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Theorem cxploglim 20818
Description: The logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 18-Sep-2014.)
Assertion
Ref Expression
cxploglim  |-  ( A  e.  RR+  ->  ( n  e.  RR+  |->  ( ( log `  n )  /  ( n  ^ c  A ) ) )  ~~> r  0 )
Distinct variable group:    A, n

Proof of Theorem cxploglim
Dummy variables  m  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rpre 10620 . . . 4  |-  ( A  e.  RR+  ->  A  e.  RR )
2 reefcl 12691 . . . 4  |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR )
31, 2syl 16 . . 3  |-  ( A  e.  RR+  ->  ( exp `  A )  e.  RR )
4 efgt1 12719 . . 3  |-  ( A  e.  RR+  ->  1  < 
( exp `  A
) )
5 cxp2limlem 20816 . . 3  |-  ( ( ( exp `  A
)  e.  RR  /\  1  <  ( exp `  A
) )  ->  (
m  e.  RR+  |->  ( m  /  ( ( exp `  A )  ^ c  m ) ) )  ~~> r  0 )
63, 4, 5syl2anc 644 . 2  |-  ( A  e.  RR+  ->  ( m  e.  RR+  |->  ( m  /  ( ( exp `  A )  ^ c  m ) ) )  ~~> r  0 )
7 reefcl 12691 . . . . . . . 8  |-  ( z  e.  RR  ->  ( exp `  z )  e.  RR )
87adantl 454 . . . . . . 7  |-  ( ( A  e.  RR+  /\  z  e.  RR )  ->  ( exp `  z )  e.  RR )
9 1re 9092 . . . . . . 7  |-  1  e.  RR
10 ifcl 3777 . . . . . . 7  |-  ( ( ( exp `  z
)  e.  RR  /\  1  e.  RR )  ->  if ( 1  <_ 
( exp `  z
) ,  ( exp `  z ) ,  1 )  e.  RR )
118, 9, 10sylancl 645 . . . . . 6  |-  ( ( A  e.  RR+  /\  z  e.  RR )  ->  if ( 1  <_  ( exp `  z ) ,  ( exp `  z
) ,  1 )  e.  RR )
129a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  n  e.  RR+ )  ->  1  e.  RR )
138adantr 453 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  n  e.  RR+ )  ->  ( exp `  z
)  e.  RR )
14 rpre 10620 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
1514adantl 454 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  n  e.  RR+ )  ->  n  e.  RR )
16 maxlt 10782 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( exp `  z )  e.  RR  /\  n  e.  RR )  ->  ( if ( 1  <_  ( exp `  z ) ,  ( exp `  z
) ,  1 )  <  n  <->  ( 1  <  n  /\  ( exp `  z )  < 
n ) ) )
1712, 13, 15, 16syl3anc 1185 . . . . . . . . 9  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  n  e.  RR+ )  ->  ( if ( 1  <_  ( exp `  z ) ,  ( exp `  z ) ,  1 )  < 
n  <->  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )
18 simprrr 743 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  z
)  <  n )
19 reeflog 20477 . . . . . . . . . . . . . . 15  |-  ( n  e.  RR+  ->  ( exp `  ( log `  n
) )  =  n )
2019ad2antrl 710 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  ( log `  n ) )  =  n )
2118, 20breqtrrd 4240 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  z
)  <  ( exp `  ( log `  n
) ) )
22 simplr 733 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
z  e.  RR )
2314ad2antrl 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  n  e.  RR )
24 simprrl 742 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
1  <  n )
2523, 24rplogcld 20526 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  RR+ )
2625rpred 10650 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  RR )
27 eflt 12720 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR  /\  ( log `  n )  e.  RR )  -> 
( z  <  ( log `  n )  <->  ( exp `  z )  <  ( exp `  ( log `  n
) ) ) )
2822, 26, 27syl2anc 644 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( z  <  ( log `  n )  <->  ( exp `  z )  <  ( exp `  ( log `  n
) ) ) )
2921, 28mpbird 225 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
z  <  ( log `  n ) )
30 breq2 4218 . . . . . . . . . . . . . . 15  |-  ( m  =  ( log `  n
)  ->  ( z  <  m  <->  z  <  ( log `  n ) ) )
31 id 21 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  ( log `  n
)  ->  m  =  ( log `  n ) )
32 oveq2 6091 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  ( log `  n
)  ->  ( ( exp `  A )  ^ c  m )  =  ( ( exp `  A
)  ^ c  ( log `  n ) ) )
3331, 32oveq12d 6101 . . . . . . . . . . . . . . . . 17  |-  ( m  =  ( log `  n
)  ->  ( m  /  ( ( exp `  A )  ^ c  m ) )  =  ( ( log `  n
)  /  ( ( exp `  A )  ^ c  ( log `  n ) ) ) )
3433fveq2d 5734 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( log `  n
