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Theorem chebbnd1 21119
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 10025 . . . . 5  |-  2  e.  RR
2 pnfxr 10669 . . . . 5  |-  +oo  e.  RR*
3 icossre 10947 . . . . 5  |-  ( ( 2  e.  RR  /\  +oo 
e.  RR* )  ->  (
2 [,)  +oo )  C_  RR )
41, 2, 3mp2an 654 . . . 4  |-  ( 2 [,)  +oo )  C_  RR
54a1i 11 . . 3  |-  (  T. 
->  ( 2 [,)  +oo )  C_  RR )
6 elicopnf 10956 . . . . . . . . . 10  |-  ( 2  e.  RR  ->  (
x  e.  ( 2 [,)  +oo )  <->  ( x  e.  RR  /\  2  <_  x ) ) )
71, 6ax-mp 8 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  <->  ( x  e.  RR  /\  2  <_  x ) )
87simplbi 447 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR )
9 0re 9047 . . . . . . . . . 10  |-  0  e.  RR
109a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  e.  RR )
11 1re 9046 . . . . . . . . . 10  |-  1  e.  RR
1211a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  e.  RR )
13 0lt1 9506 . . . . . . . . . 10  |-  0  <  1
1413a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  1 )
151a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  e.  RR )
16 1lt2 10098 . . . . . . . . . . 11  |-  1  <  2
1716a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  <  2 )
187simprbi 451 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  <_  x )
1912, 15, 8, 17, 18ltletrd 9186 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  <  x )
2010, 12, 8, 14, 19lttrd 9187 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  x )
218, 20elrpd 10602 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
228, 19rplogcld 20477 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  RR+ )
2321, 22rpdivcld 10621 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
24 ppinncl 20910 . . . . . . . 8  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
257, 24sylbi 188 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  NN )
2625nnrpd 10603 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  RR+ )
2723, 26rpdivcld 10621 . . . . 5  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+ )
2827rpcnd 10606 . . . 4  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  CC )
2928adantl 453 . . 3  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  CC )
30 8re 10034 . . . 4  |-  8  e.  RR
3130a1i 11 . . 3  |-  (  T. 
->  8  e.  RR )
32 2rp 10573 . . . . . . . 8  |-  2  e.  RR+
33 relogcl 20426 . . . . . . . 8  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3432, 33ax-mp 8 . . . . . . 7  |-  ( log `  2 )  e.  RR
35 ere 12646 . . . . . . . . 9  |-  _e  e.  RR
361, 35remulcli 9060 . . . . . . . 8  |-  ( 2  x.  _e )  e.  RR
37 2pos 10038 . . . . . . . . . 10  |-  0  <  2
38 epos 12761 . . . . . . . . . 10  |-  0  <  _e
391, 35, 37, 38mulgt0ii 9162 . . . . . . . . 9  |-  0  <  ( 2  x.  _e )
4036, 39gt0ne0ii 9519 . . . . . . . 8  |-  ( 2  x.  _e )  =/=  0
4136, 40rereccli 9735 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  e.  RR
4234, 41resubcli 9319 . . . . . 6  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR
431recni 9058 . . . . . . . . . . 11  |-  2  e.  CC
4443mulid1i 9048 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
45 egt2lt3 12760 . . . . . . . . . . . . 13  |-  ( 2  <  _e  /\  _e  <  3 )
4645simpli 445 . . . . . . . . . . . 12  |-  2  <  _e
4711, 1, 35lttri 9155 . . . . . . . . . . . 12  |-  ( ( 1  <  2  /\  2  <  _e )  ->  1  <  _e )
4816, 46, 47mp2an 654 . . . . . . . . . . 11  |-  1  <  _e
4911, 35, 1ltmul2i 9888 . . . . . . . . . . . 12  |-  ( 0  <  2  ->  (
