Step |
Hyp |
Ref |
Expression |
1 |
|
2vmadivsum.1 |
|- ( ph -> A e. RR+ ) |
2 |
|
2vmadivsum.2 |
|- ( ph -> A. y e. ( 1 [,) +oo ) ( abs ` ( sum_ i e. ( 1 ... ( |_ ` y ) ) ( ( Lam ` i ) / i ) - ( log ` y ) ) ) <_ A ) |
3 |
|
vmalogdivsum2 |
|- ( x e. ( 1 (,) +oo ) |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) ) e. O(1) |
4 |
3
|
a1i |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) ) e. O(1) ) |
5 |
|
fzfid |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( 1 ... ( |_ ` x ) ) e. Fin ) |
6 |
|
elfznn |
|- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. NN ) |
7 |
6
|
adantl |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. NN ) |
8 |
|
vmacl |
|- ( n e. NN -> ( Lam ` n ) e. RR ) |
9 |
7 8
|
syl |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( Lam ` n ) e. RR ) |
10 |
9 7
|
nndivred |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( Lam ` n ) / n ) e. RR ) |
11 |
|
fzfid |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 ... ( |_ ` ( x / n ) ) ) e. Fin ) |
12 |
|
elfznn |
|- ( m e. ( 1 ... ( |_ ` ( x / n ) ) ) -> m e. NN ) |
13 |
12
|
adantl |
|- ( ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ) -> m e. NN ) |
14 |
|
vmacl |
|- ( m e. NN -> ( Lam ` m ) e. RR ) |
15 |
13 14
|
syl |
|- ( ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ) -> ( Lam ` m ) e. RR ) |
16 |
15 13
|
nndivred |
|- ( ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ) -> ( ( Lam ` m ) / m ) e. RR ) |
17 |
11 16
|
fsumrecl |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) e. RR ) |
18 |
10 17
|
remulcld |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) e. RR ) |
19 |
5 18
|
fsumrecl |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) e. RR ) |
20 |
|
elioore |
|- ( x e. ( 1 (,) +oo ) -> x e. RR ) |
21 |
20
|
adantl |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> x e. RR ) |
22 |
|
eliooord |
|- ( x e. ( 1 (,) +oo ) -> ( 1 < x /\ x < +oo ) ) |
23 |
22
|
adantl |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( 1 < x /\ x < +oo ) ) |
24 |
23
|
simpld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> 1 < x ) |
25 |
21 24
|
rplogcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( log ` x ) e. RR+ ) |
26 |
19 25
|
rerpdivcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) / ( log ` x ) ) e. RR ) |
27 |
|
1rp |
|- 1 e. RR+ |
28 |
27
|
a1i |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> 1 e. RR+ ) |
29 |
|
1red |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> 1 e. RR ) |
30 |
29 21 24
|
ltled |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> 1 <_ x ) |
31 |
21 28 30
|
rpgecld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> x e. RR+ ) |
32 |
31
|
relogcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( log ` x ) e. RR ) |
33 |
32
|
rehalfcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( log ` x ) / 2 ) e. RR ) |
34 |
26 33
|
resubcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) e. RR ) |
35 |
34
|
recnd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) e. CC ) |
36 |
31
|
adantr |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> x e. RR+ ) |
37 |
7
|
nnrpd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. RR+ ) |
38 |
36 37
|
rpdivcld |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR+ ) |
39 |
38
|
relogcld |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` ( x / n ) ) e. RR ) |
40 |
10 39
|
remulcld |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) e. RR ) |
41 |
5 40
|
fsumrecl |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) e. RR ) |
42 |
41 25
|
rerpdivcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) / ( log ` x ) ) e. RR ) |
43 |
42 33
|
resubcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) e. RR ) |
44 |
43
|
recnd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) e. CC ) |
45 |
19
|
recnd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) e. CC ) |
46 |
41
|
recnd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) e. CC ) |
47 |
32
|
recnd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( log ` x ) e. CC ) |
48 |
25
|
rpne0d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( log ` x ) =/= 0 ) |
49 |
