Step |
Hyp |
Ref |
Expression |
1 |
|
pntpbnd.r |
⊢ 𝑅 = ( 𝑎 ∈ ℝ+ ↦ ( ( ψ ‘ 𝑎 ) − 𝑎 ) ) |
2 |
|
pntpbnd1.e |
⊢ ( 𝜑 → 𝐸 ∈ ( 0 (,) 1 ) ) |
3 |
|
pntpbnd1.x |
⊢ 𝑋 = ( exp ‘ ( 2 / 𝐸 ) ) |
4 |
|
pntpbnd1.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑋 (,) +∞ ) ) |
5 |
|
pntpbnd1.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |
6 |
|
pntpbnd1.2 |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ℕ ∀ 𝑗 ∈ ℤ ( abs ‘ Σ 𝑦 ∈ ( 𝑖 ... 𝑗 ) ( ( 𝑅 ‘ 𝑦 ) / ( 𝑦 · ( 𝑦 + 1 ) ) ) ) ≤ 𝐴 ) |
7 |
|
pntpbnd1.c |
⊢ 𝐶 = ( 𝐴 + 2 ) |
8 |
|
pntpbnd1.k |
⊢ ( 𝜑 → 𝐾 ∈ ( ( exp ‘ ( 𝐶 / 𝐸 ) ) [,) +∞ ) ) |
9 |
|
pntpbnd1.3 |
⊢ ( 𝜑 → ¬ ∃ 𝑦 ∈ ℕ ( ( 𝑌 < 𝑦 ∧ 𝑦 ≤ ( 𝐾 · 𝑌 ) ) ∧ ( abs ‘ ( ( 𝑅 ‘ 𝑦 ) / 𝑦 ) ) ≤ 𝐸 ) ) |
10 |
|
2div2e1 |
⊢ ( 2 / 2 ) = 1 |
11 |
|
2re |
⊢ 2 ∈ ℝ |
12 |
11
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℝ ) |
13 |
|
ioossre |
⊢ ( 0 (,) 1 ) ⊆ ℝ |
14 |
13 2
|
sselid |
⊢ ( 𝜑 → 𝐸 ∈ ℝ ) |
15 |
|
eliooord |
⊢ ( 𝐸 ∈ ( 0 (,) 1 ) → ( 0 < 𝐸 ∧ 𝐸 < 1 ) ) |
16 |
2 15
|
syl |
⊢ ( 𝜑 → ( 0 < 𝐸 ∧ 𝐸 < 1 ) ) |
17 |
16
|
simpld |
⊢ ( 𝜑 → 0 < 𝐸 ) |
18 |
14 17
|
elrpd |
⊢ ( 𝜑 → 𝐸 ∈ ℝ+ ) |
19 |
|
2rp |
⊢ 2 ∈ ℝ+ |
20 |
19
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℝ+ ) |
21 |
7
|
oveq1i |
⊢ ( 𝐶 − 𝐴 ) = ( ( 𝐴 + 2 ) − 𝐴 ) |
22 |
5
|
rpcnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
23 |
|
2cn |
⊢ 2 ∈ ℂ |
24 |
|
pncan2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 2 ∈ ℂ ) → ( ( 𝐴 + 2 ) − 𝐴 ) = 2 ) |
25 |
22 23 24
|
sylancl |
⊢ ( 𝜑 → ( ( 𝐴 + 2 ) − 𝐴 ) = 2 ) |
26 |
21 25
|
eqtrid |
⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = 2 ) |
27 |
26
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐶 − 𝐴 ) / 𝐸 ) = ( 2 / 𝐸 ) ) |
28 |
|
rpaddcl |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 2 ∈ ℝ+ ) → ( 𝐴 + 2 ) ∈ ℝ+ ) |
29 |
5 19 28
|
sylancl |
⊢ ( 𝜑 → ( 𝐴 + 2 ) ∈ ℝ+ ) |
30 |
7 29
|
eqeltrid |
⊢ ( 𝜑 → 𝐶 ∈ ℝ+ ) |
31 |
30
|
rpcnd |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
32 |
14
|
recnd |
⊢ ( 𝜑 → 𝐸 ∈ ℂ ) |
33 |
18
|
rpne0d |
⊢ ( 𝜑 → 𝐸 ≠ 0 ) |
34 |
31 22 32 33
|
divsubdird |
⊢ ( 𝜑 → ( ( 𝐶 − 𝐴 ) / 𝐸 ) = ( ( 𝐶 / 𝐸 ) − ( 𝐴 / 𝐸 ) ) ) |
35 |
27 34
|
eqtr3d |
⊢ ( 𝜑 → ( 2 / 𝐸 ) = ( ( 𝐶 / 𝐸 ) − ( 𝐴 / 𝐸 ) ) ) |
36 |
30 18
|
rpdivcld |
⊢ ( 𝜑 → ( 𝐶 / 𝐸 ) ∈ ℝ+ ) |
37 |
36
|
rpred |
⊢ ( 𝜑 → ( 𝐶 / 𝐸 ) ∈ ℝ ) |
38 |
5
|
rpred |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
39 |
38 18
|
rerpdivcld |
⊢ ( 𝜑 → ( 𝐴 / 𝐸 ) ∈ ℝ ) |
40 |
|
resubcl |
⊢ ( ( ( 𝐶 / 𝐸 ) ∈ ℝ ∧ 2 ∈ ℝ ) → ( ( 𝐶 / 𝐸 ) − 2 ) ∈ ℝ ) |
41 |
37 11 40
|
sylancl |
⊢ ( 𝜑 → ( ( 𝐶 / 𝐸 ) − 2 ) ∈ ℝ ) |
42 |
37
|
reefcld |
⊢ ( 𝜑 → ( exp ‘ ( 𝐶 / 𝐸 ) ) ∈ ℝ ) |
43 |
|
elicopnf |
⊢ ( ( exp ‘ ( 𝐶 / 𝐸 ) ) ∈ ℝ → ( 𝐾 ∈ ( ( exp ‘ ( 𝐶 / 𝐸 ) ) [,) +∞ ) ↔ ( 𝐾 ∈ ℝ ∧ ( exp ‘ ( 𝐶 / 𝐸 ) ) ≤ 𝐾 ) ) ) |
44 |
42 43
|
syl |
⊢ ( 𝜑 → ( 𝐾 ∈ ( ( exp ‘ ( 𝐶 / 𝐸 ) ) [,) +∞ ) ↔ ( 𝐾 ∈ ℝ ∧ ( exp ‘ ( 𝐶 / 𝐸 ) ) ≤ 𝐾 ) ) ) |
45 |
8 44
|
mpbid |
⊢ ( 𝜑 → ( 𝐾 ∈ ℝ ∧ ( exp ‘ ( 𝐶 / 𝐸 ) ) ≤ 𝐾 ) ) |
46 |
45
|
simpld |
⊢ ( 𝜑 → 𝐾 ∈ ℝ ) |
47 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
48 |
|
1re |
⊢ 1 ∈ ℝ |
49 |
48
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
50 |
|
0lt1 |
⊢ 0 < 1 |
51 |
50
|
a1i |
⊢ ( 𝜑 → 0 < 1 ) |
52 |
|
efgt1 |
⊢ ( ( 𝐶 / 𝐸 ) ∈ ℝ+ → 1 < ( exp ‘ ( 𝐶 / 𝐸 ) ) ) |
53 |
36 52
|
syl |
⊢ ( 𝜑 → 1 < ( exp ‘ ( 𝐶 / 𝐸 ) ) ) |
54 |
45
|
simprd |
⊢ ( 𝜑 → ( exp ‘ ( 𝐶 / 𝐸 ) ) ≤ 𝐾 ) |
55 |
49 42 46 53 54
|
ltletrd |
⊢ ( 𝜑 → 1 < 𝐾 ) |
56 |
47 49 46 51 55
|
lttrd |
⊢ ( 𝜑 → 0 < 𝐾 ) |
57 |
46 56
|
elrpd |
⊢ ( 𝜑 → 𝐾 ∈ ℝ+ ) |
58 |
57
|
relogcld |
⊢ ( 𝜑 → ( log ‘ 𝐾 ) ∈ ℝ ) |
59 |
|
resubcl |
⊢ ( ( ( log ‘ 𝐾 ) ∈ ℝ ∧ 2 ∈ ℝ ) → ( ( log ‘ 𝐾 ) − 2 ) ∈ ℝ ) |
60 |
58 11 59
|
sylancl |
⊢ ( 𝜑 → ( ( log ‘ 𝐾 ) − 2 ) ∈ ℝ ) |
61 |
57
|
reeflogd |
⊢ ( 𝜑 → ( exp ‘ ( log ‘ 𝐾 ) ) = 𝐾 ) |
62 |
54 61
|
breqtrrd |
⊢ ( 𝜑 → ( exp ‘ ( 𝐶 / 𝐸 ) ) ≤ ( exp ‘ ( log ‘ 𝐾 ) ) ) |
63 |
|
efle |
⊢ ( ( ( 𝐶 / 𝐸 ) ∈ ℝ ∧ ( log ‘ 𝐾 ) ∈ ℝ ) → ( ( 𝐶 / 𝐸 ) ≤ ( log ‘ 𝐾 ) ↔ ( exp ‘ ( 𝐶 / 𝐸 ) ) ≤ ( exp ‘ ( log ‘ 𝐾 ) ) ) ) |
64 |
37 58 63
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐶 / 𝐸 ) ≤ ( log ‘ 𝐾 ) ↔ ( exp ‘ ( 𝐶 / 𝐸 ) ) ≤ ( exp ‘ ( log ‘ 𝐾 ) ) ) ) |
