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Theorem cxploglim 23051
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 11225 . . . 4  |-  ( A  e.  RR+  ->  A  e.  RR )
2 reefcl 13683 . . . 4  |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR )
31, 2syl 16 . . 3  |-  ( A  e.  RR+  ->  ( exp `  A )  e.  RR )
4 efgt1 13711 . . 3  |-  ( A  e.  RR+  ->  1  < 
( exp `  A
) )
5 cxp2limlem 23049 . . 3  |-  ( ( ( exp `  A
)  e.  RR  /\  1  <  ( exp `  A
) )  ->  (
m  e.  RR+  |->  ( m  /  ( ( exp `  A )  ^c 
m ) ) )  ~~> r  0 )
63, 4, 5syl2anc 661 . 2  |-  ( A  e.  RR+  ->  ( m  e.  RR+  |->  ( m  /  ( ( exp `  A )  ^c 
m ) ) )  ~~> r  0 )
7 reefcl 13683 . . . . . . . 8  |-  ( z  e.  RR  ->  ( exp `  z )  e.  RR )
87adantl 466 . . . . . . 7  |-  ( ( A  e.  RR+  /\  z  e.  RR )  ->  ( exp `  z )  e.  RR )
9 1re 9594 . . . . . . 7  |-  1  e.  RR
10 ifcl 3981 . . . . . . 7  |-  ( ( ( exp `  z
)  e.  RR  /\  1  e.  RR )  ->  if ( 1  <_ 
( exp `  z
) ,  ( exp `  z ) ,  1 )  e.  RR )
118, 9, 10sylancl 662 . . . . . 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 465 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  n  e.  RR+ )  ->  ( exp `  z
)  e.  RR )
14 rpre 11225 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
1514adantl 466 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  n  e.  RR+ )  ->  n  e.  RR )
16 maxlt 11392 . . . . . . . . . 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 1228 . . . . . . . . 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 764 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  z
)  <  n )
19 reeflog 22709 . . . . . . . . . . . . . . 15  |-  ( n  e.  RR+  ->  ( exp `  ( log `  n
) )  =  n )
2019ad2antrl 727 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  ( log `  n ) )  =  n )
2118, 20breqtrrd 4473 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  z
)  <  ( exp `  ( log `  n
) ) )
22 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
z  e.  RR )
2314ad2antrl 727 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  n  e.  RR )
24 simprrl 763 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
1  <  n )
2523, 24rplogcld 22758 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  RR+ )
2625rpred 11255 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  RR )
27 eflt 13712 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR  /\  ( log `  n )  e.  RR )  -> 
( z  <  ( log `  n )  <->  ( exp `  z )  <  ( exp `  ( log `  n
) ) ) )
2822, 26, 27syl2anc 661 . . . . . . . . . . . . 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 232 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
z  <  ( log `  n ) )
30 breq2 4451 . . . . . . . . . . . . . . 15  |-  ( m  =  ( log `  n
)  ->  ( z  <  m  <->  z  <  ( log `  n ) ) )
31 id 22 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  ( log `  n
)  ->  m  =  ( log `  n ) )
32 oveq2 6291 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  ( log `  n
)  ->  ( ( exp `  A )  ^c  m )  =  ( ( exp `  A
)  ^c  ( log `  n ) ) )
3331, 32oveq12d 6301 . . . . . . . . . . . . . . . . 17  |-  ( m  =  ( log `  n
)  ->  ( m  /  ( ( exp `  A )  ^c 
m ) )  =  ( ( log `  n
)  /  ( ( exp `  A )  ^c  ( log `  n ) ) ) )
3433fveq2d 5869 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( log `  n
)  ->  ( abs `  ( m  /  (
( exp `  A
)  ^c  m ) ) )  =  ( abs `  (
( log `  n
)  /  ( ( exp `  A )  ^c  ( log `  n ) ) ) ) )
3534breq1d 4457 . . . . . . . . . . . . . . 15  |-  ( m  =  ( log `  n
)  ->  ( ( abs `  ( m  / 
( ( exp `  A
)  ^c  m ) ) )  < 
x  <->  ( abs `  (
( log `  n
)  /  ( ( exp `  A )  ^c  ( log `  n ) ) ) )  <  x ) )
3630, 35imbi12d 320 . . . . . . . . . . . . . 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 3210 . . . . . . . . . . . . 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 41 . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  A  e.  RR )
4140relogefd 22757 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  ( exp `  A ) )  =  A )
4241oveq2d 6299 . . . . . . . . . . . . . . . . 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 11257 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  CC )
44 rpcn 11227 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  RR+  ->  A  e.  CC )
4544ad2antrr 725 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  A  e.  CC )
4643, 45mulcomd 9616 . . . . . . . . . . . . . . . . 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 2508 . . . . . . . . . . . . . . . 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 5869 . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  A
)  e.  RR )
5049recnd 9621 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  A
)  e.  CC )
51 efne0 13692 . . . . . . . . . . . . . . . . 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 22837 . . . . . . . . . . . . . . 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 11227 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  RR+  ->  n  e.  CC )
5554ad2antrl 727 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  n  e.  CC )
56 rpne0 11234 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  RR+  ->  n  =/=  0 )
