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Theorem chebbnd1 20637
Description: The Chebyshev bound: The function π ( x ) is eventually lower bounded by a positive constant times  x  /  log ( x ). Alternatively stated, the function  ( x  /  log ( x ) )  / π ( x ) is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.)
Assertion
Ref Expression
chebbnd1  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  e.  O ( 1 )

Proof of Theorem chebbnd1
StepHypRef Expression
1 2re 9831 . . . . 5  |-  2  e.  RR
2 pnfxr 10471 . . . . 5  |-  +oo  e.  RR*
3 icossre 10746 . . . . 5  |-  ( ( 2  e.  RR  /\  +oo 
e.  RR* )  ->  (
2 [,)  +oo )  C_  RR )
41, 2, 3mp2an 653 . . . 4  |-  ( 2 [,)  +oo )  C_  RR
54a1i 10 . . 3  |-  (  T. 
->  ( 2 [,)  +oo )  C_  RR )
6 elicopnf 10755 . . . . . . . . . 10  |-  ( 2  e.  RR  ->  (
x  e.  ( 2 [,)  +oo )  <->  ( x  e.  RR  /\  2  <_  x ) ) )
71, 6ax-mp 8 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  <->  ( x  e.  RR  /\  2  <_  x ) )
87simplbi 446 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR )
9 0re 8854 . . . . . . . . . 10  |-  0  e.  RR
109a1i 10 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  e.  RR )
11 1re 8853 . . . . . . . . . 10  |-  1  e.  RR
1211a1i 10 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  e.  RR )
13 0lt1 9312 . . . . . . . . . 10  |-  0  <  1
1413a1i 10 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  1 )
151a1i 10 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  e.  RR )
16 1lt2 9902 . . . . . . . . . . 11  |-  1  <  2
1716a1i 10 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  <  2 )
187simprbi 450 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  <_  x )
1912, 15, 8, 17, 18ltletrd 8992 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  <  x )
2010, 12, 8, 14, 19lttrd 8993 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  x )
218, 20elrpd 10404 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
228, 19rplogcld 19996 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  RR+ )
2321, 22rpdivcld 10423 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
24 ppinncl 20428 . . . . . . . 8  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
257, 24sylbi 187 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  NN )
2625nnrpd 10405 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  RR+ )
2723, 26rpdivcld 10423 . . . . 5  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+ )
2827rpcnd 10408 . . . 4  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  CC )
2928adantl 452 . . 3  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  CC )
30 8re 9840 . . . 4  |-  8  e.  RR
3130a1i 10 . . 3  |-  (  T. 
->  8  e.  RR )
32 2rp 10375 . . . . . . . 8  |-  2  e.  RR+
33 relogcl 19948 . . . . . . . 8  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3432, 33ax-mp 8 . . . . . . 7  |-  ( log `  2 )  e.  RR
35 ere 12386 . . . . . . . . 9  |-  _e  e.  RR
361, 35remulcli 8867 . . . . . . . 8  |-  ( 2  x.  _e )  e.  RR
37 2pos 9844 . . . . . . . . . 10  |-  0  <  2
38 epos 12501 . . . . . . . . . 10  |-  0  <  _e
391, 35, 37, 38mulgt0ii 8968 . . . . . . . . 9  |-  0  <  ( 2  x.  _e )
4036, 39gt0ne0ii 9325 . . . . . . . 8  |-  ( 2  x.  _e )  =/=  0
4136, 40rereccli 9541 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  e.  RR
4234, 41resubcli 9125 . . . . . 6  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR
431recni 8865 . . . . . . . . . . 11  |-  2  e.  CC
4443mulid1i 8855 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
45 egt2lt3 12500 . . . . . . . . . . . . 13  |-  ( 2  <  _e  /\  _e  <  3 )
4645simpli 444 . . . . . . . . . . . 12  |-  2  <  _e
4711, 1, 35lttri 8961 . . . . . . . . . . . 12  |-  ( ( 1  <  2  /\  2  <  _e )  ->  1  <  _e )
