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Theorem chebbnd1 24173
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 10679 . . . . 5  |-  2  e.  RR
2 pnfxr 11412 . . . . 5  |- +oo  e.  RR*
3 icossre 11715 . . . . 5  |-  ( ( 2  e.  RR  /\ +oo  e.  RR* )  ->  (
2 [,) +oo )  C_  RR )
41, 2, 3mp2an 676 . . . 4  |-  ( 2 [,) +oo )  C_  RR
54a1i 11 . . 3  |-  ( T. 
->  ( 2 [,) +oo )  C_  RR )
6 elicopnf 11730 . . . . . . . . . 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 461 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR )
9 0red 9643 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  0  e.  RR )
10 1re 9641 . . . . . . . . . 10  |-  1  e.  RR
1110a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  1  e.  RR )
12 0lt1 10135 . . . . . . . . . 10  |-  0  <  1
1312a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  0  <  1 )
141a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  2  e.  RR )
15 1lt2 10776 . . . . . . . . . . 11  |-  1  <  2
1615a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  1  <  2 )
177simprbi 465 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  2  <_  x )
1811, 14, 8, 16, 17ltletrd 9794 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  1  < 
x )
199, 11, 8, 13, 18lttrd 9795 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  0  < 
x )
208, 19elrpd 11338 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
218, 18rplogcld 23443 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  RR+ )
2220, 21rpdivcld 11358 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  e.  RR+ )
23 ppinncl 23964 . . . . . . . 8  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
247, 23sylbi 198 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  NN )
2524nnrpd 11339 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  RR+ )
2622, 25rpdivcld 11358 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR+ )
2726rpcnd 11343 . . . 4  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  CC )
2827adantl 467 . . 3  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  CC )
29 8re 10694 . . . 4  |-  8  e.  RR
3029a1i 11 . . 3  |-  ( T. 
->  8  e.  RR )
31 2rp 11307 . . . . . . . 8  |-  2  e.  RR+
32 relogcl 23390 . . . . . . . 8  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3331, 32ax-mp 5 . . . . . . 7  |-  ( log `  2 )  e.  RR
34 ere 14121 . . . . . . . . 9  |-  _e  e.  RR
351, 34remulcli 9656 . . . . . . . 8  |-  ( 2  x.  _e )  e.  RR
36 2pos 10701 . . . . . . . . . 10  |-  0  <  2
37 epos 14237 . . . . . . . . . 10  |-  0  <  _e
381, 34, 36, 37mulgt0ii 9767 . . . . . . . . 9  |-  0  <  ( 2  x.  _e )
3935, 38gt0ne0ii 10149 . . . . . . . 8  |-  ( 2  x.  _e )  =/=  0
4035, 39rereccli 10371 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  e.  RR
4133, 40resubcli 9935 . . . . . 6  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR
42 2t1e2 10758 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
43 egt2lt3 14236 . . . . . . . . . . . . 13  |-  ( 2  <  _e  /\  _e  <  3 )
4443simpli 459 . . . . . . . . . . . 12  |-  2  <  _e
4510, 1, 34lttri 9759 . . . . . . . . . . . 12  |-  ( ( 1  <  2  /\  2  <  _e )  ->  1  <  _e )
4615, 44, 45mp2an 676 . . . . . . . . . . 11  |-  1  <  _e
4710, 34, 1ltmul2i 10528 . . . . . . . . . . . 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 211 . . . . . . . . . 10  |-  ( 2  x.  1 )  < 
( 2  x.  _e )
5042, 49eqbrtrri 4447 . . . . . . . . 9  |-  2  <  ( 2  x.  _e )
511, 35, 36, 38ltrecii 10523 . . . . . . . . 9  |-  ( 2  <  ( 2  x.  _e )  <->  ( 1  /  ( 2  x.  _e ) )  < 
( 1  /  2
) )
5250, 51mpbi 211 . . . . . . . 8  |-  ( 1  /  ( 2  x.  _e ) )  < 
( 1  /  2
)
5343simpri 463 . . . . . . . . . . . 12  |-  _e  <  3
54 3lt4 10779 . . . . . . . . . . . 12  |-  3  <  4
55 3re 10683 . . . . . . . . . . . . 13  |-  3  e.  RR
56 4re 10686 . . . . . . . . . . . . 13  |-  4  e.  RR
5734, 55, 56lttri 9759 . . . . . . . . . . . 12  |-  ( ( _e  <  3  /\  3  <  4 )  ->  _e  <  4
)
5853, 54, 57mp2an 676 . . . . . . . . . . 11  |-  _e  <  4
59 epr 14238 . . . . . . . . . . . 12  |-  _e  e.  RR+
60 4pos 10705 . . . . . . . . . . . . 13  |-  0  <  4
6156, 60elrpii 11305 . . . . . . . . . . . 12  |-  4  e.  RR+
