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Theorem chebbnd1 22837
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 10492 . . . . 5  |-  2  e.  RR
2 pnfxr 11193 . . . . 5  |- +oo  e.  RR*
3 icossre 11477 . . . . 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 11486 . . . . . . . . . 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 9488 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  0  e.  RR )
10 1re 9486 . . . . . . . . . 10  |-  1  e.  RR
1110a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  1  e.  RR )
12 0lt1 9963 . . . . . . . . . 10  |-  0  <  1
1312a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  0  <  1 )
141a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  2  e.  RR )
15 1lt2 10589 . . . . . . . . . . 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 9632 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  1  < 
x )
199, 11, 8, 13, 18lttrd 9633 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  0  < 
x )
208, 19elrpd 11126 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
218, 18rplogcld 22194 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  RR+ )
2220, 21rpdivcld 11145 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  e.  RR+ )
23 ppinncl 22628 . . . . . . . 8  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
247, 23sylbi 195 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  NN )
2524nnrpd 11127 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  RR+ )
2622, 25rpdivcld 11145 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR+ )
2726rpcnd 11130 . . . 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 10507 . . . 4  |-  8  e.  RR
3029a1i 11 . . 3  |-  ( T. 
->  8  e.  RR )
31 2rp 11097 . . . . . . . 8  |-  2  e.  RR+
32 relogcl 22143 . . . . . . . 8  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3331, 32ax-mp 5 . . . . . . 7  |-  ( log `  2 )  e.  RR
34 ere 13476 . . . . . . . . 9  |-  _e  e.  RR
351, 34remulcli 9501 . . . . . . . 8  |-  ( 2  x.  _e )  e.  RR
36 2pos 10514 . . . . . . . . . 10  |-  0  <  2
37 epos 13591 . . . . . . . . . 10  |-  0  <  _e
381, 34, 36, 37mulgt0ii 9608 . . . . . . . . 9  |-  0  <  ( 2  x.  _e )
3935, 38gt0ne0ii 9977 . . . . . . . 8  |-  ( 2  x.  _e )  =/=  0
4035, 39rereccli 10197 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  e.  RR
4133, 40resubcli 9772 . . . . . 6  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR
42 2t1e2 10571 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
43 egt2lt3 13590 . . . . . . . . . . . . 13  |-  ( 2  <  _e  /\  _e  <  3 )
4443simpli 458 . . . . . . . . . . . 12  |-  2  <  _e
4510, 1, 34lttri 9601 . . . . . . . . . . . 12  |-  ( ( 1  <  2  /\  2  <  _e )  ->  1  <  _e )
4615, 44, 45mp2an 672 . . . . . . . . . . 11  |-  1  <  _e
4710, 34, 1ltmul2i 10355 . . . . . . . . . . . 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 4411 . . . . . . . . 9  |-  2  <  ( 2  x.  _e )
511, 35, 36, 38ltrecii 10350 . . . . . . . . 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 10592 . . . . . . . . . . . 12  |-  3  <  4
55 3re 10496 . . . . . . . . . . . . 13  |-  3  e.  RR
56 4re 10499 . . . . . . . . . . . . 13  |-  4  e.  RR
5734, 55, 56lttri 9601 . . . . . . . . . . . 12  |-  ( ( _e  <  3  /\  3  <  4 )  ->  _e  <  4
)
5853, 54, 57mp2an 672 . . . . . . . . . . 11  |-  _e  <  4
59 epr 13592 . . . . . . . . . . . 12  |-  _e  e.  RR+
60 4pos 10518 . . . . . . . . . . . . 13  |-  0  <  4
6156, 60elrpii 11095 . . . . . . . . . . . 12  |-  4  e.  RR+
62 logltb 22164 . . . . . . . . . . . 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 22151 . . . . . . . . . 10  |-  ( log `  _e )  =  1
66 sq2 12063 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
6766fveq2i 5792 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( log `  4 )
68 2z 10779 . . . . . . . . . . . 12  |-  2  e.  ZZ
69 relogexp 22160 . . . . . . . . . . . 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 2482 . . . . . . . . . 10  |-  ( log `  4 )  =  ( 2  x.  ( log `  2 ) )
7264, 65, 713brtr3i 4417 . . . . . . . . 9  |-  1  <  ( 2  x.  ( log `  2 ) )
731, 36pm3.2i 455 . . . . . . . . . 10  |-  ( 2  e.  RR  /\  0  <  2 )
74 ltdivmul 10305 . . . . . . . . . 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 1315 . . . . . . . . 9  |-  ( ( 1  /  2 )  <  ( log `  2
)  <->  1  <  (
2  x.  ( log `  2 ) ) )
7672, 75mpbir 209 . . . . . . . 8  |-  ( 1  /  2 )  < 
( log `  2
)
77 halfre 10641 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
7840, 77, 33lttri 9601 . . . . . . . 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 9991 . . . . . . 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 11095 . . . . 5  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR+
83 rerpdivcl 11119 . . . . 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 11098 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR )
