| Step |
Hyp |
Ref |
Expression |
| 1 |
|
logdivsum.1 |
⊢ 𝐹 = ( 𝑦 ∈ ℝ+ ↦ ( Σ 𝑖 ∈ ( 1 ... ( ⌊ ‘ 𝑦 ) ) ( ( log ‘ 𝑖 ) / 𝑖 ) − ( ( ( log ‘ 𝑦 ) ↑ 2 ) / 2 ) ) ) |
| 2 |
|
mulog2sumlem.1 |
⊢ ( 𝜑 → 𝐹 ⇝𝑟 𝐿 ) |
| 3 |
|
mulog2sumlem2.t |
⊢ 𝑇 = ( ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) + ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) |
| 4 |
|
mulog2sumlem2.r |
⊢ 𝑅 = ( ( ( 1 / 2 ) + ( γ + ( abs ‘ 𝐿 ) ) ) + Σ 𝑚 ∈ ( 1 ... 2 ) ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) |
| 5 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
| 6 |
|
2re |
⊢ 2 ∈ ℝ |
| 7 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∈ Fin ) |
| 8 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℝ+ ) |
| 9 |
|
elfznn |
⊢ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) → 𝑛 ∈ ℕ ) |
| 10 |
9
|
nnrpd |
⊢ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) → 𝑛 ∈ ℝ+ ) |
| 11 |
|
rpdivcl |
⊢ ( ( 𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) → ( 𝑥 / 𝑛 ) ∈ ℝ+ ) |
| 12 |
8 10 11
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 𝑥 / 𝑛 ) ∈ ℝ+ ) |
| 13 |
12
|
relogcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( log ‘ ( 𝑥 / 𝑛 ) ) ∈ ℝ ) |
| 14 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑥 ∈ ℝ+ ) |
| 15 |
13 14
|
rerpdivcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ∈ ℝ ) |
| 16 |
7 15
|
fsumrecl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ∈ ℝ ) |
| 17 |
|
remulcl |
⊢ ( ( 2 ∈ ℝ ∧ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ∈ ℝ ) → ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) ∈ ℝ ) |
| 18 |
6 16 17
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) ∈ ℝ ) |
| 19 |
|
halfre |
⊢ ( 1 / 2 ) ∈ ℝ |
| 20 |
|
emre |
⊢ γ ∈ ℝ |
| 21 |
|
rlimcl |
⊢ ( 𝐹 ⇝𝑟 𝐿 → 𝐿 ∈ ℂ ) |
| 22 |
2 21
|
syl |
⊢ ( 𝜑 → 𝐿 ∈ ℂ ) |
| 23 |
22
|
abscld |
⊢ ( 𝜑 → ( abs ‘ 𝐿 ) ∈ ℝ ) |
| 24 |
|
readdcl |
⊢ ( ( γ ∈ ℝ ∧ ( abs ‘ 𝐿 ) ∈ ℝ ) → ( γ + ( abs ‘ 𝐿 ) ) ∈ ℝ ) |
| 25 |
20 23 24
|
sylancr |
⊢ ( 𝜑 → ( γ + ( abs ‘ 𝐿 ) ) ∈ ℝ ) |
| 26 |
|
readdcl |
⊢ ( ( ( 1 / 2 ) ∈ ℝ ∧ ( γ + ( abs ‘ 𝐿 ) ) ∈ ℝ ) → ( ( 1 / 2 ) + ( γ + ( abs ‘ 𝐿 ) ) ) ∈ ℝ ) |
| 27 |
19 25 26
|
sylancr |
⊢ ( 𝜑 → ( ( 1 / 2 ) + ( γ + ( abs ‘ 𝐿 ) ) ) ∈ ℝ ) |
| 28 |
|
fzfid |
⊢ ( 𝜑 → ( 1 ... 2 ) ∈ Fin ) |
| 29 |
|
epr |
⊢ e ∈ ℝ+ |
| 30 |
|
elfznn |
⊢ ( 𝑚 ∈ ( 1 ... 2 ) → 𝑚 ∈ ℕ ) |
| 31 |
30
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → 𝑚 ∈ ℕ ) |
| 32 |
31
|
nnrpd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → 𝑚 ∈ ℝ+ ) |
| 33 |
|
rpdivcl |
⊢ ( ( e ∈ ℝ+ ∧ 𝑚 ∈ ℝ+ ) → ( e / 𝑚 ) ∈ ℝ+ ) |
| 34 |
29 32 33
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( e / 𝑚 ) ∈ ℝ+ ) |
| 35 |
34
|
relogcld |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( log ‘ ( e / 𝑚 ) ) ∈ ℝ ) |
| 36 |
35 31
|
nndivred |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ∈ ℝ ) |
| 37 |
28 36
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑚 ∈ ( 1 ... 2 ) ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ∈ ℝ ) |
| 38 |
27 37
|
readdcld |
⊢ ( 𝜑 → ( ( ( 1 / 2 ) + ( γ + ( abs ‘ 𝐿 ) ) ) + Σ 𝑚 ∈ ( 1 ... 2 ) ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) ∈ ℝ ) |
| 39 |
4 38
|
eqeltrid |
⊢ ( 𝜑 → 𝑅 ∈ ℝ ) |
| 40 |
|
remulcl |
⊢ ( ( 𝑅 ∈ ℝ ∧ 2 ∈ ℝ ) → ( 𝑅 · 2 ) ∈ ℝ ) |
| 41 |
39 6 40
|
sylancl |
⊢ ( 𝜑 → ( 𝑅 · 2 ) ∈ ℝ ) |
| 42 |
41
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 𝑅 · 2 ) ∈ ℝ ) |
| 43 |
6
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 2 ∈ ℝ ) |
| 44 |
|
rpssre |
⊢ ℝ+ ⊆ ℝ |
| 45 |
|
2cnd |
⊢ ( 𝜑 → 2 ∈ ℂ ) |
| 46 |
|
o1const |
⊢ ( ( ℝ+ ⊆ ℝ ∧ 2 ∈ ℂ ) → ( 𝑥 ∈ ℝ+ ↦ 2 ) ∈ 𝑂(1) ) |
| 47 |
44 45 46
|
sylancr |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ 2 ) ∈ 𝑂(1) ) |
| 48 |
|
logfacrlim2 |
⊢ ( 𝑥 ∈ ℝ+ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) ⇝𝑟 1 |
| 49 |
|
rlimo1 |
⊢ ( ( 𝑥 ∈ ℝ+ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) ⇝𝑟 1 → ( 𝑥 ∈ ℝ+ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) ∈ 𝑂(1) ) |
| 50 |
48 49
|
mp1i |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) ∈ 𝑂(1) ) |
| 51 |
43 16 47 50
|
o1mul2 |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) ) ∈ 𝑂(1) ) |
| 52 |
41
|
recnd |
⊢ ( 𝜑 → ( 𝑅 · 2 ) ∈ ℂ ) |
| 53 |
|
o1const |
⊢ ( ( ℝ+ ⊆ ℝ ∧ ( 𝑅 · 2 ) ∈ ℂ ) → ( 𝑥 ∈ ℝ+ ↦ ( 𝑅 · 2 ) ) ∈ 𝑂(1) ) |
| 54 |
44 52 53
|
sylancr |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ ( 𝑅 · 2 ) ) ∈ 𝑂(1) ) |
| 55 |
18 42 51 54
|
o1add2 |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ) ∈ 𝑂(1) ) |
| 56 |
18 42
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ∈ ℝ ) |
| 57 |
9
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑛 ∈ ℕ ) |
| 58 |
|
mucl |
⊢ ( 𝑛 ∈ ℕ → ( μ ‘ 𝑛 ) ∈ ℤ ) |
| 59 |
57 58
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( μ ‘ 𝑛 ) ∈ ℤ ) |
| 60 |
59
|
zred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( μ ‘ 𝑛 ) ∈ ℝ ) |
| 61 |
60 57
|
nndivred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( μ ‘ 𝑛 ) / 𝑛 ) ∈ ℝ ) |
| 62 |
61
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( μ ‘ 𝑛 ) / 𝑛 ) ∈ ℂ ) |
| 63 |
13
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( log ‘ ( 𝑥 / 𝑛 ) ) ∈ ℂ ) |
| 64 |
63
|
sqcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) ∈ ℂ ) |
| 65 |
64
|
halfcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ∈ ℂ ) |
| 66 |
|
remulcl |
⊢ ( ( γ ∈ ℝ ∧ ( log ‘ ( 𝑥 / 𝑛 ) ) ∈ ℝ ) → ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ∈ ℝ ) |
| 67 |
20 13 66
|
sylancr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ∈ ℝ ) |
| 68 |
67
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ∈ ℂ ) |
| 69 |
22
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝐿 ∈ ℂ ) |
| 70 |
68 69
|
subcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ∈ ℂ ) |
| 71 |
65 70
|
addcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) + ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ∈ ℂ ) |
| 72 |
3 71
|
eqeltrid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑇 ∈ ℂ ) |
| 73 |
62 72
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) ∈ ℂ ) |
| 74 |
7 73
|
fsumcl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) ∈ ℂ ) |
| 75 |
|
relogcl |
⊢ ( 𝑥 ∈ ℝ+ → ( log ‘ 𝑥 ) ∈ ℝ ) |
| 76 |
75
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( log ‘ 𝑥 ) ∈ ℝ ) |
| 77 |
76
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( log ‘ 𝑥 ) ∈ ℂ ) |
| 78 |
74 77
|
subcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − ( log ‘ 𝑥 ) ) ∈ ℂ ) |
| 79 |
78
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( abs ‘ ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − ( log ‘ 𝑥 ) ) ) ∈ ℝ ) |
| 80 |
79
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( abs ‘ ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − ( log ‘ 𝑥 ) ) ) ∈ ℝ ) |
| 81 |
56
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ∈ ℝ ) |
| 82 |
56
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ∈ ℂ ) |
| 83 |
82
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( abs ‘ ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ) ∈ ℝ ) |
| 84 |
83
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( abs ‘ ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ) ∈ ℝ ) |
