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Theorem chebbnd1 23783
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 10626 . . . . 5  |-  2  e.  RR
2 pnfxr 11346 . . . . 5  |- +oo  e.  RR*
3 icossre 11630 . . . . 5  |-  ( ( 2  e.  RR  /\ +oo  e.  RR* )  ->  (
2 [,) +oo )  C_  RR )
41, 2, 3mp2an 672 . . . 4  |-  ( 2 [,) +oo )  C_  RR
54a1i 11 . . 3  |-  ( T. 
->  ( 2 [,) +oo )  C_  RR )
6 elicopnf 11645 . . . . . . . . . 10  |-  ( 2  e.  RR  ->  (
x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x ) ) )
71, 6ax-mp 5 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x ) )
87simplbi 460 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR )
9 0red 9614 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  0  e.  RR )
10 1re 9612 . . . . . . . . . 10  |-  1  e.  RR
1110a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  1  e.  RR )
12 0lt1 10096 . . . . . . . . . 10  |-  0  <  1
1312a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  0  <  1 )
141a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  2  e.  RR )
15 1lt2 10723 . . . . . . . . . . 11  |-  1  <  2
1615a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  1  <  2 )
177simprbi 464 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  2  <_  x )
1811, 14, 8, 16, 17ltletrd 9759 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  1  < 
x )
199, 11, 8, 13, 18lttrd 9760 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  0  < 
x )
208, 19elrpd 11279 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
218, 18rplogcld 23140 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  RR+ )
2220, 21rpdivcld 11298 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  e.  RR+ )
23 ppinncl 23574 . . . . . . . 8  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
247, 23sylbi 195 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  NN )
2524nnrpd 11280 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  RR+ )
2622, 25rpdivcld 11298 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR+ )
2726rpcnd 11283 . . . 4  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  CC )
2827adantl 466 . . 3  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  CC )
29 8re 10641 . . . 4  |-  8  e.  RR
3029a1i 11 . . 3  |-  ( T. 
->  8  e.  RR )
31 2rp 11250 . . . . . . . 8  |-  2  e.  RR+
32 relogcl 23089 . . . . . . . 8  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3331, 32ax-mp 5 . . . . . . 7  |-  ( log `  2 )  e.  RR
34 ere 13836 . . . . . . . . 9  |-  _e  e.  RR
351, 34remulcli 9627 . . . . . . . 8  |-  ( 2  x.  _e )  e.  RR
36 2pos 10648 . . . . . . . . . 10  |-  0  <  2
37 epos 13952 . . . . . . . . . 10  |-  0  <  _e
381, 34, 36, 37mulgt0ii 9735 . . . . . . . . 9  |-  0  <  ( 2  x.  _e )
3935, 38gt0ne0ii 10110 . . . . . . . 8  |-  ( 2  x.  _e )  =/=  0
4035, 39rereccli 10330 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  e.  RR
4133, 40resubcli 9900 . . . . . 6  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR
42 2t1e2 10705 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
43 egt2lt3 13951 . . . . . . . . . . . . 13  |-  ( 2  <  _e  /\  _e  <  3 )
4443simpli 458 . . . . . . . . . . . 12  |-  2  <  _e
4510, 1, 34lttri 9727 . . . . . . . . . . . 12  |-  ( ( 1  <  2  /\  2  <  _e )  ->  1  <  _e )
4615, 44, 45mp2an 672 . . . . . . . . . . 11  |-  1  <  _e
4710, 34, 1ltmul2i 10487 . . . . . . . . . . . 12  |-  ( 0  <  2  ->  (
1  <  _e  <->  ( 2  x.  1 )  < 
