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Theorem cxp2lim 22255
Description: Any power grows slower than any exponential with base greater than  1. (Contributed by Mario Carneiro, 18-Sep-2014.)
Assertion
Ref Expression
cxp2lim  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
n  e.  RR+  |->  ( ( n  ^c  A )  /  ( B  ^c  n ) ) )  ~~> r  0 )
Distinct variable groups:    A, n    B, n

Proof of Theorem cxp2lim
StepHypRef Expression
1 1re 9373 . . . . . . . 8  |-  1  e.  RR
2 elicopnf 11373 . . . . . . . 8  |-  ( 1  e.  RR  ->  (
n  e.  ( 1 [,) +oo )  <->  ( n  e.  RR  /\  1  <_  n ) ) )
31, 2ax-mp 5 . . . . . . 7  |-  ( n  e.  ( 1 [,) +oo )  <->  ( n  e.  RR  /\  1  <_  n ) )
43simplbi 457 . . . . . 6  |-  ( n  e.  ( 1 [,) +oo )  ->  n  e.  RR )
5 0red 9375 . . . . . . 7  |-  ( n  e.  ( 1 [,) +oo )  ->  0  e.  RR )
61a1i 11 . . . . . . 7  |-  ( n  e.  ( 1 [,) +oo )  ->  1  e.  RR )
7 0lt1 9850 . . . . . . . 8  |-  0  <  1
87a1i 11 . . . . . . 7  |-  ( n  e.  ( 1 [,) +oo )  ->  0  <  1 )
93simprbi 461 . . . . . . 7  |-  ( n  e.  ( 1 [,) +oo )  ->  1  <_  n )
105, 6, 4, 8, 9ltletrd 9519 . . . . . 6  |-  ( n  e.  ( 1 [,) +oo )  ->  0  < 
n )
114, 10elrpd 11013 . . . . 5  |-  ( n  e.  ( 1 [,) +oo )  ->  n  e.  RR+ )
1211ssriv 3348 . . . 4  |-  ( 1 [,) +oo )  C_  RR+
13 resmpt 5144 . . . 4  |-  ( ( 1 [,) +oo )  C_  RR+  ->  ( ( n  e.  RR+  |->  ( ( n  ^c  A )  /  ( B  ^c  n ) ) )  |`  (
1 [,) +oo )
)  =  ( n  e.  ( 1 [,) +oo )  |->  ( ( n  ^c  A )  /  ( B  ^c  n ) ) ) )
1412, 13ax-mp 5 . . 3  |-  ( ( n  e.  RR+  |->  ( ( n  ^c  A )  /  ( B  ^c  n ) ) )  |`  (
1 [,) +oo )
)  =  ( n  e.  ( 1 [,) +oo )  |->  ( ( n  ^c  A )  /  ( B  ^c  n ) ) )
15 0red 9375 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  0  e.  RR )
1612a1i 11 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1 [,) +oo )  C_  RR+ )
17 rpre 10985 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  n  e.  RR )
1817adantl 463 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  ->  n  e.  RR )
19 rpge0 10991 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  <_  n )
2019adantl 463 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
0  <_  n )
21 simpl2 985 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  ->  B  e.  RR )
22 0red 9375 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
0  e.  RR )
231a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
1  e.  RR )
247a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
0  <  1 )
25 simpl3 986 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
1  <  B )
2622, 23, 21, 24, 25lttrd 9520 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
0  <  B )
2721, 26elrpd 11013 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  ->  B  e.  RR+ )
2827, 18rpcxpcld 22060 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( B  ^c 
n )  e.  RR+ )
29 simp1 981 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  A  e.  RR )
30 ifcl 3819 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  if ( 1  <_  A ,  A , 
1 )  e.  RR )
3129, 1, 30sylancl 655 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  if ( 1  <_  A ,  A ,  1 )  e.  RR )
321a1i 11 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  e.  RR )
337a1i 11 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  0  <  1 )
34 max1 11145 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  RR  /\  A  e.  RR )  ->  1  <_  if (
1  <_  A ,  A ,  1 ) )
351, 29, 34sylancr 656 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  <_  if ( 1  <_  A ,  A , 
1 ) )