65 |
62 64
|
mpbird |
⊢ ( 𝜑 → ( 𝐶 / 𝐸 ) ≤ ( log ‘ 𝐾 ) ) |
66 |
37 58 12 65
|
lesub1dd |
⊢ ( 𝜑 → ( ( 𝐶 / 𝐸 ) − 2 ) ≤ ( ( log ‘ 𝐾 ) − 2 ) ) |
67 |
|
fzfid |
⊢ ( 𝜑 → ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ∈ Fin ) |
68 |
|
ioossre |
⊢ ( 𝑋 (,) +∞ ) ⊆ ℝ |
69 |
68 4
|
sselid |
⊢ ( 𝜑 → 𝑌 ∈ ℝ ) |
70 |
|
rerpdivcl |
⊢ ( ( 2 ∈ ℝ ∧ 𝐸 ∈ ℝ+ ) → ( 2 / 𝐸 ) ∈ ℝ ) |
71 |
11 18 70
|
sylancr |
⊢ ( 𝜑 → ( 2 / 𝐸 ) ∈ ℝ ) |
72 |
71
|
reefcld |
⊢ ( 𝜑 → ( exp ‘ ( 2 / 𝐸 ) ) ∈ ℝ ) |
73 |
3 72
|
eqeltrid |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
74 |
|
efgt0 |
⊢ ( ( 2 / 𝐸 ) ∈ ℝ → 0 < ( exp ‘ ( 2 / 𝐸 ) ) ) |
75 |
71 74
|
syl |
⊢ ( 𝜑 → 0 < ( exp ‘ ( 2 / 𝐸 ) ) ) |
76 |
75 3
|
breqtrrdi |
⊢ ( 𝜑 → 0 < 𝑋 ) |
77 |
73
|
rexrd |
⊢ ( 𝜑 → 𝑋 ∈ ℝ* ) |
78 |
|
elioopnf |
⊢ ( 𝑋 ∈ ℝ* → ( 𝑌 ∈ ( 𝑋 (,) +∞ ) ↔ ( 𝑌 ∈ ℝ ∧ 𝑋 < 𝑌 ) ) ) |
79 |
77 78
|
syl |
⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝑋 (,) +∞ ) ↔ ( 𝑌 ∈ ℝ ∧ 𝑋 < 𝑌 ) ) ) |
80 |
4 79
|
mpbid |
⊢ ( 𝜑 → ( 𝑌 ∈ ℝ ∧ 𝑋 < 𝑌 ) ) |
81 |
80
|
simprd |
⊢ ( 𝜑 → 𝑋 < 𝑌 ) |
82 |
47 73 69 76 81
|
lttrd |
⊢ ( 𝜑 → 0 < 𝑌 ) |
83 |
47 69 82
|
ltled |
⊢ ( 𝜑 → 0 ≤ 𝑌 ) |
84 |
|
flge0nn0 |
⊢ ( ( 𝑌 ∈ ℝ ∧ 0 ≤ 𝑌 ) → ( ⌊ ‘ 𝑌 ) ∈ ℕ0 ) |
85 |
69 83 84
|
syl2anc |
⊢ ( 𝜑 → ( ⌊ ‘ 𝑌 ) ∈ ℕ0 ) |
86 |
|
nn0p1nn |
⊢ ( ( ⌊ ‘ 𝑌 ) ∈ ℕ0 → ( ( ⌊ ‘ 𝑌 ) + 1 ) ∈ ℕ ) |
87 |
85 86
|
syl |
⊢ ( 𝜑 → ( ( ⌊ ‘ 𝑌 ) + 1 ) ∈ ℕ ) |
88 |
|
elfzuz |
⊢ ( 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) → 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) |
89 |
|
eluznn |
⊢ ( ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) → 𝑛 ∈ ℕ ) |
90 |
87 88 89
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → 𝑛 ∈ ℕ ) |
91 |
90
|
peano2nnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝑛 + 1 ) ∈ ℕ ) |
92 |
91
|
nnrecred |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 1 / ( 𝑛 + 1 ) ) ∈ ℝ ) |
93 |
67 92
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ∈ ℝ ) |
94 |
58
|
recnd |
⊢ ( 𝜑 → ( log ‘ 𝐾 ) ∈ ℂ ) |
95 |
|
2cnd |
⊢ ( 𝜑 → 2 ∈ ℂ ) |
96 |
69 82
|
elrpd |
⊢ ( 𝜑 → 𝑌 ∈ ℝ+ ) |
97 |
96
|
relogcld |
⊢ ( 𝜑 → ( log ‘ 𝑌 ) ∈ ℝ ) |
98 |
97
|
recnd |
⊢ ( 𝜑 → ( log ‘ 𝑌 ) ∈ ℂ ) |
99 |
94 95 98
|
pnpcan2d |
⊢ ( 𝜑 → ( ( ( log ‘ 𝐾 ) + ( log ‘ 𝑌 ) ) − ( 2 + ( log ‘ 𝑌 ) ) ) = ( ( log ‘ 𝐾 ) − 2 ) ) |
100 |
57 96
|
relogmuld |
⊢ ( 𝜑 → ( log ‘ ( 𝐾 · 𝑌 ) ) = ( ( log ‘ 𝐾 ) + ( log ‘ 𝑌 ) ) ) |
101 |
58 97
|
readdcld |
⊢ ( 𝜑 → ( ( log ‘ 𝐾 ) + ( log ‘ 𝑌 ) ) ∈ ℝ ) |
102 |
100 101
|
eqeltrd |
⊢ ( 𝜑 → ( log ‘ ( 𝐾 · 𝑌 ) ) ∈ ℝ ) |
103 |
|
fzfid |
⊢ ( 𝜑 → ( 0 ... ( ⌊ ‘ 𝑌 ) ) ∈ Fin ) |
104 |
|
elfznn0 |
⊢ ( 𝑛 ∈ ( 0 ... ( ⌊ ‘ 𝑌 ) ) → 𝑛 ∈ ℕ0 ) |
105 |
104
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 0 ... ( ⌊ ‘ 𝑌 ) ) ) → 𝑛 ∈ ℕ0 ) |
106 |
|
nn0p1nn |
⊢ ( 𝑛 ∈ ℕ0 → ( 𝑛 + 1 ) ∈ ℕ ) |
107 |
105 106
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 0 ... ( ⌊ ‘ 𝑌 ) ) ) → ( 𝑛 + 1 ) ∈ ℕ ) |
108 |
107
|
nnrecred |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 0 ... ( ⌊ ‘ 𝑌 ) ) ) → ( 1 / ( 𝑛 + 1 ) ) ∈ ℝ ) |
109 |
103 108
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( 0 ... ( ⌊ ‘ 𝑌 ) ) ( 1 / ( 𝑛 + 1 ) ) ∈ ℝ ) |
110 |
109 93
|
readdcld |
⊢ ( 𝜑 → ( Σ 𝑛 ∈ ( 0 ... ( ⌊ ‘ 𝑌 ) ) ( 1 / ( 𝑛 + 1 ) ) + Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) ∈ ℝ ) |
111 |
|
readdcl |
⊢ ( ( 2 ∈ ℝ ∧ ( log ‘ 𝑌 ) ∈ ℝ ) → ( 2 + ( log ‘ 𝑌 ) ) ∈ ℝ ) |
112 |
11 97 111
|
sylancr |
⊢ ( 𝜑 → ( 2 + ( log ‘ 𝑌 ) ) ∈ ℝ ) |
113 |
112 93
|
readdcld |
⊢ ( 𝜑 → ( ( 2 + ( log ‘ 𝑌 ) ) + Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) ∈ ℝ ) |
114 |
46 69
|
remulcld |
⊢ ( 𝜑 → ( 𝐾 · 𝑌 ) ∈ ℝ ) |
115 |
69
|
recnd |
⊢ ( 𝜑 → 𝑌 ∈ ℂ ) |
116 |
115
|
mulid2d |
⊢ ( 𝜑 → ( 1 · 𝑌 ) = 𝑌 ) |
117 |
49 46 55
|
ltled |
⊢ ( 𝜑 → 1 ≤ 𝐾 ) |
118 |
|
lemul1 |
⊢ ( ( 1 ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ ( 𝑌 ∈ ℝ ∧ 0 < 𝑌 ) ) → ( 1 ≤ 𝐾 ↔ ( 1 · 𝑌 ) ≤ ( 𝐾 · 𝑌 ) ) ) |
119 |
49 46 69 82 118
|
syl112anc |
⊢ ( 𝜑 → ( 1 ≤ 𝐾 ↔ ( 1 · 𝑌 ) ≤ ( 𝐾 · 𝑌 ) ) ) |
120 |
117 119
|
mpbid |
⊢ ( 𝜑 → ( 1 · 𝑌 ) ≤ ( 𝐾 · 𝑌 ) ) |
121 |
116 120
|
eqbrtrrd |
⊢ ( 𝜑 → 𝑌 ≤ ( 𝐾 · 𝑌 ) ) |
122 |
47 69 114 83 121
|
letrd |
