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Theorem chtppilimlem2 24361
Description: Lemma for chtppilim 24362. (Contributed by Mario Carneiro, 22-Sep-2014.)
Hypotheses
Ref Expression
chtppilim.1  |-  ( ph  ->  A  e.  RR+ )
chtppilim.2  |-  ( ph  ->  A  <  1 )
Assertion
Ref Expression
chtppilimlem2  |-  ( ph  ->  E. z  e.  RR  A. x  e.  ( 2 [,) +oo ) ( z  <_  x  ->  ( ( A ^ 2 )  x.  ( (π `  x )  x.  ( log `  x ) ) )  <  ( theta `  x ) ) )
Distinct variable groups:    x, z, A    ph, x, z

Proof of Theorem chtppilimlem2
StepHypRef Expression
1 simpr 467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  ( 2 [,) +oo ) )
2 2re 10707 . . . . . . . . . 10  |-  2  e.  RR
3 elicopnf 11759 . . . . . . . . . 10  |-  ( 2  e.  RR  ->  (
x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x ) ) )
42, 3ax-mp 5 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x ) )
51, 4sylib 201 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  e.  RR  /\  2  <_  x ) )
65simpld 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  RR )
7 0red 9670 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  0  e.  RR )
82a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  2  e.  RR )
9 2pos 10729 . . . . . . . . 9  |-  0  <  2
109a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  0  <  2 )
115simprd 469 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  2  <_  x )
127, 8, 6, 10, 11ltletrd 9821 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  0  <  x )
136, 12elrpd 11367 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  RR+ )
14 chtppilim.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR+ )
1514rpred 11370 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
1615adantr 471 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  A  e.  RR )
1713, 16rpcxpcld 23724 . . . . 5  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  ^c  A )  e.  RR+ )
18 ppinncl 24150 . . . . . . 7  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
195, 18syl 17 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (π `  x )  e.  NN )
2019nnrpd 11368 . . . . 5  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (π `  x )  e.  RR+ )
2117, 20rpdivcld 11387 . . . 4  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  ^c  A )  /  (π `  x ) )  e.  RR+ )
2221ralrimiva 2814 . . 3  |-  ( ph  ->  A. x  e.  ( 2 [,) +oo )
( ( x  ^c  A )  /  (π `  x ) )  e.  RR+ )
23 chtppilim.2 . . . 4  |-  ( ph  ->  A  <  1 )
24 1re 9668 . . . . 5  |-  1  e.  RR
25 difrp 11366 . . . . 5  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
2615, 24, 25sylancl 673 . . . 4  |-  ( ph  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
2723, 26mpbid 215 . . 3  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
28 ovex 6343 . . . . . . 7  |-  ( 2 [,) +oo )  e. 
_V
2928a1i 11 . . . . . 6  |-  ( ph  ->  ( 2 [,) +oo )  e.  _V )
3024a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  1  e.  RR )
31 1lt2 10805 . . . . . . . . . . 11  |-  1  <  2
3231a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <  2 )
3330, 8, 6, 32, 11ltletrd 9821 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <  x )
346, 33rplogcld 23627 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  ( log `  x )  e.  RR+ )
3513, 34rpdivcld 11387 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
3635, 20rpdivcld 11387 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+ )
3727adantr 471 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  -  A )  e.  RR+ )
3837rpred 11370 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  -  A )  e.  RR )
3913, 38rpcxpcld 23724 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  ^c  ( 1  -  A ) )  e.  RR+ )
4034, 39rpdivcld 11387 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) )  e.  RR+ )
41 eqidd 2463 . . . . . 6  |-  ( ph  ->  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) ) )
42 eqidd 2463 . . . . . 6  |-  ( ph  ->  ( x  e.  ( 2 [,) +oo )  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) ) )
4329, 36, 40, 41, 42offval2 6575 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  oF  x.  (