3615, 32, 31, 33, 35ltletrd 9519 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  0  <  if ( 1  <_  A ,  A , 
1 ) )
3731, 36elrpd 11013 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  if ( 1  <_  A ,  A ,  1 )  e.  RR+ )
3837rprecred 11026 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1  /  if ( 1  <_  A ,  A ,  1 ) )  e.  RR )
3938adantr 462 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( 1  /  if ( 1  <_  A ,  A ,  1 ) )  e.  RR )
4028, 39rpcxpcld 22060 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( B  ^c  n )  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  e.  RR+ )
4131recnd 9400 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  if ( 1  <_  A ,  A ,  1 )  e.  CC )
4241adantr 462 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  ->  if ( 1  <_  A ,  A ,  1 )  e.  CC )
4318, 20, 40, 42divcxpd 22052 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( n  / 
( ( B  ^c  n )  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) )  ^c  if ( 1  <_  A ,  A , 
1 ) )  =  ( ( n  ^c  if ( 1  <_  A ,  A , 
1 ) )  / 
( ( ( B  ^c  n )  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  ^c  if ( 1  <_  A ,  A ,  1 ) ) ) )
4437adantr 462 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  ->  if ( 1  <_  A ,  A ,  1 )  e.  RR+ )
4544rpne0d 11020 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  ->  if ( 1  <_  A ,  A ,  1 )  =/=  0 )
4642, 45recid2d 10091 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( 1  /  if ( 1  <_  A ,  A ,  1 ) )  x.  if ( 1  <_  A ,  A ,  1 ) )  =  1 )
4746oveq2d 6096 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( B  ^c  n )  ^c  ( ( 1  /  if ( 1  <_  A ,  A ,  1 ) )  x.  if ( 1  <_  A ,  A ,  1 ) ) )  =  ( ( B  ^c  n )  ^c  1 ) )
4828, 39, 42cxpmuld 22064 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( B  ^c  n )  ^c  ( ( 1  /  if ( 1  <_  A ,  A ,  1 ) )  x.  if ( 1  <_  A ,  A ,  1 ) ) )  =  ( ( ( B  ^c 
n )  ^c 
( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  ^c  if ( 1  <_  A ,  A ,  1 ) ) )
4928rpcnd 11017 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( B  ^c 
n )  e.  CC )
5049cxp1d 22036 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( B  ^c  n )  ^c  1 )  =  ( B  ^c 
n ) )
5147, 48, 503eqtr3d 2473 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( ( B  ^c  n )  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  ^c  if ( 1  <_  A ,  A ,  1 ) )  =  ( B  ^c  n ) )
5251oveq2d 6096 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( n  ^c  if ( 1  <_  A ,  A , 
1 ) )  / 
( ( ( B  ^c  n )  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  ^c  if ( 1  <_  A ,  A ,  1 ) ) )  =  ( ( n  ^c  if ( 1  <_  A ,  A ,  1 ) )  /  ( B  ^c  n ) ) )
5343, 52eqtrd 2465 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( n  / 
( ( B  ^c  n )  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) )  ^c  if ( 1  <_  A ,  A , 
1 ) )  =  ( ( n  ^c  if ( 1  <_  A ,  A , 
1 ) )  / 
( B  ^c 
n ) ) )
5453mpteq2dva 4366 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
n  e.  RR+  |->  ( ( n  /  ( ( B  ^c  n )  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) )  ^c  if ( 1  <_  A ,  A , 
1 ) ) )  =  ( n  e.  RR+  |->  ( ( n  ^c  if ( 1  <_  A ,  A ,  1 ) )  /  ( B  ^c  n ) ) ) )
55 ovex 6105 . . . . . . . 8  |-  ( n  /  ( ( B  ^c  n )  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) )  e.  _V
5655a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( n  /  (
( B  ^c 
n )  ^c 
( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) )  e. 