⊢ ( 𝜑 → 0 ≤ ( 𝐾 · 𝑌 ) ) |
123 |
|
flge0nn0 |
⊢ ( ( ( 𝐾 · 𝑌 ) ∈ ℝ ∧ 0 ≤ ( 𝐾 · 𝑌 ) ) → ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ∈ ℕ0 ) |
124 |
114 122 123
|
syl2anc |
⊢ ( 𝜑 → ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ∈ ℕ0 ) |
125 |
|
nn0p1nn |
⊢ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ∈ ℕ0 → ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ∈ ℕ ) |
126 |
124 125
|
syl |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ∈ ℕ ) |
127 |
126
|
nnrpd |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ∈ ℝ+ ) |
128 |
127
|
relogcld |
⊢ ( 𝜑 → ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ∈ ℝ ) |
129 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
130 |
114
|
flcld |
⊢ ( 𝜑 → ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ∈ ℤ ) |
131 |
130
|
peano2zd |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ∈ ℤ ) |
132 |
|
elfznn |
⊢ ( 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) → 𝑘 ∈ ℕ ) |
133 |
132
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) → 𝑘 ∈ ℕ ) |
134 |
|
nnrecre |
⊢ ( 𝑘 ∈ ℕ → ( 1 / 𝑘 ) ∈ ℝ ) |
135 |
134
|
recnd |
⊢ ( 𝑘 ∈ ℕ → ( 1 / 𝑘 ) ∈ ℂ ) |
136 |
133 135
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) → ( 1 / 𝑘 ) ∈ ℂ ) |
137 |
|
oveq2 |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( 1 / 𝑘 ) = ( 1 / ( 𝑛 + 1 ) ) ) |
138 |
129 129 131 136 137
|
fsumshftm |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) = Σ 𝑛 ∈ ( ( 1 − 1 ) ... ( ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) − 1 ) ) ( 1 / ( 𝑛 + 1 ) ) ) |
139 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
140 |
139
|
a1i |
⊢ ( 𝜑 → ( 1 − 1 ) = 0 ) |
141 |
130
|
zcnd |
⊢ ( 𝜑 → ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ∈ ℂ ) |
142 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
143 |
|
pncan |
⊢ ( ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) − 1 ) = ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) |
144 |
141 142 143
|
sylancl |
⊢ ( 𝜑 → ( ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) − 1 ) = ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) |
145 |
140 144
|
oveq12d |
⊢ ( 𝜑 → ( ( 1 − 1 ) ... ( ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) − 1 ) ) = ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) |
146 |
145
|
sumeq1d |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( ( 1 − 1 ) ... ( ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) − 1 ) ) ( 1 / ( 𝑛 + 1 ) ) = Σ 𝑛 ∈ ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) |
147 |
|
reflcl |
⊢ ( 𝑌 ∈ ℝ → ( ⌊ ‘ 𝑌 ) ∈ ℝ ) |
148 |
69 147
|
syl |
⊢ ( 𝜑 → ( ⌊ ‘ 𝑌 ) ∈ ℝ ) |
149 |
148
|
ltp1d |
⊢ ( 𝜑 → ( ⌊ ‘ 𝑌 ) < ( ( ⌊ ‘ 𝑌 ) + 1 ) ) |
150 |
|
fzdisj |
⊢ ( ( ⌊ ‘ 𝑌 ) < ( ( ⌊ ‘ 𝑌 ) + 1 ) → ( ( 0 ... ( ⌊ ‘ 𝑌 ) ) ∩ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) = ∅ ) |
151 |
149 150
|
syl |
⊢ ( 𝜑 → ( ( 0 ... ( ⌊ ‘ 𝑌 ) ) ∩ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) = ∅ ) |
152 |
|
flwordi |
⊢ ( ( 𝑌 ∈ ℝ ∧ ( 𝐾 · 𝑌 ) ∈ ℝ ∧ 𝑌 ≤ ( 𝐾 · 𝑌 ) ) → ( ⌊ ‘ 𝑌 ) ≤ ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) |
153 |
69 114 121 152
|
syl3anc |
⊢ ( 𝜑 → ( ⌊ ‘ 𝑌 ) ≤ ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) |
154 |
|
elfz2nn0 |
⊢ ( ( ⌊ ‘ 𝑌 ) ∈ ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ↔ ( ( ⌊ ‘ 𝑌 ) ∈ ℕ0 ∧ ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ∈ ℕ0 ∧ ( ⌊ ‘ 𝑌 ) ≤ ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) |
155 |
85 124 153 154
|
syl3anbrc |
⊢ ( 𝜑 → ( ⌊ ‘ 𝑌 ) ∈ ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) |
156 |
|
fzsplit |
⊢ ( ( ⌊ ‘ 𝑌 ) ∈ ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) → ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) = ( ( 0 ... ( ⌊ ‘ 𝑌 ) ) ∪ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) ) |
157 |
155 156
|
syl |
⊢ ( 𝜑 → ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) = ( ( 0 ... ( ⌊ ‘ 𝑌 ) ) ∪ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) ) |
158 |
|
fzfid |
⊢ ( 𝜑 → ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ∈ Fin ) |
159 |
|
elfznn0 |