( 2  x.  _e ) ) )
4836, 47ax-mp 5 . . . . . . . . . . 11  |-  ( 1  <  _e  <->  ( 2  x.  1 )  < 
( 2  x.  _e ) )
4946, 48mpbi 208 . . . . . . . . . 10  |-  ( 2  x.  1 )  < 
( 2  x.  _e )
5042, 49eqbrtrri 4477 . . . . . . . . 9  |-  2  <  ( 2  x.  _e )
511, 35, 36, 38ltrecii 10482 . . . . . . . . 9  |-  ( 2  <  ( 2  x.  _e )  <->  ( 1  /  ( 2  x.  _e ) )  < 
( 1  /  2
) )
5250, 51mpbi 208 . . . . . . . 8  |-  ( 1  /  ( 2  x.  _e ) )  < 
( 1  /  2
)
5343simpri 462 . . . . . . . . . . . 12  |-  _e  <  3
54 3lt4 10726 . . . . . . . . . . . 12  |-  3  <  4
55 3re 10630 . . . . . . . . . . . . 13  |-  3  e.  RR
56 4re 10633 . . . . . . . . . . . . 13  |-  4  e.  RR
5734, 55, 56lttri 9727 . . . . . . . . . . . 12  |-  ( ( _e  <  3  /\  3  <  4 )  ->  _e  <  4
)
5853, 54, 57mp2an 672 . . . . . . . . . . 11  |-  _e  <  4
59 epr 13953 . . . . . . . . . . . 12  |-  _e  e.  RR+
60 4pos 10652 . . . . . . . . . . . . 13  |-  0  <  4
6156, 60elrpii 11248 . . . . . . . . . . . 12  |-  4  e.  RR+
62 logltb 23110 . . . . . . . . . . . 12  |-  ( ( _e  e.  RR+  /\  4  e.  RR+ )  ->  (
_e  <  4  <->  ( log `  _e )  <  ( log `  4 ) ) )
6359, 61, 62mp2an 672 . . . . . . . . . . 11  |-  ( _e 
<  4  <->  ( log `  _e )  <  ( log `  4 ) )
6458, 63mpbi 208 . . . . . . . . . 10  |-  ( log `  _e )  <  ( log `  4 )
65 loge 23097 . . . . . . . . . 10  |-  ( log `  _e )  =  1
66 sq2 12267 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
6766fveq2i 5875 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( log `  4 )
68 2z 10917 . . . . . . . . . . . 12  |-  2  e.  ZZ
69 relogexp 23106 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR+  /\  2  e.  ZZ )  ->  ( log `  ( 2 ^ 2 ) )  =  ( 2  x.  ( log `  2 ) ) )
7031, 68, 69mp2an 672 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( 2  x.  ( log `  2 ) )
7167, 70eqtr3i 2488 . . . . . . . . . 10  |-  ( log `  4 )  =  ( 2  x.  ( log `  2 ) )
7264, 65, 713brtr3i 4483 . . . . . . . . 9  |-  1  <  ( 2  x.  ( log `  2 ) )
731, 36pm3.2i 455 . . . . . . . . . 10  |-  ( 2  e.  RR  /\  0  <  2 )
74 ltdivmul 10438 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( log `  2 )  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( 1  /  2 )  < 
( log `  2
)  <->  1  <  (
2  x.  ( log `  2 ) ) ) )
7510, 33, 73, 74mp3an 1324 . . . . . . . . 9  |-  ( ( 1  /  2 )  <  ( log `  2
)  <->  1  <  (
2  x.  ( log `  2 ) ) )
7672, 75mpbir 209 . . . . . . . 8  |-  ( 1  /  2 )  < 
( log `  2
)
77 halfre 10775 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
7840, 77, 33lttri 9727 . . . . . . . 8  |-  ( ( ( 1  /  (
2  x.  _e ) )  <  ( 1  /  2 )  /\  ( 1  /  2
)  <  ( log `  2 ) )  -> 
( 1  /  (
2  x.  _e ) )  <  ( log `  2 ) )
7952, 76, 78mp2an 672 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  < 
( log `  2
)
8040, 33posdifi 10124 . . . . . . 7  |-  ( ( 1  /  ( 2  x.  _e ) )  <  ( log `  2
)  <->  0  <  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
8179, 80mpbi 208 . . . . . 6  |-  0  <  ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )
8241, 81elrpii 11248 . . . . 5  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR+
83 rerpdivcl 11272 . . . . 5  |-  ( ( 2  e.  RR  /\  ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )  e.  RR+ )  ->  (
2  /  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) ) )  e.  RR )
841, 82, 83mp2an 672 . . . 4  |-  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) )  e.  RR
8584a1i 11 . . 3  |-  ( T. 
