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Theorem cxploglim 20268
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
Dummy variables  m  x  y  z are mutually distinct and distinct from all other variables.

Proof of Theorem cxploglim
StepHypRef Expression
1 rpre 10357 . . . 4  |-  ( A  e.  RR+  ->  A  e.  RR )
2 reefcl 12364 . . . 4  |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR )
31, 2syl 17 . . 3  |-  ( A  e.  RR+  ->  ( exp `  A )  e.  RR )
4 efgt1 12392 . . 3  |-  ( A  e.  RR+  ->  1  < 
( exp `  A
) )
5 cxp2limlem 20266 . . 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 12364 . . . . . . . 8  |-  ( z  e.  RR  ->  ( exp `  z )  e.  RR )
87adantl 454 . . . . . . 7  |-  ( ( A  e.  RR+  /\  z  e.  RR )  ->  ( exp `  z )  e.  RR )
9 1re 8834 . . . . . . 7  |-  1  e.  RR
10 ifcl 3604 . . . . . . 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 12 . . . . . . . . . 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 10357 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
1514adantl 454 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  n  e.  RR+ )  ->  n  e.  RR )
16 maxlt 10517 . . . . . . . . . 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 1184 . . . . . . . . 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 19930 . . . . . . . . . . . . . . 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 4052 . . . . . . . . . . . . 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 19976 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  RR+ )
2625rpred 10387 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  RR )
27 eflt 12393 . . . . . . . . . . . . . 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 4030 . . . . . . . . . . . . . . 15  |-  ( m  =  ( log `  n
)  ->  ( z  <  m  <->  z  <  ( log `  n ) ) )
31 id 21 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  ( log `  n
)  ->  m  =  ( log `  n ) )
32 oveq2 5829 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  ( log `  n
)  ->  ( ( exp `  A )  ^ c  m )  =  ( ( exp `  A
)  ^ c  ( log `  n ) ) )
3331, 32oveq12d 5839 . . . . . . . . . . . . . . . . 17  |-  ( m  =  ( log `  n
)  ->  ( m  /  ( ( exp `  A )  ^ c  m ) )  =  ( ( log `  n
)  /  ( ( exp `  A )  ^ c  ( log `  n ) ) ) )
3433fveq2d 5491 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( log `  n
)  ->  ( abs `  ( m  /  (
( exp `  A
)  ^ c  m ) ) )  =  ( abs `  (
( log `  n
)  /  ( ( exp `  A )  ^ c  ( log `  n ) ) ) ) )
3534breq1d 4036 . . . . . . . . . . . . . . 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 2883 . . . . . . . . . . . . 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 17 . . . . . . . . . . . 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 39 . . . . . . . . . . 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 19975 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  ( exp `  A ) )  =  A )
4241oveq2d 5837 . . . . . . . . . . . . . . . . 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 10389 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  CC )
44 rpcn 10359 . . . . . . . . . . . . . . . . . . 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 8853 . . . . . . . . . . . . . . . . 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 2318 . . . . . . . . . . . . . . . 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 5491 . . . . . . . . . . . . . . 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 8858 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  A
)  e.  CC )
51 efne0 12373 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( exp `  A )  =/=  0 )
5245, 51syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  A
)  =/=  0 )
5350, 52, 43cxpefd 20055 . . . . . . . . . . . . . . 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 10359 . . . . . . . . . . . . . . . . 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 10366 . . . . . . . . . . . . . . . . 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 20055 . . . . . . . . . . . . . . 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 2328 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( ( exp `  A
)  ^ c  ( log `  n ) )  =  ( n  ^ c  A ) )
6059oveq2d 5837 . . . . . . . . . . . . 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 5491 . . . . . . . . . . . 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 4036 . . . . . . . . . . 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 74 . . . . . . 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 2636 . . . . . 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 4029 . . . . . . . . 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 2566 . . . . . . 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 2887 . . . . . 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 1355 . . . . 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 2670 . . . 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 2625 . . 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 12381 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  ( exp `  A )  e.  RR+ )
