Step |
Hyp |
Ref |
Expression |
1 |
|
rpre |
|- ( A e. RR+ -> A e. RR ) |
2 |
|
reefcl |
|- ( A e. RR -> ( exp ` A ) e. RR ) |
3 |
1 2
|
syl |
|- ( A e. RR+ -> ( exp ` A ) e. RR ) |
4 |
|
efgt1 |
|- ( A e. RR+ -> 1 < ( exp ` A ) ) |
5 |
|
cxp2limlem |
|- ( ( ( exp ` A ) e. RR /\ 1 < ( exp ` A ) ) -> ( m e. RR+ |-> ( m / ( ( exp ` A ) ^c m ) ) ) ~~>r 0 ) |
6 |
3 4 5
|
syl2anc |
|- ( A e. RR+ -> ( m e. RR+ |-> ( m / ( ( exp ` A ) ^c m ) ) ) ~~>r 0 ) |
7 |
|
reefcl |
|- ( z e. RR -> ( exp ` z ) e. RR ) |
8 |
7
|
adantl |
|- ( ( A e. RR+ /\ z e. RR ) -> ( exp ` z ) e. RR ) |
9 |
|
1re |
|- 1 e. RR |
10 |
|
ifcl |
|- ( ( ( exp ` z ) e. RR /\ 1 e. RR ) -> if ( 1 <_ ( exp ` z ) , ( exp ` z ) , 1 ) e. RR ) |
11 |
8 9 10
|
sylancl |
|- ( ( A e. RR+ /\ z e. RR ) -> if ( 1 <_ ( exp ` z ) , ( exp ` z ) , 1 ) e. RR ) |
12 |
|
rpre |
|- ( n e. RR+ -> n e. RR ) |
13 |
|
maxlt |
|- ( ( 1 e. RR /\ ( exp ` z ) e. RR /\ n e. RR ) -> ( if ( 1 <_ ( exp ` z ) , ( exp ` z ) , 1 ) < n <-> ( 1 < n /\ ( exp ` z ) < n ) ) ) |
14 |
9 8 12 13
|
mp3an3an |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ n e. RR+ ) -> ( if ( 1 <_ ( exp ` z ) , ( exp ` z ) , 1 ) < n <-> ( 1 < n /\ ( exp ` z ) < n ) ) ) |
15 |
|
simprrr |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( exp ` z ) < n ) |
16 |
|
reeflog |
|- ( n e. RR+ -> ( exp ` ( log ` n ) ) = n ) |
17 |
16
|
ad2antrl |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( exp ` ( log ` n ) ) = n ) |
18 |
15 17
|
breqtrrd |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( exp ` z ) < ( exp ` ( log ` n ) ) ) |
19 |
|
simplr |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> z e. RR ) |
20 |
12
|
ad2antrl |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> n e. RR ) |
21 |
|
simprrl |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> 1 < n ) |
22 |
20 21
|
rplogcld |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( log ` n ) e. RR+ ) |
23 |
22
|
rpred |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( log ` n ) e. RR ) |
24 |
|
eflt |
|- ( ( z e. RR /\ ( log ` n ) e. RR ) -> ( z < ( log ` n ) <-> ( exp ` z ) < ( exp ` ( log ` n ) ) ) ) |
25 |
19 23 24
|
syl2anc |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( z < ( log ` n ) <-> ( exp ` z ) < ( exp ` ( log ` n ) ) ) ) |
26 |
18 25
|
mpbird |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> z < ( log ` n ) ) |
27 |
|
breq2 |
|- ( m = ( log ` n ) -> ( z < m <-> z < ( log ` n ) ) ) |
28 |
|
id |
|- ( m = ( log ` n ) -> m = ( log ` n ) ) |
29 |
|
oveq2 |
|- ( m = ( log ` n ) -> ( ( exp ` A ) ^c m ) = ( ( exp ` A ) ^c ( log ` n ) ) ) |
30 |
28 29
|
oveq12d |
|- ( m = ( log ` n ) -> ( m / ( ( exp ` A ) ^c m ) ) = ( ( log ` n ) / ( ( exp ` A ) ^c ( log ` n ) ) ) ) |
