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Theorem cxploglim 20769
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 10574 . . . 4  |-  ( A  e.  RR+  ->  A  e.  RR )
2 reefcl 12644 . . . 4  |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR )
31, 2syl 16 . . 3  |-  ( A  e.  RR+  ->  ( exp `  A )  e.  RR )
4 efgt1 12672 . . 3  |-  ( A  e.  RR+  ->  1  < 
( exp `  A
) )
5 cxp2limlem 20767 . . 3  |-  ( ( ( exp `  A
)  e.  RR  /\  1  <  ( exp `  A
) )  ->  (
m  e.  RR+  |->  ( m  /  ( ( exp `  A )  ^ c  m ) ) )  ~~> r  0 )
63, 4, 5syl2anc 643 . 2  |-  ( A  e.  RR+  ->  ( m  e.  RR+  |->  ( m  /  ( ( exp `  A )  ^ c  m ) ) )  ~~> r  0 )
7 reefcl 12644 . . . . . . . 8  |-  ( z  e.  RR  ->  ( exp `  z )  e.  RR )
87adantl 453 . . . . . . 7  |-  ( ( A  e.  RR+  /\  z  e.  RR )  ->  ( exp `  z )  e.  RR )
9 1re 9046 . . . . . . 7  |-  1  e.  RR
10 ifcl 3735 . . . . . . 7  |-  ( ( ( exp `  z
)  e.  RR  /\  1  e.  RR )  ->  if ( 1  <_ 
( exp `  z
) ,  ( exp `  z ) ,  1 )  e.  RR )
118, 9, 10sylancl 644 . . . . . 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 452 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  n  e.  RR+ )  ->  ( exp `  z
)  e.  RR )
14 rpre 10574 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
1514adantl 453 . . . . . . . . . 10  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  n  e.  RR+ )  ->  n  e.  RR )
16 maxlt 10736 . . . . . . . . . 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 742 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  z
)  <  n )
19 reeflog 20428 . . . . . . . . . . . . . . 15  |-  ( n  e.  RR+  ->  ( exp `  ( log `  n
) )  =  n )
2019ad2antrl 709 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  ( log `  n ) )  =  n )
2118, 20breqtrrd 4198 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  z
)  <  ( exp `  ( log `  n
) ) )
22 simplr 732 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
z  e.  RR )
2314ad2antrl 709 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  n  e.  RR )
24 simprrl 741 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
1  <  n )
2523, 24rplogcld 20477 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  RR+ )
2625rpred 10604 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  RR )
27 eflt 12673 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR  /\  ( log `  n )  e.  RR )  -> 
( z  <  ( log `  n )  <->  ( exp `  z )  <  ( exp `  ( log `  n
) ) ) )
2822, 26, 27syl2anc 643 . . . . . . . . . . . . 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 224 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
z  <  ( log `  n ) )
30 breq2 4176 . . . . . . . . . . . . . . 15  |-  ( m  =  ( log `  n
)  ->  ( z  <  m  <->  z  <  ( log `  n ) ) )
31 id 20 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  ( log `  n
)  ->  m  =  ( log `  n ) )
32 oveq2 6048 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  ( log `  n
)  ->  ( ( exp `  A )  ^ c  m )  =  ( ( exp `  A
)  ^ c  ( log `  n ) ) )
3331, 32oveq12d 6058 . . . . . . . . . . . . . . . . 17  |-  ( m  =  ( log `  n
)  ->  ( m  /  ( ( exp `  A )  ^ c  m ) )  =  ( ( log `  n
)  /  ( ( exp `  A )  ^ c  ( log `  n ) ) ) )
3433fveq2d 5691 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( log `  n
)  ->  ( abs `  ( m  /  (
( exp `  A
)  ^ c  m ) ) )  =  ( abs `  (
( log `  n
)  /  ( ( exp `  A )  ^ c  ( log `  n ) ) ) ) )
3534breq1d 4182 . . . . . . . . . . . . . . 15  |-  ( m  =  ( log `  n
)  ->  ( ( abs `  ( m  / 
( ( exp `  A
)  ^ c  m ) ) )  < 
x  <->  ( abs `  (
( log `  n
)  /  ( ( exp `  A )  ^ c  ( log `  n ) ) ) )  <  x ) )
3630, 35imbi12d 312 . . . . . . . . . . . . . 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 3008 . . . . . . . . . . . . 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 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 707 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  A  e.  RR )
4140relogefd 20476 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  ( exp `  A ) )  =  A )
4241oveq2d 6056 . . . . . . . . . . . . . . . . 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 10606 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( log `  n
)  e.  CC )
44 rpcn 10576 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  RR+  ->  A  e.  CC )
4544ad2antrr 707 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  A  e.  CC )
4643, 45mulcomd 9065 . . . . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . . . . 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 5691 . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  A
)  e.  RR )