⊢ ( 𝑛 ∈ ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) → 𝑛 ∈ ℕ0 ) |
160 |
159
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → 𝑛 ∈ ℕ0 ) |
161 |
160 106
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝑛 + 1 ) ∈ ℕ ) |
162 |
161
|
nnrecred |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 1 / ( 𝑛 + 1 ) ) ∈ ℝ ) |
163 |
162
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 1 / ( 𝑛 + 1 ) ) ∈ ℂ ) |
164 |
151 157 158 163
|
fsumsplit |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( 0 ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) = ( Σ 𝑛 ∈ ( 0 ... ( ⌊ ‘ 𝑌 ) ) ( 1 / ( 𝑛 + 1 ) ) + Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) ) |
165 |
138 146 164
|
3eqtrd |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) = ( Σ 𝑛 ∈ ( 0 ... ( ⌊ ‘ 𝑌 ) ) ( 1 / ( 𝑛 + 1 ) ) + Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) ) |
166 |
165 110
|
eqeltrd |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) ∈ ℝ ) |
167 |
|
fllep1 |
⊢ ( ( 𝐾 · 𝑌 ) ∈ ℝ → ( 𝐾 · 𝑌 ) ≤ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) |
168 |
114 167
|
syl |
⊢ ( 𝜑 → ( 𝐾 · 𝑌 ) ≤ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) |
169 |
57 96
|
rpmulcld |
⊢ ( 𝜑 → ( 𝐾 · 𝑌 ) ∈ ℝ+ ) |
170 |
169 127
|
logled |
⊢ ( 𝜑 → ( ( 𝐾 · 𝑌 ) ≤ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ↔ ( log ‘ ( 𝐾 · 𝑌 ) ) ≤ ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) ) |
171 |
168 170
|
mpbid |
⊢ ( 𝜑 → ( log ‘ ( 𝐾 · 𝑌 ) ) ≤ ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) |
172 |
|
emre |
⊢ γ ∈ ℝ |
173 |
172
|
a1i |
⊢ ( 𝜑 → γ ∈ ℝ ) |
174 |
166 128
|
resubcld |
⊢ ( 𝜑 → ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) ∈ ℝ ) |
175 |
|
0re |
⊢ 0 ∈ ℝ |
176 |
|
emgt0 |
⊢ 0 < γ |
177 |
175 172 176
|
ltleii |
⊢ 0 ≤ γ |
178 |
177
|
a1i |
⊢ ( 𝜑 → 0 ≤ γ ) |
179 |
|
harmonicbnd |
⊢ ( ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ∈ ℕ → ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) ∈ ( γ [,] 1 ) ) |
180 |
126 179
|
syl |
⊢ ( 𝜑 → ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) ∈ ( γ [,] 1 ) ) |
181 |
172 48
|
elicc2i |
⊢ ( ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) ∈ ( γ [,] 1 ) ↔ ( ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) ∈ ℝ ∧ γ ≤ ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) ∧ ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) ≤ 1 ) ) |
182 |
181
|
simp2bi |
⊢ ( ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) ∈ ( γ [,] 1 ) → γ ≤ ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) ) |
183 |
180 182
|
syl |
⊢ ( 𝜑 → γ ≤ ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) ) |
184 |
47 173 174 178 183
|
letrd |
⊢ ( 𝜑 → 0 ≤ ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) ) |
185 |
166 128
|
subge0d |
⊢ ( 𝜑 → ( 0 ≤ ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ) ↔ ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ≤ Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) ) ) |
186 |
184 185
|
mpbid |
⊢ ( 𝜑 → ( log ‘ ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ≤ Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) ) |
187 |
102 128 166 171 186
|
letrd |
⊢ ( 𝜑 → ( log ‘ ( 𝐾 · 𝑌 ) ) ≤ Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) + 1 ) ) ( 1 / 𝑘 ) ) |
188 |
187 165
|
breqtrd |
⊢ ( 𝜑 → ( log ‘ ( 𝐾 · 𝑌 ) ) ≤ ( Σ 𝑛 ∈ ( 0 ... ( ⌊ ‘ 𝑌 ) ) ( 1 / ( 𝑛 + 1 ) ) + Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) ) |
189 |
69
|
flcld |
⊢ ( 𝜑 → ( ⌊ ‘ 𝑌 ) ∈ ℤ ) |
190 |
189
|
peano2zd |
⊢ ( 𝜑 → ( ( ⌊ ‘ 𝑌 ) + 1 ) ∈ ℤ ) |
191 |
|
elfznn |
⊢ ( 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) → 𝑘 ∈ ℕ ) |
192 |
191
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) → 𝑘 ∈ ℕ ) |
193 |
192 135
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) → ( 1 / 𝑘 ) ∈ ℂ ) |
194 |
129 129 190 193 137
|
fsumshftm |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) = Σ 𝑛 ∈ ( ( 1 − 1 ) ... ( ( ( ⌊ ‘ 𝑌 ) + 1 ) − 1 ) ) ( 1 / ( 𝑛 + 1 ) ) ) |
195 |
148
|
recnd |
⊢ ( 𝜑 → ( ⌊ ‘ 𝑌 ) ∈ ℂ ) |
196 |
|
pncan |
⊢ ( ( ( ⌊ ‘ 𝑌 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( ( ⌊ ‘ 𝑌 ) + 1 ) − 1 ) = ( ⌊ ‘ 𝑌 ) ) |