x  e.  ( 2 [,) +oo )  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( x  /  ( log `  x
) )  /  (π `  x ) )  x.  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) ) ) )
4435rpcnd 11372 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  /  ( log `  x ) )  e.  CC )
4540rpcnd 11372 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) )  e.  CC )
4620rpcnne0d 11379 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(π `  x )  e.  CC  /\  (π `  x
)  =/=  0 ) )
47 div23 10317 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  e.  CC  /\  (
( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) )  e.  CC  /\  ( (π `  x )  e.  CC  /\  (π `  x
)  =/=  0 ) )  ->  ( (
( x  /  ( log `  x ) )  x.  ( ( log `  x )  /  (
x  ^c  ( 1  -  A ) ) ) )  / 
(π `  x ) )  =  ( ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  x.  ( ( log `  x )  /  (
x  ^c  ( 1  -  A ) ) ) ) )
4844, 45, 46, 47syl3anc 1276 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( x  / 
( log `  x
) )  x.  (
( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) )  /  (π `  x ) )  =  ( ( ( x  /  ( log `  x
) )  /  (π `  x ) )  x.  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) ) )
4934rpcnne0d 11379 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
5039rpcnne0d 11379 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  ^c 
( 1  -  A
) )  e.  CC  /\  ( x  ^c 
( 1  -  A
) )  =/=  0
) )
516recnd 9695 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  CC )
52 dmdcan 10345 . . . . . . . . . 10  |-  ( ( ( ( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 )  /\  ( ( x  ^c  ( 1  -  A ) )  e.  CC  /\  ( x  ^c  ( 1  -  A ) )  =/=  0 )  /\  x  e.  CC )  ->  ( ( ( log `  x )  /  (
x  ^c  ( 1  -  A ) ) )  x.  (
x  /  ( log `  x ) ) )  =  ( x  / 
( x  ^c 
( 1  -  A
) ) ) )
5349, 50, 51, 52syl3anc 1276 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) )  x.  ( x  /  ( log `  x
) ) )  =  ( x  /  (
x  ^c  ( 1  -  A ) ) ) )
5444, 45mulcomd 9690 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  /  ( log `  x ) )  x.  ( ( log `  x )  /  (
x  ^c  ( 1  -  A ) ) ) )  =  ( ( ( log `  x )  /  (
x  ^c  ( 1  -  A ) ) )  x.  (
x  /  ( log `  x ) ) ) )
5513rpcnne0d 11379 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
56 ax-1cn 9623 . . . . . . . . . . . . 13  |-  1  e.  CC
5756a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  1  e.  CC )
5837rpcnd 11372 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  -  A )  e.  CC )
59 cxpsub 23676 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  1  e.  CC  /\  ( 1  -  A
)  e.  CC )  ->  ( x  ^c  ( 1  -  ( 1  -  A
) ) )  =  ( ( x  ^c  1 )  / 
( x  ^c 
( 1  -  A
) ) ) )
6055, 57, 58, 59syl3anc 1276 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  ^c  ( 1  -  ( 1  -  A ) ) )  =  ( ( x  ^c  1 )  /  ( x  ^c  ( 1  -  A ) ) ) )
6116recnd 9695 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  A  e.  CC )
62 nncan 9929 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  (
1  -  A ) )  =  A )
6356, 61, 62sylancr 674 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  -  ( 1  -  A ) )  =  A )
6463oveq2d 6331 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  ^c  ( 1  -  ( 1  -  A ) ) )  =  ( x  ^c  A ) )
6560, 64eqtr3d 2498 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  ^c 
1 )  /  (
x  ^c  ( 1  -  A ) ) )  =  ( x  ^c  A ) )
6651cxp1d 23700 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  ^c  1 )  =  x )
6766oveq1d 6330 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  ^c 
1 )  /  (
x  ^c  ( 1  -  A ) ) )  =  ( x  /  ( x  ^c  ( 1  -  A ) ) ) )
6865, 67eqtr3d 2498 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  ^c  A )  =  ( x  /  ( x  ^c  ( 1  -  A ) ) ) )
6953, 54, 683eqtr4d 2506 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  /  ( log `  x ) )  x.  ( ( log `  x )  /  (
x  ^c  ( 1  -  A ) ) ) )  =  ( x  ^c  A ) )
7069oveq1d 6330 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( x  / 
( log `  x
) )  x.  (
( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) )  /  (π `  x ) )  =  ( ( x  ^c  A )  /  (π `  x ) ) )
7148, 70eqtr3d 2498 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( x  / 
( log `  x