62 logltb 23414 . . . . . . . . . . . 12  |-  ( ( _e  e.  RR+  /\  4  e.  RR+ )  ->  (
_e  <  4  <->  ( log `  _e )  <  ( log `  4 ) ) )
6359, 61, 62mp2an 676 . . . . . . . . . . 11  |-  ( _e 
<  4  <->  ( log `  _e )  <  ( log `  4 ) )
6458, 63mpbi 211 . . . . . . . . . 10  |-  ( log `  _e )  <  ( log `  4 )
65 loge 23401 . . . . . . . . . 10  |-  ( log `  _e )  =  1
66 sq2 12368 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
6766fveq2i 5884 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( log `  4 )
68 2z 10969 . . . . . . . . . . . 12  |-  2  e.  ZZ
69 relogexp 23410 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR+  /\  2  e.  ZZ )  ->  ( log `  ( 2 ^ 2 ) )  =  ( 2  x.  ( log `  2 ) ) )
7031, 68, 69mp2an 676 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( 2  x.  ( log `  2 ) )
7167, 70eqtr3i 2460 . . . . . . . . . 10  |-  ( log `  4 )  =  ( 2  x.  ( log `  2 ) )
7264, 65, 713brtr3i 4453 . . . . . . . . 9  |-  1  <  ( 2  x.  ( log `  2 ) )
731, 36pm3.2i 456 . . . . . . . . . 10  |-  ( 2  e.  RR  /\  0  <  2 )
74 ltdivmul 10479 . . . . . . . . . 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 1360 . . . . . . . . 9  |-  ( ( 1  /  2 )  <  ( log `  2
)  <->  1  <  (
2  x.  ( log `  2 ) ) )
7672, 75mpbir 212 . . . . . . . 8  |-  ( 1  /  2 )  < 
( log `  2
)
77 halfre 10828 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
7840, 77, 33lttri 9759 . . . . . . . 8  |-  ( ( ( 1  /  (
2  x.  _e ) )  <  ( 1  /  2 )  /\  ( 1  /  2
)  <  ( log `  2 ) )  -> 
( 1  /  (
2  x.  _e ) )  <  ( log `  2 ) )
7952, 76, 78mp2an 676 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  < 
( log `  2
)
8040, 33posdifi 10163 . . . . . . 7  |-  ( ( 1  /  ( 2  x.  _e ) )  <  ( log `  2
)  <->  0  <  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
8179, 80mpbi 211 . . . . . 6  |-  0  <  ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )
8241, 81elrpii 11305 . . . . 5  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR+
83 rerpdivcl 11330 . . . . 5  |-  ( ( 2  e.  RR  /\  ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )  e.  RR+ )  ->  (
2  /  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) ) )  e.  RR )
841, 82, 83mp2an 676 . . . 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 11308 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR )
87 rpge0 11314 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  0  <_ 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) )
8886, 87absidd 13463 . . . . . . 7  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( abs `  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )
8926, 88syl 17 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( abs `  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )
9089adantr 466 . . . . 5  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( abs `  (
( x  /  ( log `  x ) )  /  (π `  x ) ) )  =  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )
91 eqid 2429 . . . . . . . . . 10  |-  ( |_
`  ( x  / 
2 ) )  =  ( |_ `  (
x  /  2 ) )
9291chebbnd1lem3 24172 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  8  <_  x )  -> 
( ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) )  /  2
)  <  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
938, 92sylan 473 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) )  /  2
)  <  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
941recni 9654 . . . . . . . . . 10  |-  2  e.  CC
95 2ne0 10702 . . . . . . . . . 10  |-  2  =/=  0
9641recni 9654 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  CC
9741, 81gt0ne0ii 10149 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  =/=  0
98 recdiv 10312 . . . . . . . . . 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 677 . . . . . . . . 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 11343 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  e.  CC )
10224nncnd 10625 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  CC )
10322rpne0d 11346 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  =/=  0
)
10424nnne0d 10654 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  =/=  0 )