78 rpre 10357 . . . . . . . . 9  |-  ( m  e.  RR+  ->  m  e.  RR )
7978adantl 454 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  m  e.  RR )
8077, 79rpcxpcld 20073 . . . . . . 7  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
( exp `  A
)  ^ c  m )  e.  RR+ )
8175, 80rpdivcld 10404 . . . . . 6  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
m  /  ( ( exp `  A )  ^ c  m ) )  e.  RR+ )
8281rpcnd 10389 . . . . 5  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
m  /  ( ( exp `  A )  ^ c  m ) )  e.  CC )
8382ralrimiva 2629 . . . 4  |-  ( A  e.  RR+  ->  A. m  e.  RR+  ( m  / 
( ( exp `  A
)  ^ c  m ) )  e.  CC )
84 rpssre 10361 . . . . 5  |-  RR+  C_  RR
8584a1i 12 . . . 4  |-  ( A  e.  RR+  ->  RR+  C_  RR )
8683, 85rlim0lt 11979 . . 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 19928 . . . . . . . 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 20073 . . . . . . 7  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
n  ^ c  A
)  e.  RR+ )
9288, 91rerpdivcld 10414 . . . . . 6  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
( log `  n
)  /  ( n  ^ c  A ) )  e.  RR )
9392recnd 8858 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
( log `  n
)  /  ( n  ^ c  A ) )  e.  CC )
9493ralrimiva 2629 . . . 4  |-  ( A  e.  RR+  ->  A. n  e.  RR+  ( ( log `  n )  /  (
n  ^ c  A
) )  e.  CC )
9594, 85rlim0lt 11979 . . 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 16 1  |-  ( A  e.  RR+  ->  ( n  e.  RR+  |->  ( ( log `  n )  /  ( n  ^ c  A ) ) )  ~~> r  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1625    e. wcel 1687    =/= wne 2449   A.wral 2546   E.wrex 2547    C_ wss 3155   ifcif 3568   class class class wbr 4026    e. cmpt 4080   ` cfv 5223  (class class class)co 5821   CCcc 8732   RRcr 8733   0cc0 8734   1c1 8735    x. cmul 8739    < clt 8864    <_ cle 8865    / cdiv 9420   RR+crp 10351   abscabs 11715    ~~> r crli 11955   expce 12339   logclog 19908    ^ c ccxp 19909
This theorem is referenced by:  cxploglim2  20269  logfacrlim  20459  chtppilimlem2  20619  chpchtlim  20624  dchrvmasumlema  20645  logdivsum  20678
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-inf2 7339  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812  ax-addf 8813  ax-mulf 8814
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-iin 3911  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-se 4354  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-isom 5232  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-of 6041  df-1st 6085  df-2nd 6086  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-1o 6476  df-2o 6477  df-oadd 6480  df-er 6657  df-map 6771  df-pm 6772  df-ixp 6815  df-en 6861  df-dom 6862  df-sdom 6863  df-fin 6864  df-fi 7162  df-sup 7191  df-oi 7222  df-card 7569  df-cda 7791  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-div 9421  df-nn 9744  df-2 9801  df-3 9802  df-4 9803  df-5 9804  df-6 9805  df-7 9806  df-8 9807  df-9 9808  df-10 9809  df-n0 9963  df-z 10022  df-dec 10122  df-uz 10228  df-q 10314  df-rp 10352  df-xneg 10449  df-xadd 10450  df-xmul 10451  df-ioo 10656  df-ioc 10657  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101  df-fac 11285  df-bc 11312  df-hash 11334  df-shft 11558  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-limsup 11941  df-clim 11958  df-rlim 11959  df-sum 12155  df-ef 12345  df-sin 12347  df-cos 12348  df-pi 12350  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-starv 13219  df-sca 13220  df-vsca 13221  df-tset 13223  df-ple 13224  df-ds 13226  df-hom 13228  df-cco 13229  df-rest 13323  df-topn 13324  df-topgen 13340  df-pt 13341  df-prds 13344  df-xrs 13399  df-0g 13400  df-gsum 13401  df-qtop 13406  df-imas 13407  df-xps 13409  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-submnd 14412  df-mulg 14488  df-cntz 14789  df-cmn 15087  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cld 16752  df-ntr 16753  df-cls 16754  df-nei 16831  df-lp 16864  df-perf 16865  df-cn 16953  df-cnp 16954  df-haus 17039  df-tx 17253  df-hmeo 17442  df-fbas 17516  df-fg 17517  df-fil 17537  df-fm 17629  df-flim 17630  df-flf 17631  df-xms 17881  df-ms 17882  df-tms 17883  df-cncf 18378  df-limc 19212  df-dv 19213  df-log 19910  df-cxp 19911
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