45 46 47 48
|
divsubdird |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) ) / ( log ` x ) ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) / ( log ` x ) ) - ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) / ( log ` x ) ) ) ) |
50 |
10
|
recnd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( Lam ` n ) / n ) e. CC ) |
51 |
17
|
recnd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) e. CC ) |
52 |
39
|
recnd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` ( x / n ) ) e. CC ) |
53 |
50 51 52
|
subdid |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) = ( ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) - ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) ) ) |
54 |
53
|
sumeq2dv |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) - ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) ) ) |
55 |
18
|
recnd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) e. CC ) |
56 |
40
|
recnd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) e. CC ) |
57 |
5 55 56
|
fsumsub |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) - ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) ) ) |
58 |
54 57
|
eqtrd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) ) ) |
59 |
58
|
oveq1d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) / ( log ` x ) ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) ) / ( log ` x ) ) ) |
60 |
26
|
recnd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) / ( log ` x ) ) e. CC ) |
61 |
42
|
recnd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) / ( log ` x ) ) e. CC ) |
62 |
33
|
recnd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( log ` x ) / 2 ) e. CC ) |
63 |
60 61 62
|
nnncan2d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) / ( log ` x ) ) - ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) / ( log ` x ) ) ) ) |
64 |
49 59 63
|
3eqtr4d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) / ( log ` x ) ) = ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) ) ) |
65 |
64
|
mpteq2dva |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) / ( log ` x ) ) ) = ( x e. ( 1 (,) +oo ) |-> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) ) ) ) |
66 |
|
1red |
|- ( ph -> 1 e. RR ) |
67 |
5 10
|
fsumrecl |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) e. RR ) |
68 |
67 25
|
rerpdivcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) e. RR ) |
69 |
1
|
rpred |
|- ( ph -> A e. RR ) |
70 |
69
|
adantr |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> A e. RR ) |
71 |
|
ioossre |
|- ( 1 (,) +oo ) C_ RR |
72 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
73 |
|
o1const |
|- ( ( ( 1 (,) +oo ) C_ RR /\ 1 e. CC ) -> ( x e. ( 1 (,) +oo ) |-> 1 ) e. O(1) ) |
74 |
71 72 73
|
sylancr |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> 1 ) e. O(1) ) |
75 |
68
|
recnd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) e. CC ) |
76 |
|
1cnd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> 1 e. CC ) |
77 |
67
|
recnd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) e. CC ) |
78 |
77 47 47 48
|
divsubdird |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) - ( log ` x ) ) / ( log ` x ) ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) - ( ( log ` x ) / ( log ` x ) ) ) ) |
79 |
77 47
|
subcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) - ( log ` x ) ) e. CC ) |
80 |
79 47 48
|
divrecd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) - ( log ` x ) ) / ( log ` x ) ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) - ( log ` x ) ) x. ( 1 / ( log ` x ) ) ) ) |
81 |
47 48
|
dividd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( log ` x ) / ( log ` x ) ) = 1 ) |
82 |
81
|
oveq2d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) - ( ( log ` x ) / ( log ` x ) ) ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) - 1 ) ) |
83 |
78 80 82
|
3eqtr3d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) - ( log ` x ) ) x. ( 1 / ( log ` x ) ) ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) - 1 ) ) |
84 |
83
|
mpteq2dva |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) - ( log ` x ) ) x. ( 1 / ( log ` x ) ) ) ) = ( x e. ( 1 (,) +oo ) |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) - 1 ) ) ) |