197 |
195 142 196
|
sylancl |
⊢ ( 𝜑 → ( ( ( ⌊ ‘ 𝑌 ) + 1 ) − 1 ) = ( ⌊ ‘ 𝑌 ) ) |
198 |
140 197
|
oveq12d |
⊢ ( 𝜑 → ( ( 1 − 1 ) ... ( ( ( ⌊ ‘ 𝑌 ) + 1 ) − 1 ) ) = ( 0 ... ( ⌊ ‘ 𝑌 ) ) ) |
199 |
198
|
sumeq1d |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( ( 1 − 1 ) ... ( ( ( ⌊ ‘ 𝑌 ) + 1 ) − 1 ) ) ( 1 / ( 𝑛 + 1 ) ) = Σ 𝑛 ∈ ( 0 ... ( ⌊ ‘ 𝑌 ) ) ( 1 / ( 𝑛 + 1 ) ) ) |
200 |
194 199
|
eqtrd |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) = Σ 𝑛 ∈ ( 0 ... ( ⌊ ‘ 𝑌 ) ) ( 1 / ( 𝑛 + 1 ) ) ) |
201 |
200 109
|
eqeltrd |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) ∈ ℝ ) |
202 |
87
|
nnrpd |
⊢ ( 𝜑 → ( ( ⌊ ‘ 𝑌 ) + 1 ) ∈ ℝ+ ) |
203 |
202
|
relogcld |
⊢ ( 𝜑 → ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ∈ ℝ ) |
204 |
|
readdcl |
⊢ ( ( 1 ∈ ℝ ∧ ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ∈ ℝ ) → ( 1 + ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ∈ ℝ ) |
205 |
48 203 204
|
sylancr |
⊢ ( 𝜑 → ( 1 + ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ∈ ℝ ) |
206 |
|
harmonicbnd |
⊢ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ∈ ℕ → ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ∈ ( γ [,] 1 ) ) |
207 |
87 206
|
syl |
⊢ ( 𝜑 → ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ∈ ( γ [,] 1 ) ) |
208 |
172 48
|
elicc2i |
⊢ ( ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ∈ ( γ [,] 1 ) ↔ ( ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ∈ ℝ ∧ γ ≤ ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ∧ ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ≤ 1 ) ) |
209 |
208
|
simp3bi |
⊢ ( ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ∈ ( γ [,] 1 ) → ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ≤ 1 ) |
210 |
207 209
|
syl |
⊢ ( 𝜑 → ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ≤ 1 ) |
211 |
201 203 49
|
lesubaddd |
⊢ ( 𝜑 → ( ( Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) − ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ≤ 1 ↔ Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) ≤ ( 1 + ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ) ) |
212 |
210 211
|
mpbid |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) ≤ ( 1 + ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ) |
213 |
|
readdcl |
⊢ ( ( 1 ∈ ℝ ∧ ( log ‘ 𝑌 ) ∈ ℝ ) → ( 1 + ( log ‘ 𝑌 ) ) ∈ ℝ ) |
214 |
48 97 213
|
sylancr |
⊢ ( 𝜑 → ( 1 + ( log ‘ 𝑌 ) ) ∈ ℝ ) |
215 |
|
peano2re |
⊢ ( ( ⌊ ‘ 𝑌 ) ∈ ℝ → ( ( ⌊ ‘ 𝑌 ) + 1 ) ∈ ℝ ) |
216 |
148 215
|
syl |
⊢ ( 𝜑 → ( ( ⌊ ‘ 𝑌 ) + 1 ) ∈ ℝ ) |
217 |
12 69
|
remulcld |
⊢ ( 𝜑 → ( 2 · 𝑌 ) ∈ ℝ ) |
218 |
|
epr |
⊢ e ∈ ℝ+ |
219 |
|
rpmulcl |
⊢ ( ( e ∈ ℝ+ ∧ 𝑌 ∈ ℝ+ ) → ( e · 𝑌 ) ∈ ℝ+ ) |
220 |
218 96 219
|
sylancr |
⊢ ( 𝜑 → ( e · 𝑌 ) ∈ ℝ+ ) |
221 |
220
|
rpred |
⊢ ( 𝜑 → ( e · 𝑌 ) ∈ ℝ ) |
222 |
|
flle |
⊢ ( 𝑌 ∈ ℝ → ( ⌊ ‘ 𝑌 ) ≤ 𝑌 ) |
223 |
69 222
|
syl |
⊢ ( 𝜑 → ( ⌊ ‘ 𝑌 ) ≤ 𝑌 ) |
224 |
20 18
|
rpdivcld |
⊢ ( 𝜑 → ( 2 / 𝐸 ) ∈ ℝ+ ) |
225 |
|
efgt1 |
⊢ ( ( 2 / 𝐸 ) ∈ ℝ+ → 1 < ( exp ‘ ( 2 / 𝐸 ) ) ) |
226 |
224 225
|
syl |
⊢ ( 𝜑 → 1 < ( exp ‘ ( 2 / 𝐸 ) ) ) |
227 |
226 3
|
breqtrrdi |
⊢ ( 𝜑 → 1 < 𝑋 ) |
228 |
49 73 69 227 81
|
lttrd |
⊢ ( 𝜑 → 1 < 𝑌 ) |
229 |
49 69 228
|
ltled |
⊢ ( 𝜑 → 1 ≤ 𝑌 ) |
230 |
148 49 69 69 223 229
|
le2addd |
⊢ ( 𝜑 → ( ( ⌊ ‘ 𝑌 ) + 1 ) ≤ ( 𝑌 + 𝑌 ) ) |
231 |
115
|
2timesd |
⊢ ( 𝜑 → ( 2 · 𝑌 ) = ( 𝑌 + 𝑌 ) ) |
232 |
230 231
|
breqtrrd |
⊢ ( 𝜑 → ( ( ⌊ ‘ 𝑌 ) + 1 ) ≤ ( 2 · 𝑌 ) ) |
233 |
|
ere |
⊢ e ∈ ℝ |
234 |
|
egt2lt3 |
⊢ ( 2 < e ∧ e < 3 ) |
235 |
234
|
simpli |
⊢ 2 < e |
236 |
11 233 235
|
ltleii |
⊢ 2 ≤ e |
237 |
236
|
a1i |
⊢ ( 𝜑 → 2 ≤ e ) |
238 |
233
|
a1i |
⊢ ( 𝜑 → e ∈ ℝ ) |
239 |
|
lemul1 |
⊢ ( ( 2 ∈ ℝ ∧ e ∈ ℝ ∧ ( 𝑌 ∈ ℝ ∧ 0 < 𝑌 ) ) → ( 2 ≤ e ↔ ( 2 · 𝑌 ) ≤ ( e · 𝑌 ) ) ) |
240 |
12 238 69 82 239
|
syl112anc |
⊢ ( 𝜑 → ( 2 ≤ e ↔ ( 2 · 𝑌 ) ≤ ( e · 𝑌 ) ) ) |
241 |
237 240
|
mpbid |
⊢ ( 𝜑 → ( 2 · 𝑌 ) ≤ ( e · 𝑌 ) ) |
242 |
216 217 221 232 241
|
letrd |
⊢ ( 𝜑 → ( ( ⌊ ‘ 𝑌 ) + 1 ) ≤ ( e · 𝑌 ) ) |
243 |
202 220
|
logled |
⊢ ( 𝜑 → ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ≤ ( e · 𝑌 ) ↔ ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ≤ ( log ‘ ( e · 𝑌 ) ) ) ) |
