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Theorem cxploglim 20288
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 10376 . . . 4  |-  ( A  e.  RR+  ->  A  e.  RR )
2 reefcl 12384 . . . 4  |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR )
31, 2syl 15 . . 3  |-  ( A  e.  RR+  ->  ( exp `  A )  e.  RR )
4 efgt1 12412 . . 3  |-  ( A  e.  RR+  ->  1  < 
( exp `  A
) )
5 cxp2limlem 20286 . . 3  |-  ( ( ( exp `  A
)  e.  RR  /\  1  <  ( exp `  A
) )  ->  (
m  e.  RR+  |->  ( m  /  ( ( exp `  A )  ^ c  m ) ) )  ~~> r  0 )
63, 4, 5syl2anc 642 . 2  |-  ( A  e.  RR+  ->  ( m  e.  RR+  |->  ( m  /  ( ( exp `  A )  ^ c  m ) ) )  ~~> r  0 )
7 reefcl 12384 . . . . . . . 8  |-  ( z  e.  RR  ->  ( exp `  z )  e.  RR )
87adantl 452 . . . . . . 7  |-  ( ( A  e.  RR+  /\  z  e.  RR )  ->  ( exp `  z )  e.  RR )
9 1re 8853 . . . . . . 7  |-  1  e.  RR
10 ifcl 3614 . . . . . . 7  |-  ( ( ( exp `  z
)  e.  RR  /\  1  e.  RR )  ->  if ( 1  <_ 
( exp `  z
) ,  ( exp `  z ) ,  1 )  e.  RR )
118, 9, 10sylancl 643 . . . . . 6  |-  ( ( A  e.  RR+  /\  z  e.  RR )  ->  if ( 1  <_  ( exp `  z ) ,  ( exp `  z
) ,  1 )  e.  RR )
129a1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  n  e.  RR+ )  ->  1  e.  RR )
138adantr 451 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  n  e.  RR+ )  ->  ( exp `  z
)  e.  RR )
14 rpre 10376 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
1514adantl 452 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  n  e.  RR+ )  ->  n  e.  RR )
16 maxlt 10537 . . . . . . . . . 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 1182 . . . . . . . . 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 741 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  z
)  <  n )
19 reeflog 19950 . . . . . . . . . . . . . . 15  |-  ( n  e.  RR+  ->  ( exp `  ( log `  n
) )  =  n )
2019ad2antrl 708 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  ( log `  n ) )  =  n )
2118, 20breqtrrd 4065 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  z
)  <  ( exp `  ( log `  n
) ) )
22 simplr 731 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
z  e.  RR )
2314ad2antrl 708 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  n  e.  RR )
24 simprrl 740 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
1  <  n )
2523, 24rplogcld 19996 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  RR+ )
2625rpred 10406 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  RR )
27 eflt 12413 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR  /\  ( log `  n )  e.  RR )  -> 
( z  <  ( log `  n )  <->  ( exp `  z )  <  ( exp `  ( log `  n
) ) ) )
2822, 26, 27syl2anc 642 . . . . . . . . . . . . 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 223 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
z  <  ( log `  n ) )
30 breq2 4043 . . . . . . . . . . . . . . 15  |-  ( m  =  ( log `  n
)  ->  ( z  <  m  <->  z  <  ( log `  n ) ) )
31 id 19 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  ( log `  n
)  ->  m  =  ( log `  n ) )
32 oveq2 5882 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  ( log `  n
)  ->  ( ( exp `  A )  ^ c  m )  =  ( ( exp `  A
)  ^ c  ( log `  n ) ) )
3331, 32oveq12d 5892 . . . . . . . . . . . . . . . . 17  |-  ( m  =  ( log `  n
)  ->  ( m  /  ( ( exp `  A )  ^ c  m ) )  =  ( ( log `  n
)  /  ( ( exp `  A )  ^ c  ( log `  n ) ) ) )
3433fveq2d 5545 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( log `  n
)  ->  ( abs `  ( m  /  (
( exp `  A
)  ^ c  m ) ) )  =  ( abs `  (
( log `  n
)  /  ( ( exp `  A )  ^ c  ( log `  n ) ) ) ) )
3534breq1d 4049 . . . . . . . . . . . . . . 15  |-  ( m  =  ( log `  n
)  ->  ( ( abs `  ( m  / 
( ( exp `  A
)  ^ c  m ) ) )  < 
x  <->  ( abs `  (
( log `  n
)  /  ( ( exp `  A )  ^ c  ( log `  n ) ) ) )  <  x ) )
3630, 35imbi12d 311 . . . . . . . . . . . . . 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 2893 . . . . . . . . . . . . 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 15 . . . . . . . . . . . 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 37 . . . . . . . . . . 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 706 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  A  e.  RR )
4140relogefd 19995 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  ( exp `  A ) )  =  A )
4241oveq2d 5890 . . . . . . . . . . . . . . . . 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 10408 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  CC )
44 rpcn 10378 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  RR+  ->  A  e.  CC )
4544ad2antrr 706 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  A  e.  CC )
4643, 45mulcomd 8872 . . . . . . . . . . . . . . . . 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 2328 . . . . . . . . . . . . . . . 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 5545 . . . . . . . . . . . . . . 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 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  A