85 |
67 32
|
resubcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) - ( log ` x ) ) e. RR ) |
86 |
29 25
|
rerpdivcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( 1 / ( log ` x ) ) e. RR ) |
87 |
31
|
ex |
|- ( ph -> ( x e. ( 1 (,) +oo ) -> x e. RR+ ) ) |
88 |
87
|
ssrdv |
|- ( ph -> ( 1 (,) +oo ) C_ RR+ ) |
89 |
|
vmadivsum |
|- ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) - ( log ` x ) ) ) e. O(1) |
90 |
89
|
a1i |
|- ( ph -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) - ( log ` x ) ) ) e. O(1) ) |
91 |
88 90
|
o1res2 |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) - ( log ` x ) ) ) e. O(1) ) |
92 |
|
divlogrlim |
|- ( x e. ( 1 (,) +oo ) |-> ( 1 / ( log ` x ) ) ) ~~>r 0 |
93 |
|
rlimo1 |
|- ( ( x e. ( 1 (,) +oo ) |-> ( 1 / ( log ` x ) ) ) ~~>r 0 -> ( x e. ( 1 (,) +oo ) |-> ( 1 / ( log ` x ) ) ) e. O(1) ) |
94 |
92 93
|
mp1i |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> ( 1 / ( log ` x ) ) ) e. O(1) ) |
95 |
85 86 91 94
|
o1mul2 |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) - ( log ` x ) ) x. ( 1 / ( log ` x ) ) ) ) e. O(1) ) |
96 |
84 95
|
eqeltrrd |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) - 1 ) ) e. O(1) ) |
97 |
75 76 96
|
o1dif |
|- ( ph -> ( ( x e. ( 1 (,) +oo ) |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) ) e. O(1) <-> ( x e. ( 1 (,) +oo ) |-> 1 ) e. O(1) ) ) |
98 |
74 97
|
mpbird |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) ) e. O(1) ) |
99 |
69
|
recnd |
|- ( ph -> A e. CC ) |
100 |
|
o1const |
|- ( ( ( 1 (,) +oo ) C_ RR /\ A e. CC ) -> ( x e. ( 1 (,) +oo ) |-> A ) e. O(1) ) |
101 |
71 99 100
|
sylancr |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> A ) e. O(1) ) |
102 |
68 70 98 101
|
o1mul2 |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) x. A ) ) e. O(1) ) |
103 |
68 70
|
remulcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) x. A ) e. RR ) |
104 |
17 39
|
resubcld |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) e. RR ) |
105 |
10 104
|
remulcld |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) e. RR ) |
106 |
5 105
|
fsumrecl |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) e. RR ) |
107 |
106
|
recnd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) e. CC ) |
108 |
107 47 48
|
divcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) / ( log ` x ) ) e. CC ) |
109 |
107
|
abscld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( abs ` sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) e. RR ) |
110 |
67 70
|
remulcld |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) x. A ) e. RR ) |
111 |
105
|
recnd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) e. CC ) |
112 |
111
|
abscld |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) e. RR ) |
113 |
5 112
|
fsumrecl |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( abs ` ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) e. RR ) |
114 |
5 111
|
fsumabs |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( abs ` sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) <_ sum_ n e. ( 1 ... ( |_ ` x ) ) ( abs ` ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) ) |
115 |
70
|
adantr |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> A e. RR ) |
116 |
10 115
|
remulcld |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( Lam ` n ) / n ) x. A ) e. RR ) |
117 |
104
|
recnd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) e. CC ) |
118 |
50 117
|
absmuld |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) = ( ( abs ` ( ( Lam ` n ) / n ) ) x. ( abs ` ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) ) |
119 |
|
vmage0 |
|- ( n e. NN -> 0 <_ ( Lam ` n ) ) |
120 |
7 119
|
syl |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> 0 <_ ( Lam ` n ) ) |
121 |
9 37 120
|
divge0d |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> 0 <_ ( ( Lam ` n ) / n ) ) |
122 |
10 121
|
absidd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( Lam ` n ) / n ) ) = ( ( Lam ` n ) / n ) ) |
123 |
122
|
