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Theorem chtppilimlem2 21121
Description: Lemma for chtppilim 21122. (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 448 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  ( 2 [,)  +oo ) )
2 2re 10025 . . . . . . . . . 10  |-  2  e.  RR
3 elicopnf 10956 . . . . . . . . . 10  |-  ( 2  e.  RR  ->  (
x  e.  ( 2 [,)  +oo )  <->  ( x  e.  RR  /\  2  <_  x ) ) )
42, 3ax-mp 8 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  <->  ( x  e.  RR  /\  2  <_  x ) )
51, 4sylib 189 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  e.  RR  /\  2  <_  x ) )
65simpld 446 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR )
7 0re 9047 . . . . . . . . 9  |-  0  e.  RR
87a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  e.  RR )
92a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  e.  RR )
10 2pos 10038 . . . . . . . . 9  |-  0  <  2
1110a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <  2 )
125simprd 450 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  <_  x )
138, 9, 6, 11, 12ltletrd 9186 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <  x )
146, 13elrpd 10602 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR+ )
15 chtppilim.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR+ )
1615rpred 10604 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
1716adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  A  e.  RR )
1814, 17rpcxpcld 20574 . . . . 5  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  A
)  e.  RR+ )
19 ppinncl 20910 . . . . . . 7  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
205, 19syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  NN )
2120nnrpd 10603 . . . . 5  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  RR+ )
2218, 21rpdivcld 10621 . . . 4  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c  A )  /  (π `  x ) )  e.  RR+ )
2322ralrimiva 2749 . . 3  |-  ( ph  ->  A. x  e.  ( 2 [,)  +oo )
( ( x  ^ c  A )  /  (π `  x ) )  e.  RR+ )
24 chtppilim.2 . . . 4  |-  ( ph  ->  A  <  1 )
25 1re 9046 . . . . 5  |-  1  e.  RR
26 difrp 10601 . . . . 5  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
2716, 25, 26sylancl 644 . . . 4  |-  ( ph  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
2824, 27mpbid 202 . . 3  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
29 ovex 6065 . . . . . . 7  |-  ( 2 [,)  +oo )  e.  _V
3029a1i 11 . . . . . 6  |-  ( ph  ->  ( 2 [,)  +oo )  e.  _V )
3125a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  RR )
32 1lt2 10098 . . . . . . . . . . 11  |-  1  <  2
3332a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <  2 )
3431, 9, 6, 33, 12ltletrd 9186 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <  x )
356, 34rplogcld 20477 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( log `  x )  e.  RR+ )
3614, 35rpdivcld 10621 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
3736, 21rpdivcld 10621 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+ )
3828adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  -  A )  e.  RR+ )
3938rpred 10604 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  -  A )  e.  RR )
4014, 39rpcxpcld 20574 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  ( 1  -  A ) )  e.  RR+ )
4135, 40rpdivcld 10621 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) )  e.  RR+ )
42 eqidd 2405 . . . . . 6  |-  ( ph  ->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) ) )
43 eqidd 2405 . . . . . 6  |-  ( ph  ->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( ( log `  x )  /  ( x  ^ c  ( 1  -  A ) ) ) ) )
4430, 37, 41, 42, 43offval2 6281 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  o F  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 ) ) ) ) ) )
4536rpcnd 10606 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  /  ( log `  x ) )  e.  CC )
4641rpcnd 10606 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) )  e.  CC )
4721rpcnne0d 10613 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(π `  x )  e.  CC  /\  (π `  x
)  =/=  0 ) )
48 div23 9653 . . . . . . . 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 ) ) ) ) )
4945, 46, 47, 48syl3anc 1184 . . . . . . 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 ) ) ) ) )
5035rpcnne0d 10613 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
5140rpcnne0d 10613 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c 
( 1  -  A
) )  e.  CC  /\  ( x  ^ c 
( 1  -  A
) )  =/=  0
) )
526recnd 9070 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  CC )
53 dmdcan 9680 . . . . . . . . . 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
) ) ) )
5450, 51, 52, 53syl3anc 1184 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) )  x.  ( x  /  ( log `  x
) ) )  =  ( x  /  (
x  ^ c  ( 1  -  A ) ) ) )
5545, 46mulcomd 9065 . . . . . . . . 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 ) ) ) )
5614rpcnne0d 10613 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
57 ax-1cn 9004 . . . . . . . . . . . . 13  |-  1  e.  CC
5857a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  CC )
5938rpcnd 10606 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  -  A )  e.  CC )
60 cxpsub 20526 . . . . . . . . . . . 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
) ) ) )
6156, 58, 59, 60syl3anc 1184 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  ( 1  -  ( 1  -  A ) ) )  =  ( ( x  ^ c  1 )  /  ( x  ^ c  ( 1  -  A ) ) ) )
6217recnd 9070 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  A  e.  CC )
63 nncan 9286 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  (
1  -  A ) )  =  A )
6457, 62, 63sylancr 645 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  -  ( 1  -  A ) )  =  A )
6564oveq2d 6056 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  ( 1  -  ( 1  -  A ) ) )  =  ( x  ^ c  A ) )
6661, 65eqtr3d 2438 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c 
1 )  /  (
x  ^ c  ( 1  -  A ) ) )  =  ( x  ^ c  A
) )