) )  /  (π `  x ) )  x.  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) )  =  ( ( x  ^c  A )  /  (π `  x ) ) )
7271mpteq2dva 4503 . . . . 5  |-  ( ph  ->  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( x  /  ( log `  x
) )  /  (π `  x ) )  x.  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  ^c  A )  /  (π `  x ) ) ) )
7343, 72eqtrd 2496 . . . 4  |-  ( ph  ->  ( ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  oF  x.  (
x  e.  ( 2 [,) +oo )  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  ^c  A )  /  (π `  x ) ) ) )
74 chebbnd1 24359 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  e.  O(1)
7513ex 440 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
)
7675ssrdv 3450 . . . . . 6  |-  ( ph  ->  ( 2 [,) +oo )  C_  RR+ )
77 cxploglim 23952 . . . . . . 7  |-  ( ( 1  -  A )  e.  RR+  ->  ( x  e.  RR+  |->  ( ( log `  x )  /  ( x  ^c  ( 1  -  A ) ) ) )  ~~> r  0 )
7827, 77syl 17 . . . . . 6  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) )  ~~> r  0 )
7976, 78rlimres2 13674 . . . . 5  |-  ( ph  ->  ( x  e.  ( 2 [,) +oo )  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) )  ~~> r  0 )
80 o1rlimmul 13731 . . . . 5  |-  ( ( ( x  e.  ( 2 [,) +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  e.  O(1)  /\  ( x  e.  ( 2 [,) +oo )  |->  ( ( log `  x )  /  ( x  ^c  ( 1  -  A ) ) ) )  ~~> r  0 )  ->  ( ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  oF  x.  ( x  e.  ( 2 [,) +oo )  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) ) )  ~~> r  0 )
8174, 79, 80sylancr 674 . . . 4  |-  ( ph  ->  ( ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  oF  x.  (
x  e.  ( 2 [,) +oo )  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) ) )  ~~> r  0 )
8273, 81eqbrtrrd 4439 . . 3  |-  ( ph  ->  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  ^c  A )  /  (π `  x ) ) )  ~~> r  0 )
8322, 27, 82rlimi 13626 . 2  |-  ( ph  ->  E. z  e.  RR  A. x  e.  ( 2 [,) +oo ) ( z  <_  x  ->  ( abs `  ( ( ( x  ^c  A )  /  (π `  x ) )  - 
0 ) )  < 
( 1  -  A
) ) )
8421rpcnd 11372 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  ^c  A )  /  (π `  x ) )  e.  CC )
8584subid1d 10001 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( x  ^c  A )  /  (π `  x ) )  - 
0 )  =  ( ( x  ^c  A )  /  (π `  x ) ) )
8685fveq2d 5892 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  ( abs `  ( ( ( x  ^c  A )  /  (π `  x
) )  -  0 ) )  =  ( abs `  ( ( x  ^c  A )  /  (π `  x
) ) ) )
8721rpred 11370 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  ^c  A )  /  (π `  x ) )  e.  RR )
8821rpge0d 11374 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  0  <_  ( ( x  ^c  A )  /  (π `  x ) ) )
8987, 88absidd 13533 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  ( abs `  ( ( x  ^c  A )  /  (π `  x ) ) )  =  ( ( x  ^c  A )  /  (π `  x
) ) )
9086, 89eqtrd 2496 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  ( abs `  ( ( ( x  ^c  A )  /  (π `  x
) )  -  0 ) )  =  ( ( x  ^c  A )  /  (π `  x ) ) )
9190breq1d 4426 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( abs `  (
( ( x  ^c  A )  /  (π `  x ) )  - 
0 ) )  < 
( 1  -  A
)  <->  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )
9214adantr 471 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,) +oo )  /\  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  A  e.  RR+ )
9323adantr 471 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,) +oo )  /\  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  A  <  1
)
94 simprl 769 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,) +oo )  /\  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  x  e.  ( 2 [,) +oo )
)
95 simprr 771 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,) +oo )  /\  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) )
9692, 93, 94, 95chtppilimlem1 24360 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( 2 [,) +oo )  /\  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  ( ( A ^ 2 )  x.  ( (π `  x )  x.  ( log `  x
) ) )  < 
( theta `  x )
)
9796expr 624 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( x  ^c  A )  /  (π `  x ) )  < 
( 1  -  A