oveq1d |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( ( Lam ` n ) / n ) ) x. ( abs ` ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) = ( ( ( Lam ` n ) / n ) x. ( abs ` ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) ) |
124 |
118 123
|
eqtrd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) = ( ( ( Lam ` n ) / n ) x. ( abs ` ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) ) |
125 |
117
|
abscld |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) e. RR ) |
126 |
|
fveq2 |
|- ( i = m -> ( Lam ` i ) = ( Lam ` m ) ) |
127 |
|
id |
|- ( i = m -> i = m ) |
128 |
126 127
|
oveq12d |
|- ( i = m -> ( ( Lam ` i ) / i ) = ( ( Lam ` m ) / m ) ) |
129 |
128
|
cbvsumv |
|- sum_ i e. ( 1 ... ( |_ ` y ) ) ( ( Lam ` i ) / i ) = sum_ m e. ( 1 ... ( |_ ` y ) ) ( ( Lam ` m ) / m ) |
130 |
|
fveq2 |
|- ( y = ( x / n ) -> ( |_ ` y ) = ( |_ ` ( x / n ) ) ) |
131 |
130
|
oveq2d |
|- ( y = ( x / n ) -> ( 1 ... ( |_ ` y ) ) = ( 1 ... ( |_ ` ( x / n ) ) ) ) |
132 |
131
|
sumeq1d |
|- ( y = ( x / n ) -> sum_ m e. ( 1 ... ( |_ ` y ) ) ( ( Lam ` m ) / m ) = sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) |
133 |
129 132
|
syl5eq |
|- ( y = ( x / n ) -> sum_ i e. ( 1 ... ( |_ ` y ) ) ( ( Lam ` i ) / i ) = sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) |
134 |
|
fveq2 |
|- ( y = ( x / n ) -> ( log ` y ) = ( log ` ( x / n ) ) ) |
135 |
133 134
|
oveq12d |
|- ( y = ( x / n ) -> ( sum_ i e. ( 1 ... ( |_ ` y ) ) ( ( Lam ` i ) / i ) - ( log ` y ) ) = ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) |
136 |
135
|
fveq2d |
|- ( y = ( x / n ) -> ( abs ` ( sum_ i e. ( 1 ... ( |_ ` y ) ) ( ( Lam ` i ) / i ) - ( log ` y ) ) ) = ( abs ` ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) |
137 |
136
|
breq1d |
|- ( y = ( x / n ) -> ( ( abs ` ( sum_ i e. ( 1 ... ( |_ ` y ) ) ( ( Lam ` i ) / i ) - ( log ` y ) ) ) <_ A <-> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) <_ A ) ) |
138 |
2
|
ad2antrr |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> A. y e. ( 1 [,) +oo ) ( abs ` ( sum_ i e. ( 1 ... ( |_ ` y ) ) ( ( Lam ` i ) / i ) - ( log ` y ) ) ) <_ A ) |
139 |
38
|
rpred |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR ) |
140 |
7
|
nncnd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. CC ) |
141 |
140
|
mulid2d |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 x. n ) = n ) |
142 |
|
fznnfl |
|- ( x e. RR -> ( n e. ( 1 ... ( |_ ` x ) ) <-> ( n e. NN /\ n <_ x ) ) ) |
143 |
21 142
|
syl |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( n e. ( 1 ... ( |_ ` x ) ) <-> ( n e. NN /\ n <_ x ) ) ) |
144 |
143
|
simplbda |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n <_ x ) |
145 |
141 144
|
eqbrtrd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 x. n ) <_ x ) |
146 |
|
1red |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> 1 e. RR ) |
147 |
21
|
adantr |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> x e. RR ) |
148 |
146 147 37
|
lemuldivd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( 1 x. n ) <_ x <-> 1 <_ ( x / n ) ) ) |
149 |
145 148
|
mpbid |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> 1 <_ ( x / n ) ) |
150 |
|
1re |
|- 1 e. RR |
151 |
|
elicopnf |
|- ( 1 e. RR -> ( ( x / n ) e. ( 1 [,) +oo ) <-> ( ( x / n ) e. RR /\ 1 <_ ( x / n ) ) ) ) |
152 |
150 151
|
ax-mp |
|- ( ( x / n ) e. ( 1 [,) +oo ) <-> ( ( x / n ) e. RR /\ 1 <_ ( x / n ) ) ) |
153 |
139 149 152
|
sylanbrc |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. ( 1 [,) +oo ) ) |
154 |
137 138 153
|
rspcdva |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) <_ A ) |
155 |
125 115 10 121 154
|
lemul2ad |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( Lam ` n ) / n ) x. ( abs ` ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) <_ ( ( ( Lam ` n ) / n ) x. A ) ) |
156 |
124 155
|
eqbrtrd |
|- ( ( ( ph /\ x e. ( 1 (,) +oo ) ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) <_ ( ( ( Lam ` n ) / n ) x. A ) ) |
157 |
5 112 116 156
|
fsumle |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( abs ` ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) <_ sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. A ) ) |
158 |
99
|
adantr |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> A e. CC ) |