| 85 |
59
|
zcnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( μ ‘ 𝑛 ) ∈ ℂ ) |
| 86 |
|
fzfid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ∈ Fin ) |
| 87 |
|
elfznn |
⊢ ( 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) → 𝑚 ∈ ℕ ) |
| 88 |
|
nnrp |
⊢ ( 𝑚 ∈ ℕ → 𝑚 ∈ ℝ+ ) |
| 89 |
|
rpdivcl |
⊢ ( ( ( 𝑥 / 𝑛 ) ∈ ℝ+ ∧ 𝑚 ∈ ℝ+ ) → ( ( 𝑥 / 𝑛 ) / 𝑚 ) ∈ ℝ+ ) |
| 90 |
12 88 89
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑥 / 𝑛 ) / 𝑚 ) ∈ ℝ+ ) |
| 91 |
90
|
relogcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ∈ ℝ ) |
| 92 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → 𝑚 ∈ ℕ ) |
| 93 |
91 92
|
nndivred |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ∈ ℝ ) |
| 94 |
93
|
recnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ∈ ℂ ) |
| 95 |
87 94
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ∈ ℂ ) |
| 96 |
86 95
|
fsumcl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ∈ ℂ ) |
| 97 |
72 96
|
subcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ∈ ℂ ) |
| 98 |
57
|
nncnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑛 ∈ ℂ ) |
| 99 |
57
|
nnne0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑛 ≠ 0 ) |
| 100 |
85 97 98 99
|
div23d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) / 𝑛 ) = ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 101 |
62 72 96
|
subdid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) = ( ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 102 |
100 101
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) / 𝑛 ) = ( ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 103 |
102
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) / 𝑛 ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 104 |
62 96
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ∈ ℂ ) |
| 105 |
7 73 104
|
fsumsub |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) = ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 106 |
103 105
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) / 𝑛 ) = ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 107 |
106
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) / 𝑛 ) = ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 108 |
86 62 95
|
fsummulc2 |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) = Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) |
| 109 |
85
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( μ ‘ 𝑛 ) ∈ ℂ ) |
| 110 |
98 99
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 𝑛 ∈ ℂ ∧ 𝑛 ≠ 0 ) ) |
| 111 |
110
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝑛 ∈ ℂ ∧ 𝑛 ≠ 0 ) ) |
| 112 |
|
div23 |
⊢ ( ( ( μ ‘ 𝑛 ) ∈ ℂ ∧ ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ∈ ℂ ∧ ( 𝑛 ∈ ℂ ∧ 𝑛 ≠ 0 ) ) → ( ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) / 𝑛 ) = ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) |
| 113 |
|
divass |
⊢ ( ( ( μ ‘ 𝑛 ) ∈ ℂ ∧ ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ∈ ℂ ∧ ( 𝑛 ∈ ℂ ∧ 𝑛 ≠ 0 ) ) → ( ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) / 𝑛 ) = ( ( μ ‘ 𝑛 ) · ( ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) / 𝑛 ) ) ) |
| 114 |
112 113
|
eqtr3d |
⊢ ( ( ( μ ‘ 𝑛 ) ∈ ℂ ∧ ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ∈ ℂ ∧ ( 𝑛 ∈ ℂ ∧ 𝑛 ≠ 0 ) ) → ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) = ( ( μ ‘ 𝑛 ) · ( ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) / 𝑛 ) ) ) |
| 115 |
109 94 111 114
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) = ( ( μ ‘ 𝑛 ) · ( ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) / 𝑛 ) ) ) |
| 116 |
91
|
recnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ∈ ℂ ) |
| 117 |
92
|
nnrpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → 𝑚 ∈ ℝ+ ) |
| 118 |
|
rpcnne0 |
⊢ ( 𝑚 ∈ ℝ+ → ( 𝑚 ∈ ℂ ∧ 𝑚 ≠ 0 ) ) |
| 119 |
117 118
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝑚 ∈ ℂ ∧ 𝑚 ≠ 0 ) ) |
| 120 |
|
divdiv1 |
⊢ ( ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ∈ ℂ ∧ ( 𝑚 ∈ ℂ ∧ 𝑚 ≠ 0 ) ∧ ( 𝑛 ∈ ℂ ∧ 𝑛 ≠ 0 ) ) → ( ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) / 𝑛 ) = ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / ( 𝑚 · 𝑛 ) ) ) |
| 121 |
116 119 111 120
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) / 𝑛 ) = ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / ( 𝑚 · 𝑛 ) ) ) |
| 122 |
|
rpre |
⊢ ( 𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ ) |
| 123 |
122
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℝ ) |
| 124 |
123
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑥 ∈ ℝ ) |
| 125 |
124
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑥 ∈ ℂ ) |
| 126 |
125
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → 𝑥 ∈ ℂ ) |
| 127 |
|
divdiv1 |
⊢ ( ( 𝑥 ∈ ℂ ∧ ( 𝑛 ∈ ℂ ∧ 𝑛 ≠ 0 ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑚 ≠ 0 ) ) → ( ( 𝑥 / 𝑛 ) / 𝑚 ) = ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) |
| 128 |
126 111 119 127
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑥 / 𝑛 ) / 𝑚 ) = ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) |
| 129 |
128
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) = ( log ‘ ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) ) |
| 130 |
92
|
nncnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → 𝑚 ∈ ℂ ) |
| 131 |
98
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → 𝑛 ∈ ℂ ) |
| 132 |
130 131
|
mulcomd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝑚 · 𝑛 ) = ( 𝑛 · 𝑚 ) ) |
| 133 |
129 132
|
oveq12d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / ( 𝑚 · 𝑛 ) ) = ( ( log ‘ ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) / ( 𝑛 · 𝑚 ) ) ) |
| 134 |
121 133
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) / 𝑛 ) = ( ( log ‘ ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) / ( 𝑛 · 𝑚 ) ) ) |
| 135 |
134
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( ( μ ‘ 𝑛 ) · ( ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) / 𝑛 ) ) = ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) / ( 𝑛 · 𝑚 ) ) ) ) |
| 136 |
115 135
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) = ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) / ( 𝑛 · 𝑚 ) ) ) ) |
| 137 |
87 136
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) = ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) / ( 𝑛 · 𝑚 ) ) ) ) |
| 138 |
137
|
sumeq2dv |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) = Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) / ( 𝑛 · 𝑚 ) ) ) ) |
| 139 |
108 138
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) = Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) / ( 𝑛 · 𝑚 ) ) ) ) |
| 140 |
139
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) / ( 𝑛 · 𝑚 ) ) ) ) |
| 141 |
|
oveq2 |
⊢ ( 𝑘 = ( 𝑛 · 𝑚 ) → ( 𝑥 / 𝑘 ) = ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) |
| 142 |
141
|
fveq2d |
⊢ ( 𝑘 = ( 𝑛 · 𝑚 ) → ( log ‘ ( 𝑥 / 𝑘 ) ) = ( log ‘ ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) ) |
| 143 |
|
id |
⊢ ( 𝑘 = ( 𝑛 · 𝑚 ) → 𝑘 = ( 𝑛 · 𝑚 ) ) |
| 144 |
142 143
|
oveq12d |