_V )
5718recnd 9400 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  ->  n  e.  CC )
5838recnd 9400 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1  /  if ( 1  <_  A ,  A ,  1 ) )  e.  CC )
5958adantr 462 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( 1  /  if ( 1  <_  A ,  A ,  1 ) )  e.  CC )
6057, 59mulcomd 9395 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( n  x.  (
1  /  if ( 1  <_  A ,  A ,  1 ) ) )  =  ( ( 1  /  if ( 1  <_  A ,  A ,  1 ) )  x.  n ) )
6160oveq2d 6096 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( B  ^c 
( n  x.  (
1  /  if ( 1  <_  A ,  A ,  1 ) ) ) )  =  ( B  ^c 
( ( 1  /  if ( 1  <_  A ,  A ,  1 ) )  x.  n ) ) )
6227, 18, 59cxpmuld 22064 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( B  ^c 
( n  x.  (
1  /  if ( 1  <_  A ,  A ,  1 ) ) ) )  =  ( ( B  ^c  n )  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) )
6327, 39, 57cxpmuld 22064 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( B  ^c 
( ( 1  /  if ( 1  <_  A ,  A ,  1 ) )  x.  n ) )  =  ( ( B  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  ^c 
n ) )
6461, 62, 633eqtr3d 2473 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( B  ^c  n )  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  =  ( ( B  ^c 
( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  ^c 
n ) )
6564oveq2d 6096 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( n  /  (
( B  ^c 
n )  ^c 
( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) )  =  ( n  /  (
( B  ^c 
( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  ^c 
n ) ) )
6665mpteq2dva 4366 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
n  e.  RR+  |->  ( n  /  ( ( B  ^c  n )  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) ) )  =  ( n  e.  RR+  |->  ( n  /  (
( B  ^c 
( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  ^c 
n ) ) ) )
67 simp2 982 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  B  e.  RR )
68 simp3 983 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  <  B )
6915, 32, 67, 33, 68lttrd 9520 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  0  <  B )
7067, 69elrpd 11013 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  B  e.  RR+ )
7170, 38rpcxpcld 22060 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( B  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  e.  RR+ )
7271rpred 11015 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( B  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  e.  RR )
73581cxpd 22037 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  =  1 )
74 0le1 9851 . . . . . . . . . . . . 13  |-  0  <_  1
7574a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  0  <_  1 )
7670rpge0d 11019 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  0  <_  B )
7737rpreccld 11025 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1  /  if ( 1  <_  A ,  A ,  1 ) )  e.  RR+ )
7832, 75, 67, 76, 77cxplt2d 22056 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1  <  B  <->  ( 1  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  <  ( B  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) ) )
7968, 78mpbid 210 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  <  ( B  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) )
8073, 79eqbrtrrd 4302 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  <  ( B  ^c 
( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) )
81 cxp2limlem 22254 . . . . . . . . 9  |-  ( ( ( B  ^c 
( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  e.  RR  /\  1  <  ( B  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) )  ->  (
n  e.  RR+  |->  ( n  /  ( ( B  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  ^c  n ) ) )  ~~> r  0 )
8272, 80, 81syl2anc 654 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
n  e.  RR+  |->  ( n  /  ( ( B  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) )  ^c  n ) ) )  ~~> r  0 )
8366, 82eqbrtrd 4300 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
n  e.  RR+  |->  ( n  /  ( ( B  ^c  n )  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) ) )  ~~> r  0 )
8456, 83, 37rlimcxp 22252 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
n  e.  RR+  |->  ( ( n  /  ( ( B  ^c  n )  ^c  ( 1  /  if ( 1  <_  A ,  A ,  1 ) ) ) )  ^c  if ( 1  <_  A ,  A , 
1 ) ) )  ~~> r  0 )
8554, 84eqbrtrrd 4302 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
n  e.  RR+  |->  ( ( n  ^c  if ( 1  <_  A ,  A ,  1 ) )  /  ( B  ^c  n ) ) )  ~~> r  0 )
8616, 85rlimres2 13023 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
n  e.  ( 1 [,) +oo )  |->  ( ( n  ^c  if ( 1  <_  A ,  A ,  1 ) )  /  ( B  ^c  n ) ) )  ~~> r  0 )
87 simpr 458 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  ->  n  e.  RR+ )
8831adantr 462 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  ->  if ( 1  <_  A ,  A ,  1 )  e.  RR )
8987, 88rpcxpcld 22060 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( n  ^c  if ( 1  <_  A ,  A ,  1 ) )  e.  RR+ )