6752cxp1d 20550 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  1 )  =  x )
6867oveq1d 6055 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c 
1 )  /  (
x  ^ c  ( 1  -  A ) ) )  =  ( x  /  ( x  ^ c  ( 1  -  A ) ) ) )
6966, 68eqtr3d 2438 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  A
)  =  ( x  /  ( x  ^ c  ( 1  -  A ) ) ) )
7054, 55, 693eqtr4d 2446 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  /  ( log `  x ) )  x.  ( ( log `  x )  /  (
x  ^ c  ( 1  -  A ) ) ) )  =  ( x  ^ c  A ) )
7170oveq1d 6055 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( x  / 
( log `  x
) )  x.  (
( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) ) )  /  (π `  x ) )  =  ( ( x  ^ c  A )  /  (π `  x ) ) )
7249, 71eqtr3d 2438 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( x  / 
( log `  x
) )  /  (π `  x ) )  x.  ( ( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) ) )  =  ( ( x  ^ c  A )  /  (π `  x ) ) )
7372mpteq2dva 4255 . . . . 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 ) ) ) )
7444, 73eqtrd 2436 . . . 4  |-  ( ph  ->  ( ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  o F  x.  (
x  e.  ( 2 [,)  +oo )  |->  ( ( log `  x )  /  ( x  ^ c  ( 1  -  A ) ) ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  ^ c  A
)  /  (π `  x
) ) ) )
75 chebbnd1 21119 . . . . 5  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  e.  O ( 1 )
7614ex 424 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( 2 [,)  +oo )  ->  x  e.  RR+ )
)
7776ssrdv 3314 . . . . . 6  |-  ( ph  ->  ( 2 [,)  +oo )  C_  RR+ )
78 cxploglim 20769 . . . . . . 7  |-  ( ( 1  -  A )  e.  RR+  ->  ( x  e.  RR+  |->  ( ( log `  x )  /  ( x  ^ c  ( 1  -  A ) ) ) )  ~~> r  0 )
7928, 78syl 16 . . . . . 6  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) ) )  ~~> r  0 )
8077, 79rlimres2 12310 . . . . 5  |-  ( ph  ->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) ) )  ~~> r  0 )
81 o1rlimmul 12367 . . . . 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 ) ) )  o F  x.  ( x  e.  (
2 [,)  +oo )  |->  ( ( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) ) ) )  ~~> r  0 )
8275, 80, 81sylancr 645 . . . 4  |-  ( ph  ->  ( ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  o F  x.  (
x  e.  ( 2 [,)  +oo )  |->  ( ( log `  x )  /  ( x  ^ c  ( 1  -  A ) ) ) ) )  ~~> r  0 )
8374, 82eqbrtrrd 4194 . . 3  |-  ( ph  ->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  ^ c  A )  /  (π `  x ) ) )  ~~> r  0 )
8423, 28, 83rlimi 12262 . 2  |-  ( ph  ->  E. z  e.  RR  A. x  e.  ( 2 [,)  +oo ) ( z  <_  x  ->  ( abs `  ( ( ( x  ^ c  A
)  /  (π `  x
) )  -  0 ) )  <  (
1  -  A ) ) )
8522rpcnd 10606 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c  A )  /  (π `  x ) )  e.  CC )
8685subid1d 9356 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( x  ^ c  A )  /  (π `  x ) )  - 
0 )  =  ( ( x  ^ c  A )  /  (π `  x ) ) )
8786fveq2d 5691 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( abs `  ( ( ( x  ^ c  A
)  /  (π `  x
) )  -  0 ) )  =  ( abs `  ( ( x  ^ c  A
)  /  (π `  x
) ) ) )
8822rpred 10604 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c  A )  /  (π `  x ) )  e.  RR )
8922rpge0d 10608 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <_  ( ( x  ^ c  A )  /  (π `  x ) ) )
9088, 89absidd 12180 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( abs `  ( ( x  ^ c  A )  /  (π `  x ) ) )  =  ( ( x  ^ c  A
)  /  (π `  x
) ) )
9187, 90eqtrd 2436 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( abs `  ( ( ( x  ^ c  A
)  /  (π `  x
) )  -  0 ) )  =  ( ( x  ^ c  A )  /  (π `  x ) ) )
9291breq1d 4182 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( abs `  (
( ( x  ^ c  A )  /  (π `  x ) )  - 
0 ) )  < 
( 1  -  A
)  <->  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )
9315adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,)  +oo )  /\  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  A  e.  RR+ )
9424adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,)  +oo )  /\  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  A  <  1
)
95 simprl 733 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,)  +oo )  /\  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  x  e.  ( 2 [,)  +oo )
)
96 simprr 734 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,)  +oo )  /\  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) )
9793, 94, 95, 96chtppilimlem1 21120 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( 2 [,)  +oo )  /\  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  ( ( A ^ 2 )  x.  ( (π `  x )  x.  ( log `  x
) ) )  < 
( theta `  x )
)
9897expr 599 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( x  ^ c  A )  /  (π `  x ) )  < 
( 1  -  A
)  ->  ( ( A ^ 2 )  x.  ( (π `  x )  x.  ( log `  x
) ) )  < 
( theta `  x )
) )
9992, 98sylbid 207 . . . . 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 )
) )
10099imim2d 50 . . . 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 ) ) ) )
101100ralimdva 2744 . . 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 )
) ) )
102101reximdv 2777 . 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 )
) ) )
10384, 102mpd 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   _Vcvv 2916   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040    o Fcof 6262   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    +oocpnf 9073    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   2c2 10005   RR+crp 10568   [,)cico 10874   ^cexp 11337   abscabs 11994    ~~> r crli 12234   O (
1 )co1 12235   logclog 20405    ^ c ccxp 20406   thetaccht 20826  πcppi 20829
This theorem is referenced by:  chtppilim  21122
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-o1 12239  df-lo1 12240  df-sum 12435  df-ef 12625  df-e 12626  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  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  df-cht 20832  df-ppi 20835
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