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Theorem chtppilimlem2 20639
Description: Lemma for chtppilim 20640. (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 447 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  ( 2 [,)  +oo ) )
2 2re 9831 . . . . . . . . . 10  |-  2  e.  RR
3 elicopnf 10755 . . . . . . . . . 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 188 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  e.  RR  /\  2  <_  x ) )
65simpld 445 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR )
7 0re 8854 . . . . . . . . 9  |-  0  e.  RR
87a1i 10 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  e.  RR )
92a1i 10 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  e.  RR )
10 2pos 9844 . . . . . . . . 9  |-  0  <  2
1110a1i 10 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <  2 )
125simprd 449 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  <_  x )
138, 9, 6, 11, 12ltletrd 8992 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <  x )
146, 13elrpd 10404 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR+ )
15 chtppilim.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR+ )
1615rpred 10406 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
1716adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  A  e.  RR )
1814, 17rpcxpcld 20093 . . . . 5  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  A
)  e.  RR+ )
19 ppinncl 20428 . . . . . . 7  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
205, 19syl 15 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  NN )
2120nnrpd 10405 . . . . 5  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  RR+ )
2218, 21rpdivcld 10423 . . . 4  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c  A )  /  (π `  x ) )  e.  RR+ )
2322ralrimiva 2639 . . 3  |-  ( ph  ->  A. x  e.  ( 2 [,)  +oo )
( ( x  ^ c  A )  /  (π `  x ) )  e.  RR+ )
24 chtppilim.2 . . . 4  |-  ( ph  ->  A  <  1 )
25 1re 8853 . . . . 5  |-  1  e.  RR
26 difrp 10403 . . . . 5  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
2716, 25, 26sylancl 643 . . . 4  |-  ( ph  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
2824, 27mpbid 201 . . 3  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
29 ovex 5899 . . . . . . 7  |-  ( 2 [,)  +oo )  e.  _V
3029a1i 10 . . . . . 6  |-  ( ph  ->  ( 2 [,)  +oo )  e.  _V )
3125a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  RR )
32 1lt2 9902 . . . . . . . . . . 11  |-  1  <  2
3332a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <  2 )
3431, 9, 6, 33, 12ltletrd 8992 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <  x )
356, 34rplogcld 19996 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( log `  x )  e.  RR+ )
3614, 35rpdivcld 10423 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
3736, 21rpdivcld 10423 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+ )
3828adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  -  A )  e.  RR+ )
3938rpred 10406 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  -  A )  e.  RR )
4014, 39rpcxpcld 20093 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  ( 1  -  A ) )  e.  RR+ )
4135, 40rpdivcld 10423 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) )  e.  RR+ )
42 eqidd 2297 . . . . . 6  |-  ( ph  ->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) ) )
43 eqidd 2297 . . . . . 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 6111 . . . . 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 10408 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  /  ( log `  x ) )  e.  CC )
4641rpcnd 10408 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) )  e.  CC )
4721rpcnne0d 10415 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(π `  x )  e.  CC  /\  (π `  x
)  =/=  0 ) )
48 div23 9459 . . . . . . . 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 1182 . . . . . . 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 10415 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
5140rpcnne0d 10415 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c 
( 1  -  A
) )  e.  CC  /\  ( x  ^ c 
( 1  -  A
) )  =/=  0
) )
526recnd 8877 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  CC )
53 dmdcan 9486 . . . . . . . . . 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 1182 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) )  x.  ( x  /  ( log `  x
) ) )  =  ( x  /  (
x  ^ c  ( 1  -  A ) ) ) )
5545, 46mulcomd 8872 . . . . . . . . 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 10415 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
57 ax-1cn 8811 . . . . . . . . . . . . 13  |-  1  e.  CC
5857a1i 10 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  CC )
5938rpcnd 10408 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  -  A )  e.  CC )
60 cxpsub 20045 . . . . . . . . . . . 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 1182 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  ( 1  -  ( 1  -  A ) ) )  =  ( ( x  ^ c  1 )  /  ( x  ^ c  ( 1  -  A ) ) ) )
6217recnd 8877 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  A  e.  CC )
63 nncan 9092 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  (
1  -  A ) )  =  A )
6457, 62, 63sylancr 644 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  -  ( 1  -  A ) )  =  A )