5049recnd 9070 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( exp `  A
)  e.  CC )
51 efne0 12653 . . . . . . . . . . . . . . . . 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 20556 . . . . . . . . . . . . . . 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 10576 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  RR+  ->  n  e.  CC )
5554ad2antrl 709 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  n  e.  CC )
56 rpne0 10583 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  RR+  ->  n  =/=  0 )
5756ad2antrl 709 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  ->  n  =/=  0 )
5855, 57, 45cxpefd 20556 . . . . . . . . . . . . . . 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 2446 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR+  /\  z  e.  RR )  /\  ( n  e.  RR+  /\  ( 1  < 
n  /\  ( exp `  z )  <  n
) ) )  -> 
( ( exp `  A
)  ^ c  ( log `  n ) )  =  ( n  ^ c  A ) )
6059oveq2d 6056 . . . . . . . . . . . . 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 5691 . . . . . . . . . . . 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 4182 . . . . . . . . . . 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 206 . . . . . . . . . 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 599 . . . . . . . . 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 207 . . . . . . . 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 2756 . . . . . 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 4175 . . . . . . . . 9  |-  ( y  =  if ( 1  <_  ( exp `  z
) ,  ( exp `  z ) ,  1 )  ->  ( y  <  n  <->  if ( 1  <_ 
( exp `  z
) ,  ( exp `  z ) ,  1 )  <  n ) )
6968imbi1d 309 . . . . . . . 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 2686 . . . . . . 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 3012 . . . . . 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 1369 . . . . 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 2790 . . . 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 2745 . . 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 448 . . . . . . 7  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  m  e.  RR+ )
761adantr 452 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  A  e.  RR )
7776rpefcld 12661 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  ( exp `  A )  e.  RR+ )
78 rpre 10574 . . . . . . . . 9  |-  ( m  e.  RR+  ->  m  e.  RR )
7978adantl 453 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  m  e.  RR )
8077, 79rpcxpcld 20574 . . . . . . 7  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
( exp `  A
)  ^ c  m )  e.  RR+ )
8175, 80rpdivcld 10621 . . . . . 6  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
m  /  ( ( exp `  A )  ^ c  m ) )  e.  RR+ )
8281rpcnd 10606 . . . . 5  |-  ( ( A  e.  RR+  /\  m  e.  RR+ )  ->  (
m  /  ( ( exp `  A )  ^ c  m ) )  e.  CC )
8382ralrimiva 2749 . . . 4  |-  ( A  e.  RR+  ->  A. m  e.  RR+  ( m  / 
( ( exp `  A
)  ^ c  m ) )  e.  CC )
84 rpssre 10578 . . . . 5  |-  RR+  C_  RR
8584a1i 11 . . . 4  |-  ( A  e.  RR+  ->  RR+  C_  RR )
8683, 85rlim0lt 12258 . . 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 20426 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
8887adantl 453 . . . . . . 7  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  ( log `  n )  e.  RR )
89 simpr 448 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  n  e.  RR+ )
901adantr 452 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  A  e.  RR )
9189, 90rpcxpcld 20574 . . . . . . 7  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
n  ^ c  A
)  e.  RR+ )
9288, 91rerpdivcld 10631 . . . . . 6  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
( log `  n
)  /  ( n  ^ c  A ) )  e.  RR )
9392recnd 9070 . . . . 5  |-  ( ( A  e.  RR+  /\  n  e.  RR+ )  ->  (
( log `  n
)  /  ( n  ^ c  A ) )  e.  CC )
9493ralrimiva 2749 . . . 4  |-  ( A  e.  RR+  ->  A. n  e.  RR+  ( ( log `  n )  /  (
n  ^ c  A
) )  e.  CC )
9594, 85rlim0lt 12258 . . 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 260 . 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667    C_ wss 3280   ifcif 3699   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    < clt 9076    <_ cle 9077    / cdiv 9633   RR+crp 10568   abscabs 11994    ~~> r crli 12234   expce 12619   logclog 20405    ^ c ccxp 20406
This theorem is referenced by:  cxploglim2  20770  logfacrlim  20961  chtppilimlem2  21121  chpchtlim  21126  dchrvmasumlema  21147  logdivsum  21180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408
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