⊢ ( 𝑘 = ( 𝑛 · 𝑚 ) → ( ( log ‘ ( 𝑥 / 𝑘 ) ) / 𝑘 ) = ( ( log ‘ ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) / ( 𝑛 · 𝑚 ) ) ) |
| 145 |
144
|
oveq2d |
⊢ ( 𝑘 = ( 𝑛 · 𝑚 ) → ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / 𝑘 ) ) / 𝑘 ) ) = ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) / ( 𝑛 · 𝑚 ) ) ) ) |
| 146 |
8
|
rpred |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℝ ) |
| 147 |
|
ssrab2 |
⊢ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ⊆ ℕ |
| 148 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∧ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ) ) → 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ) |
| 149 |
147 148
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∧ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ) ) → 𝑛 ∈ ℕ ) |
| 150 |
149 58
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∧ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ) ) → ( μ ‘ 𝑛 ) ∈ ℤ ) |
| 151 |
150
|
zred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∧ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ) ) → ( μ ‘ 𝑛 ) ∈ ℝ ) |
| 152 |
|
elfznn |
⊢ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) → 𝑘 ∈ ℕ ) |
| 153 |
152
|
adantr |
⊢ ( ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∧ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ) → 𝑘 ∈ ℕ ) |
| 154 |
153
|
nnrpd |
⊢ ( ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∧ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ) → 𝑘 ∈ ℝ+ ) |
| 155 |
|
rpdivcl |
⊢ ( ( 𝑥 ∈ ℝ+ ∧ 𝑘 ∈ ℝ+ ) → ( 𝑥 / 𝑘 ) ∈ ℝ+ ) |
| 156 |
8 154 155
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∧ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ) ) → ( 𝑥 / 𝑘 ) ∈ ℝ+ ) |
| 157 |
156
|
relogcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∧ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ) ) → ( log ‘ ( 𝑥 / 𝑘 ) ) ∈ ℝ ) |
| 158 |
152
|
ad2antrl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∧ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ) ) → 𝑘 ∈ ℕ ) |
| 159 |
157 158
|
nndivred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∧ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ) ) → ( ( log ‘ ( 𝑥 / 𝑘 ) ) / 𝑘 ) ∈ ℝ ) |
| 160 |
151 159
|
remulcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∧ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ) ) → ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / 𝑘 ) ) / 𝑘 ) ) ∈ ℝ ) |
| 161 |
160
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∧ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ) ) → ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / 𝑘 ) ) / 𝑘 ) ) ∈ ℂ ) |
| 162 |
145 146 161
|
dvdsflsumcom |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) Σ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / 𝑘 ) ) / 𝑘 ) ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / ( 𝑛 · 𝑚 ) ) ) / ( 𝑛 · 𝑚 ) ) ) ) |
| 163 |
140 162
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) = Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) Σ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / 𝑘 ) ) / 𝑘 ) ) ) |
| 164 |
163
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) = Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) Σ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / 𝑘 ) ) / 𝑘 ) ) ) |
| 165 |
|
oveq2 |
⊢ ( 𝑘 = 1 → ( 𝑥 / 𝑘 ) = ( 𝑥 / 1 ) ) |
| 166 |
165
|
fveq2d |
⊢ ( 𝑘 = 1 → ( log ‘ ( 𝑥 / 𝑘 ) ) = ( log ‘ ( 𝑥 / 1 ) ) ) |
| 167 |
|
id |
⊢ ( 𝑘 = 1 → 𝑘 = 1 ) |
| 168 |
166 167
|
oveq12d |
⊢ ( 𝑘 = 1 → ( ( log ‘ ( 𝑥 / 𝑘 ) ) / 𝑘 ) = ( ( log ‘ ( 𝑥 / 1 ) ) / 1 ) ) |
| 169 |
|
fzfid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( 1 ... ( ⌊ ‘ 𝑥 ) ) ∈ Fin ) |
| 170 |
|
fz1ssnn |
⊢ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ⊆ ℕ |
| 171 |
170
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( 1 ... ( ⌊ ‘ 𝑥 ) ) ⊆ ℕ ) |
| 172 |
123
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → 𝑥 ∈ ℝ ) |
| 173 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → 1 ≤ 𝑥 ) |
| 174 |
|
flge1nn |
⊢ ( ( 𝑥 ∈ ℝ ∧ 1 ≤ 𝑥 ) → ( ⌊ ‘ 𝑥 ) ∈ ℕ ) |
| 175 |
172 173 174
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( ⌊ ‘ 𝑥 ) ∈ ℕ ) |
| 176 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
| 177 |
175 176
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( ⌊ ‘ 𝑥 ) ∈ ( ℤ≥ ‘ 1 ) ) |
| 178 |
|
eluzfz1 |
⊢ ( ( ⌊ ‘ 𝑥 ) ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) |
| 179 |
177 178
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → 1 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) |
| 180 |
152
|
nnrpd |
⊢ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) → 𝑘 ∈ ℝ+ ) |
| 181 |
8 180 155
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 𝑥 / 𝑘 ) ∈ ℝ+ ) |
| 182 |
181
|
relogcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( log ‘ ( 𝑥 / 𝑘 ) ) ∈ ℝ ) |
| 183 |
170
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 1 ... ( ⌊ ‘ 𝑥 ) ) ⊆ ℕ ) |
| 184 |
183
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑘 ∈ ℕ ) |
| 185 |
182 184
|
nndivred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( log ‘ ( 𝑥 / 𝑘 ) ) / 𝑘 ) ∈ ℝ ) |
| 186 |
185
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( log ‘ ( 𝑥 / 𝑘 ) ) / 𝑘 ) ∈ ℂ ) |
| 187 |
186
|
adantlrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( log ‘ ( 𝑥 / 𝑘 ) ) / 𝑘 ) ∈ ℂ ) |
| 188 |
168 169 171 179 187
|
musumsum |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) Σ 𝑛 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘 } ( ( μ ‘ 𝑛 ) · ( ( log ‘ ( 𝑥 / 𝑘 ) ) / 𝑘 ) ) = ( ( log ‘ ( 𝑥 / 1 ) ) / 1 ) ) |
| 189 |
8
|
rpcnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℂ ) |
| 190 |
189
|
div1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 𝑥 / 1 ) = 𝑥 ) |
| 191 |
190
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( log ‘ ( 𝑥 / 1 ) ) = ( log ‘ 𝑥 ) ) |
| 192 |
191
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( log ‘ ( 𝑥 / 1 ) ) / 1 ) = ( ( log ‘ 𝑥 ) / 1 ) ) |
| 193 |
77
|
div1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( log ‘ 𝑥 ) / 1 ) = ( log ‘ 𝑥 ) ) |
| 194 |
192 193
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( log ‘ ( 𝑥 / 1 ) ) / 1 ) = ( log ‘ 𝑥 ) ) |
| 195 |
194
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( ( log ‘ ( 𝑥 / 1 ) ) / 1 ) = ( log ‘ 𝑥 ) ) |
| 196 |
164 188 195
|
3eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) = ( log ‘ 𝑥 ) ) |
| 197 |
196
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) = ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − ( log ‘ 𝑥 ) ) ) |
| 198 |
107 197
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) / 𝑛 ) = ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − ( log ‘ 𝑥 ) ) ) |
| 199 |
198
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( abs ‘ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) / 𝑛 ) ) = ( abs ‘ ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − ( log ‘ 𝑥 ) ) ) ) |
| 200 |
|
ere |
⊢ e ∈ ℝ |
| 201 |
200
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → e ∈ ℝ ) |
| 202 |
|
1re |
⊢ 1 ∈ ℝ |
| 203 |
|
1lt2 |
⊢ 1 < 2 |
| 204 |
|
egt2lt3 |
⊢ ( 2 < e ∧ e < 3 ) |
| 205 |
204
|