5756ad2antrl 727 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  n  =/=  0 )
5855, 57, 45cxpefd 22837 . . . . . . . . . . . . . . 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 2518 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( ( exp `  A
)  ^c  ( log `  n ) )  =  ( n  ^c  A ) )
6059oveq2d 6299 . . . . . . . . . . . . 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 5869 . . . . . . . . . . . 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 4457 . . . . . . . . . . 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 214 . . . . . . . . . 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 615 . . . . . . . . 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 215 . . . . . . . 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 78 . . . . . . 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 2882 . . . . . 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 4450 . . . . . . . . 9  |-  ( y  =  if ( 1  <_  ( exp `  z
) ,  ( exp `  z ) ,  1 )  ->  ( y  <  n  <->  if ( 1  <_ 
( exp `  z
) ,  ( exp `  z ) ,  1 )  <  n ) )
6968imbi1d 317 . . . . . . . 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 2903 . . . . . . 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 3214 . . . . . 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, 71syl6an 545 . . . . 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 2955 . . . 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 2874 . . 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 461 . . . . . . 7  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  m  e.  RR+ )
761adantr 465 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  A  e.  RR )
7776rpefcld 13700 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  ( exp `  A )  e.  RR+ )
78 rpre 11225 . . . . . . . . 9  |-  ( m  e.  RR+  ->  m  e.  RR )
7978adantl 466 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  m  e.  RR )
8077, 79rpcxpcld 22855 . . . . . . 7  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
( exp `  A
)  ^c  m )  e.  RR+ )
8175, 80rpdivcld 11272 . . . . . 6  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
m  /  ( ( exp `  A )  ^c  m ) )  e.  RR+ )
8281rpcnd 11257 . . . . 5  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
m  /  ( ( exp `  A )  ^c  m ) )  e.  CC )
8382ralrimiva 2878 . . . 4  |-  ( A  e.  RR+  ->  A. m  e.  RR+  ( m  / 
( ( exp `  A
)  ^c  m ) )  e.  CC )
84 rpssre 11229 . . . . 5  |-  RR+  C_  RR
8584a1i 11 . . . 4  |-  ( A  e.  RR+  ->  RR+  C_  RR )
8683, 85rlim0lt 13294 . . 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 22707 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
8887adantl 466 . . . . . . 7  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  ( log `  n )  e.  RR )
89 simpr 461 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  n  e.  RR+ )
901adantr 465 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  A  e.  RR )
9189, 90rpcxpcld 22855 . . . . . . 7  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
n  ^c  A )  e.  RR+ )
9288, 91rerpdivcld 11282 . . . . . 6  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
( log `  n
)  /  ( n  ^c  A ) )  e.  RR )
9392recnd 9621 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
( log `  n
)  /  ( n  ^c  A ) )  e.  CC )
9493ralrimiva 2878 . . . 4  |-  ( A  e.  RR+  ->  A. n  e.  RR+  ( ( log `  n )  /  (
n  ^c  A ) )  e.  CC )
9594, 85rlim0lt 13294 . . 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 268 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815    C_ wss 3476   ifcif 3939   class class class wbr 4447    |-> cmpt 4505   ` cfv 5587  (class class class)co 6283   CCcc 9489   RRcr 9490   0cc0 9491   1c1 9492    x. cmul 9496    < clt 9627    <_ cle 9628    / cdiv 10205   RR+crp 11219   abscabs 13029    ~~> r crli 13270   expce 13658   logclog 22686    ^c ccxp 22687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-ioc 11533  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-mod 11964  df-seq 12075  df-exp 12134  df-fac 12321  df-bc 12348  df-hash 12373  df-shft 12862  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-limsup 13256  df-clim 13273  df-rlim 13274  df-sum 13471  df-ef 13664  df-sin 13666  df-cos 13667  df-pi 13669  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-rest 14677  df-topn 14678  df-0g 14696  df-gsum 14697  df-topgen 14698  df-pt 14699  df-prds 14702  df-xrs 14756  df-qtop 14761  df-imas 14762  df-xps 14764  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-fbas 18203  df-fg 18204  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cld 19302  df-ntr 19303  df-cls 19304  df-nei 19381  df-lp 19419  df-perf 19420  df-cn 19510  df-cnp 19511  df-haus 19598  df-tx 19814  df-hmeo 20007  df-fil 20098  df-fm 20190  df-flim 20191  df-flf 20192  df-xms 20574  df-ms 20575  df-tms 20576  df-cncf 21133  df-limc 22021  df-dv 22022  df-log 22688  df-cxp 22689
This theorem is referenced by:  cxploglim2  23052  logfacrlim  23243  chtppilimlem2  23403  chpchtlim  23408  dchrvmasumlema  23429  logdivsum  23462
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