)  ->  ( ( A ^ 2 )  x.  ( (π `  x )  x.  ( log `  x
) ) )  < 
( theta `  x )
) )
9891, 97sylbid 223 . . . . 5  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( abs `  (
( ( x  ^c  A )  /  (π `  x ) )  - 
0 ) )  < 
( 1  -  A
)  ->  ( ( A ^ 2 )  x.  ( (π `  x )  x.  ( log `  x
) ) )  < 
( theta `  x )
) )
9998imim2d 54 . . . 4  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( z  <_  x  ->  ( abs `  (
( ( x  ^c  A )  /  (π `  x ) )  - 
0 ) )  < 
( 1  -  A
) )  ->  (
z  <_  x  ->  ( ( A ^ 2 )  x.  ( (π `  x )  x.  ( log `  x ) ) )  <  ( theta `  x ) ) ) )
10099ralimdva 2808 . . 3  |-  ( ph  ->  ( A. x  e.  ( 2 [,) +oo ) ( z  <_  x  ->  ( abs `  (
( ( x  ^c  A )  /  (π `  x ) )  - 
0 ) )  < 
( 1  -  A
) )  ->  A. x  e.  ( 2 [,) +oo ) ( z  <_  x  ->  ( ( A ^ 2 )  x.  ( (π `  x )  x.  ( log `  x
) ) )  < 
( theta `  x )
) ) )
101100reximdv 2873 . 2  |-  ( ph  ->  ( E. z  e.  RR  A. x  e.  ( 2 [,) +oo ) ( z  <_  x  ->  ( abs `  (
( ( x  ^c  A )  /  (π `  x ) )  - 
0 ) )  < 
( 1  -  A
) )  ->  E. z  e.  RR  A. x  e.  ( 2 [,) +oo ) ( z  <_  x  ->  ( ( A ^ 2 )  x.  ( (π `  x )  x.  ( log `  x
) ) )  < 
( theta `  x )
) ) )
10283, 101mpd 15 1  |-  ( ph  ->  E. z  e.  RR  A. x  e.  ( 2 [,) +oo ) ( z  <_  x  ->  ( ( A ^ 2 )  x.  ( (π `  x )  x.  ( log `  x ) ) )  <  ( theta `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750   _Vcvv 3057   class class class wbr 4416    |-> cmpt 4475   ` cfv 5601  (class class class)co 6315    oFcof 6556   CCcc 9563   RRcr 9564   0cc0 9565   1c1 9566    x. cmul 9570   +oocpnf 9698    < clt 9701    <_ cle 9702    - cmin 9886    / cdiv 10297   NNcn 10637   2c2 10687   RR+crp 11331   [,)cico 11666   ^cexp 12304   abscabs 13346    ~~> r crli 13598   O(1)co1 13599   logclog 23553    ^c ccxp 23554   thetaccht 24066  πcppi 24069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643  ax-addf 9644  ax-mulf 9645
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558  df-om 6720  df-1st 6820  df-2nd 6821  df-supp 6942  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-2o 7209  df-oadd 7212  df-er 7389  df-map 7500  df-pm 7501  df-ixp 7549  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-fsupp 7910  df-fi 7951  df-sup 7982  df-inf 7983  df-oi 8051  df-card 8399  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-7 10701  df-8 10702  df-9 10703  df-10 10704  df-n0 10899  df-z 10967  df-dec 11081  df-uz 11189  df-q 11294  df-rp 11332  df-xneg 11438  df-xadd 11439  df-xmul 11440  df-ioo 11668  df-ioc 11669  df-ico 11670  df-icc 11671  df-fz 11814  df-fzo 11947  df-fl 12060  df-mod 12129  df-seq 12246  df-exp 12305  df-fac 12492  df-bc 12520  df-hash 12548  df-shft 13179  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-limsup 13575  df-clim 13601  df-rlim 13602  df-o1 13603  df-lo1 13604  df-sum 13802  df-ef 14170  df-e 14171  df-sin 14172  df-cos 14173  df-pi 14175  df-dvds 14355  df-gcd 14518  df-prm 14672  df-pc 14836  df-struct 15172  df-ndx 15173  df-slot 15174  df-base 15175  df-sets 15176  df-ress 15177  df-plusg 15252  df-mulr 15253  df-starv 15254  df-sca 15255  df-vsca 15256  df-ip 15257  df-tset 15258  df-ple 15259  df-ds 15261  df-unif 15262  df-hom 15263  df-cco 15264  df-rest 15370  df-topn 15371  df-0g 15389  df-gsum 15390  df-topgen 15391  df-pt 15392  df-prds 15395  df-xrs 15449  df-qtop 15455  df-imas 15456  df-xps 15459  df-mre 15541  df-mrc 15542  df-acs 15544  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-submnd 16632  df-mulg 16725  df-cntz 17020  df-cmn 17481  df-psmet 19011  df-xmet 19012  df-met 19013  df-bl 19014  df-mopn 19015  df-fbas 19016  df-fg 19017  df-cnfld 19020  df-top 19970  df-bases 19971  df-topon 19972  df-topsp 19973  df-cld 20083  df-ntr 20084  df-cls 20085  df-nei 20163  df-lp 20201  df-perf 20202  df-cn 20292  df-cnp 20293  df-haus 20380  df-tx 20626  df-hmeo 20819  df-fil 20910  df-fm 21002  df-flim 21003  df-flf 21004  df-xms 21384  df-ms 21385  df-tms 21386  df-cncf 21959  df-limc 22870  df-dv 22871  df-log 23555  df-cxp 23556  df-cht 24072  df-ppi 24075
This theorem is referenced by:  chtppilim  24362
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