simpli |
⊢ 2 < e |
| 206 |
202 6 200
|
lttri |
⊢ ( ( 1 < 2 ∧ 2 < e ) → 1 < e ) |
| 207 |
203 205 206
|
mp2an |
⊢ 1 < e |
| 208 |
202 200 207
|
ltleii |
⊢ 1 ≤ e |
| 209 |
201 208
|
jctir |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( e ∈ ℝ ∧ 1 ≤ e ) ) |
| 210 |
39
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑅 ∈ ℝ ) |
| 211 |
19
|
a1i |
⊢ ( 𝜑 → ( 1 / 2 ) ∈ ℝ ) |
| 212 |
|
1rp |
⊢ 1 ∈ ℝ+ |
| 213 |
|
rphalfcl |
⊢ ( 1 ∈ ℝ+ → ( 1 / 2 ) ∈ ℝ+ ) |
| 214 |
212 213
|
ax-mp |
⊢ ( 1 / 2 ) ∈ ℝ+ |
| 215 |
|
rpge0 |
⊢ ( ( 1 / 2 ) ∈ ℝ+ → 0 ≤ ( 1 / 2 ) ) |
| 216 |
214 215
|
mp1i |
⊢ ( 𝜑 → 0 ≤ ( 1 / 2 ) ) |
| 217 |
20
|
a1i |
⊢ ( 𝜑 → γ ∈ ℝ ) |
| 218 |
|
0re |
⊢ 0 ∈ ℝ |
| 219 |
|
emgt0 |
⊢ 0 < γ |
| 220 |
218 20 219
|
ltleii |
⊢ 0 ≤ γ |
| 221 |
220
|
a1i |
⊢ ( 𝜑 → 0 ≤ γ ) |
| 222 |
22
|
absge0d |
⊢ ( 𝜑 → 0 ≤ ( abs ‘ 𝐿 ) ) |
| 223 |
217 23 221 222
|
addge0d |
⊢ ( 𝜑 → 0 ≤ ( γ + ( abs ‘ 𝐿 ) ) ) |
| 224 |
211 25 216 223
|
addge0d |
⊢ ( 𝜑 → 0 ≤ ( ( 1 / 2 ) + ( γ + ( abs ‘ 𝐿 ) ) ) ) |
| 225 |
|
log1 |
⊢ ( log ‘ 1 ) = 0 |
| 226 |
31
|
nncnd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → 𝑚 ∈ ℂ ) |
| 227 |
226
|
mulid2d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( 1 · 𝑚 ) = 𝑚 ) |
| 228 |
32
|
rpred |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → 𝑚 ∈ ℝ ) |
| 229 |
6
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → 2 ∈ ℝ ) |
| 230 |
200
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → e ∈ ℝ ) |
| 231 |
|
elfzle2 |
⊢ ( 𝑚 ∈ ( 1 ... 2 ) → 𝑚 ≤ 2 ) |
| 232 |
231
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → 𝑚 ≤ 2 ) |
| 233 |
6 200 205
|
ltleii |
⊢ 2 ≤ e |
| 234 |
233
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → 2 ≤ e ) |
| 235 |
228 229 230 232 234
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → 𝑚 ≤ e ) |
| 236 |
227 235
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( 1 · 𝑚 ) ≤ e ) |
| 237 |
|
1red |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → 1 ∈ ℝ ) |
| 238 |
237 230 32
|
lemuldivd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( ( 1 · 𝑚 ) ≤ e ↔ 1 ≤ ( e / 𝑚 ) ) ) |
| 239 |
236 238
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → 1 ≤ ( e / 𝑚 ) ) |
| 240 |
|
logleb |
⊢ ( ( 1 ∈ ℝ+ ∧ ( e / 𝑚 ) ∈ ℝ+ ) → ( 1 ≤ ( e / 𝑚 ) ↔ ( log ‘ 1 ) ≤ ( log ‘ ( e / 𝑚 ) ) ) ) |
| 241 |
212 34 240
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( 1 ≤ ( e / 𝑚 ) ↔ ( log ‘ 1 ) ≤ ( log ‘ ( e / 𝑚 ) ) ) ) |
| 242 |
239 241
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( log ‘ 1 ) ≤ ( log ‘ ( e / 𝑚 ) ) ) |
| 243 |
225 242
|
eqbrtrrid |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → 0 ≤ ( log ‘ ( e / 𝑚 ) ) ) |
| 244 |
35 32 243
|
divge0d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → 0 ≤ ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) |
| 245 |
28 36 244
|
fsumge0 |
⊢ ( 𝜑 → 0 ≤ Σ 𝑚 ∈ ( 1 ... 2 ) ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) |
| 246 |
27 37 224 245
|
addge0d |
⊢ ( 𝜑 → 0 ≤ ( ( ( 1 / 2 ) + ( γ + ( abs ‘ 𝐿 ) ) ) + Σ 𝑚 ∈ ( 1 ... 2 ) ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) ) |
| 247 |
246 4
|
breqtrrdi |
⊢ ( 𝜑 → 0 ≤ 𝑅 ) |
| 248 |
247
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 0 ≤ 𝑅 ) |
| 249 |
210 248
|
jca |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ) ) |
| 250 |
85 97
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ∈ ℂ ) |
| 251 |
|
remulcl |
⊢ ( ( 2 ∈ ℝ ∧ ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ∈ ℝ ) → ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) ∈ ℝ ) |
| 252 |
6 15 251
|
sylancr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) ∈ ℝ ) |
| 253 |
6
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 2 ∈ ℝ ) |
| 254 |
|
0le2 |
⊢ 0 ≤ 2 |
| 255 |
254
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 0 ≤ 2 ) |
| 256 |
98
|
mulid2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 1 · 𝑛 ) = 𝑛 ) |
| 257 |
|
fznnfl |
⊢ ( 𝑥 ∈ ℝ → ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ↔ ( 𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑥 ) ) ) |
| 258 |
123 257
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ↔ ( 𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑥 ) ) ) |
| 259 |
258
|
simplbda |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑛 ≤ 𝑥 ) |
| 260 |
256 259
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 1 · 𝑛 ) ≤ 𝑥 ) |
| 261 |
|
1red |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 1 ∈ ℝ ) |
| 262 |
57
|
nnrpd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑛 ∈ ℝ+ ) |
| 263 |
261 124 262
|
lemuldivd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( 1 · 𝑛 ) ≤ 𝑥 ↔ 1 ≤ ( 𝑥 / 𝑛 ) ) ) |
| 264 |
260 263
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 1 ≤ ( 𝑥 / 𝑛 ) ) |
| 265 |
|
logleb |
⊢ ( ( 1 ∈ ℝ+ ∧ ( 𝑥 / 𝑛 ) ∈ ℝ+ ) → ( 1 ≤ ( 𝑥 / 𝑛 ) ↔ ( log ‘ 1 ) ≤ ( log ‘ ( 𝑥 / 𝑛 ) ) ) ) |
| 266 |
212 12 265
|
sylancr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 1 ≤ ( 𝑥 / 𝑛 ) ↔ ( log ‘ 1 ) ≤ ( log ‘ ( 𝑥 / 𝑛 ) ) ) ) |
| 267 |
264 266
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( log ‘ 1 ) ≤ ( log ‘ ( 𝑥 / 𝑛 ) ) ) |
| 268 |
225 267
|
eqbrtrrid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 0 ≤ ( log ‘ ( 𝑥 / 𝑛 ) ) ) |
| 269 |
|
rpregt0 |
⊢ ( 𝑥 ∈ ℝ+ → ( 𝑥 ∈ ℝ ∧ 0 < 𝑥 ) ) |
| 270 |
269
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 𝑥 ∈ ℝ ∧ 0 < 𝑥 ) ) |
| 271 |
|
divge0 |
⊢ ( ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ∈ ℝ ∧ 0 ≤ ( log ‘ ( 𝑥 / 𝑛 ) ) ) ∧ ( 𝑥 ∈ ℝ ∧ 0 < 𝑥 ) ) → 0 ≤ ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) |
| 272 |
13 268 270 271
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 0 ≤ ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) |
| 273 |
253 15 255 272
|
mulge0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 0 ≤ ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) ) |
| 274 |
250
|
abscld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( abs ‘ ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) ∈ ℝ ) |
| 275 |
274
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → ( abs ‘ ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) ∈ ℝ ) |
| 276 |
97
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ∈ ℂ ) |
| 277 |
276
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ∈ ℝ ) |
| 278 |
262
|
rpred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 𝑛 ∈ ℝ ) |
| 279 |
252 278
|
remulcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) · 𝑛 ) ∈ ℝ ) |
| 280 |
279
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → ( ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) · 𝑛 ) ∈ ℝ ) |
| 281 |
85 97
|
absmuld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( abs ‘ ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) = ( ( abs ‘ ( μ ‘ 𝑛 ) ) · ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) ) |
| 282 |
85
|
abscld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( abs ‘ ( μ ‘ 𝑛 ) ) ∈ ℝ ) |
| 283 |
97
|
abscld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ∈ ℝ ) |
| 284 |
97
|
absge0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 0 ≤ ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 285 |