4816, 46, 47mp2an 653 . . . . . . . . . . 11  |-  1  <  _e
4911, 35, 1ltmul2i 9694 . . . . . . . . . . . 12  |-  ( 0  <  2  ->  (
1  <  _e  <->  ( 2  x.  1 )  < 
( 2  x.  _e ) ) )
5037, 49ax-mp 8 . . . . . . . . . . 11  |-  ( 1  <  _e  <->  ( 2  x.  1 )  < 
( 2  x.  _e ) )
5148, 50mpbi 199 . . . . . . . . . 10  |-  ( 2  x.  1 )  < 
( 2  x.  _e )
5244, 51eqbrtrri 4060 . . . . . . . . 9  |-  2  <  ( 2  x.  _e )
531, 36, 37, 39ltrecii 9689 . . . . . . . . 9  |-  ( 2  <  ( 2  x.  _e )  <->  ( 1  /  ( 2  x.  _e ) )  < 
( 1  /  2
) )
5452, 53mpbi 199 . . . . . . . 8  |-  ( 1  /  ( 2  x.  _e ) )  < 
( 1  /  2
)
5545simpri 448 . . . . . . . . . . . 12  |-  _e  <  3
56 3lt4 9905 . . . . . . . . . . . 12  |-  3  <  4
57 3re 9833 . . . . . . . . . . . . 13  |-  3  e.  RR
58 4re 9835 . . . . . . . . . . . . 13  |-  4  e.  RR
5935, 57, 58lttri 8961 . . . . . . . . . . . 12  |-  ( ( _e  <  3  /\  3  <  4 )  ->  _e  <  4
)
6055, 56, 59mp2an 653 . . . . . . . . . . 11  |-  _e  <  4
61 epr 12502 . . . . . . . . . . . 12  |-  _e  e.  RR+
62 4pos 9848 . . . . . . . . . . . . 13  |-  0  <  4
6358, 62elrpii 10373 . . . . . . . . . . . 12  |-  4  e.  RR+
64 logltb 19969 . . . . . . . . . . . 12  |-  ( ( _e  e.  RR+  /\  4  e.  RR+ )  ->  (
_e  <  4  <->  ( log `  _e )  <  ( log `  4 ) ) )
6561, 63, 64mp2an 653 . . . . . . . . . . 11  |-  ( _e 
<  4  <->  ( log `  _e )  <  ( log `  4 ) )
6660, 65mpbi 199 . . . . . . . . . 10  |-  ( log `  _e )  <  ( log `  4 )
67 loge 19956 . . . . . . . . . 10  |-  ( log `  _e )  =  1
68 sq2 11215 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
6968fveq2i 5544 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( log `  4 )
70 2z 10070 . . . . . . . . . . . 12  |-  2  e.  ZZ
71 relogexp 19965 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR+  /\  2  e.  ZZ )  ->  ( log `  ( 2 ^ 2 ) )  =  ( 2  x.  ( log `  2 ) ) )
7232, 70, 71mp2an 653 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( 2  x.  ( log `  2 ) )
7369, 72eqtr3i 2318 . . . . . . . . . 10  |-  ( log `  4 )  =  ( 2  x.  ( log `  2 ) )
7466, 67, 733brtr3i 4066 . . . . . . . . 9  |-  1  <  ( 2  x.  ( log `  2 ) )
751, 37pm3.2i 441 . . . . . . . . . 10  |-  ( 2  e.  RR  /\  0  <  2 )
76 ltdivmul 9644 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( log `  2 )  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( 1  /  2 )  < 
( log `  2
)  <->  1  <  (
2  x.  ( log `  2 ) ) ) )
7711, 34, 75, 76mp3an 1277 . . . . . . . . 9  |-  ( ( 1  /  2 )  <  ( log `  2
)  <->  1  <  (
2  x.  ( log `  2 ) ) )
7874, 77mpbir 200 . . . . . . . 8  |-  ( 1  /  2 )  < 
( log `  2
)
79 rehalfcl 9954 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  (
1  /  2 )  e.  RR )
8011, 79ax-mp 8 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
8141, 80, 34lttri 8961 . . . . . . . 8  |-  ( ( ( 1  /  (
2  x.  _e ) )  <  ( 1  /  2 )  /\  ( 1  /  2
)  <  ( log `  2 ) )  -> 
( 1  /  (
2  x.  _e ) )  <  ( log `  2 ) )
8254, 78, 81mp2an 653 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  < 
( log `  2
)
8341, 34posdifi 9339 . . . . . . 7  |-  ( ( 1  /  ( 2  x.  _e ) )  <  ( log `  2
)  <->  0  <  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
8482, 83mpbi 199 . . . . . 6  |-  0  <  ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )
8542, 84elrpii 10373 . . . . 5  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR+
86 rerpdivcl 10397 . . . . 5  |-  ( ( 2  e.  RR  /\  ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )  e.  RR+ )  ->  (
2  /  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) ) )  e.  RR )
871, 85, 86mp2an 653 . . . 4  |-  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) )  e.  RR
8887a1i 10 . . 3  |-  (  T. 