)  e.  RR )
5049recnd 8877 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  A
)  e.  CC )
51 efne0 12393 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( exp `  A )  =/=  0 )
5245, 51syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  A
)  =/=  0 )
5350, 52, 43cxpefd 20075 . . . . . . . . . . . . . . 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 10378 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  RR+  ->  n  e.  CC )
5554ad2antrl 708 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  n  e.  CC )
56 rpne0 10385 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  RR+  ->  n  =/=  0 )
5756ad2antrl 708 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  n  =/=  0 )
5855, 57, 45cxpefd 20075 . . . . . . . . . . . . . . 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 2338 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( ( exp `  A
)  ^ c  ( log `  n ) )  =  ( n  ^ c  A ) )
6059oveq2d 5890 . . . . . . . . . . . . 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 5545 . . . . . . . . . . . 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 4049 . . . . . . . . . . 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 205 . . . . . . . . . 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 598 . . . . . . . . 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 206 . . . . . . . 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 72 . . . . . . 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 2646 . . . . . 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 4042 . . . . . . . . 9  |-  ( y  =  if ( 1  <_  ( exp `  z
) ,  ( exp `  z ) ,  1 )  ->  ( y  <  n  <->  if ( 1  <_ 
( exp `  z
) ,  ( exp `  z ) ,  1 )  <  n ) )
6968imbi1d 308 . . . . . . . 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 2576 . . . . . . 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 2897 . . . . . 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 1353 . . . . 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 2680 . . . 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 2635 . . 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 447 . . . . . . 7  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  m  e.  RR+ )
761adantr 451 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  A  e.  RR )
7776rpefcld 12401 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  ( exp `  A )  e.  RR+ )
78 rpre 10376 . . . . . . . . 9  |-  ( m  e.  RR+  ->  m  e.  RR )
7978adantl 452 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  m  e.  RR )
8077, 79rpcxpcld 20093 . . . . . . 7  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
( exp `  A
)  ^ c  m )  e.  RR+ )
8175, 80rpdivcld 10423 . . . . . 6  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
m  /  ( ( exp `  A )  ^ c  m ) )  e.  RR+ )
8281rpcnd 10408 . . . . 5  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
m  /  ( ( exp `  A )  ^ c  m ) )  e.  CC )
8382ralrimiva 2639 . . . 4  |-  ( A  e.  RR+  ->  A. m  e.  RR+  ( m  / 
( ( exp `  A
)  ^ c  m ) )  e.  CC )
84 rpssre 10380 . . . . 5  |-  RR+  C_  RR
8584a1i 10 . . . 4  |-  ( A  e.  RR+  ->  RR+  C_  RR )
8683, 85rlim0lt 11999 . . 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 19948 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
8887adantl 452 . . . . . . 7  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  ( log `  n )  e.  RR )
89 simpr 447 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  n  e.  RR+ )
901adantr 451 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  A  e.  RR )
9189, 90rpcxpcld 20093 . . . . . . 7  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
n  ^ c  A
)  e.  RR+ )
9288, 91rerpdivcld 10433 . . . . . 6  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
( log `  n
)  /  ( n  ^ c  A ) )  e.  RR )
9392recnd 8877 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
( log `  n
)  /  ( n  ^ c  A ) )  e.  CC )
9493ralrimiva 2639 . . . 4  |-  ( A  e.  RR+  ->  A. n  e.  RR+  ( ( log `  n )  /  (
n  ^ c  A
) )  e.  CC )
9594, 85rlim0lt 11999 . . 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 259 . 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 14 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   ifcif 3578   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    < clt 8883    <_ cle 8884    / cdiv 9439   RR+crp 10370   abscabs 11735    ~~> r crli 11975   expce 12359   logclog 19928    ^ c ccxp 19929
This theorem is referenced by:  cxploglim2  20289  logfacrlim  20479  chtppilimlem2  20639  chpchtlim  20644  dchrvmasumlema  20665  logdivsum  20698
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931
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