|
mule1 |
⊢ ( 𝑛 ∈ ℕ → ( abs ‘ ( μ ‘ 𝑛 ) ) ≤ 1 ) |
| 286 |
57 285
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( abs ‘ ( μ ‘ 𝑛 ) ) ≤ 1 ) |
| 287 |
282 261 283 284 286
|
lemul1ad |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( abs ‘ ( μ ‘ 𝑛 ) ) · ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) ≤ ( 1 · ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) ) |
| 288 |
283
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ∈ ℂ ) |
| 289 |
288
|
mulid2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 1 · ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) = ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 290 |
287 289
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( abs ‘ ( μ ‘ 𝑛 ) ) · ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) ≤ ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 291 |
281 290
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( abs ‘ ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) ≤ ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 292 |
291
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → ( abs ‘ ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) ≤ ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 293 |
2
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → 𝐹 ⇝𝑟 𝐿 ) |
| 294 |
12
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → ( 𝑥 / 𝑛 ) ∈ ℝ+ ) |
| 295 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → e ≤ ( 𝑥 / 𝑛 ) ) |
| 296 |
1 293 294 295
|
mulog2sumlem1 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → ( abs ‘ ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) − ( ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) + ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ) ) ≤ ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / ( 𝑥 / 𝑛 ) ) ) ) |
| 297 |
72 96
|
abssubd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) = ( abs ‘ ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) − 𝑇 ) ) ) |
| 298 |
297
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) = ( abs ‘ ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) − 𝑇 ) ) ) |
| 299 |
3
|
oveq2i |
⊢ ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) − 𝑇 ) = ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) − ( ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) + ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ) |
| 300 |
299
|
fveq2i |
⊢ ( abs ‘ ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) − 𝑇 ) ) = ( abs ‘ ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) − ( ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) + ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ) ) |
| 301 |
298 300
|
eqtrdi |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) = ( abs ‘ ( Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) − ( ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) + ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ) ) ) |
| 302 |
|
2cnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 2 ∈ ℂ ) |
| 303 |
15
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ∈ ℂ ) |
| 304 |
302 303 98
|
mulassd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) · 𝑛 ) = ( 2 · ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) · 𝑛 ) ) ) |
| 305 |
|
rpcnne0 |
⊢ ( 𝑥 ∈ ℝ+ → ( 𝑥 ∈ ℂ ∧ 𝑥 ≠ 0 ) ) |
| 306 |
305
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 𝑥 ∈ ℂ ∧ 𝑥 ≠ 0 ) ) |
| 307 |
|
divdiv2 |
⊢ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ∈ ℂ ∧ ( 𝑥 ∈ ℂ ∧ 𝑥 ≠ 0 ) ∧ ( 𝑛 ∈ ℂ ∧ 𝑛 ≠ 0 ) ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) / ( 𝑥 / 𝑛 ) ) = ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) · 𝑛 ) / 𝑥 ) ) |
| 308 |
63 306 110 307
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) / ( 𝑥 / 𝑛 ) ) = ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) · 𝑛 ) / 𝑥 ) ) |
| 309 |
|
div23 |
⊢ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ ( 𝑥 ∈ ℂ ∧ 𝑥 ≠ 0 ) ) → ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) · 𝑛 ) / 𝑥 ) = ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) · 𝑛 ) ) |
| 310 |
63 98 306 309
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) · 𝑛 ) / 𝑥 ) = ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) · 𝑛 ) ) |
| 311 |
308 310
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) / ( 𝑥 / 𝑛 ) ) = ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) · 𝑛 ) ) |
| 312 |
311
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / ( 𝑥 / 𝑛 ) ) ) = ( 2 · ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) · 𝑛 ) ) ) |
| 313 |
304 312
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) · 𝑛 ) = ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / ( 𝑥 / 𝑛 ) ) ) ) |
| 314 |
313
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → ( ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) · 𝑛 ) = ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / ( 𝑥 / 𝑛 ) ) ) ) |
| 315 |
296 301 314
|
3brtr4d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ≤ ( ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) · 𝑛 ) ) |
| 316 |
275 277 280 292 315
|
letrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ e ≤ ( 𝑥 / 𝑛 ) ) → ( abs ‘ ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) ≤ ( ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) · 𝑛 ) ) |
| 317 |
274
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) ∈ ℝ ) |
| 318 |
283
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ∈ ℝ ) |
| 319 |
39
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → 𝑅 ∈ ℝ ) |
| 320 |
291
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) ≤ ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 321 |
72
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → 𝑇 ∈ ℂ ) |
| 322 |
321
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ 𝑇 ) ∈ ℝ ) |
| 323 |
96
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ∈ ℂ ) |
| 324 |
323
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ∈ ℝ ) |
| 325 |
322 324
|
readdcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( abs ‘ 𝑇 ) + ( abs ‘ Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ∈ ℝ ) |
| 326 |
321 323
|
abs2dif2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ≤ ( ( abs ‘ 𝑇 ) + ( abs ‘ Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
| 327 |
27
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( 1 / 2 ) + ( γ + ( abs ‘ 𝐿 ) ) ) ∈ ℝ ) |
| 328 |
37
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → Σ 𝑚 ∈ ( 1 ... 2 ) ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ∈ ℝ ) |
| 329 |
3
|
fveq2i |
⊢ ( abs ‘ 𝑇 ) = ( abs ‘ ( ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) + ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ) |
| 330 |
329 322
|
eqeltrrid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) + ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ) ∈ ℝ ) |
| 331 |
65
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ∈ ℂ ) |
| 332 |
331
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ) ∈ ℝ ) |
| 333 |
70
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ∈ ℂ ) |