31 |
30
|
fveq2d |
|- ( m = ( log ` n ) -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) = ( abs ` ( ( log ` n ) / ( ( exp ` A ) ^c ( log ` n ) ) ) ) ) |
32 |
31
|
breq1d |
|- ( m = ( log ` n ) -> ( ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x <-> ( abs ` ( ( log ` n ) / ( ( exp ` A ) ^c ( log ` n ) ) ) ) < x ) ) |
33 |
27 32
|
imbi12d |
|- ( m = ( log ` n ) -> ( ( z < m -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x ) <-> ( z < ( log ` n ) -> ( abs ` ( ( log ` n ) / ( ( exp ` A ) ^c ( log ` n ) ) ) ) < x ) ) ) |
34 |
33
|
rspcv |
|- ( ( log ` n ) e. RR+ -> ( A. m e. RR+ ( z < m -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x ) -> ( z < ( log ` n ) -> ( abs ` ( ( log ` n ) / ( ( exp ` A ) ^c ( log ` n ) ) ) ) < x ) ) ) |
35 |
22 34
|
syl |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( A. m e. RR+ ( z < m -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x ) -> ( z < ( log ` n ) -> ( abs ` ( ( log ` n ) / ( ( exp ` A ) ^c ( log ` n ) ) ) ) < x ) ) ) |
36 |
26 35
|
mpid |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( A. m e. RR+ ( z < m -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x ) -> ( abs ` ( ( log ` n ) / ( ( exp ` A ) ^c ( log ` n ) ) ) ) < x ) ) |
37 |
1
|
ad2antrr |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> A e. RR ) |
38 |
37
|
relogefd |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( log ` ( exp ` A ) ) = A ) |
39 |
38
|
oveq2d |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( ( log ` n ) x. ( log ` ( exp ` A ) ) ) = ( ( log ` n ) x. A ) ) |
40 |
22
|
rpcnd |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( log ` n ) e. CC ) |
41 |
|
rpcn |
|- ( A e. RR+ -> A e. CC ) |
42 |
41
|
ad2antrr |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> A e. CC ) |
43 |
40 42
|
mulcomd |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( ( log ` n ) x. A ) = ( A x. ( log ` n ) ) ) |
44 |
39 43
|
eqtrd |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( ( log ` n ) x. ( log ` ( exp ` A ) ) ) = ( A x. ( log ` n ) ) ) |
45 |
44
|
fveq2d |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( exp ` ( ( log ` n ) x. ( log ` ( exp ` A ) ) ) ) = ( exp ` ( A x. ( log ` n ) ) ) ) |
46 |
3
|
ad2antrr |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( exp ` A ) e. RR ) |
47 |
46
|
recnd |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( exp ` A ) e. CC ) |
48 |
|
efne0 |
|- ( A e. CC -> ( exp ` A ) =/= 0 ) |
49 |
42 48
|
syl |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( exp ` A ) =/= 0 ) |
50 |
47 49 40
|
cxpefd |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( ( exp ` A ) ^c ( log ` n ) ) = ( exp ` ( ( log ` n ) x. ( log ` ( exp ` A ) ) ) ) ) |