159 |
5 158 50
|
fsummulc1 |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) x. A ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. A ) ) |
160 |
157 159
|
breqtrrd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( abs ` ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) <_ ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) x. A ) ) |
161 |
109 113 110 114 160
|
letrd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( abs ` sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) <_ ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) x. A ) ) |
162 |
109 110 25 161
|
lediv1dd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( abs ` sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) / ( log ` x ) ) <_ ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) x. A ) / ( log ` x ) ) ) |
163 |
107 47 48
|
absdivd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) / ( log ` x ) ) ) = ( ( abs ` sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) / ( abs ` ( log ` x ) ) ) ) |
164 |
25
|
rpge0d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> 0 <_ ( log ` x ) ) |
165 |
32 164
|
absidd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( abs ` ( log ` x ) ) = ( log ` x ) ) |
166 |
165
|
oveq2d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( abs ` sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) / ( abs ` ( log ` x ) ) ) = ( ( abs ` sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) / ( log ` x ) ) ) |
167 |
163 166
|
eqtrd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) / ( log ` x ) ) ) = ( ( abs ` sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) ) / ( log ` x ) ) ) |
168 |
5 10 121
|
fsumge0 |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> 0 <_ sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) ) |
169 |
67 25 168
|
divge0d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> 0 <_ ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) ) |
170 |
1
|
adantr |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> A e. RR+ ) |
171 |
170
|
rpge0d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> 0 <_ A ) |
172 |
68 70 169 171
|
mulge0d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> 0 <_ ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) x. A ) ) |
173 |
103 172
|
absidd |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( abs ` ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) x. A ) ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) x. A ) ) |
174 |
77 158 47 48
|
div23d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) x. A ) / ( log ` x ) ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) x. A ) ) |
175 |
173 174
|
eqtr4d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( abs ` ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) x. A ) ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) x. A ) / ( log ` x ) ) ) |
176 |
162 167 175
|
3brtr4d |
|- ( ( ph /\ x e. ( 1 (,) +oo ) ) -> ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) / ( log ` x ) ) ) <_ ( abs ` ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) x. A ) ) ) |
177 |
176
|
adantrr |
|- ( ( ph /\ ( x e. ( 1 (,) +oo ) /\ 1 <_ x ) ) -> ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) / ( log ` x ) ) ) <_ ( abs ` ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( Lam ` n ) / n ) / ( log ` x ) ) x. A ) ) ) |
178 |
66 102 103 108 177
|
o1le |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) - ( log ` ( x / n ) ) ) ) / ( log ` x ) ) ) e. O(1) ) |
179 |
65 178
|
eqeltrrd |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) ) ) e. O(1) ) |
180 |
35 44 179
|
o1dif |
|- ( ph -> ( ( x e. ( 1 (,) +oo ) |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) ) e. O(1) <-> ( x e. ( 1 (,) +oo ) |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. ( log ` ( x / n ) ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) ) e. O(1) ) ) |
181 |
4 180
|
mpbird |
|- ( ph -> ( x e. ( 1 (,) +oo ) |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( Lam ` n ) / n ) x. sum_ m e. ( 1 ... ( |_ ` ( x / n ) ) ) ( ( Lam ` m ) / m ) ) / ( log ` x ) ) - ( ( log ` x ) / 2 ) ) ) e. O(1) ) |