9089, 28rpdivcld 11032 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( n  ^c  if ( 1  <_  A ,  A , 
1 ) )  / 
( B  ^c 
n ) )  e.  RR+ )
9190rpred 11015 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( n  ^c  if ( 1  <_  A ,  A , 
1 ) )  / 
( B  ^c 
n ) )  e.  RR )
9211, 91sylan2 471 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  ( ( n  ^c  if ( 1  <_  A ,  A ,  1 ) )  /  ( B  ^c  n ) )  e.  RR )
93 simpl1 984 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  ->  A  e.  RR )
9487, 93rpcxpcld 22060 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( n  ^c  A )  e.  RR+ )
9594, 28rpdivcld 11032 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( n  ^c  A )  /  ( B  ^c  n ) )  e.  RR+ )
9611, 95sylan2 471 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  ( ( n  ^c  A )  /  ( B  ^c  n ) )  e.  RR+ )
9796rpred 11015 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  ( ( n  ^c  A )  /  ( B  ^c  n ) )  e.  RR )
9811, 94sylan2 471 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  ( n  ^c  A )  e.  RR+ )
9998rpred 11015 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  ( n  ^c  A )  e.  RR )
10011, 89sylan2 471 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  ( n  ^c  if ( 1  <_  A ,  A , 
1 ) )  e.  RR+ )
101100rpred 11015 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  ( n  ^c  if ( 1  <_  A ,  A , 
1 ) )  e.  RR )
10211, 28sylan2 471 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  ( B  ^c  n )  e.  RR+ )
1034adantl 463 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  n  e.  RR )
1049adantl 463 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  1  <_  n
)
105 simpl1 984 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  A  e.  RR )
10631adantr 462 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  if ( 1  <_  A ,  A ,  1 )  e.  RR )
107 max2 11147 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  A  e.  RR )  ->  A  <_  if (
1  <_  A ,  A ,  1 ) )
1081, 105, 107sylancr 656 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  A  <_  if ( 1  <_  A ,  A ,  1 ) )
109103, 104, 105, 106, 108cxplead 22051 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  ( n  ^c  A )  <_  (
n  ^c  if ( 1  <_  A ,  A ,  1 ) ) )
11099, 101, 102, 109lediv1dd 11069 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  ( ( n  ^c  A )  /  ( B  ^c  n ) )  <_  ( ( n  ^c  if ( 1  <_  A ,  A ,  1 ) )  /  ( B  ^c  n ) ) )
111110adantrr 709 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  ( n  e.  (
1 [,) +oo )  /\  0  <_  n ) )  ->  ( (
n  ^c  A )  /  ( B  ^c  n ) )  <_  ( (
n  ^c  if ( 1  <_  A ,  A ,  1 ) )  /  ( B  ^c  n ) ) )
11296rpge0d 11019 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  ( 1 [,) +oo ) )  ->  0  <_  (
( n  ^c  A )  /  ( B  ^c  n ) ) )
113112adantrr 709 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  ( n  e.  (
1 [,) +oo )  /\  0  <_  n ) )  ->  0  <_  ( ( n  ^c  A )  /  ( B  ^c  n ) ) )
11415, 15, 86, 92, 97, 111, 113rlimsqz2 13112 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
n  e.  ( 1 [,) +oo )  |->  ( ( n  ^c  A )  /  ( B  ^c  n ) ) )  ~~> r  0 )
11514, 114syl5eqbr 4313 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
( n  e.  RR+  |->  ( ( n  ^c  A )  /  ( B  ^c  n ) ) )  |`  (
1 [,) +oo )
)  ~~> r  0 )
11695rpcnd 11017 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  n  e.  RR+ )  -> 
( ( n  ^c  A )  /  ( B  ^c  n ) )  e.  CC )
117 eqid 2433 . . . 4  |-  ( n  e.  RR+  |->  ( ( n  ^c  A )  /  ( B  ^c  n ) ) )  =  ( n  e.  RR+  |->  ( ( n  ^c  A )  /  ( B  ^c  n ) ) )
118116, 117fmptd 5855 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
n  e.  RR+  |->  ( ( n  ^c  A )  /  ( B  ^c  n ) ) ) : RR+ --> CC )
119 rpssre 10989 . . . 4  |-  RR+  C_  RR
120119a1i 11 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  RR+  C_  RR )
121118, 120, 32rlimresb 13027 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
( n  e.  RR+  |->  ( ( n  ^c  A )  /  ( B  ^c  n ) ) )  ~~> r  0  <-> 
( ( n  e.  RR+  |->  ( ( n  ^c  A )  /  ( B  ^c  n ) ) )  |`  ( 1 [,) +oo ) )  ~~> r  0 ) )
122115, 121mpbird 232 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
n  e.  RR+  |->  ( ( n  ^c  A )  /  ( B  ^c  n ) ) )  ~~> r  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755   _Vcvv 2962    C_ wss 3316   ifcif 3779   class class class wbr 4280    e. cmpt 4338    |` cres 4829  (class class class)co 6080   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    x. cmul 9275   +oocpnf 9403    < clt 9406    <_ cle 9407    / cdiv 9981   RR+crp 10979   [,)cico 11290    ~~> r crli 12947    ^c ccxp 21892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-sum 13148  df-ef 13336  df-sin 13338  df-cos 13339  df-pi 13341  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893  df-cxp 21894
This theorem is referenced by: (None)
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