . . . . 8
⊢ ((𝑥 / 𝑑) ∈ (1[,)3) → (∀𝑚 ∈ (1[,)3)(abs‘(𝐹 − 𝑇)) ≤ 𝑅 → (abs‘(𝐾 − 𝑇)) ≤ 𝑅)) |
179 | 175, 176,
178 | sylc 65 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ (𝑥 / 𝑑) < 3) →
(abs‘(𝐾 − 𝑇)) ≤ 𝑅) |
180 | 169, 179 | eqbrtrd 4675 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ (𝑥 / 𝑑) < 3) →
(abs‘(abs‘(𝐾
− 𝑇))) ≤ 𝑅) |
181 | 8, 68, 101, 102, 104, 129, 168, 180 | fsumharmonic 24738 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(abs‘Σ𝑑 ∈
(1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ≤ (Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) +
1)))) |
182 | 30 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈
ℂ) |
183 | 7, 182, 163 | fsummulc2 14516 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐶 · Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥))) |
184 | 183 | oveq1d 6665 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((𝐶 · Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))) =
(Σ𝑑 ∈
(1...(⌊‘𝑥))(𝐶 · ((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) +
1)))) |
185 | 181, 184 | breqtrrd 4681 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(abs‘Σ𝑑 ∈
(1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ≤ ((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) +
1)))) |
186 | 41 | leabsd 14153 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((𝐶 · Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) + 1))) ≤
(abs‘((𝐶 ·
Σ𝑑 ∈
(1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) +
1))))) |
187 | 62, 41, 64, 185, 186 | letrd 10194 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(abs‘Σ𝑑 ∈
(1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ≤ (abs‘((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) +
1))))) |
188 | 187 | adantrr 753 |
. 2
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘Σ𝑑 ∈
(1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ≤ (abs‘((𝐶 · Σ𝑑 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑑)) / 𝑥)) + (𝑅 · ((log‘3) +
1))))) |
189 | 1, 40, 41, 56, 188 | o1le 14383 |
1
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
Σ𝑑 ∈
(1...(⌊‘𝑥))((abs‘(𝐾 − 𝑇)) / 𝑑)) ∈ 𝑂(1)) |