51 |
|
rpcn |
|- ( n e. RR+ -> n e. CC ) |
52 |
51
|
ad2antrl |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> n e. CC ) |
53 |
|
rpne0 |
|- ( n e. RR+ -> n =/= 0 ) |
54 |
53
|
ad2antrl |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> n =/= 0 ) |
55 |
52 54 42
|
cxpefd |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( n ^c A ) = ( exp ` ( A x. ( log ` n ) ) ) ) |
56 |
45 50 55
|
3eqtr4d |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( ( exp ` A ) ^c ( log ` n ) ) = ( n ^c A ) ) |
57 |
56
|
oveq2d |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( ( log ` n ) / ( ( exp ` A ) ^c ( log ` n ) ) ) = ( ( log ` n ) / ( n ^c A ) ) ) |
58 |
57
|
fveq2d |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( abs ` ( ( log ` n ) / ( ( exp ` A ) ^c ( log ` n ) ) ) ) = ( abs ` ( ( log ` n ) / ( n ^c A ) ) ) ) |
59 |
58
|
breq1d |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( ( abs ` ( ( log ` n ) / ( ( exp ` A ) ^c ( log ` n ) ) ) ) < x <-> ( abs ` ( ( log ` n ) / ( n ^c A ) ) ) < x ) ) |
60 |
36 59
|
sylibd |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ ( n e. RR+ /\ ( 1 < n /\ ( exp ` z ) < n ) ) ) -> ( A. m e. RR+ ( z < m -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x ) -> ( abs ` ( ( log ` n ) / ( n ^c A ) ) ) < x ) ) |
61 |
60
|
expr |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ n e. RR+ ) -> ( ( 1 < n /\ ( exp ` z ) < n ) -> ( A. m e. RR+ ( z < m -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x ) -> ( abs ` ( ( log ` n ) / ( n ^c A ) ) ) < x ) ) ) |
62 |
14 61
|
sylbid |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ n e. RR+ ) -> ( if ( 1 <_ ( exp ` z ) , ( exp ` z ) , 1 ) < n -> ( A. m e. RR+ ( z < m -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x ) -> ( abs ` ( ( log ` n ) / ( n ^c A ) ) ) < x ) ) ) |
63 |
62
|
com23 |
|- ( ( ( A e. RR+ /\ z e. RR ) /\ n e. RR+ ) -> ( A. m e. RR+ ( z < m -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x ) -> ( if ( 1 <_ ( exp ` z ) , ( exp ` z ) , 1 ) < n -> ( abs ` ( ( log ` n ) / ( n ^c A ) ) ) < x ) ) ) |
64 |
63
|
ralrimdva |
|- ( ( A e. RR+ /\ z e. RR ) -> ( A. m e. RR+ ( z < m -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x ) -> A. n e. RR+ ( if ( 1 <_ ( exp ` z ) , ( exp ` z ) , 1 ) < n -> ( abs ` ( ( log ` n ) / ( n ^c A ) ) ) < x ) ) ) |
65 |
|
breq1 |
|- ( y = if ( 1 <_ ( exp ` z ) , ( exp ` z ) , 1 ) -> ( y < n <-> if ( 1 <_ ( exp ` z ) , ( exp ` z ) , 1 ) < n ) ) |
66 |
65
|
rspceaimv |
|- ( ( if ( 1 <_ ( exp ` z ) , ( exp ` z ) , 1 ) e. RR /\ A. n e. RR+ ( if ( 1 <_ ( exp ` z ) , ( exp ` z ) , 1 ) < n -> ( abs ` ( ( log ` n ) / ( n ^c A ) ) ) < x ) ) -> E. y e. RR A. n e. RR+ ( y < n -> ( abs ` ( ( log ` n ) / ( n ^c A ) ) ) < x ) ) |