| 334 |
333
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ∈ ℝ ) |
| 335 |
332 334
|
readdcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( abs ‘ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ) + ( abs ‘ ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ) ∈ ℝ ) |
| 336 |
331 333
|
abstrid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) + ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ) ≤ ( ( abs ‘ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ) + ( abs ‘ ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ) ) |
| 337 |
19
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( 1 / 2 ) ∈ ℝ ) |
| 338 |
25
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( γ + ( abs ‘ 𝐿 ) ) ∈ ℝ ) |
| 339 |
13
|
resqcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) ∈ ℝ ) |
| 340 |
339
|
rehalfcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ∈ ℝ ) |
| 341 |
13
|
sqge0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 0 ≤ ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) ) |
| 342 |
|
2pos |
⊢ 0 < 2 |
| 343 |
6 342
|
pm3.2i |
⊢ ( 2 ∈ ℝ ∧ 0 < 2 ) |
| 344 |
343
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 2 ∈ ℝ ∧ 0 < 2 ) ) |
| 345 |
|
divge0 |
⊢ ( ( ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) ) ∧ ( 2 ∈ ℝ ∧ 0 < 2 ) ) → 0 ≤ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ) |
| 346 |
339 341 344 345
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 0 ≤ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ) |
| 347 |
340 346
|
absidd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( abs ‘ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ) = ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ) |
| 348 |
347
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ) = ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ) |
| 349 |
12
|
rpred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 𝑥 / 𝑛 ) ∈ ℝ ) |
| 350 |
|
ltle |
⊢ ( ( ( 𝑥 / 𝑛 ) ∈ ℝ ∧ e ∈ ℝ ) → ( ( 𝑥 / 𝑛 ) < e → ( 𝑥 / 𝑛 ) ≤ e ) ) |
| 351 |
349 200 350
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( 𝑥 / 𝑛 ) < e → ( 𝑥 / 𝑛 ) ≤ e ) ) |
| 352 |
351
|
imp |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( 𝑥 / 𝑛 ) ≤ e ) |
| 353 |
12
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( 𝑥 / 𝑛 ) ∈ ℝ+ ) |
| 354 |
|
logleb |
⊢ ( ( ( 𝑥 / 𝑛 ) ∈ ℝ+ ∧ e ∈ ℝ+ ) → ( ( 𝑥 / 𝑛 ) ≤ e ↔ ( log ‘ ( 𝑥 / 𝑛 ) ) ≤ ( log ‘ e ) ) ) |
| 355 |
353 29 354
|
sylancl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( 𝑥 / 𝑛 ) ≤ e ↔ ( log ‘ ( 𝑥 / 𝑛 ) ) ≤ ( log ‘ e ) ) ) |
| 356 |
352 355
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( log ‘ ( 𝑥 / 𝑛 ) ) ≤ ( log ‘ e ) ) |
| 357 |
|
loge |
⊢ ( log ‘ e ) = 1 |
| 358 |
356 357
|
breqtrdi |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( log ‘ ( 𝑥 / 𝑛 ) ) ≤ 1 ) |
| 359 |
|
0le1 |
⊢ 0 ≤ 1 |
| 360 |
359
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 0 ≤ 1 ) |
| 361 |
13 261 268 360
|
le2sqd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) ≤ 1 ↔ ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) ≤ ( 1 ↑ 2 ) ) ) |
| 362 |
361
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) ≤ 1 ↔ ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) ≤ ( 1 ↑ 2 ) ) ) |
| 363 |
358 362
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) ≤ ( 1 ↑ 2 ) ) |
| 364 |
|
sq1 |
⊢ ( 1 ↑ 2 ) = 1 |
| 365 |
363 364
|
breqtrdi |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) ≤ 1 ) |
| 366 |
339
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) ∈ ℝ ) |
| 367 |
|
1red |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → 1 ∈ ℝ ) |
| 368 |
343
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( 2 ∈ ℝ ∧ 0 < 2 ) ) |
| 369 |
|
lediv1 |
⊢ ( ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) ∈ ℝ ∧ 1 ∈ ℝ ∧ ( 2 ∈ ℝ ∧ 0 < 2 ) ) → ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) ≤ 1 ↔ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ≤ ( 1 / 2 ) ) ) |
| 370 |
366 367 368 369
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) ≤ 1 ↔ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ≤ ( 1 / 2 ) ) ) |
| 371 |
365 370
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ≤ ( 1 / 2 ) ) |
| 372 |
348 371
|
eqbrtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ) ≤ ( 1 / 2 ) ) |
| 373 |
69
|
abscld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( abs ‘ 𝐿 ) ∈ ℝ ) |
| 374 |
67 373
|
readdcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) + ( abs ‘ 𝐿 ) ) ∈ ℝ ) |
| 375 |
374
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) + ( abs ‘ 𝐿 ) ) ∈ ℝ ) |
| 376 |
68
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ∈ ℂ ) |
| 377 |
22
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → 𝐿 ∈ ℂ ) |
| 378 |
376 377
|
abs2dif2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ≤ ( ( abs ‘ ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ) + ( abs ‘ 𝐿 ) ) ) |
| 379 |
20
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → γ ∈ ℝ ) |
| 380 |
220
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 0 ≤ γ ) |
| 381 |
379 13 380 268
|
mulge0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 0 ≤ ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ) |
| 382 |
67 381
|
absidd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( abs ‘ ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ) = ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ) |
| 383 |
382
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ) = ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ) |
| 384 |
383
|
oveq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( abs ‘ ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ) + ( abs ‘ 𝐿 ) ) = ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) + ( abs ‘ 𝐿 ) ) ) |
| 385 |
378 384
|
breqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ≤ ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) + ( abs ‘ 𝐿 ) ) ) |
| 386 |
67
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ∈ ℝ ) |
| 387 |
20
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → γ ∈ ℝ ) |
| 388 |
377
|
abscld |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ 𝐿 ) ∈ ℝ ) |
| 389 |
13
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( log ‘ ( 𝑥 / 𝑛 ) ) ∈ ℝ ) |
| 390 |
387 219
|
jctir |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( γ ∈ ℝ ∧ 0 < γ ) ) |
| 391 |
|
lemul2 |
⊢ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ∈ ℝ ∧ 1 ∈ ℝ ∧ ( γ ∈ ℝ ∧ 0 < γ ) ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) ≤ 1 ↔ ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ≤ ( γ · 1 ) ) ) |
| 392 |
389 367 390 391
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( log ‘ ( 𝑥 / 𝑛 ) ) ≤ 1 ↔ ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ≤ ( γ · 1 ) ) ) |
| 393 |
358 392
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ≤ ( γ · 1 ) ) |
| 394 |
20
|
recni |
⊢ γ ∈ ℂ |
| 395 |
394
|
mulid1i |
⊢ ( γ · 1 ) = γ |
| 396 |
393 395
|
breqtrdi |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) ≤ γ ) |
| 397 |
386 387 388 396
|