1  <  _e  <->  ( 2  x.  1 )  < 
( 2  x.  _e ) ) )
5037, 49ax-mp 8 . . . . . . . . . . 11  |-  ( 1  <  _e  <->  ( 2  x.  1 )  < 
( 2  x.  _e ) )
5148, 50mpbi 200 . . . . . . . . . 10  |-  ( 2  x.  1 )  < 
( 2  x.  _e )
5244, 51eqbrtrri 4193 . . . . . . . . 9  |-  2  <  ( 2  x.  _e )
531, 36, 37, 39ltrecii 9883 . . . . . . . . 9  |-  ( 2  <  ( 2  x.  _e )  <->  ( 1  /  ( 2  x.  _e ) )  < 
( 1  /  2
) )
5452, 53mpbi 200 . . . . . . . 8  |-  ( 1  /  ( 2  x.  _e ) )  < 
( 1  /  2
)
5545simpri 449 . . . . . . . . . . . 12  |-  _e  <  3
56 3lt4 10101 . . . . . . . . . . . 12  |-  3  <  4
57 3re 10027 . . . . . . . . . . . . 13  |-  3  e.  RR
58 4re 10029 . . . . . . . . . . . . 13  |-  4  e.  RR
5935, 57, 58lttri 9155 . . . . . . . . . . . 12  |-  ( ( _e  <  3  /\  3  <  4 )  ->  _e  <  4
)
6055, 56, 59mp2an 654 . . . . . . . . . . 11  |-  _e  <  4
61 epr 12762 . . . . . . . . . . . 12  |-  _e  e.  RR+
62 4pos 10042 . . . . . . . . . . . . 13  |-  0  <  4
6358, 62elrpii 10571 . . . . . . . . . . . 12  |-  4  e.  RR+
64 logltb 20447 . . . . . . . . . . . 12  |-  ( ( _e  e.  RR+  /\  4  e.  RR+ )  ->  (
_e  <  4  <->  ( log `  _e )  <  ( log `  4 ) ) )
6561, 63, 64mp2an 654 . . . . . . . . . . 11  |-  ( _e 
<  4  <->  ( log `  _e )  <  ( log `  4 ) )
6660, 65mpbi 200 . . . . . . . . . 10  |-  ( log `  _e )  <  ( log `  4 )
67 loge 20434 . . . . . . . . . 10  |-  ( log `  _e )  =  1
68 sq2 11432 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
6968fveq2i 5690 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( log `  4 )
70 2z 10268 . . . . . . . . . . . 12  |-  2  e.  ZZ
71 relogexp 20443 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR+  /\  2  e.  ZZ )  ->  ( log `  ( 2 ^ 2 ) )  =  ( 2  x.  ( log `  2 ) ) )
7232, 70, 71mp2an 654 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( 2  x.  ( log `  2 ) )
7369, 72eqtr3i 2426 . . . . . . . . . 10  |-  ( log `  4 )  =  ( 2  x.  ( log `  2 ) )
7466, 67, 733brtr3i 4199 . . . . . . . . 9  |-  1  <  ( 2  x.  ( log `  2 ) )
751, 37pm3.2i 442 . . . . . . . . . 10  |-  ( 2  e.  RR  /\  0  <  2 )
76 ltdivmul 9838 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( log `  2 )  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( 1  /  2 )  < 
( log `  2
)  <->  1  <  (
2  x.  ( log `  2 ) ) ) )
7711, 34, 75, 76mp3an 1279 . . . . . . . . 9  |-  ( ( 1  /  2 )  <  ( log `  2
)  <->  1  <  (
2  x.  ( log `  2 ) ) )
7874, 77mpbir 201 . . . . . . . 8  |-  ( 1  /  2 )  < 
( log `  2
)
7911rehalfcli 10172 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
8041, 79, 34lttri 9155 . . . . . . . 8  |-  ( ( ( 1  /  (
2  x.  _e ) )  <  ( 1  /  2 )  /\  ( 1  /  2
)  <  ( log `  2 ) )  -> 
( 1  /  (
2  x.  _e ) )  <  ( log `  2 ) )
8154, 78, 80mp2an 654 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  < 
( log `  2
)
8241, 34posdifi 9533 . . . . . . 7  |-  ( ( 1  /  ( 2  x.  _e ) )  <  ( log `  2
)  <->  0  <  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
8381, 82mpbi 200 . . . . . 6  |-  0  <  ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )
8442, 83elrpii 10571 . . . . 5  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR+
85 rerpdivcl 10595 . . . . 5  |-  ( ( 2  e.  RR  /\  ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )  e.  RR+ )  ->  (
2  /  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) ) )  e.  RR )
861, 84, 85mp2an 654 . . . 4  |-  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) )  e.  RR
8786a1i 11 . . 3  |-  (  T. 
