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Theorem chebbnd1 23380
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 10596 . . . . 5  |-  2  e.  RR
2 pnfxr 11312 . . . . 5  |- +oo  e.  RR*
3 icossre 11596 . . . . 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 11611 . . . . . . . . . 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 9588 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  0  e.  RR )
10 1re 9586 . . . . . . . . . 10  |-  1  e.  RR
1110a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  1  e.  RR )
12 0lt1 10066 . . . . . . . . . 10  |-  0  <  1
1312a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  0  <  1 )
141a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  2  e.  RR )
15 1lt2 10693 . . . . . . . . . . 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 9732 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  1  < 
x )
199, 11, 8, 13, 18lttrd 9733 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  0  < 
x )
208, 19elrpd 11245 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
218, 18rplogcld 22737 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  RR+ )
2220, 21rpdivcld 11264 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  e.  RR+ )
23 ppinncl 23171 . . . . . . . 8  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
247, 23sylbi 195 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  NN )
2524nnrpd 11246 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  RR+ )
2622, 25rpdivcld 11264 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR+ )
2726rpcnd 11249 . . . 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 10611 . . . 4  |-  8  e.  RR
3029a1i 11 . . 3  |-  ( T. 
->  8  e.  RR )
31 2rp 11216 . . . . . . . 8  |-  2  e.  RR+
32 relogcl 22686 . . . . . . . 8  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3331, 32ax-mp 5 . . . . . . 7  |-  ( log `  2 )  e.  RR
34 ere 13677 . . . . . . . . 9  |-  _e  e.  RR
351, 34remulcli 9601 . . . . . . . 8  |-  ( 2  x.  _e )  e.  RR
36 2pos 10618 . . . . . . . . . 10  |-  0  <  2
37 epos 13792 . . . . . . . . . 10  |-  0  <  _e
381, 34, 36, 37mulgt0ii 9708 . . . . . . . . 9  |-  0  <  ( 2  x.  _e )
3935, 38gt0ne0ii 10080 . . . . . . . 8  |-  ( 2  x.  _e )  =/=  0
4035, 39rereccli 10300 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  e.  RR
4133, 40resubcli 9872 . . . . . 6  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR
42 2t1e2 10675 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
43 egt2lt3 13791 . . . . . . . . . . . . 13  |-  ( 2  <  _e  /\  _e  <  3 )
4443simpli 458 . . . . . . . . . . . 12  |-  2  <  _e
4510, 1, 34lttri 9701 . . . . . . . . . . . 12  |-  ( ( 1  <  2  /\  2  <  _e )  ->  1  <  _e )
4615, 44, 45mp2an 672 . . . . . . . . . . 11  |-  1  <  _e
4710, 34, 1ltmul2i 10458 . . . . . . . . . . . 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 4463 . . . . . . . . 9  |-  2  <  ( 2  x.  _e )
511, 35, 36, 38ltrecii 10453 . . . . . . . . 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 10696 . . . . . . . . . . . 12  |-  3  <  4
55 3re 10600 . . . . . . . . . . . . 13  |-  3  e.  RR
56 4re 10603 . . . . . . . . . . . . 13  |-  4  e.  RR
5734, 55, 56lttri 9701 . . . . . . . . . . . 12  |-  ( ( _e  <  3  /\  3  <  4 )  ->  _e  <  4
)
5853, 54, 57mp2an 672 . . . . . . . . . . 11  |-  _e  <  4
59 epr 13793 . . . . . . . . . . . 12  |-  _e  e.  RR+
60 4pos 10622 . . . . . . . . . . . . 13  |-  0  <  4
6156, 60elrpii 11214 . . . . . . . . . . . 12  |-  4  e.  RR+
62 logltb 22707 . . . . . . . . . . . 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 22694 . . . . . . . . . 10  |-  ( log `  _e )  =  1
66 sq2 12221 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
6766fveq2i 5862 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( log `  4 )
68 2z 10887 . . . . . . . . . . . 12  |-  2  e.  ZZ
69 relogexp 22703 . . . . . . . . . . . 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 2493 . . . . . . . . . 10  |-  ( log `  4 )  =  ( 2  x.  ( log `  2 ) )
7264, 65, 713brtr3i 4469 . . . . . . . . 9  |-  1  <  ( 2  x.  ( log `  2 ) )
731, 36pm3.2i 455 . . . . . . . . . 10  |-  ( 2  e.  RR  /\  0  <  2 )
74 ltdivmul 10408 . . . . . . . . . 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 1319 . . . . . . . . 9  |-  ( ( 1  /  2 )  <  ( log `  2
)  <->  1  <  (
2  x.  ( log `  2 ) ) )
7672, 75mpbir 209 . . . . . . . 8  |-  ( 1  /  2 )  < 
( log `  2
)
77 halfre 10745 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
