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Theorem chebbnd1 22680
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 10387 . . . . 5  |-  2  e.  RR
2 pnfxr 11088 . . . . 5  |- +oo  e.  RR*
3 icossre 11372 . . . . 5  |-  ( ( 2  e.  RR  /\ +oo  e.  RR* )  ->  (
2 [,) +oo )  C_  RR )
41, 2, 3mp2an 667 . . . 4  |-  ( 2 [,) +oo )  C_  RR
54a1i 11 . . 3  |-  ( T. 
->  ( 2 [,) +oo )  C_  RR )
6 elicopnf 11381 . . . . . . . . . 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 457 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR )
9 0red 9383 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  0  e.  RR )
10 1re 9381 . . . . . . . . . 10  |-  1  e.  RR
1110a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  1  e.  RR )
12 0lt1 9858 . . . . . . . . . 10  |-  0  <  1
1312a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  0  <  1 )
141a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  2  e.  RR )
15 1lt2 10484 . . . . . . . . . . 11  |-  1  <  2
1615a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  1  <  2 )
177simprbi 461 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  2  <_  x )
1811, 14, 8, 16, 17ltletrd 9527 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  1  < 
x )
199, 11, 8, 13, 18lttrd 9528 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  0  < 
x )
208, 19elrpd 11021 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
218, 18rplogcld 22037 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  RR+ )
2220, 21rpdivcld 11040 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  e.  RR+ )
23 ppinncl 22471 . . . . . . . 8  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
247, 23sylbi 195 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  NN )
2524nnrpd 11022 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  RR+ )
2622, 25rpdivcld 11040 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR+ )
2726rpcnd 11025 . . . 4  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  CC )
2827adantl 463 . . 3  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  CC )
29 8re 10402 . . . 4  |-  8  e.  RR
3029a1i 11 . . 3  |-  ( T. 
->  8  e.  RR )
31 2rp 10992 . . . . . . . 8  |-  2  e.  RR+
32 relogcl 21986 . . . . . . . 8  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3331, 32ax-mp 5 . . . . . . 7  |-  ( log `  2 )  e.  RR
34 ere 13370 . . . . . . . . 9  |-  _e  e.  RR
351, 34remulcli 9396 . . . . . . . 8  |-  ( 2  x.  _e )  e.  RR
36 2pos 10409 . . . . . . . . . 10  |-  0  <  2
37 epos 13485 . . . . . . . . . 10  |-  0  <  _e
381, 34, 36, 37mulgt0ii 9503 . . . . . . . . 9  |-  0  <  ( 2  x.  _e )
3935, 38gt0ne0ii 9872 . . . . . . . 8  |-  ( 2  x.  _e )  =/=  0
4035, 39rereccli 10092 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  e.  RR
4133, 40resubcli 9667 . . . . . 6  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR
42 2t1e2 10466 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
43 egt2lt3 13484 . . . . . . . . . . . . 13  |-  ( 2  <  _e  /\  _e  <  3 )
4443simpli 455 . . . . . . . . . . . 12  |-  2  <  _e
4510, 1, 34lttri 9496 . . . . . . . . . . . 12  |-  ( ( 1  <  2  /\  2  <  _e )  ->  1  <  _e )
4615, 44, 45mp2an 667 . . . . . . . . . . 11  |-  1  <  _e
4710, 34, 1ltmul2i 10250 . . . . . . . . . . . 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 4310 . . . . . . . . 9  |-  2  <  ( 2  x.  _e )
511, 35, 36, 38ltrecii 10245 . . . . . . . . 9  |-  ( 2  <  ( 2  x.  _e )  <->  ( 1  /  ( 2  x.  _e ) )  < 
( 1  /  2
) )
5250, 51mpbi 208 . . . . . . . 8  |-  ( 1  /  ( 2  x.  _e ) )  < 
( 1  /  2
)
5343simpri 459 . . . . . . . . . . . 12  |-  _e  <  3
54 3lt4 10487 . . . . . . . . . . . 12  |-  3  <  4
55 3re 10391 . . . . . . . . . . . . 13  |-  3  e.  RR
56 4re 10394 . . . . . . . . . . . . 13  |-  4  e.  RR
5734, 55, 56lttri 9496 . . . . . . . . . . . 12  |-  ( ( _e  <  3  /\  3  <  4 )  ->  _e  <  4
)
5853, 54, 57mp2an 667 . . . . . . . . . . 11  |-  _e  <  4
59 epr 13486 . . . . . . . . . . . 12  |-  _e  e.  RR+
60 4pos 10413 . . . . . . . . . . . . 13  |-  0  <  4
6156, 60elrpii 10990 . . . . . . . . . . . 12  |-  4  e.  RR+
62 logltb 22007 . . . . . . . . . . . 12  |-  ( ( _e  e.  RR+  /\  4  e.  RR+ )  ->  (
_e  <  4  <->  ( log `  _e )  <  ( log `  4 ) ) )
6359, 61, 62mp2an 667 . . . . . . . . . . 11  |-  ( _e 
<  4  <->  ( log `  _e )  <  ( log `  4 ) )
6458, 63mpbi 208 . . . . . . . . . 10  |-  ( log `  _e )  <  ( log `  4 )
65 loge 21994 . . . . . . . . . 10  |-  ( log `  _e )  =  1
66 sq2 11958 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
6766fveq2i 5691 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( log `  4 )
68 2z 10674 . . . . . . . . . . . 12  |-  2  e.  ZZ
69 relogexp 22003 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR+  /\  2  e.  ZZ )  ->  ( log `  ( 2 ^ 2 ) )  =  ( 2  x.  ( log `  2 ) ) )
7031, 68, 69mp2an 667 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( 2  x.  ( log `  2 ) )
7167, 70eqtr3i 2463 . . . . . . . . . 10  |-  ( log `  4 )  =  ( 2  x.  ( log `  2 ) )
7264, 65, 713brtr3i 4316 . . . . . . . . 9  |-  1  <  ( 2  x.  ( log `  2 ) )
731, 36pm3.2i 452 . . . . . . . . . 10  |-  ( 2  e.  RR  /\  0  <  2 )
74 ltdivmul 10200 . . . . . . . . . 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 1309 . . . . . . . . 9  |-  ( ( 1  /  2 )  <  ( log `  2
)  <->  1  <  (
2  x.  ( log `  2 ) ) )
7672, 75mpbir 209 . . . . . . . 8  |-  ( 1  /  2 )  < 
( log `  2
)
77 halfre 10536 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
7840, 77, 33lttri 9496 . . . . . . . 8  |-  ( ( ( 1  /  (
2  x.  _e ) )  <  ( 1  /  2 )  /\  ( 1  /  2
)  <  ( log `  2 ) )  -> 
( 1  /  (
2  x.  _e ) )  <  ( log `  2 ) )
7952, 76, 78mp2an 667 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  < 
( log `  2
)
