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Theorem chtppilimlem2 22682
Description: Lemma for chtppilim 22683. (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 458 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  ( 2 [,) +oo ) )
2 2re 10387 . . . . . . . . . 10  |-  2  e.  RR
3 elicopnf 11381 . . . . . . . . . 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 196 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  e.  RR  /\  2  <_  x ) )
65simpld 456 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  RR )
7 0red 9383 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  0  e.  RR )
82a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  2  e.  RR )
9 2pos 10409 . . . . . . . . 9  |-  0  <  2
109a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  0  <  2 )
115simprd 460 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  2  <_  x )
127, 8, 6, 10, 11ltletrd 9527 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  0  <  x )
136, 12elrpd 11021 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  RR+ )
14 chtppilim.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR+ )
1514rpred 11023 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
1615adantr 462 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  A  e.  RR )
1713, 16rpcxpcld 22134 . . . . 5  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  ^c  A )  e.  RR+ )
18 ppinncl 22471 . . . . . . 7  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
195, 18syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (π `  x )  e.  NN )
2019nnrpd 11022 . . . . 5  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (π `  x )  e.  RR+ )
2117, 20rpdivcld 11040 . . . 4  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  ^c  A )  /  (π `  x ) )  e.  RR+ )
2221ralrimiva 2797 . . 3  |-  ( ph  ->  A. x  e.  ( 2 [,) +oo )
( ( x  ^c  A )  /  (π `  x ) )  e.  RR+ )
23 chtppilim.2 . . . 4  |-  ( ph  ->  A  <  1 )
24 1re 9381 . . . . 5  |-  1  e.  RR
25 difrp 11020 . . . . 5  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
2615, 24, 25sylancl 657 . . . 4  |-  ( ph  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
2723, 26mpbid 210 . . 3  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
28 ovex 6115 . . . . . . 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 10484 . . . . . . . . . . 11  |-  1  <  2
3231a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <  2 )
3330, 8, 6, 32, 11ltletrd 9527 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <  x )
346, 33rplogcld 22037 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  ( log `  x )  e.  RR+ )
3513, 34rpdivcld 11040 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
3635, 20rpdivcld 11040 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+ )
3727adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  -  A )  e.  RR+ )
3837rpred 11023 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  -  A )  e.  RR )
3913, 38rpcxpcld 22134 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  ^c  ( 1  -  A ) )  e.  RR+ )
4034, 39rpdivcld 11040 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) )  e.  RR+ )
41 eqidd 2442 . . . . . 6  |-  ( ph  ->  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) ) )
42 eqidd 2442 . . . . . 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 6335 . . . . 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 11025 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  /  ( log `  x ) )  e.  CC )
4540rpcnd 11025 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) )  e.  CC )
4620rpcnne0d 11032 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(π `  x )  e.  CC  /\  (π `  x
)  =/=  0 ) )
47 div23 10009 . . . . . . . 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 1213 . . . . . . 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 11032 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
5039rpcnne0d 11032 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  ^c 
( 1  -  A
) )  e.  CC  /\  ( x  ^c 
( 1  -  A
) )  =/=  0
) )
516recnd 9408 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  CC )
52 dmdcan 10037 . . . . . . . . . 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 1213 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) )  x.  ( x  /  ( log `  x
) ) )  =  ( x  /  (
x  ^c  ( 1  -  A ) ) ) )
5444, 45mulcomd 9403 . . . . . . . . 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 11032 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
56 ax-1cn 9336 . . . . . . . . . . . . 13  |-  1  e.  CC
5756a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  1  e.  CC )
5837rpcnd 11025 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  -  A )  e.  CC )
59 cxpsub 22086 . . . . . . . . . . . 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 1213 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  ^c  ( 1  -  ( 1  -  A ) ) )  =  ( ( x  ^c  1 )  /  ( x  ^c  ( 1  -  A ) ) ) )
6116recnd 9408 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  A  e.  CC )
62 nncan 9634 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  (
1  -  A ) )  =  A )
6356, 61, 62sylancr 658 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  -  ( 1  -  A ) )  =  A )
6463oveq2d 6106 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  ^c  ( 1  -  ( 1  -  A ) ) )  =  ( x  ^c  A ) )