244 |
242 243
|
mpbid |
⊢ ( 𝜑 → ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ≤ ( log ‘ ( e · 𝑌 ) ) ) |
245 |
|
relogmul |
⊢ ( ( e ∈ ℝ+ ∧ 𝑌 ∈ ℝ+ ) → ( log ‘ ( e · 𝑌 ) ) = ( ( log ‘ e ) + ( log ‘ 𝑌 ) ) ) |
246 |
218 96 245
|
sylancr |
⊢ ( 𝜑 → ( log ‘ ( e · 𝑌 ) ) = ( ( log ‘ e ) + ( log ‘ 𝑌 ) ) ) |
247 |
|
loge |
⊢ ( log ‘ e ) = 1 |
248 |
247
|
oveq1i |
⊢ ( ( log ‘ e ) + ( log ‘ 𝑌 ) ) = ( 1 + ( log ‘ 𝑌 ) ) |
249 |
246 248
|
eqtrdi |
⊢ ( 𝜑 → ( log ‘ ( e · 𝑌 ) ) = ( 1 + ( log ‘ 𝑌 ) ) ) |
250 |
244 249
|
breqtrd |
⊢ ( 𝜑 → ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ≤ ( 1 + ( log ‘ 𝑌 ) ) ) |
251 |
203 214 49 250
|
leadd2dd |
⊢ ( 𝜑 → ( 1 + ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ≤ ( 1 + ( 1 + ( log ‘ 𝑌 ) ) ) ) |
252 |
|
df-2 |
⊢ 2 = ( 1 + 1 ) |
253 |
252
|
oveq1i |
⊢ ( 2 + ( log ‘ 𝑌 ) ) = ( ( 1 + 1 ) + ( log ‘ 𝑌 ) ) |
254 |
142
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℂ ) |
255 |
254 254 98
|
addassd |
⊢ ( 𝜑 → ( ( 1 + 1 ) + ( log ‘ 𝑌 ) ) = ( 1 + ( 1 + ( log ‘ 𝑌 ) ) ) ) |
256 |
253 255
|
eqtrid |
⊢ ( 𝜑 → ( 2 + ( log ‘ 𝑌 ) ) = ( 1 + ( 1 + ( log ‘ 𝑌 ) ) ) ) |
257 |
251 256
|
breqtrrd |
⊢ ( 𝜑 → ( 1 + ( log ‘ ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ) ≤ ( 2 + ( log ‘ 𝑌 ) ) ) |
258 |
201 205 112 212 257
|
letrd |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... ( ( ⌊ ‘ 𝑌 ) + 1 ) ) ( 1 / 𝑘 ) ≤ ( 2 + ( log ‘ 𝑌 ) ) ) |
259 |
200 258
|
eqbrtrrd |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( 0 ... ( ⌊ ‘ 𝑌 ) ) ( 1 / ( 𝑛 + 1 ) ) ≤ ( 2 + ( log ‘ 𝑌 ) ) ) |
260 |
109 112 93 259
|
leadd1dd |
⊢ ( 𝜑 → ( Σ 𝑛 ∈ ( 0 ... ( ⌊ ‘ 𝑌 ) ) ( 1 / ( 𝑛 + 1 ) ) + Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) ≤ ( ( 2 + ( log ‘ 𝑌 ) ) + Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) ) |
261 |
102 110 113 188 260
|
letrd |
⊢ ( 𝜑 → ( log ‘ ( 𝐾 · 𝑌 ) ) ≤ ( ( 2 + ( log ‘ 𝑌 ) ) + Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) ) |
262 |
100 261
|
eqbrtrrd |
⊢ ( 𝜑 → ( ( log ‘ 𝐾 ) + ( log ‘ 𝑌 ) ) ≤ ( ( 2 + ( log ‘ 𝑌 ) ) + Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) ) |
263 |
101 112 93
|
lesubadd2d |
⊢ ( 𝜑 → ( ( ( ( log ‘ 𝐾 ) + ( log ‘ 𝑌 ) ) − ( 2 + ( log ‘ 𝑌 ) ) ) ≤ Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ↔ ( ( log ‘ 𝐾 ) + ( log ‘ 𝑌 ) ) ≤ ( ( 2 + ( log ‘ 𝑌 ) ) + Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) ) ) |
264 |
262 263
|
mpbird |
⊢ ( 𝜑 → ( ( ( log ‘ 𝐾 ) + ( log ‘ 𝑌 ) ) − ( 2 + ( log ‘ 𝑌 ) ) ) ≤ Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) |
265 |
99 264
|
eqbrtrrd |
⊢ ( 𝜑 → ( ( log ‘ 𝐾 ) − 2 ) ≤ Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) |
266 |
92
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 1 / ( 𝑛 + 1 ) ) ∈ ℂ ) |
267 |
67 32 266
|
fsummulc2 |
⊢ ( 𝜑 → ( 𝐸 · Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) = Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 𝐸 · ( 1 / ( 𝑛 + 1 ) ) ) ) |
268 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → 𝐸 ∈ ℝ ) |
269 |
268
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → 𝐸 ∈ ℂ ) |
270 |
91
|
nncnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝑛 + 1 ) ∈ ℂ ) |
271 |
91
|
nnne0d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝑛 + 1 ) ≠ 0 ) |
272 |
269 270 271
|
divrecd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝐸 / ( 𝑛 + 1 ) ) = ( 𝐸 · ( 1 / ( 𝑛 + 1 ) ) ) ) |
273 |
268 91
|
nndivred |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝐸 / ( 𝑛 + 1 ) ) ∈ ℝ ) |
274 |
272 273
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝐸 · ( 1 / ( 𝑛 + 1 ) ) ) ∈ ℝ ) |
275 |
67 274
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 𝐸 · ( 1 / ( 𝑛 + 1 ) ) ) ∈ ℝ ) |
276 |
90
|
nnrpd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → 𝑛 ∈ ℝ+ ) |
277 |
1
|
pntrf |
⊢ 𝑅 : ℝ+ ⟶ ℝ |
278 |
277
|
ffvelrni |
⊢ ( 𝑛 ∈ ℝ+ → ( 𝑅 ‘ 𝑛 ) ∈ ℝ ) |
279 |
276 278
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝑅 ‘ 𝑛 ) ∈ ℝ ) |
280 |
90 91
|
nnmulcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝑛 · ( 𝑛 + 1 ) ) ∈ ℕ ) |
281 |
279 280
|