->  ( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )  e.  RR )
86 rpre 11251 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR )
87 rpge0 11257 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  0  <_ 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) )
8886, 87absidd 13266 . . . . . . 7  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( abs `  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )
8926, 88syl 16 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( abs `  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )
9089adantr 465 . . . . 5  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( abs `  (
( x  /  ( log `  x ) )  /  (π `  x ) ) )  =  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )
91 eqid 2457 . . . . . . . . . 10  |-  ( |_
`  ( x  / 
2 ) )  =  ( |_ `  (
x  /  2 ) )
9291chebbnd1lem3 23782 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  8  <_  x )  -> 
( ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) )  /  2
)  <  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
938, 92sylan 471 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) )  /  2
)  <  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
941recni 9625 . . . . . . . . . 10  |-  2  e.  CC
95 2ne0 10649 . . . . . . . . . 10  |-  2  =/=  0
9641recni 9625 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  CC
9741, 81gt0ne0ii 10110 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  =/=  0
98 recdiv 10271 . . . . . . . . . 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 ) )
9994, 95, 96, 97, 98mp4an 673 . . . . . . . . 9  |-  ( 1  /  ( 2  / 
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )  =  ( ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )  /  2 )
10099a1i 11 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( 1  /  (
2  /  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) ) ) )  =  ( ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )  /  2 ) )
10122rpcnd 11283 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  e.  CC )
10224nncnd 10572 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  CC )
10322rpne0d 11286 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  =/=  0
)
10424nnne0d 10601 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  =/=  0 )
105101, 102, 103, 104recdivd 10358 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( 1  /  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  =  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )
106102, 101, 103divrecd 10344 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  =  ( (π `  x
)  x.  ( 1  /  ( x  / 
( log `  x
) ) ) ) )
10720rpcnne0d 11290 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  e.  CC  /\  x  =/=  0 ) )
10821rpcnne0d 11290 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( log `  x )  e.  CC  /\  ( log `  x )  =/=  0 ) )
109 recdiv 10271 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( ( log `  x )  e.  CC  /\  ( log `  x
)  =/=  0 ) )  ->  ( 1  /  ( x  / 
( log `  x
) ) )  =  ( ( log `  x
)  /  x ) )
110107, 108, 109syl2anc 661 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  ( 1  /  ( x  / 
( log `  x
) ) )  =  ( ( log `  x
)  /  x ) )
111110oveq2d 6312 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  (
1  /  ( x  /  ( log `  x
) ) ) )  =  ( (π `  x
)  x.  ( ( log `  x )  /  x ) ) )
112105, 106, 1113eqtrd 2502 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( 1  /  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  =  ( (π `  x
)  x.  ( ( log `  x )  /  x ) ) )
113112adantr 465 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( 1  /  (
( x  /  ( log `  x ) )  /  (π `  x ) ) )  =  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
11493, 100, 1133brtr4d 4486 . . . . . . 7  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( 1  /  (
2  /  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) ) ) )  <  ( 1  /  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) ) )
11526adantr 465 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( x  / 
( log `  x
) )  /  (π `  x ) )  e.  RR+ )
116 elrp 11247 . . . . . . . . 9  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  <->  ( ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR  /\  0  <  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) ) )
1171, 41, 36, 81divgt0ii 10483 . . . . . . . . . 10  |-  0  <  ( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
118 ltrec 10446 . . . . . . . . . 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 ) ) ) ) )
11984, 117, 118mpanr12 685 . . . . . . . . 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 ) ) ) ) )
120116, 119sylbi 195 . . . . . . . 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 ) ) ) ) )
121115, 120syl 16 . . . . . . 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 ) ) ) ) )
122114, 121mpbird 232 . . . . . 6  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( x  / 
( log `  x
) )  /  (π `  x ) )  < 
( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )
123115rpred 11281 . . . . . . 7  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( x  / 
( log `  x
) )  /  (π `  x ) )  e.  RR )
124 ltle 9690 . . . . . . 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 ) ) ) ) ) )
125123, 84, 124sylancl 662 . . . . . 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 ) ) ) ) ) )
126122, 125mpd 15 . . . . 5  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( x  / 
( log `  x
) )  /  (π `  x ) )  <_ 
( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )
12790, 126eqbrtrd 4476 . . . 4  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( abs `  (
( x  /  ( log `  x ) )  /  (π `  x ) ) )  <_  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) ) )
128127adantl 466 . . 3  |-  ( ( T.  /\  ( x  e.  ( 2 [,) +oo )  /\  8  <_  x ) )  -> 
( abs `  (
( x  /  ( log `  x ) )  /  (π `  x ) ) )  <_  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) ) )
1295, 28, 30, 85, 128elo1d 13371 . 2  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  e.  O(1) )
130129trud 1404 1  |-  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  e.  O(1)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395   T. wtru 1396    e. wcel 1819    =/= wne 2652    C_ wss 3471   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646    - cmin 9824    / cdiv 10227   NNcn 10556   2c2 10606   3c3 10607   4c4 10608   8c8 10612   ZZcz 10885   RR+crp 11245   [,)cico 11556   |_cfl 11930   ^cexp 12169   abscabs 13079   O(1)co1 13321   _eceu 13810   logclog 23068  πcppi 23493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-o1 13325  df-lo1 13326  df-sum 13521  df-ef 13815  df-e 13816  df-sin 13817  df-cos 13818  df-pi 13820  df-dvds 13999  df-gcd 14157  df-prm 14230  df-pc 14373  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-limc 22396  df-dv 22397  df-log 23070  df-ppi 23499
This theorem is referenced by:  chtppilimlem2  23785  chto1lb  23789
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