6564oveq2d 5890 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  ( 1  -  ( 1  -  A ) ) )  =  ( x  ^ c  A ) )
6661, 65eqtr3d 2330 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c 
1 )  /  (
x  ^ c  ( 1  -  A ) ) )  =  ( x  ^ c  A
) )
6752cxp1d 20069 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  1 )  =  x )
6867oveq1d 5889 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c 
1 )  /  (
x  ^ c  ( 1  -  A ) ) )  =  ( x  /  ( x  ^ c  ( 1  -  A ) ) ) )
6966, 68eqtr3d 2330 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  ^ c  A
)  =  ( x  /  ( x  ^ c  ( 1  -  A ) ) ) )
7054, 55, 693eqtr4d 2338 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  /  ( log `  x ) )  x.  ( ( log `  x )  /  (
x  ^ c  ( 1  -  A ) ) ) )  =  ( x  ^ c  A ) )
7170oveq1d 5889 . . . . . . 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 2330 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( x  / 
( log `  x
) )  /  (π `  x ) )  x.  ( ( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) ) )  =  ( ( x  ^ c  A )  /  (π `  x ) ) )
7372mpteq2dva 4122 . . . . 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 2328 . . . 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 20637 . . . . 5  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  e.  O ( 1 )
7614ex 423 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( 2 [,)  +oo )  ->  x  e.  RR+ )
)
7776ssrdv 3198 . . . . . 6  |-  ( ph  ->  ( 2 [,)  +oo )  C_  RR+ )
78 cxploglim 20288 . . . . . . 7  |-  ( ( 1  -  A )  e.  RR+  ->  ( x  e.  RR+  |->  ( ( log `  x )  /  ( x  ^ c  ( 1  -  A ) ) ) )  ~~> r  0 )
7928, 78syl 15 . . . . . 6  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) ) )  ~~> r  0 )
8077, 79rlimres2 12051 . . . . 5  |-  ( ph  ->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( log `  x
)  /  ( x  ^ c  ( 1  -  A ) ) ) )  ~~> r  0 )
81 o1rlimmul 12108 . . . . 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 644 . . . 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 4061 . . 3  |-  ( ph  ->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  ^ c  A )  /  (π `  x ) ) )  ~~> r  0 )
8423, 28, 83rlimi 12003 . 2  |-  ( ph  ->  E. z  e.  RR  A. x  e.  ( 2 [,)  +oo ) ( z  <_  x  ->  ( abs `  ( ( ( x  ^ c  A
)  /  (π `  x
) )  -  0 ) )  <  (
1  -  A ) ) )
8522rpcnd 10408 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c  A )  /  (π `  x ) )  e.  CC )
8685subid1d 9162 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( x  ^ c  A )  /  (π `  x ) )  - 
0 )  =  ( ( x  ^ c  A )  /  (π `  x ) ) )
8786fveq2d 5545 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( abs `  ( ( ( x  ^ c  A
)  /  (π `  x
) )  -  0 ) )  =  ( abs `  ( ( x  ^ c  A
)  /  (π `  x
) ) ) )
8822rpred 10406 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  ^ c  A )  /  (π `  x ) )  e.  RR )
8922rpge0d 10410 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <_  ( ( x  ^ c  A )  /  (π `  x ) ) )
9088, 89absidd 11921 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( abs `  ( ( x  ^ c  A )  /  (π `  x ) ) )  =  ( ( x  ^ c  A
)  /  (π `  x
) ) )
9187, 90eqtrd 2328 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( abs `  ( ( ( x  ^ c  A
)  /  (π `  x
) )  -  0 ) )  =  ( ( x  ^ c  A )  /  (π `  x ) ) )
9291breq1d 4049 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( abs `  (
( ( x  ^ c  A )  /  (π `  x ) )  - 
0 ) )  < 
( 1  -  A
)  <->  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )
9315adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,)  +oo )  /\  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  A  e.  RR+ )
9424adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,)  +oo )  /\  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  A  <  1
)
95 simprl 732 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,)  +oo )  /\  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  x  e.  ( 2 [,)  +oo )
)
96 simprr 733 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 2 [,)  +oo )  /\  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) )
9793, 94, 95, 96chtppilimlem1 20638 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( 2 [,)  +oo )  /\  ( ( x  ^ c  A )  /  (π `  x ) )  <  ( 1  -  A ) ) )  ->  ( ( A ^ 2 )  x.  ( (π `  x )  x.  ( log `  x
) ) )  < 
( theta `  x )
)
9897expr 598 . . . . . 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 206 . . . . 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 48 . . . 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 2634 . . 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 2667 . 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 14 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    +oocpnf 8880    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   RR+crp 10370   [,)cico 10674   ^cexp 11120   abscabs 11735    ~~> r crli 11975   O (
1 )co1 11976   logclog 19928    ^ c ccxp 19929   thetaccht 20344  πcppi 20347
This theorem is referenced by:  chtppilim  20640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-o1 11980  df-lo1 11981  df-sum 12175  df-ef 12365  df-e 12366  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931  df-cht 20350  df-ppi 20353
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