nndivred |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ( 𝑅 ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 + 1 ) ) ) ∈ ℝ ) |
282 |
281
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ( 𝑅 ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 + 1 ) ) ) ∈ ℂ ) |
283 |
282
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 + 1 ) ) ) ) ∈ ℝ ) |
284 |
67 283
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 + 1 ) ) ) ) ∈ ℝ ) |
285 |
279 90
|
nndivred |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ∈ ℝ ) |
286 |
285
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ∈ ℂ ) |
287 |
286
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) ∈ ℝ ) |
288 |
91
|
nnrpd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝑛 + 1 ) ∈ ℝ+ ) |
289 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ¬ ∃ 𝑦 ∈ ℕ ( ( 𝑌 < 𝑦 ∧ 𝑦 ≤ ( 𝐾 · 𝑌 ) ) ∧ ( abs ‘ ( ( 𝑅 ‘ 𝑦 ) / 𝑦 ) ) ≤ 𝐸 ) ) |
290 |
|
elfzle1 |
⊢ ( 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) → ( ( ⌊ ‘ 𝑌 ) + 1 ) ≤ 𝑛 ) |
291 |
290
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ( ⌊ ‘ 𝑌 ) + 1 ) ≤ 𝑛 ) |
292 |
69
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → 𝑌 ∈ ℝ ) |
293 |
292
|
flcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ⌊ ‘ 𝑌 ) ∈ ℤ ) |
294 |
90
|
nnzd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → 𝑛 ∈ ℤ ) |
295 |
|
zltp1le |
⊢ ( ( ( ⌊ ‘ 𝑌 ) ∈ ℤ ∧ 𝑛 ∈ ℤ ) → ( ( ⌊ ‘ 𝑌 ) < 𝑛 ↔ ( ( ⌊ ‘ 𝑌 ) + 1 ) ≤ 𝑛 ) ) |
296 |
293 294 295
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ( ⌊ ‘ 𝑌 ) < 𝑛 ↔ ( ( ⌊ ‘ 𝑌 ) + 1 ) ≤ 𝑛 ) ) |
297 |
291 296
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ⌊ ‘ 𝑌 ) < 𝑛 ) |
298 |
|
fllt |
⊢ ( ( 𝑌 ∈ ℝ ∧ 𝑛 ∈ ℤ ) → ( 𝑌 < 𝑛 ↔ ( ⌊ ‘ 𝑌 ) < 𝑛 ) ) |
299 |
292 294 298
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝑌 < 𝑛 ↔ ( ⌊ ‘ 𝑌 ) < 𝑛 ) ) |
300 |
297 299
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → 𝑌 < 𝑛 ) |
301 |
|
elfzle2 |
⊢ ( 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) → 𝑛 ≤ ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) |
302 |
301
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → 𝑛 ≤ ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) |
303 |
114
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝐾 · 𝑌 ) ∈ ℝ ) |
304 |
|
flge |
⊢ ( ( ( 𝐾 · 𝑌 ) ∈ ℝ ∧ 𝑛 ∈ ℤ ) → ( 𝑛 ≤ ( 𝐾 · 𝑌 ) ↔ 𝑛 ≤ ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) |
305 |
303 294 304
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝑛 ≤ ( 𝐾 · 𝑌 ) ↔ 𝑛 ≤ ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) |
306 |
302 305
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → 𝑛 ≤ ( 𝐾 · 𝑌 ) ) |
307 |
|
breq2 |
⊢ ( 𝑦 = 𝑛 → ( 𝑌 < 𝑦 ↔ 𝑌 < 𝑛 ) ) |
308 |
|
breq1 |
⊢ ( 𝑦 = 𝑛 → ( 𝑦 ≤ ( 𝐾 · 𝑌 ) ↔ 𝑛 ≤ ( 𝐾 · 𝑌 ) ) ) |
309 |
307 308
|
anbi12d |
⊢ ( 𝑦 = 𝑛 → ( ( 𝑌 < 𝑦 ∧ 𝑦 ≤ ( 𝐾 · 𝑌 ) ) ↔ ( 𝑌 < 𝑛 ∧ 𝑛 ≤ ( 𝐾 · 𝑌 ) ) ) ) |
310 |
|
fveq2 |
⊢ ( 𝑦 = 𝑛 → ( 𝑅 ‘ 𝑦 ) = ( 𝑅 ‘ 𝑛 ) ) |
311 |
|
id |
⊢ ( 𝑦 = 𝑛 → 𝑦 = 𝑛 ) |
312 |
310 311
|
oveq12d |
⊢ ( 𝑦 = 𝑛 → ( ( 𝑅 ‘ 𝑦 ) / 𝑦 ) = ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) |
313 |
312
|
fveq2d |
⊢ ( 𝑦 = 𝑛 → ( abs ‘ ( ( 𝑅 ‘ 𝑦 ) / 𝑦 ) ) = ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) ) |
314 |
313
|
breq1d |
⊢ ( 𝑦 = 𝑛 → ( ( abs ‘ ( ( 𝑅 ‘ 𝑦 ) / 𝑦 ) ) ≤ 𝐸 ↔ ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) ≤ 𝐸 ) ) |
315 |
309 314
|
anbi12d |
⊢ ( 𝑦 = 𝑛 → ( ( ( 𝑌 < 𝑦 ∧ 𝑦 ≤ ( 𝐾 · 𝑌 ) ) ∧ ( abs ‘ ( ( 𝑅 ‘ 𝑦 ) / 𝑦 ) ) ≤ 𝐸 ) ↔ ( ( 𝑌 < 𝑛 ∧ 𝑛 ≤ ( 𝐾 · 𝑌 ) ) ∧ ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) ≤ 𝐸 ) ) ) |
316 |
315
|
rspcev |
⊢ ( ( 𝑛 ∈ ℕ ∧ ( ( 𝑌 < 𝑛 ∧ 𝑛 ≤ ( 𝐾 · 𝑌 ) ) ∧ ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) ≤ 𝐸 ) ) → ∃ 𝑦 ∈ ℕ ( ( 𝑌 < 𝑦 ∧ 𝑦 ≤ ( 𝐾 · 𝑌 ) ) ∧ ( abs ‘ ( ( 𝑅 ‘ 𝑦 ) / 𝑦 ) ) ≤ 𝐸 ) ) |
317 |
316
|
expr |
⊢ ( ( 𝑛 ∈ ℕ ∧ ( 𝑌 < 𝑛 ∧ 𝑛 ≤ ( 𝐾 · 𝑌 ) ) ) → ( ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) ≤ 𝐸 → ∃ 𝑦 ∈ ℕ ( ( 𝑌 < 𝑦 ∧ 𝑦 ≤ ( 𝐾 · 𝑌 ) ) ∧ ( abs ‘ ( ( 𝑅 ‘ 𝑦 ) / 𝑦 ) ) ≤ 𝐸 ) ) ) |
318 |
90 300 306 317
|
syl12anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) ≤ 𝐸 → ∃ 𝑦 ∈ ℕ ( ( 𝑌 < 𝑦 ∧ 𝑦 ≤ ( 𝐾 · 𝑌 ) ) ∧ ( abs ‘ ( ( 𝑅 ‘ 𝑦 ) / 𝑦 ) ) ≤ 𝐸 ) ) ) |
319 |
289 318
|
