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Theorem chebbnd1 21158
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 10061 . . . . 5  |-  2  e.  RR
2 pnfxr 10705 . . . . 5  |-  +oo  e.  RR*
3 icossre 10983 . . . . 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 10992 . . . . . . . . . 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 9083 . . . . . . . . . 10  |-  0  e.  RR
109a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  e.  RR )
11 1re 9082 . . . . . . . . . 10  |-  1  e.  RR
1211a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  e.  RR )
13 0lt1 9542 . . . . . . . . . 10  |-  0  <  1
1413a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  1 )
151a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  e.  RR )
16 1lt2 10134 . . . . . . . . . . 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 9222 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  <  x )
2010, 12, 8, 14, 19lttrd 9223 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  x )
218, 20elrpd 10638 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
228, 19rplogcld 20516 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  RR+ )
2321, 22rpdivcld 10657 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
24 ppinncl 20949 . . . . . . . 8  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
257, 24sylbi 188 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  NN )
2625nnrpd 10639 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  RR+ )
2723, 26rpdivcld 10657 . . . . 5  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+ )
2827rpcnd 10642 . . . 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 10070 . . . 4  |-  8  e.  RR
3130a1i 11 . . 3  |-  (  T. 
->  8  e.  RR )
32 2rp 10609 . . . . . . . 8  |-  2  e.  RR+
33 relogcl 20465 . . . . . . . 8  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3432, 33ax-mp 8 . . . . . . 7  |-  ( log `  2 )  e.  RR
35 ere 12683 . . . . . . . . 9  |-  _e  e.  RR
361, 35remulcli 9096 . . . . . . . 8  |-  ( 2  x.  _e )  e.  RR
37 2pos 10074 . . . . . . . . . 10  |-  0  <  2
38 epos 12798 . . . . . . . . . 10  |-  0  <  _e
391, 35, 37, 38mulgt0ii 9198 . . . . . . . . 9  |-  0  <  ( 2  x.  _e )
4036, 39gt0ne0ii 9555 . . . . . . . 8  |-  ( 2  x.  _e )  =/=  0
4136, 40rereccli 9771 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  e.  RR
4234, 41resubcli 9355 . . . . . 6  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR
431recni 9094 . . . . . . . . . . 11  |-  2  e.  CC
4443mulid1i 9084 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
45 egt2lt3 12797 . . . . . . . . . . . . 13  |-  ( 2  <  _e  /\  _e  <  3 )
4645simpli 445 . . . . . . . . . . . 12  |-  2  <  _e
4711, 1, 35lttri 9191 . . . . . . . . . . . 12  |-  ( ( 1  <  2  /\  2  <  _e )  ->  1  <  _e )
4816, 46, 47mp2an 654 . . . . . . . . . . 11  |-  1  <  _e
4911, 35, 1ltmul2i 9924 . . . . . . . . . . . 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 4225 . . . . . . . . 9  |-  2  <  ( 2  x.  _e )
531, 36, 37, 39ltrecii 9919 . . . . . . . . 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 10137 . . . . . . . . . . . 12  |-  3  <  4
57 3re 10063 . . . . . . . . . . . . 13  |-  3  e.  RR
58 4re 10065 . . . . . . . . . . . . 13  |-  4  e.  RR
5935, 57, 58lttri 9191 . . . . . . . . . . . 12  |-  ( ( _e  <  3  /\  3  <  4 )  ->  _e  <  4
)
6055, 56, 59mp2an 654 . . . . . . . . . . 11  |-  _e  <  4
61 epr 12799 . . . . . . . . . . . 12  |-  _e  e.  RR+
62 4pos 10078 . . . . . . . . . . . . 13  |-  0  <  4
6358, 62elrpii 10607 . . . . . . . . . . . 12  |-  4  e.  RR+
64 logltb 20486 . . . . . . . . . . . 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 20473 . . . . . . . . . 10  |-  ( log `  _e )  =  1
68 sq2 11469 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
6968fveq2i 5723 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( log `  4 )
70 2z 10304 . . . . . . . . . . . 12  |-  2  e.  ZZ
71 relogexp 20482 . . . . . . . . . . . 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 2457 . . . . . . . . . 10  |-  ( log `  4 )  =  ( 2  x.  ( log `  2 ) )
7466, 67, 733brtr3i 4231 . . . . . . . . 9  |-  1  <  ( 2  x.  ( log `  2 ) )
751, 37pm3.2i 442 . . . . . . . . . 10  |-  ( 2  e.  RR  /\  0  <  2 )
76 ltdivmul 9874 . . . . . . . . . 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 10208 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
8041, 79, 34lttri 9191 . . . . . . . 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 9569 . . . . . . 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 10607 . . . . 5  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR+
85 rerpdivcl 10631 . . . . 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 10610 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR )
89 rpge0 10616 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  0  <_ 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) )
9088, 89absidd 12217 . . . . . . 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 2435 . . . . . . . . . 10  |-  ( |_
`  ( x  / 
2 ) )  =  ( |_ `  (
x  /  2 ) )
9493chebbnd1lem3 21157 . . . . . . . . 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 10075 . . . . . . . . . 10  |-  2  =/=  0
9742recni 9094 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  CC
9842, 83gt0ne0ii 9555 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  =/=  0
99 recdiv 9712 . . . . . . . . . 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 10642 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  CC )
10325nncnd 10008 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  CC )
10423rpne0d 10645 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  =/=  0 )
10525nnne0d 10036 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  =/=  0
)
106102, 103, 104, 105recdivd 9799 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
1  /  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  =  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
107103, 102, 104divrecd 9785 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  =  ( (π `  x )  x.  (
1  /  ( x  /  ( log `  x
) ) ) ) )
10821rpcnne0d 10649 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
10922rpcnne0d 10649 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
110 recdiv 9712 . . . . . . . . . . . 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 6089 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( 1  /  (
x  /  ( log `  x ) ) ) )  =  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
113106, 107, 1123eqtrd 2471 . . . . . . . . 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 4234 . . . . . . 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 10606 . . . . . . . . 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 9920 . . . . . . . . . 10  |-  0  <  ( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
119 ltrec 9883 . . . . . . . . . 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 10640 . . . . . . 7  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR )
125 ltle 9155 . . . . . . 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 4224 . . . 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 12322 . 2  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  e.  O ( 1 ) )
131130trud 1332 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 1325    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    +oocpnf 9109   RR*cxr 9111    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   NNcn 9992   2c2 10041   3c3 10042   4c4 10043   8c8 10047   ZZcz 10274   RR+crp 10604   [,)cico 10910   |_cfl 11193   ^cexp 11374   abscabs 12031   O ( 1 )co1 12272   _eceu 12657   logclog 20444  πcppi 20868
This theorem is referenced by:  chtppilimlem2  21160  chto1lb  21164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-o1 12276  df-lo1 12277  df-sum 12472  df-ef 12662  df-e 12663  df-sin 12664  df-cos 12665  df-pi 12667  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446  df-ppi 20874
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