105101, 102, 103, 104recdivd 10399 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( 1  /  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  =  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )
106102, 101, 103divrecd 10385 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  =  ( (π `  x
)  x.  ( 1  /  ( x  / 
( log `  x
) ) ) ) )
10720rpcnne0d 11350 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  e.  CC  /\  x  =/=  0 ) )
10821rpcnne0d 11350 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( log `  x )  e.  CC  /\  ( log `  x )  =/=  0 ) )
109 recdiv 10312 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( ( log `  x )  e.  CC  /\  ( log `  x
)  =/=  0 ) )  ->  ( 1  /  ( x  / 
( log `  x
) ) )  =  ( ( log `  x
)  /  x ) )
110107, 108, 109syl2anc 665 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  ( 1  /  ( x  / 
( log `  x
) ) )  =  ( ( log `  x
)  /  x ) )
111110oveq2d 6321 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  (
1  /  ( x  /  ( log `  x
) ) ) )  =  ( (π `  x
)  x.  ( ( log `  x )  /  x ) ) )
112105, 106, 1113eqtrd 2474 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( 1  /  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  =  ( (π `  x
)  x.  ( ( log `  x )  /  x ) ) )
113112adantr 466 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( 1  /  (
( x  /  ( log `  x ) )  /  (π `  x ) ) )  =  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
11493, 100, 1133brtr4d 4456 . . . . . . 7  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( 1  /  (
2  /  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) ) ) )  <  ( 1  /  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) ) )
11526adantr 466 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( x  / 
( log `  x
) )  /  (π `  x ) )  e.  RR+ )
116 elrp 11304 . . . . . . . . 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 10524 . . . . . . . . . 10  |-  0  <  ( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
118 ltrec 10487 . . . . . . . . . 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 689 . . . . . . . . 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 198 . . . . . . . 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 17 . . . . . . 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 235 . . . . . 6  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( x  / 
( log `  x
) )  /  (π `  x ) )  < 
( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )
123115rpred 11341 . . . . . . 7  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( x  / 
( log `  x
) )  /  (π `  x ) )  e.  RR )
124 ltle 9721 . . . . . . 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 666 . . . . . 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 4446 . . . 4  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( abs `  (
( x  /  ( log `  x ) )  /  (π `  x ) ) )  <_  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) ) )
128127adantl 467 . . 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 13578 . 2  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  e.  O(1) )
130129trud 1446 1  |-  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  e.  O(1)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   T. wtru 1438    e. wcel 1870    =/= wne 2625    C_ wss 3442   class class class wbr 4426    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543   +oocpnf 9671   RR*cxr 9673    < clt 9674    <_ cle 9675    - cmin 9859    / cdiv 10268   NNcn 10609   2c2 10659   3c3 10660   4c4 10661   8c8 10665   ZZcz 10937   RR+crp 11302   [,)cico 11637   |_cfl 12023   ^cexp 12269   abscabs 13276   O(1)co1 13528   _eceu 14093   logclog 23369  πcppi 23883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-o1 13532  df-lo1 13533  df-sum 13731  df-ef 14099  df-e 14100  df-sin 14101  df-cos 14102  df-pi 14104  df-dvds 14284  df-gcd 14443  df-prm 14594  df-pc 14750  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-haus 20262  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cncf 21806  df-limc 22698  df-dv 22699  df-log 23371  df-ppi 23889
This theorem is referenced by:  chtppilimlem2  24175  chto1lb  24179
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