leadd1dd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) + ( abs ‘ 𝐿 ) ) ≤ ( γ + ( abs ‘ 𝐿 ) ) ) |
| 398 |
334 375 338 385 397
|
letrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ≤ ( γ + ( abs ‘ 𝐿 ) ) ) |
| 399 |
332 334 337 338 372 398
|
le2addd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( abs ‘ ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) ) + ( abs ‘ ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ) ≤ ( ( 1 / 2 ) + ( γ + ( abs ‘ 𝐿 ) ) ) ) |
| 400 |
330 335 327 336 399
|
letrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( ( ( ( log ‘ ( 𝑥 / 𝑛 ) ) ↑ 2 ) / 2 ) + ( ( γ · ( log ‘ ( 𝑥 / 𝑛 ) ) ) − 𝐿 ) ) ) ≤ ( ( 1 / 2 ) + ( γ + ( abs ‘ 𝐿 ) ) ) ) |
| 401 |
329 400
|
eqbrtrid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ 𝑇 ) ≤ ( ( 1 / 2 ) + ( γ + ( abs ‘ 𝐿 ) ) ) ) |
| 402 |
87 93
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ∈ ℝ ) |
| 403 |
86 402
|
fsumrecl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ∈ ℝ ) |
| 404 |
403
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ∈ ℝ ) |
| 405 |
87 91
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ∈ ℝ ) |
| 406 |
87 130
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → 𝑚 ∈ ℂ ) |
| 407 |
406
|
mulid2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( 1 · 𝑚 ) = 𝑚 ) |
| 408 |
|
fznnfl |
⊢ ( ( 𝑥 / 𝑛 ) ∈ ℝ → ( 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ↔ ( 𝑚 ∈ ℕ ∧ 𝑚 ≤ ( 𝑥 / 𝑛 ) ) ) ) |
| 409 |
349 408
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ↔ ( 𝑚 ∈ ℕ ∧ 𝑚 ≤ ( 𝑥 / 𝑛 ) ) ) ) |
| 410 |
409
|
simplbda |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → 𝑚 ≤ ( 𝑥 / 𝑛 ) ) |
| 411 |
407 410
|
eqbrtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( 1 · 𝑚 ) ≤ ( 𝑥 / 𝑛 ) ) |
| 412 |
|
1red |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → 1 ∈ ℝ ) |
| 413 |
349
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( 𝑥 / 𝑛 ) ∈ ℝ ) |
| 414 |
117
|
rpregt0d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝑚 ∈ ℝ ∧ 0 < 𝑚 ) ) |
| 415 |
87 414
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( 𝑚 ∈ ℝ ∧ 0 < 𝑚 ) ) |
| 416 |
|
lemuldiv |
⊢ ( ( 1 ∈ ℝ ∧ ( 𝑥 / 𝑛 ) ∈ ℝ ∧ ( 𝑚 ∈ ℝ ∧ 0 < 𝑚 ) ) → ( ( 1 · 𝑚 ) ≤ ( 𝑥 / 𝑛 ) ↔ 1 ≤ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ) |
| 417 |
412 413 415 416
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( ( 1 · 𝑚 ) ≤ ( 𝑥 / 𝑛 ) ↔ 1 ≤ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ) |
| 418 |
411 417
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → 1 ≤ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) |
| 419 |
87 90
|
sylan2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( ( 𝑥 / 𝑛 ) / 𝑚 ) ∈ ℝ+ ) |
| 420 |
|
logleb |
⊢ ( ( 1 ∈ ℝ+ ∧ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ∈ ℝ+ ) → ( 1 ≤ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ↔ ( log ‘ 1 ) ≤ ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ) ) |
| 421 |
212 419 420
|
sylancr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( 1 ≤ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ↔ ( log ‘ 1 ) ≤ ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ) ) |
| 422 |
418 421
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( log ‘ 1 ) ≤ ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ) |
| 423 |
225 422
|
eqbrtrrid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → 0 ≤ ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ) |
| 424 |
|
divge0 |
⊢ ( ( ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ∈ ℝ ∧ 0 ≤ ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ) ∧ ( 𝑚 ∈ ℝ ∧ 0 < 𝑚 ) ) → 0 ≤ ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) |
| 425 |
405 423 415 424
|
syl21anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → 0 ≤ ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) |
| 426 |
86 402 425
|
fsumge0 |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → 0 ≤ Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) |
| 427 |
426
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → 0 ≤ Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) |
| 428 |
404 427
|
absidd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) = Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) |
| 429 |
|
fzfid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ∈ Fin ) |
| 430 |
349
|
flcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) → ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ∈ ℤ ) |
| 431 |
430
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ∈ ℤ ) |
| 432 |
|
2z |
⊢ 2 ∈ ℤ |
| 433 |
432
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → 2 ∈ ℤ ) |
| 434 |
349
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( 𝑥 / 𝑛 ) ∈ ℝ ) |
| 435 |
200
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → e ∈ ℝ ) |
| 436 |
|
3re |
⊢ 3 ∈ ℝ |
| 437 |
436
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → 3 ∈ ℝ ) |
| 438 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( 𝑥 / 𝑛 ) < e ) |
| 439 |
204
|
simpri |
⊢ e < 3 |
| 440 |
439
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → e < 3 ) |
| 441 |
434 435 437 438 440
|
lttrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( 𝑥 / 𝑛 ) < 3 ) |
| 442 |
|
3z |
⊢ 3 ∈ ℤ |
| 443 |
|
fllt |
⊢ ( ( ( 𝑥 / 𝑛 ) ∈ ℝ ∧ 3 ∈ ℤ ) → ( ( 𝑥 / 𝑛 ) < 3 ↔ ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) < 3 ) ) |
| 444 |
434 442 443
|
sylancl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( 𝑥 / 𝑛 ) < 3 ↔ ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) < 3 ) ) |
| 445 |
441 444
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) < 3 ) |
| 446 |
|
df-3 |
⊢ 3 = ( 2 + 1 ) |
| 447 |
445 446
|
breqtrdi |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) < ( 2 + 1 ) ) |
| 448 |
|
zleltp1 |
⊢ ( ( ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ∈ ℤ ∧ 2 ∈ ℤ ) → ( ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ≤ 2 ↔ ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) < ( 2 + 1 ) ) ) |
| 449 |
431 432 448
|
sylancl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ≤ 2 ↔ ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) < ( 2 + 1 ) ) ) |
| 450 |
447 449
|
mpbird |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ≤ 2 ) |
| 451 |
|
eluz2 |
⊢ ( 2 ∈ ( ℤ≥ ‘ ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ↔ ( ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ∈ ℤ ∧ 2 ∈ ℤ ∧ ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ≤ 2 ) ) |
| 452 |
431 433 450 451
|
syl3anbrc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → 2 ∈ ( ℤ≥ ‘ ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) |
| 453 |
|
fzss2 |
⊢ ( 2 ∈ ( ℤ≥ ‘ ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) → ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ⊆ ( 1 ... 2 ) ) |
| 454 |
452 453
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ⊆ ( 1 ... 2 ) ) |
| 455 |
454
|
sselda |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → 𝑚 ∈ ( 1 ... 2 ) ) |
| 456 |
36
|
ad5ant15 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ∈ ℝ ) |
| 457 |
455 456
|
syldan |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ∈ ℝ ) |
| 458 |
429 457
|