87 rpge0 11104 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  0  <_ 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) )
8886, 87absidd 13011 . . . . . . 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 2451 . . . . . . . . . 10  |-  ( |_
`  ( x  / 
2 ) )  =  ( |_ `  (
x  /  2 ) )
9291chebbnd1lem3 22836 . . . . . . . . 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 9499 . . . . . . . . . 10  |-  2  e.  CC
95 2ne0 10515 . . . . . . . . . 10  |-  2  =/=  0
9641recni 9499 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  CC
9741, 81gt0ne0ii 9977 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  =/=  0
98 recdiv 10138 . . . . . . . . . 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 11130 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  e.  CC )
10224nncnd 10439 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  CC )
10322rpne0d 11133 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  =/=  0
)
10424nnne0d 10467 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  =/=  0 )
105101, 102, 103, 104recdivd 10225 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( 1  /  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  =  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )
106102, 101, 103divrecd 10211 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  =  ( (π `  x
)  x.  ( 1  /  ( x  / 
( log `  x
) ) ) ) )
10720rpcnne0d 11137 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  e.  CC  /\  x  =/=  0 ) )
10821rpcnne0d 11137 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( log `  x )  e.  CC  /\  ( log `  x )  =/=  0 ) )
109 recdiv 10138 . . . . . . . . . . . 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 6206 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  (
1  /  ( x  /  ( log `  x
) ) ) )  =  ( (π `  x
)  x.  ( ( log `  x )  /  x ) ) )
112105, 106, 1113eqtrd 2496 . . . . . . . . 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 4420 . . . . . . 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 11094 . . . . . . . . 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 10351 . . . . . . . . . 10  |-  0  <  ( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
118 ltrec 10314 . . . . . . . . . 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 11128 . . . . . . 7  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( x  / 
( log `  x
) )  /  (π `  x ) )  e.  RR )
124 ltle 9564 . . . . . . 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 4410 . . . 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 13116 . 2  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  e.  O(1) )
130129trud 1379 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 1370   T. wtru 1371    e. wcel 1758    =/= wne 2644    C_ wss 3426   class class class wbr 4390    |-> cmpt 4448   ` cfv 5516  (class class class)co 6190   CCcc 9381   RRcr 9382   0cc0 9383   1c1 9384    x. cmul 9388   +oocpnf 9516   RR*cxr 9518    < clt 9519    <_ cle 9520    - cmin 9696    / cdiv 10094   NNcn 10423   2c2 10472   3c3 10473   4c4 10474   8c8 10478   ZZcz 10747   RR+crp 11092   [,)cico 11403   |_cfl 11741   ^cexp 11966   abscabs 12825   O(1)co1 13066   _eceu 13450   logclog 22122  πcppi 22547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-fi 7762  df-sup 7792  df-oi 7825  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-q 11055  df-rp 11093  df-xneg 11190  df-xadd 11191  df-xmul 11192  df-ioo 11405  df-ioc 11406  df-ico 11407  df-icc 11408  df-fz 11539  df-fzo 11650  df-fl 11743  df-mod 11810  df-seq 11908  df-exp 11967  df-fac 12153  df-bc 12180  df-hash 12205  df-shft 12658  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-limsup 13051  df-clim 13068  df-rlim 13069  df-o1 13070  df-lo1 13071  df-sum 13266  df-ef 13455  df-e 13456  df-sin 13457  df-cos 13458  df-pi 13460  df-dvds 13638  df-gcd 13793  df-prm 13866  df-pc 14006  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-hom 14364  df-cco 14365  df-rest 14463  df-topn 14464  df-0g 14482  df-gsum 14483  df-topgen 14484  df-pt 14485  df-prds 14488  df-xrs 14542  df-qtop 14547  df-imas 14548  df-xps 14550  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-submnd 15567  df-mulg 15650  df-cntz 15937  df-cmn 16383  df-psmet 17918  df-xmet 17919  df-met 17920  df-bl 17921  df-mopn 17922  df-fbas 17923  df-fg 17924  df-cnfld 17928  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-cld 18739  df-ntr 18740  df-cls 18741  df-nei 18818  df-lp 18856  df-perf 18857  df-cn 18947  df-cnp 18948  df-haus 19035  df-tx 19251  df-hmeo 19444  df-fil 19535  df-fm 19627  df-flim 19628  df-flf 19629  df-xms 20011  df-ms 20012  df-tms 20013  df-cncf 20570  df-limc 21457  df-dv 21458  df-log 22124  df-ppi 22553
This theorem is referenced by:  chtppilimlem2  22839  chto1lb  22843
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