67 |
11 64 66
|
syl6an |
|- ( ( A e. RR+ /\ z e. RR ) -> ( A. m e. RR+ ( z < m -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x ) -> E. y e. RR A. n e. RR+ ( y < n -> ( abs ` ( ( log ` n ) / ( n ^c A ) ) ) < x ) ) ) |
68 |
67
|
rexlimdva |
|- ( A e. RR+ -> ( E. z e. RR A. m e. RR+ ( z < m -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x ) -> E. y e. RR A. n e. RR+ ( y < n -> ( abs ` ( ( log ` n ) / ( n ^c A ) ) ) < x ) ) ) |
69 |
68
|
ralimdv |
|- ( A e. RR+ -> ( A. x e. RR+ E. z e. RR A. m e. RR+ ( z < m -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x ) -> A. x e. RR+ E. y e. RR A. n e. RR+ ( y < n -> ( abs ` ( ( log ` n ) / ( n ^c A ) ) ) < x ) ) ) |
70 |
|
simpr |
|- ( ( A e. RR+ /\ m e. RR+ ) -> m e. RR+ ) |
71 |
1
|
adantr |
|- ( ( A e. RR+ /\ m e. RR+ ) -> A e. RR ) |
72 |
71
|
rpefcld |
|- ( ( A e. RR+ /\ m e. RR+ ) -> ( exp ` A ) e. RR+ ) |
73 |
|
rpre |
|- ( m e. RR+ -> m e. RR ) |
74 |
73
|
adantl |
|- ( ( A e. RR+ /\ m e. RR+ ) -> m e. RR ) |
75 |
72 74
|
rpcxpcld |
|- ( ( A e. RR+ /\ m e. RR+ ) -> ( ( exp ` A ) ^c m ) e. RR+ ) |
76 |
70 75
|
rpdivcld |
|- ( ( A e. RR+ /\ m e. RR+ ) -> ( m / ( ( exp ` A ) ^c m ) ) e. RR+ ) |
77 |
76
|
rpcnd |
|- ( ( A e. RR+ /\ m e. RR+ ) -> ( m / ( ( exp ` A ) ^c m ) ) e. CC ) |
78 |
77
|
ralrimiva |
|- ( A e. RR+ -> A. m e. RR+ ( m / ( ( exp ` A ) ^c m ) ) e. CC ) |
79 |
|
rpssre |
|- RR+ C_ RR |
80 |
79
|
a1i |
|- ( A e. RR+ -> RR+ C_ RR ) |
81 |
78 80
|
rlim0lt |
|- ( A e. RR+ -> ( ( m e. RR+ |-> ( m / ( ( exp ` A ) ^c m ) ) ) ~~>r 0 <-> A. x e. RR+ E. z e. RR A. m e. RR+ ( z < m -> ( abs ` ( m / ( ( exp ` A ) ^c m ) ) ) < x ) ) ) |
82 |
|
relogcl |
|- ( n e. RR+ -> ( log ` n ) e. RR ) |
83 |
82
|
adantl |
|- ( ( A e. RR+ /\ n e. RR+ ) -> ( log ` n ) e. RR ) |
84 |
|
simpr |
|- ( ( A e. RR+ /\ n e. RR+ ) -> n e. RR+ ) |
85 |
1
|
adantr |
|- ( ( A e. RR+ /\ n e. RR+ ) -> A e. RR ) |
86 |
84 85
|
rpcxpcld |
|- ( ( A e. RR+ /\ n e. RR+ ) -> ( n ^c A ) e. RR+ ) |
87 |
83 86
|
rerpdivcld |
|- ( ( A e. RR+ /\ n e. RR+ ) -> ( ( log ` n ) / ( n ^c A ) ) e. RR ) |
88 |
87
|
recnd |
|- ( ( A e. RR+ /\ n e. RR+ ) -> ( ( log ` n ) / ( n ^c A ) ) e. CC ) |
89 |
88
|
ralrimiva |
|- ( A e. RR+ -> A. n e. RR+ ( ( log ` n ) / ( n ^c A ) ) e. CC ) |
90 |
89 80
|
rlim0lt |
|- ( A e. RR+ -> ( ( n e. RR+ |-> ( ( log ` n ) / ( n ^c A ) ) ) ~~>r 0 <-> A. x e. RR+ E. y e. RR A. n e. RR+ ( y < n -> ( abs ` ( ( log ` n ) / ( n ^c A ) ) ) < x ) ) ) |
91 |
69 81 90
|
3imtr4d |
|- ( A e. RR+ -> ( ( m e. RR+ |-> ( m / ( ( exp ` A ) ^c m ) ) ) ~~>r 0 -> ( n e. RR+ |-> ( ( log ` n ) / ( n ^c A ) ) ) ~~>r 0 ) ) |
92 |
6 91
|
mpd |
|- ( A e. RR+ -> ( n e. RR+ |-> ( ( log ` n ) / ( n ^c A ) ) ) ~~>r 0 ) |