)  ->  ( abs `  ( m  /  (
( exp `  A
)  ^ c  m ) ) )  =  ( abs `  (
( log `  n
)  /  ( ( exp `  A )  ^ c  ( log `  n ) ) ) ) )
3534breq1d 4224 . . . . . . . . . . . . . . 15  |-  ( m  =  ( log `  n
)  ->  ( ( abs `  ( m  / 
( ( exp `  A
)  ^ c  m ) ) )  < 
x  <->  ( abs `  (
( log `  n
)  /  ( ( exp `  A )  ^ c  ( log `  n ) ) ) )  <  x ) )
3630, 35imbi12d 313 . . . . . . . . . . . . . 14  |-  ( m  =  ( log `  n
)  ->  ( (
z  <  m  ->  ( abs `  ( m  /  ( ( exp `  A )  ^ c  m ) ) )  <  x )  <->  ( z  <  ( log `  n
)  ->  ( abs `  ( ( log `  n
)  /  ( ( exp `  A )  ^ c  ( log `  n ) ) ) )  <  x ) ) )
3736rspcv 3050 . . . . . . . . . . . . 13  |-  ( ( 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 ) ) )
3825, 37syl 16 . . . . . . . . . . . 12  |-  ( ( ( 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 ) ) )
3929, 38mpid 40 . . . . . . . . . . 11  |-  ( ( ( 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 ) )
401ad2antrr 708 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  A  e.  RR )
4140relogefd 20525 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  ( exp `  A ) )  =  A )
4241oveq2d 6099 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( ( log `  n
)  x.  ( log `  ( exp `  A
) ) )  =  ( ( log `  n
)  x.  A ) )
4325rpcnd 10652 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  CC )
44 rpcn 10622 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  RR+  ->  A  e.  CC )
4544ad2antrr 708 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  A  e.  CC )
4643, 45mulcomd 9111 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( ( log `  n
)  x.  A )  =  ( A  x.  ( log `  n ) ) )
4742, 46eqtrd 2470 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( ( log `  n
)  x.  ( log `  ( exp `  A
) ) )  =  ( A  x.  ( log `  n ) ) )
4847fveq2d 5734 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) ) ) )
493ad2antrr 708 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  A
)  e.  RR )
5049recnd 9116 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  A
)  e.  CC )
51 efne0 12700 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( exp `  A )  =/=  0 )
5245, 51syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  A
)  =/=  0 )
5350, 52, 43cxpefd 20605 . . . . . . . . . . . . . . 15  |-  ( ( ( 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
) ) ) ) )
54 rpcn 10622 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  RR+  ->  n  e.  CC )
5554ad2antrl 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  n  e.  CC )
56 rpne0 10629 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  RR+  ->  n  =/=  0 )
5756ad2antrl 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  n  =/=  0 )
5855, 57, 45cxpefd 20605 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( n  ^ c  A )  =  ( exp `  ( A  x.  ( log `  n
) ) ) )
5948, 53, 583eqtr4d 2480 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( ( exp `  A
)  ^ c  ( log `  n ) )  =  ( n  ^ c  A ) )
6059oveq2d 6099 . . . . . . . . . . . . 13  |-  ( ( ( 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
) ) )
6160fveq2d 5734 . . . . . . . . . . . 12  |-  ( ( ( 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 ) ) ) )
6261breq1d 4224 . . . . . . . . . . 11  |-  ( ( ( 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
) )
6339, 62sylibd 207 . . . . . . . . . 10  |-  ( ( ( 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
) )
6463expr 600 . . . . . . . . 9  |-  ( ( ( 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
) ) )
6517, 64sylbid 208 . . . . . . . 8  |-  ( ( ( 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 ) ) )
6665com23 75 . . . . . . 7  |-  ( ( ( 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 ) ) )
6766ralrimdva 2798 . . . . . 6  |-  ( ( 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 ) ) )
68 breq1 4217 . . . . . . . . 9  |-  ( y  =  if ( 1  <_  ( exp `  z
) ,  ( exp `  z ) ,  1 )  ->  ( y  <  n  <->  if ( 1  <_ 
( exp `  z
) ,  ( exp `  z ) ,  1 )  <  n ) )
6968imbi1d 310 . . . . . . . 8  |-  ( y  =  if ( 1  <_  ( exp `  z
) ,  ( exp `  z ) ,  1 )  ->  ( (
