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Theorem chebbnd1 20621
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 9815 . . . . 5  |-  2  e.  RR
2 pnfxr 10455 . . . . 5  |-  +oo  e.  RR*
3 icossre 10730 . . . . 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 10739 . . . . . . . . . 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 8838 . . . . . . . . . 10  |-  0  e.  RR
109a1i 10 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  e.  RR )
11 1re 8837 . . . . . . . . . 10  |-  1  e.  RR
1211a1i 10 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  e.  RR )
13 0lt1 9296 . . . . . . . . . 10  |-  0  <  1
1413a1i 10 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  1 )
151a1i 10 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  e.  RR )
16 1lt2 9886 . . . . . . . . . . 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 8976 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  <  x )
2010, 12, 8, 14, 19lttrd 8977 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  x )
218, 20elrpd 10388 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
228, 19rplogcld 19980 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  RR+ )
2321, 22rpdivcld 10407 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
24 ppinncl 20412 . . . . . . . 8  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
257, 24sylbi 187 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  NN )
2625nnrpd 10389 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  RR+ )
2723, 26rpdivcld 10407 . . . . 5  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+ )
2827rpcnd 10392 . . . 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 9824 . . . 4  |-  8  e.  RR
3130a1i 10 . . 3  |-  (  T. 
->  8  e.  RR )
32 2rp 10359 . . . . . . . 8  |-  2  e.  RR+
33 relogcl 19932 . . . . . . . 8  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3432, 33ax-mp 8 . . . . . . 7  |-  ( log `  2 )  e.  RR
35 ere 12370 . . . . . . . . 9  |-  _e  e.  RR
361, 35remulcli 8851 . . . . . . . 8  |-  ( 2  x.  _e )  e.  RR
37 2pos 9828 . . . . . . . . . 10  |-  0  <  2
38 epos 12485 . . . . . . . . . 10  |-  0  <  _e
391, 35, 37, 38mulgt0ii 8952 . . . . . . . . 9  |-  0  <  ( 2  x.  _e )
4036, 39gt0ne0ii 9309 . . . . . . . 8  |-  ( 2  x.  _e )  =/=  0
4136, 40rereccli 9525 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  e.  RR
4234, 41resubcli 9109 . . . . . 6  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR
431recni 8849 . . . . . . . . . . 11  |-  2  e.  CC
4443mulid1i 8839 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
45 egt2lt3 12484 . . . . . . . . . . . . 13  |-  ( 2  <  _e  /\  _e  <  3 )
4645simpli 444 . . . . . . . . . . . 12  |-  2  <  _e
4711, 1, 35lttri 8945 . . . . . . . . . . . 12  |-  ( ( 1  <  2  /\  2  <  _e )  ->  1  <  _e )
4816, 46, 47mp2an 653 . . . . . . . . . . 11  |-  1  <  _e
4911, 35, 1ltmul2i 9678 . . . . . . . . . . . 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 4044 . . . . . . . . 9  |-  2  <  ( 2  x.  _e )
531, 36, 37, 39ltrecii 9673 . . . . . . . . 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 9889 . . . . . . . . . . . 12  |-  3  <  4
57 3re 9817 . . . . . . . . . . . . 13  |-  3  e.  RR
58 4re 9819 . . . . . . . . . . . . 13  |-  4  e.  RR
5935, 57, 58lttri 8945 . . . . . . . . . . . 12  |-  ( ( _e  <  3  /\  3  <  4 )  ->  _e  <  4
)
6055, 56, 59mp2an 653 . . . . . . . . . . 11  |-  _e  <  4
61 epr 12486 . . . . . . . . . . . 12  |-  _e  e.  RR+
62 4pos 9832 . . . . . . . . . . . . 13  |-  0  <  4
6358, 62elrpii 10357 . . . . . . . . . . . 12  |-  4  e.  RR+
64 logltb 19953 . . . . . . . . . . . 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 19940 . . . . . . . . . 10  |-  ( log `  _e )  =  1
68 sq2 11199 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
6968fveq2i 5528 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( log `  4 )
70 2z 10054 . . . . . . . . . . . 12  |-  2  e.  ZZ
71 relogexp 19949 . . . . . . . . . . . 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 2305 . . . . . . . . . 10  |-  ( log `  4 )  =  ( 2  x.  ( log `  2 ) )
7466, 67, 733brtr3i 4050 . . . . . . . . 9  |-  1  <  ( 2  x.  ( log `  2 ) )
751, 37pm3.2i 441 . . . . . . . . . 10  |-  ( 2  e.  RR  /\  0  <  2 )
76 ltdivmul 9628 . . . . . . . . . 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 9938 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  (
1  /  2 )  e.  RR )
8011, 79ax-mp 8 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
8141, 80, 34lttri 8945 . . . . . . . 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 9323 . . . . . . 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 10357 . . . . 5  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR+
86 rerpdivcl 10381 . . . . 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 10360 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR )
90 rpge0 10366 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  0  <_ 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) )
9189, 90absidd 11905 . . . . . . 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 2283 . . . . . . . . . 10  |-  ( |_
`  ( x  / 
2 ) )  =  ( |_ `  (
x  /  2 ) )
9594chebbnd1lem3 20620 . . . . . . . . 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 9829 . . . . . . . . . 10  |-  2  =/=  0
9842recni 8849 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  CC
9942, 84gt0ne0ii 9309 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  =/=  0
100 recdiv 9466 . . . . . . . . . 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 10392 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  CC )
10425nncnd 9762 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  CC )
10523rpne0d 10395 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  =/=  0 )
10625nnne0d 9790 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  =/=  0
)
107103, 104, 105, 106recdivd 9553 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
1  /  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  =  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
108104, 103, 105divrecd 9539 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  =  ( (π `  x )  x.  (
1  /  ( x  /  ( log `  x
) ) ) ) )
10921rpcnne0d 10399 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
11022rpcnne0d 10399 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
111 recdiv 9466 . . . . . . . . . . . 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 5874 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( 1  /  (
x  /  ( log `  x ) ) ) )  =  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
114107, 108, 1133eqtrd 2319 . . . . . . . . 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 4053 . . . . . . 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 10356 . . . . . . . . 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 9674 . . . . . . . . . 10  |-  0  <  ( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
120 ltrec 9637 . . . . . . . . . 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 10390 . . . . . . 7  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR )
126 ltle 8910 . . . . . . 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 4043 . . . 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 12010 . 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 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   3c3 9796   4c4 9797   8c8 9801   ZZcz 10024   RR+crp 10354   [,)cico 10658   |_cfl 10924   ^cexp 11104   abscabs 11719   O ( 1 )co1 11960   _eceu 12344   logclog 19912  πcppi 20331
This theorem is referenced by:  chtppilimlem2  20623  chto1lb  20627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-o1 11964  df-lo1 11965  df-sum 12159  df-ef 12349  df-e 12350  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-ppi 20337
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