->  ( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )  e.  RR )
88 rpre 10574 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR )
89 rpge0 10580 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  0  <_ 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) )
9088, 89absidd 12180 . . . . . . 7  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( abs `  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )
9127, 90syl 16 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( abs `  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  =  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )
9291adantr 452 . . . . 5  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  ( abs `  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  =  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )
93 eqid 2404 . . . . . . . . . 10  |-  ( |_
`  ( x  / 
2 ) )  =  ( |_ `  (
x  /  2 ) )
9493chebbnd1lem3 21118 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  8  <_  x )  -> 
( ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) )  /  2
)  <  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
958, 94sylan 458 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )  /  2 )  < 
( (π `  x )  x.  ( ( log `  x
)  /  x ) ) )
96 2ne0 10039 . . . . . . . . . 10  |-  2  =/=  0
9742recni 9058 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  CC
9842, 83gt0ne0ii 9519 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  =/=  0
99 recdiv 9676 . . . . . . . . . 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 ) )
10043, 96, 97, 98, 99mp4an 655 . . . . . . . . 9  |-  ( 1  /  ( 2  / 
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )  =  ( ( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) )  /  2 )
101100a1i 11 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
1  /  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) ) )  =  ( ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  / 
2 ) )
10223rpcnd 10606 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  CC )
10325nncnd 9972 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  CC )
10423rpne0d 10609 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  =/=  0 )
10525nnne0d 10000 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  =/=  0
)
106102, 103, 104, 105recdivd 9763 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
1  /  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  =  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
107103, 102, 104divrecd 9749 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  =  ( (π `  x )  x.  (
1  /  ( x  /  ( log `  x
) ) ) ) )
10821rpcnne0d 10613 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
10922rpcnne0d 10613 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
110 recdiv 9676 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( ( log `  x )  e.  CC  /\  ( log `  x
)  =/=  0 ) )  ->  ( 1  /  ( x  / 
( log `  x
) ) )  =  ( ( log `  x
)  /  x ) )
111108, 109, 110syl2anc 643 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
1  /  ( x  /  ( log `  x
) ) )  =  ( ( log `  x
)  /  x ) )
112111oveq2d 6056 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( 1  /  (
x  /  ( log `  x ) ) ) )  =  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
113106, 107, 1123eqtrd 2440 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
1  /  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  =  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
114113adantr 452 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
1  /  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  =  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
11595, 101, 1143brtr4d 4202 . . . . . . 7  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
1  /  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) ) )  <  ( 1  / 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) ) )
11627adantr 452 . . . . . . . 8  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+ )
117 elrp 10570 . . . . . . . . 9  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  <->  ( ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR  /\  0  <  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) ) )
1181, 42, 37, 83divgt0ii 9884 . . . . . . . . . 10  |-  0  <  ( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
119 ltrec 9847 . . . . . . . . . 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 ) ) ) ) )
12086, 118, 119mpanr12 667 . . . . . . . . 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 ) ) ) ) )
121117, 120sylbi 188 . . . . . . . 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 ) ) ) ) )
122116, 121syl 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 ) ) ) ) )
123115, 122mpbird 224 . . . . . 6  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  <  ( 2  / 
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )
124116rpred 10604 . . . . . . 7  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR )
125 ltle 9119 . . . . . . 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 ) ) ) ) ) )
126124, 86, 125sylancl 644 . . . . . 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 ) ) ) ) ) )
127123, 126mpd 15 . . . . 5  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  <_  ( 2  / 
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )
12892, 127eqbrtrd 4192 . . . 4  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  ( abs `  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  <_  ( 2  / 
( ( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) ) )
129128adantl 453 . . 3  |-  ( (  T.  /\  ( x  e.  ( 2 [,) 
+oo )  /\  8  <_  x ) )  -> 
( abs `  (
( x  /  ( log `  x ) )  /  (π `  x ) ) )  <_  ( 2  /  ( ( log `  2 )  -  ( 1  /  (
2  x.  _e ) ) ) ) )
1305, 29, 31, 87, 129elo1d 12285 . 2  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  e.  O ( 1 ) )
131130trud 1329 1  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  e.  O ( 1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    T. wtru 1322    = wceq 1649    e. wcel 1721    =/= wne 2567    C_ wss 3280   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    +oocpnf 9073   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   2c2 10005   3c3 10006   4c4 10007   8c8 10011   ZZcz 10238   RR+crp 10568   [,)cico 10874   |_cfl 11156   ^cexp 11337   abscabs 11994   O ( 1 )co1 12235   _eceu 12620   logclog 20405  πcppi 20829
This theorem is referenced by:  chtppilimlem2  21121  chto1lb  21125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-o1 12239  df-lo1 12240  df-sum 12435  df-ef 12625  df-e 12626  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-ppi 20835
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