7840, 77, 33lttri 9701 . . . . . . . 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 10094 . . . . . . 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 11214 . . . . 5  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR+
83 rerpdivcl 11238 . . . . 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 11217 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR )
87 rpge0 11223 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  0  <_ 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) )
8886, 87absidd 13205 . . . . . . 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 2462 . . . . . . . . . 10  |-  ( |_
`  ( x  / 
2 ) )  =  ( |_ `  (
x  /  2 ) )
9291chebbnd1lem3 23379 . . . . . . . . 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 9599 . . . . . . . . . 10  |-  2  e.  CC
95 2ne0 10619 . . . . . . . . . 10  |-  2  =/=  0
9641recni 9599 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  CC
9741, 81gt0ne0ii 10080 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  =/=  0
98 recdiv 10241 . . . . . . . . . 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 11249 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  e.  CC )
10224nncnd 10543 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  CC )
10322rpne0d 11252 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( log `  x
) )  =/=  0
)
10424nnne0d 10571 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  =/=  0 )
105101, 102, 103, 104recdivd 10328 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( 1  /  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  =  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )
106102, 101, 103divrecd 10314 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  =  ( (π `  x
)  x.  ( 1  /  ( x  / 
( log `  x
) ) ) ) )
10720rpcnne0d 11256 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  e.  CC  /\  x  =/=  0 ) )
10821rpcnne0d 11256 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( log `  x )  e.  CC  /\  ( log `  x )  =/=  0 ) )
109 recdiv 10241 . . . . . . . . . . . 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 6293 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  (
1  /  ( x  /  ( log `  x
) ) ) )  =  ( (π `  x
)  x.  ( ( log `  x )  /  x ) ) )
112105, 106, 1113eqtrd 2507 . . . . . . . . 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 4472 . . . . . . 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 11213 . . . . . . . . 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 10454 . . . . . . . . . 10  |-  0  <  ( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
118 ltrec 10417 . . . . . . . . . 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 11247 . . . . . . 7  |-  ( ( x  e.  ( 2 [,) +oo )  /\  8  <_  x )  -> 
( ( x  / 
( log `  x
) )  /  (π `  x ) )  e.  RR )
124 ltle 9664 . . . . . . 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 4462 . . . 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 13310 . 2  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  e.  O(1) )
130129trud 1383 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 1374   T. wtru 1375    e. wcel 1762    =/= wne 2657    C_ wss 3471   class class class wbr 4442    |-> cmpt 4500   ` cfv 5581  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    x. cmul 9488   +oocpnf 9616   RR*cxr 9618    < clt 9619    <_ cle 9620    - cmin 9796    / cdiv 10197   NNcn 10527   2c2 10576   3c3 10577   4c4 10578   8c8 10582   ZZcz 10855   RR+crp 11211   [,)cico 11522   |_cfl 11886   ^cexp 12124   abscabs 13019   O(1)co1 13260   _eceu 13651   logclog 22665  πcppi 23090
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-ioc 11525  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-fac 12311  df-bc 12338  df-hash 12363  df-shft 12852  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-o1 13264  df-lo1 13265  df-sum 13460  df-ef 13656  df-e 13657  df-sin 13658  df-cos 13659  df-pi 13661  df-dvds 13839  df-gcd 13995  df-prm 14068  df-pc 14211  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-fbas 18182  df-fg 18183  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lp 19398  df-perf 19399  df-cn 19489  df-cnp 19490  df-haus 19577  df-tx 19793  df-hmeo 19986  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-xms 20553  df-ms 20554  df-tms 20555  df-cncf 21112  df-limc 22000  df-dv 22001  df-log 22667  df-ppi 23096
This theorem is referenced by:  chtppilimlem2  23382  chto1lb  23386
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