fsumrecl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ∈ ℝ ) |
| 459 |
93
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ℕ ) → ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ∈ ℝ ) |
| 460 |
87 459
|
sylan2 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ∈ ℝ ) |
| 461 |
352
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( 𝑥 / 𝑛 ) ≤ e ) |
| 462 |
434
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( 𝑥 / 𝑛 ) ∈ ℝ ) |
| 463 |
200
|
a1i |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → e ∈ ℝ ) |
| 464 |
32
|
rpregt0d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( 𝑚 ∈ ℝ ∧ 0 < 𝑚 ) ) |
| 465 |
464
|
ad5ant15 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( 𝑚 ∈ ℝ ∧ 0 < 𝑚 ) ) |
| 466 |
|
lediv1 |
⊢ ( ( ( 𝑥 / 𝑛 ) ∈ ℝ ∧ e ∈ ℝ ∧ ( 𝑚 ∈ ℝ ∧ 0 < 𝑚 ) ) → ( ( 𝑥 / 𝑛 ) ≤ e ↔ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ≤ ( e / 𝑚 ) ) ) |
| 467 |
462 463 465 466
|
syl3anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( ( 𝑥 / 𝑛 ) ≤ e ↔ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ≤ ( e / 𝑚 ) ) ) |
| 468 |
461 467
|
mpbid |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( ( 𝑥 / 𝑛 ) / 𝑚 ) ≤ ( e / 𝑚 ) ) |
| 469 |
90
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑥 / 𝑛 ) / 𝑚 ) ∈ ℝ+ ) |
| 470 |
30 469
|
sylan2 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( ( 𝑥 / 𝑛 ) / 𝑚 ) ∈ ℝ+ ) |
| 471 |
34
|
ad5ant15 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( e / 𝑚 ) ∈ ℝ+ ) |
| 472 |
470 471
|
logled |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( ( ( 𝑥 / 𝑛 ) / 𝑚 ) ≤ ( e / 𝑚 ) ↔ ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ≤ ( log ‘ ( e / 𝑚 ) ) ) ) |
| 473 |
468 472
|
mpbid |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ≤ ( log ‘ ( e / 𝑚 ) ) ) |
| 474 |
91
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ℕ ) → ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ∈ ℝ ) |
| 475 |
30 474
|
sylan2 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ∈ ℝ ) |
| 476 |
35
|
ad5ant15 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( log ‘ ( e / 𝑚 ) ) ∈ ℝ ) |
| 477 |
|
lediv1 |
⊢ ( ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ∈ ℝ ∧ ( log ‘ ( e / 𝑚 ) ) ∈ ℝ ∧ ( 𝑚 ∈ ℝ ∧ 0 < 𝑚 ) ) → ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ≤ ( log ‘ ( e / 𝑚 ) ) ↔ ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ≤ ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) ) |
| 478 |
475 476 465 477
|
syl3anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) ≤ ( log ‘ ( e / 𝑚 ) ) ↔ ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ≤ ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) ) |
| 479 |
473 478
|
mpbid |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ≤ ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) |
| 480 |
455 479
|
syldan |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ) → ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ≤ ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) |
| 481 |
429 460 457 480
|
fsumle |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ≤ Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) |
| 482 |
|
fzfid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( 1 ... 2 ) ∈ Fin ) |
| 483 |
244
|
ad5ant15 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) ∧ 𝑚 ∈ ( 1 ... 2 ) ) → 0 ≤ ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) |
| 484 |
482 456 483 454
|
fsumless |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ≤ Σ 𝑚 ∈ ( 1 ... 2 ) ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) |
| 485 |
404 458 328 481 484
|
letrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ≤ Σ 𝑚 ∈ ( 1 ... 2 ) ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) |
| 486 |
428 485
|
eqbrtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ≤ Σ 𝑚 ∈ ( 1 ... 2 ) ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) |
| 487 |
322 324 327 328 401 486
|
le2addd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( abs ‘ 𝑇 ) + ( abs ‘ Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ≤ ( ( ( 1 / 2 ) + ( γ + ( abs ‘ 𝐿 ) ) ) + Σ 𝑚 ∈ ( 1 ... 2 ) ( ( log ‘ ( e / 𝑚 ) ) / 𝑚 ) ) ) |
| 488 |
487 4
|
breqtrrdi |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( ( abs ‘ 𝑇 ) + ( abs ‘ Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ≤ 𝑅 ) |
| 489 |
318 325 319 326 488
|
letrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ≤ 𝑅 ) |
| 490 |
317 318 319 320 489
|
letrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ) ∧ ( 𝑥 / 𝑛 ) < e ) → ( abs ‘ ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) ) ≤ 𝑅 ) |
| 491 |
8 209 249 250 252 273 316 490
|
fsumharmonic |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( abs ‘ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) / 𝑛 ) ) ≤ ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · ( ( log ‘ e ) + 1 ) ) ) ) |
| 492 |
|
2cnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 2 ∈ ℂ ) |
| 493 |
7 492 303
|
fsummulc2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) ) |
| 494 |
|
df-2 |
⊢ 2 = ( 1 + 1 ) |
| 495 |
357
|
oveq1i |
⊢ ( ( log ‘ e ) + 1 ) = ( 1 + 1 ) |
| 496 |
494 495
|
eqtr4i |
⊢ 2 = ( ( log ‘ e ) + 1 ) |
| 497 |
496
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 2 = ( ( log ‘ e ) + 1 ) ) |
| 498 |
497
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 𝑅 · 2 ) = ( 𝑅 · ( ( log ‘ e ) + 1 ) ) ) |
| 499 |
493 498
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) = ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( 2 · ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · ( ( log ‘ e ) + 1 ) ) ) ) |
| 500 |
491 499
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( abs ‘ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) / 𝑛 ) ) ≤ ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ) |
| 501 |
500
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( abs ‘ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) · ( 𝑇 − Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝑥 / 𝑛 ) ) ) ( ( log ‘ ( ( 𝑥 / 𝑛 ) / 𝑚 ) ) / 𝑚 ) ) ) / 𝑛 ) ) ≤ ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ) |
| 502 |
199 501
|
eqbrtrrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( abs ‘ ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − ( log ‘ 𝑥 ) ) ) ≤ ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ) |
| 503 |
56
|
leabsd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ≤ ( abs ‘ ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ) ) |
| 504 |
503
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ≤ ( abs ‘ ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ) ) |
| 505 |
80 81 84 502 504
|
letrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥 ) ) → ( abs ‘ ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − ( log ‘ 𝑥 ) ) ) ≤ ( abs ‘ ( ( 2 · Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( log ‘ ( 𝑥 / 𝑛 ) ) / 𝑥 ) ) + ( 𝑅 · 2 ) ) ) ) |
| 506 |
5 55 56 78 505
|
o1le |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ+ ↦ ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( ( ( μ ‘ 𝑛 ) / 𝑛 ) · 𝑇 ) − ( log ‘ 𝑥 ) ) ) ∈ 𝑂(1) ) |