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