mtod |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ¬ ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) ≤ 𝐸 ) |
320 |
287 268
|
letrid |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) ≤ 𝐸 ∨ 𝐸 ≤ ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) ) ) |
321 |
320
|
ord |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ¬ ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) ≤ 𝐸 → 𝐸 ≤ ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) ) ) |
322 |
319 321
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → 𝐸 ≤ ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) ) |
323 |
268 287 288 322
|
lediv1dd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝐸 / ( 𝑛 + 1 ) ) ≤ ( ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) / ( 𝑛 + 1 ) ) ) |
324 |
286 270 271
|
absdivd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( abs ‘ ( ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) / ( 𝑛 + 1 ) ) ) = ( ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) / ( abs ‘ ( 𝑛 + 1 ) ) ) ) |
325 |
279
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝑅 ‘ 𝑛 ) ∈ ℂ ) |
326 |
90
|
nncnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → 𝑛 ∈ ℂ ) |
327 |
90
|
nnne0d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → 𝑛 ≠ 0 ) |
328 |
325 326 270 327 271
|
divdiv1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) / ( 𝑛 + 1 ) ) = ( ( 𝑅 ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 + 1 ) ) ) ) |
329 |
328
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( abs ‘ ( ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) / ( 𝑛 + 1 ) ) ) = ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 + 1 ) ) ) ) ) |
330 |
288
|
rprege0d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ( 𝑛 + 1 ) ∈ ℝ ∧ 0 ≤ ( 𝑛 + 1 ) ) ) |
331 |
|
absid |
⊢ ( ( ( 𝑛 + 1 ) ∈ ℝ ∧ 0 ≤ ( 𝑛 + 1 ) ) → ( abs ‘ ( 𝑛 + 1 ) ) = ( 𝑛 + 1 ) ) |
332 |
330 331
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( abs ‘ ( 𝑛 + 1 ) ) = ( 𝑛 + 1 ) ) |
333 |
332
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) / ( abs ‘ ( 𝑛 + 1 ) ) ) = ( ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) / ( 𝑛 + 1 ) ) ) |
334 |
324 329 333
|
3eqtr3rd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / 𝑛 ) ) / ( 𝑛 + 1 ) ) = ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 + 1 ) ) ) ) ) |
335 |
323 272 334
|
3brtr3d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ) → ( 𝐸 · ( 1 / ( 𝑛 + 1 ) ) ) ≤ ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 + 1 ) ) ) ) ) |
336 |
67 274 283 335
|
fsumle |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 𝐸 · ( 1 / ( 𝑛 + 1 ) ) ) ≤ Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 + 1 ) ) ) ) ) |
337 |
1 2 3 4 5 6 7 8 9
|
pntpbnd1 |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( abs ‘ ( ( 𝑅 ‘ 𝑛 ) / ( 𝑛 · ( 𝑛 + 1 ) ) ) ) ≤ 𝐴 ) |
338 |
275 284 38 336 337
|
letrd |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 𝐸 · ( 1 / ( 𝑛 + 1 ) ) ) ≤ 𝐴 ) |
339 |
267 338
|
eqbrtrd |
⊢ ( 𝜑 → ( 𝐸 · Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) ≤ 𝐴 ) |
340 |
93 38 18
|
lemuldiv2d |
⊢ ( 𝜑 → ( ( 𝐸 · Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ) ≤ 𝐴 ↔ Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ≤ ( 𝐴 / 𝐸 ) ) ) |
341 |
339 340
|
mpbid |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( ( ( ⌊ ‘ 𝑌 ) + 1 ) ... ( ⌊ ‘ ( 𝐾 · 𝑌 ) ) ) ( 1 / ( 𝑛 + 1 ) ) ≤ ( 𝐴 / 𝐸 ) ) |
342 |
60 93 39 265 341
|
letrd |
⊢ ( 𝜑 → ( ( log ‘ 𝐾 ) − 2 ) ≤ ( 𝐴 / 𝐸 ) ) |
343 |
41 60 39 66 342
|
letrd |
⊢ ( 𝜑 → ( ( 𝐶 / 𝐸 ) − 2 ) ≤ ( 𝐴 / 𝐸 ) ) |
344 |
37 12 39 343
|
subled |
⊢ ( 𝜑 → ( ( 𝐶 / 𝐸 ) − ( 𝐴 / 𝐸 ) ) ≤ 2 ) |
345 |
35 344
|
eqbrtrd |
⊢ ( 𝜑 → ( 2 / 𝐸 ) ≤ 2 ) |
346 |
12 18 20 345
|
lediv23d |
⊢ ( 𝜑 → ( 2 / 2 ) ≤ 𝐸 ) |
347 |
10 346
|
eqbrtrrid |
⊢ ( 𝜑 → 1 ≤ 𝐸 ) |
348 |
16
|
simprd |
⊢ ( 𝜑 → 𝐸 < 1 ) |
349 |
|
ltnle |
⊢ ( ( 𝐸 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 𝐸 < 1 ↔ ¬ 1 ≤ 𝐸 ) ) |
350 |
14 48 349
|
sylancl |
⊢ ( 𝜑 → ( 𝐸 < 1 ↔ ¬ 1 ≤ 𝐸 ) ) |
351 |
348 350
|
mpbid |
⊢ ( 𝜑 → ¬ 1 ≤ 𝐸 ) |
352 |
347 351
|
pm2.65i |
⊢ ¬ 𝜑 |