y  <  n  ->  ( abs `  ( ( log `  n )  /  ( n  ^ c  A ) ) )  <  x )  <->  ( if ( 1  <_  ( exp `  z ) ,  ( exp `  z
) ,  1 )  <  n  ->  ( abs `  ( ( log `  n )  /  (
n  ^ c  A
) ) )  < 
x ) ) )
7069ralbidv 2727 . . . . . . 7  |-  ( y  =  if ( 1  <_  ( exp `  z
) ,  ( exp `  z ) ,  1 )  ->  ( A. n  e.  RR+  ( y  <  n  ->  ( abs `  ( ( log `  n )  /  (
n  ^ c  A
) ) )  < 
x )  <->  A. n  e.  RR+  ( if ( 1  <_  ( exp `  z ) ,  ( exp `  z ) ,  1 )  < 
n  ->  ( abs `  ( ( log `  n
)  /  ( n  ^ c  A ) ) )  <  x
) ) )
7170rspcev 3054 . . . . . 6  |-  ( ( 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 ) )
7211, 67, 71ee12an 1373 . . . . 5  |-  ( ( 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 ) ) )
7372rexlimdva 2832 . . . 4  |-  ( 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
) ) )
7473ralimdv 2787 . . 3  |-  ( 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
) ) )
75 simpr 449 . . . . . . 7  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  m  e.  RR+ )
761adantr 453 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  A  e.  RR )
7776rpefcld 12708 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  ( exp `  A )  e.  RR+ )
78 rpre 10620 . . . . . . . . 9  |-  ( m  e.  RR+  ->  m  e.  RR )
7978adantl 454 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  m  e.  RR )
8077, 79rpcxpcld 20623 . . . . . . 7  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
( exp `  A
)  ^ c  m )  e.  RR+ )
8175, 80rpdivcld 10667 . . . . . 6  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
m  /  ( ( exp `  A )  ^ c  m ) )  e.  RR+ )
8281rpcnd 10652 . . . . 5  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
m  /  ( ( exp `  A )  ^ c  m ) )  e.  CC )
8382ralrimiva 2791 . . . 4  |-  ( A  e.  RR+  ->  A. m  e.  RR+  ( m  / 
( ( exp `  A
)  ^ c  m ) )  e.  CC )
84 rpssre 10624 . . . . 5  |-  RR+  C_  RR
8584a1i 11 . . . 4  |-  ( A  e.  RR+  ->  RR+  C_  RR )
8683, 85rlim0lt 12305 . . 3  |-  ( 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
) ) )
87 relogcl 20475 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
8887adantl 454 . . . . . . 7  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  ( log `  n )  e.  RR )
89 simpr 449 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  n  e.  RR+ )
901adantr 453 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  A  e.  RR )
9189, 90rpcxpcld 20623 . . . . . . 7  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
n  ^ c  A
)  e.  RR+ )
9288, 91rerpdivcld 10677 . . . . . 6  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
( log `  n
)  /  ( n  ^ c  A ) )  e.  RR )
9392recnd 9116 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
( log `  n
)  /  ( n  ^ c  A ) )  e.  CC )
9493ralrimiva 2791 . . . 4  |-  ( A  e.  RR+  ->  A. n  e.  RR+  ( ( log `  n )  /  (
n  ^ c  A
) )  e.  CC )
9594, 85rlim0lt 12305 . . 3  |-  ( 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
) ) )
9674, 86, 953imtr4d 261 . 2  |-  ( A  e.  RR+  ->  ( ( m  e.  RR+  |->  ( m  /  ( ( exp `  A )  ^ c  m ) ) )  ~~> r  0  ->  (
n  e.  RR+  |->  ( ( log `  n )  /  ( n  ^ c  A ) ) )  ~~> r  0 ) )
976, 96mpd 15 1  |-  ( A  e.  RR+  ->  ( n  e.  RR+  |->  ( ( log `  n )  /  ( n  ^ c  A ) ) )  ~~> r  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708    C_ wss 3322   ifcif 3741   class class class wbr 4214    e. cmpt 4268   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    x. cmul 8997    < clt 9122    <_ cle 9123    / cdiv 9679   RR+crp 10614   abscabs 12041    ~~> r crli 12281   expce 12666   logclog 20454    ^ c ccxp 20455
This theorem is referenced by:  cxploglim2  20819  logfacrlim  21010  chtppilimlem2  21170  chpchtlim  21175  dchrvmasumlema  21196  logdivsum  21229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ioc 10923  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482  df-ef 12672  df-sin 12674  df-cos 12675  df-pi 12677  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cn 17293  df-cnp 17294  df-haus 17381  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-limc 19755  df-dv 19756  df-log 20456  df-cxp 20457
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