MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  chtppilimlem2 Structured version   Unicode version

Theorem chtppilimlem2 21168
Description: Lemma for chtppilim 21169. (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 10069 . . . . . . . . . 10  |-  2  e.  RR
3 elicopnf 11000 . . . . . . . . . 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 9091 . . . . . . . . 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 10082 . . . . . . . . 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 9230 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <  x )
146, 13elrpd 10646 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR+ )
15 chtppilim.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR+ )
1615rpred 10648 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
1716adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  A  e.  RR )
1814, 17rpcxpcld 20621 . . . . 5  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  A
)  e.  RR+ )
19 ppinncl 20957 . . . . . . 7  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
205, 19syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  NN )
2120nnrpd 10647 . . . . 5  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  RR+ )
2218, 21rpdivcld 10665 . . . 4  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c  A )  /  (π `  x ) )  e.  RR+ )
2322ralrimiva 2789 . . 3  |-  ( ph  ->  A. x  e.  ( 2 [,)  +oo )
( ( x  ^ c  A )  /  (π `  x ) )  e.  RR+ )
24 chtppilim.2 . . . 4  |-  ( ph  ->  A  <  1 )
25 1re 9090 . . . . 5  |-  1  e.  RR
26 difrp 10645 . . . . 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 6106 . . . . . . 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 10142 . . . . . . . . . . 11  |-  1  <  2
3332a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <  2 )
3431, 9, 6, 33, 12ltletrd 9230 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <  x )
356, 34rplogcld 20524 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( log `  x )  e.  RR+ )
3614, 35rpdivcld 10665 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
3736, 21rpdivcld 10665 . . . . . 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 10648 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  -  A )  e.  RR )
4014, 39rpcxpcld 20621 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  ( 1  -  A ) )  e.  RR+ )
4135, 40rpdivcld 10665 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) )  e.  RR+ )
42 eqidd 2437 . . . . . 6  |-  ( ph  ->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) ) )
43 eqidd 2437 . . . . . 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 6322 . . . . 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 10650 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  /  ( log `  x ) )  e.  CC )
4641rpcnd 10650 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) )  e.  CC )
4721rpcnne0d 10657 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(π `  x )  e.  CC  /\  (π `  x
)  =/=  0 ) )
48 div23 9697 . . . . . . . 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 10657 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
5140rpcnne0d 10657 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c 
( 1  -  A
) )  e.  CC  /\  ( x  ^ c 
( 1  -  A
) )  =/=  0
) )
526recnd 9114 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  CC )
53 dmdcan 9724 . . . . . . . . . 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 9109 . . . . . . . . 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 10657 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
57 ax-1cn 9048 . . . . . . . . . . . . 13  |-  1  e.  CC
5857a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  CC )
5938rpcnd 10650 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  -  A )  e.  CC )
60 cxpsub 20573 . . . . . . . . . . . 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 9114 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  A  e.  CC )
63 nncan 9330 . . . . . . . . . . . . 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 6097 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  ( 1  -  ( 1  -  A ) ) )  =  ( x  ^ c  A ) )
6661, 65eqtr3d 2470 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c 
1 )  /  (
x  ^ c  ( 1  -  A ) ) )  =  ( x  ^ c  A
) )
6752cxp1d 20597 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  1 )  =  x )
6867oveq1d 6096 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c 
1 )  /  (
x  ^ c  ( 1  -  A ) ) )  =  ( x  /  ( x  ^ c  ( 1  -  A ) ) ) )
6966, 68eqtr3d 2470 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  A
)  =  ( x  /  ( x  ^ c  ( 1  -  A ) ) ) )
7054, 55, 693eqtr4d 2478 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  /  ( log `  x ) )  x.  ( ( log `  x )  /  (
x  ^ c  ( 1  -  A ) ) ) )  =  ( x  ^ c  A ) )
7170oveq1d 6096 . . . . . . 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 2470 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( x  / 
( log `  x
) )  /  (π `  x ) )  x.  ( ( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) ) )  =  ( ( x  ^ c  A )  /  (π `  x ) ) )
7372mpteq2dva 4295 . . . . 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 2468 . . . 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 21166 . . . . 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 3354 . . . . . 6  |-  ( ph  ->  ( 2 [,)  +oo )  C_  RR+ )
78 cxploglim 20816 . . . . . . 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 12355 . . . . 5  |-  ( ph  ->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) ) )  ~~> r  0 )
81 o1rlimmul 12412 . . . . 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 4234 . . 3  |-  ( ph  ->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  ^ c  A )  /  (π `  x ) ) )  ~~> r  0 )
8423, 28, 83rlimi 12307 . 2  |-  ( ph  ->  E. z  e.  RR  A. x  e.  ( 2 [,)  +oo ) ( z  <_  x  ->  ( abs `  ( ( ( x  ^ c  A
)  /  (π `  x
) )  -  0 ) )  <  (
1  -  A ) ) )
8522rpcnd 10650 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c  A )  /  (π `  x ) )  e.  CC )
8685subid1d 9400 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( x  ^ c  A )  /  (π `  x ) )  - 
0 )  =  ( ( x  ^ c  A )  /  (π `  x ) ) )
8786fveq2d 5732 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( abs `  ( ( ( x  ^ c  A
)  /  (π `  x
) )  -  0 ) )  =  ( abs `  ( ( x  ^ c  A
)  /  (π `  x
) ) ) )
8822rpred 10648 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c  A )  /  (π `  x ) )  e.  RR )
8922rpge0d 10652 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <_  ( ( x  ^ c  A )  /  (π `  x ) ) )
9088, 89absidd 12225 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( abs `  ( ( x  ^ c  A )  /  (π `  x ) ) )  =  ( ( x  ^ c  A
)  /  (π `  x
) ) )
9187, 90eqtrd 2468 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( abs `  ( ( ( x  ^ c  A
)  /  (π `  x
) )  -  0 ) )  =  ( ( x  ^ c  A )  /  (π `  x ) ) )
9291breq1d 4222 . . . . . 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 21167 . . . . . . 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 2784 . . 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 2817 . 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 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   _Vcvv 2956   class class class wbr 4212    e. cmpt 4266   ` cfv 5454  (class class class)co 6081    o Fcof 6303   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    +oocpnf 9117    < clt 9120    <_ cle 9121    - cmin 9291    / cdiv 9677   NNcn 10000   2c2 10049   RR+crp 10612   [,)cico 10918   ^cexp 11382   abscabs 12039    ~~> r crli 12279   O (
1 )co1 12280   logclog 20452    ^ c ccxp 20453   thetaccht 20873  πcppi 20876
This theorem is referenced by:  chtppilim  21169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ioc 10921  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-o1 12284  df-lo1 12285  df-sum 12480  df-ef 12670  df-e 12671  df-sin 12672  df-cos 12673  df-pi 12675  df-dvds 12853  df-gcd 13007  df-prm 13080  df-pc 13211  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-limc 19753  df-dv 19754  df-log 20454  df-cxp 20455  df-cht 20879  df-ppi 20882
  Copyright terms: Public domain W3C validator