8040, 33posdifi 9886 . . . . . . 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 10990 . . . . 5  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR+
83 rerpdivcl 11014 . . . . 5  |-  ( ( 2  e.  RR  /\  ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )  e.  RR+ )  ->  (
2  /  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) ) )  e.  RR )
841, 82, 83mp2an 667 . . . 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 10993 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR )
87 rpge0 10999 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  0  <_ 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) )
8886, 87absidd 12905 . . . . . . 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 462 . . . . 5  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( abs `  (
( x  /  ( log `  x ) )  /  (π `  x ) ) )  =  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )
91 eqid 2441 . . . . . . . . . 10  |-  ( |_
`  ( x  / 
2 ) )  =  ( |_ `  (
x  /  2 ) )
9291chebbnd1lem3 22679 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  8  <_  x )  -> 
( ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) )  /  2
)  <  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
938, 92sylan 468 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) )  /  2
)  <  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
941recni 9394 . . . . . . . . . 10  |-  2  e.  CC
95 2ne0 10410 . . . . . . . . . 10  |-  2  =/=  0
9641recni 9394 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  CC
9741, 81gt0ne0ii 9872 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  =/=  0
98 recdiv 10033 . . . . . . . . . 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 668 . . . . . . . . 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 11025 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  e.  CC )
10224nncnd 10334 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  CC )
10322rpne0d 11028 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  =/=  0
)
10424nnne0d 10362 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  =/=  0 )
105101, 102, 103, 104recdivd 10120 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( 1  /  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  =  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )
106102, 101, 103divrecd 10106 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  =  ( (π `  x
)  x.  ( 1  /  ( x  / 
( log `  x
) ) ) ) )
10720rpcnne0d 11032 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  e.  CC  /\  x  =/=  0 ) )
10821rpcnne0d 11032 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( log `  x )  e.  CC  /\  ( log `  x )  =/=  0 ) )
109 recdiv 10033 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( ( log `  x )  e.  CC  /\  ( log `  x
)  =/=  0 ) )  ->  ( 1  /  ( x  / 
( log `  x
) ) )  =  ( ( log `  x
)  /  x ) )
110107, 108, 109syl2anc 656 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  ( 1  /  ( x  / 
( log `  x
) ) )  =  ( ( log `  x
)  /  x ) )
111110oveq2d 6106 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  (
1  /  ( x  /  ( log `  x
) ) ) )  =  ( (π `  x
)  x.  ( ( log `  x )  /  x ) ) )
112105, 106, 1113eqtrd 2477 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( 1  /  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  =  ( (π `  x
)  x.  ( ( log `  x )  /  x ) ) )
113112adantr 462 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( 1  /  (
( x  /  ( log `  x ) )  /  (π `  x ) ) )  =  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
11493, 100, 1133brtr4d 4319 . . . . . . 7  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( 1  /  (
2  /  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) ) ) )  <  ( 1  /  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) ) )
11526adantr 462 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( x  / 
( log `  x
) )  /  (π `  x ) )  e.  RR+ )
116 elrp 10989 . . . . . . . . 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 10246 . . . . . . . . . 10  |-  0  <  ( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
118 ltrec 10209 . . . . . . . . . 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 680 . . . . . . . . 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 11023 . . . . . . 7  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( x  / 
( log `  x
) )  /  (π `  x ) )  e.  RR )
124 ltle 9459 . . . . . . 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 657 . . . . . 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 4309 . . . 4  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( abs `  (
( x  /  ( log `  x ) )  /  (π `  x ) ) )  <_  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) ) )
128127adantl 463 . . 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 13010 . 2  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  e.  O(1) )
130129trud 1373 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 1364   T. wtru 1365    e. wcel 1761    =/= wne 2604    C_ wss 3325   class class class wbr 4289    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    x. cmul 9283   +oocpnf 9411   RR*cxr 9413    < clt 9414    <_ cle 9415    - cmin 9591    / cdiv 9989   NNcn 10318   2c2 10367   3c3 10368   4c4 10369   8c8 10373   ZZcz 10642   RR+crp 10987   [,)cico 11298   |_cfl 11636   ^cexp 11861   abscabs 12719   O(1)co1 12960   _eceu 13344   logclog 21965  πcppi 22390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-o1 12964  df-lo1 12965  df-sum 13160  df-ef 13349  df-e 13350  df-sin 13351  df-cos 13352  df-pi 13354  df-dvds 13532  df-gcd 13687  df-prm 13760  df-pc 13900  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967  df-ppi 22396
This theorem is referenced by:  chtppilimlem2  22682  chto1lb  22686
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