->  ( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )  e.  RR )
89 rpre 10376 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR )
90 rpge0 10382 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  0  <_ 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) )
9189, 90absidd 11921 . . . . . . 7  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( abs `  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )
9227, 91syl 15 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( abs `  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  =  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )
9392adantr 451 . . . . 5  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  ( abs `  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  =  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )
94 eqid 2296 . . . . . . . . . 10  |-  ( |_
`  ( x  / 
2 ) )  =  ( |_ `  (
x  /  2 ) )
9594chebbnd1lem3 20636 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  8  <_  x )  -> 
( ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) )  /  2
)  <  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
968, 95sylan 457 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )  /  2 )  < 
( (π `  x )  x.  ( ( log `  x
)  /  x ) ) )
97 2ne0 9845 . . . . . . . . . 10  |-  2  =/=  0
9842recni 8865 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  CC
9942, 84gt0ne0ii 9325 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  =/=  0
100 recdiv 9482 . . . . . . . . . 10  |-  ( ( ( 2  e.  CC  /\  2  =/=  0 )  /\  ( ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  CC  /\  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  =/=  0 ) )  -> 
( 1  /  (
2  /  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) ) ) )  =  ( ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )  /  2 ) )
10143, 97, 98, 99, 100mp4an 654 . . . . . . . . 9  |-  ( 1  /  ( 2  / 
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )  =  ( ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )  /  2 )
102101a1i 10 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
1  /  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) ) )  =  ( ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  / 
2 ) )
10323rpcnd 10408 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  CC )
10425nncnd 9778 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  CC )
10523rpne0d 10411 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  =/=  0 )
10625nnne0d 9806 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  =/=  0
)
107103, 104, 105, 106recdivd 9569 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
1  /  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  =  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
108104, 103, 105divrecd 9555 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  =  ( (π `  x )  x.  (
1  /  ( x  /  ( log `  x
) ) ) ) )
10921rpcnne0d 10415 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
11022rpcnne0d 10415 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
111 recdiv 9482 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( ( log `  x )  e.  CC  /\  ( log `  x
)  =/=  0 ) )  ->  ( 1  /  ( x  / 
( log `  x
) ) )  =  ( ( log `  x
)  /  x ) )
112109, 110, 111syl2anc 642 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
1  /  ( x  /  ( log `  x
) ) )  =  ( ( log `  x
)  /  x ) )
113112oveq2d 5890 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( 1  /  (
x  /  ( log `  x ) ) ) )  =  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
114107, 108, 1133eqtrd 2332 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
1  /  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  =  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
115114adantr 451 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
1  /  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  =  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
11696, 102, 1153brtr4d 4069 . . . . . . 7  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
1  /  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) ) )  <  ( 1  / 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) ) )
11727adantr 451 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+ )
118 elrp 10372 . . . . . . . . 9  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  <->  ( ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR  /\  0  <  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) ) )
1191, 42, 37, 84divgt0ii 9690 . . . . . . . . . 10  |-  0  <  ( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
120 ltrec 9653 . . . . . . . . . 10  |-  ( ( ( ( ( x  /  ( log `  x
) )  /  (π `  x ) )  e.  RR  /\  0  < 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  /\  ( ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) )  e.  RR  /\  0  < 
( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) ) )  -> 
( ( ( x  /  ( log `  x
) )  /  (π `  x ) )  < 
( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )  <->  ( 1  / 
( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )  <  (
1  /  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) ) ) )
12187, 119, 120mpanr12 666 . . . . . . . . 9  |-  ( ( ( ( x  / 
( log `  x
) )  /  (π `  x ) )  e.  RR  /\  0  < 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  ->  ( ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  <  ( 2  / 
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )  <->  ( 1  / 
( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )  <  (
1  /  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) ) ) )
122118, 121sylbi 187 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  <  ( 2  / 
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )  <->  ( 1  / 
( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )  <  (
1  /  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) ) ) )
123117, 122syl 15 . . . . . . 7  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( ( x  / 
( log `  x
) )  /  (π `  x ) )  < 
( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )  <->  ( 1  / 
( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )  <  (
1  /  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) ) ) )
124116, 123mpbird 223 . . . . . 6  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  <  ( 2  / 
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )
125117rpred 10406 . . . . . . 7  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR )
126 ltle 8926 . . . . . . 7  |-  ( ( ( ( x  / 
( log `  x
) )  /  (π `  x ) )  e.  RR  /\  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) )  e.  RR )  ->  (
( ( x  / 
( log `  x
) )  /  (π `  x ) )  < 
( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )  ->  ( (
x  /  ( log `  x ) )  / 
(π `  x ) )  <_  ( 2  / 
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) ) )
127125, 87, 126sylancl 643 . . . . . 6  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( ( x  / 
( log `  x
) )  /  (π `  x ) )  < 
( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )  ->  ( (
x  /  ( log `  x ) )  / 
(π `  x ) )  <_  ( 2  / 
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) ) )
128124, 127mpd 14 . . . . 5  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  <_  ( 2  / 
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )
12993, 128eqbrtrd 4059 . . . 4  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  ( abs `  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  <_  ( 2  / 
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )
130129adantl 452 . . 3  |-  ( (  T.  /\  ( x  e.  ( 2 [,) 
+oo )  /\  8  <_  x ) )  -> 
( abs `  (
( x  /  ( log `  x ) )  /  (π `  x ) ) )  <_  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) ) )
1315, 29, 31, 88, 130elo1d 12026 . 2  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  e.  O ( 1 ) )
132131trud 1314 1  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  e.  O ( 1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    T. wtru 1307    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   3c3 9812   4c4 9813   8c8 9817   ZZcz 10040   RR+crp 10370   [,)cico 10674   |_cfl 10940   ^cexp 11120   abscabs 11735   O ( 1 )co1 11976   _eceu 12360   logclog 19928  πcppi 20347
This theorem is referenced by:  chtppilimlem2  20639  chto1lb  20643
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-ppi 20353
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