6560, 64eqtr3d 2475 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  ^c 
1 )  /  (
x  ^c  ( 1  -  A ) ) )  =  ( x  ^c  A ) )
6651cxp1d 22110 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  ^c  1 )  =  x )
6766oveq1d 6105 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  ^c 
1 )  /  (
x  ^c  ( 1  -  A ) ) )  =  ( x  /  ( x  ^c  ( 1  -  A ) ) ) )
6865, 67eqtr3d 2475 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  ^c  A )  =  ( x  /  ( x  ^c  ( 1  -  A ) ) ) )
6953, 54, 683eqtr4d 2483 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  /  ( log `  x ) )  x.  ( ( log `  x )  /  (
x  ^c  ( 1  -  A ) ) ) )  =  ( x  ^c  A ) )
7069oveq1d 6105 . . . . . . 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 2475 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( x  / 
( log `  x
) )  /  (π `  x ) )  x.  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) )  =  ( ( x  ^c  A )  /  (π `  x ) ) )
7271mpteq2dva 4375 . . . . 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 2473 . . . 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 22680 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  e.  O(1)
7513ex 434 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
)
7675ssrdv 3359 . . . . . 6  |-  ( ph  ->  ( 2 [,) +oo )  C_  RR+ )
77 cxploglim 22330 . . . . . . 7  |-  ( ( 1  -  A )  e.  RR+  ->  ( x  e.  RR+  |->  ( ( log `  x )  /  ( x  ^c  ( 1  -  A ) ) ) )  ~~> r  0 )
7827, 77syl 16 . . . . . 6  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) )  ~~> r  0 )
7976, 78rlimres2 13035 . . . . 5  |-  ( ph  ->  ( x  e.  ( 2 [,) +oo )  |->  ( ( log `  x
)  /  ( x  ^c  ( 1  -  A ) ) ) )  ~~> r  0 )
80 o1rlimmul 13092 . . . . 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 658 . . . 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 4311 . . 3  |-  ( ph  ->  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  ^c  A )  /  (π `  x ) ) )  ~~> r  0 )
8322, 27, 82rlimi 12987 . 2  |-  ( ph  ->  E. z  e.  RR  A. x  e.  ( 2 [,) +oo ) ( z  <_  x  ->  ( abs `  ( ( ( x  ^c  A )  /  (π `  x ) )  - 
0 ) )  < 
( 1  -  A
) ) )
8421rpcnd 11025 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  ^c  A )  /  (π `  x ) )  e.  CC )
8584subid1d 9704 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( x  ^c  A )  /  (π `  x ) )  - 
0 )  =  ( ( x  ^c  A )  /  (π `  x ) ) )
8685fveq2d 5692 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  ( abs `  ( ( ( x  ^c  A )  /  (π `  x
) )  -  0 ) )  =  ( abs `  ( ( x  ^c  A )  /  (π `  x
) ) ) )
8721rpred 11023 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  ^c  A )  /  (π `  x ) )  e.  RR )
8821rpge0d 11027 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  0  <_  ( ( x  ^c  A )  /  (π `  x ) ) )
8987, 88absidd 12905 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  ( abs `  ( ( x  ^c  A )  /  (π `  x ) ) )  =  ( ( x  ^c  A )  /  (π `  x
) ) )
9086, 89eqtrd 2473 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  ( abs `  ( ( ( x  ^c  A )  /  (π `  x
) )  -  0 ) )  =  ( ( x  ^c  A )  /  (π `  x ) ) )
9190breq1d 4299 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( abs `  (
( ( x  ^c  A )  /  (π `  x ) )  - 
0 ) )  < 
( 1  -  A
)  <->  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )
9214adantr 462 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,) +oo )  /\  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  A  e.  RR+ )
9323adantr 462 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,) +oo )  /\  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  A  <  1
)
94 simprl 750 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,) +oo )  /\  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  x  e.  ( 2 [,) +oo )
)
95 simprr 751 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,) +oo )  /\  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) )
9692, 93, 94, 95chtppilimlem1 22681 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( 2 [,) +oo )  /\  ( ( x  ^c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  ( ( A ^ 2 )  x.  ( (π `  x )  x.  ( log `  x
) ) )  < 
( theta `  x )
)
9796expr 612 . . . . . 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 215 . . . . 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 52 . . . 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 2792 . . 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 2825 . 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 184    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   _Vcvv 2970   class class class wbr 4289    e. cmpt 4347   ` cfv 5415  (class class class)co 6090    oFcof 6317   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    x. cmul 9283   +oocpnf 9411    < clt 9414    <_ cle 9415    - cmin 9591    / cdiv 9989   NNcn 10318   2c2 10367   RR+crp 10987   [,)cico 11298   ^cexp 11861   abscabs 12719    ~~> r crli 12959   O(1)co1 12960   logclog 21965    ^c ccxp 21966   thetaccht 22387  πcppi 22390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-o1 12964  df-lo1 12965  df-sum 13160  df-ef 13349  df-e 13350  df-sin 13351  df-cos 13352  df-pi 13354  df-dvds 13532  df-gcd 13687  df-prm 13760  df-pc 13900  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967  df-cxp 21968  df-cht 22393  df-ppi 22396
This theorem is referenced by:  chtppilim  22683
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