MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  chebbnd1 Unicode version

Theorem chebbnd1 21033
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 10001 . . . . 5  |-  2  e.  RR
2 pnfxr 10645 . . . . 5  |-  +oo  e.  RR*
3 icossre 10923 . . . . 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 10932 . . . . . . . . . 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 9024 . . . . . . . . . 10  |-  0  e.  RR
109a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  e.  RR )
11 1re 9023 . . . . . . . . . 10  |-  1  e.  RR
1211a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  e.  RR )
13 0lt1 9482 . . . . . . . . . 10  |-  0  <  1
1413a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  1 )
151a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  e.  RR )
16 1lt2 10074 . . . . . . . . . . 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 9162 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  <  x )
2010, 12, 8, 14, 19lttrd 9163 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  x )
218, 20elrpd 10578 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
228, 19rplogcld 20391 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  RR+ )
2321, 22rpdivcld 10597 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
24 ppinncl 20824 . . . . . . . 8  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
257, 24sylbi 188 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  NN )
2625nnrpd 10579 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  RR+ )
2723, 26rpdivcld 10597 . . . . 5  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+ )
2827rpcnd 10582 . . . 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 10010 . . . 4  |-  8  e.  RR
3130a1i 11 . . 3  |-  (  T. 
->  8  e.  RR )
32 2rp 10549 . . . . . . . 8  |-  2  e.  RR+
33 relogcl 20340 . . . . . . . 8  |-  ( 2  e.  RR+  ->  ( log `  2 )  e.  RR )
3432, 33ax-mp 8 . . . . . . 7  |-  ( log `  2 )  e.  RR
35 ere 12618 . . . . . . . . 9  |-  _e  e.  RR
361, 35remulcli 9037 . . . . . . . 8  |-  ( 2  x.  _e )  e.  RR
37 2pos 10014 . . . . . . . . . 10  |-  0  <  2
38 epos 12733 . . . . . . . . . 10  |-  0  <  _e
391, 35, 37, 38mulgt0ii 9138 . . . . . . . . 9  |-  0  <  ( 2  x.  _e )
4036, 39gt0ne0ii 9495 . . . . . . . 8  |-  ( 2  x.  _e )  =/=  0
4136, 40rereccli 9711 . . . . . . 7  |-  ( 1  /  ( 2  x.  _e ) )  e.  RR
4234, 41resubcli 9295 . . . . . 6  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR
431recni 9035 . . . . . . . . . . 11  |-  2  e.  CC
4443mulid1i 9025 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
45 egt2lt3 12732 . . . . . . . . . . . . 13  |-  ( 2  <  _e  /\  _e  <  3 )
4645simpli 445 . . . . . . . . . . . 12  |-  2  <  _e
4711, 1, 35lttri 9131 . . . . . . . . . . . 12  |-  ( ( 1  <  2  /\  2  <  _e )  ->  1  <  _e )
4816, 46, 47mp2an 654 . . . . . . . . . . 11  |-  1  <  _e
4911, 35, 1ltmul2i 9864 . . . . . . . . . . . 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 4174 . . . . . . . . 9  |-  2  <  ( 2  x.  _e )
531, 36, 37, 39ltrecii 9859 . . . . . . . . 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 10077 . . . . . . . . . . . 12  |-  3  <  4
57 3re 10003 . . . . . . . . . . . . 13  |-  3  e.  RR
58 4re 10005 . . . . . . . . . . . . 13  |-  4  e.  RR
5935, 57, 58lttri 9131 . . . . . . . . . . . 12  |-  ( ( _e  <  3  /\  3  <  4 )  ->  _e  <  4
)
6055, 56, 59mp2an 654 . . . . . . . . . . 11  |-  _e  <  4
61 epr 12734 . . . . . . . . . . . 12  |-  _e  e.  RR+
62 4pos 10018 . . . . . . . . . . . . 13  |-  0  <  4
6358, 62elrpii 10547 . . . . . . . . . . . 12  |-  4  e.  RR+
64 logltb 20361 . . . . . . . . . . . 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 20348 . . . . . . . . . 10  |-  ( log `  _e )  =  1
68 sq2 11404 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
6968fveq2i 5671 . . . . . . . . . . 11  |-  ( log `  ( 2 ^ 2 ) )  =  ( log `  4 )
70 2z 10244 . . . . . . . . . . . 12  |-  2  e.  ZZ
71 relogexp 20357 . . . . . . . . . . . 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 2409 . . . . . . . . . 10  |-  ( log `  4 )  =  ( 2  x.  ( log `  2 ) )
7466, 67, 733brtr3i 4180 . . . . . . . . 9  |-  1  <  ( 2  x.  ( log `  2 ) )
751, 37pm3.2i 442 . . . . . . . . . 10  |-  ( 2  e.  RR  /\  0  <  2 )
76 ltdivmul 9814 . . . . . . . . . 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 10148 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
8041, 79, 34lttri 9131 . . . . . . . 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 9509 . . . . . . 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 10547 . . . . 5  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  RR+
85 rerpdivcl 10571 . . . . 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 10550 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  e.  RR )
89 rpge0 10556 . . . . . . . 8  |-  ( ( ( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR+  ->  0  <_ 
( ( x  / 
( log `  x
) )  /  (π `  x ) ) )
9088, 89absidd 12152 . . . . . . 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 2387 . . . . . . . . . 10  |-  ( |_
`  ( x  / 
2 ) )  =  ( |_ `  (
x  /  2 ) )
9493chebbnd1lem3 21032 . . . . . . . . 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 10015 . . . . . . . . . 10  |-  2  =/=  0
9742recni 9035 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  e.  CC
9842, 83gt0ne0ii 9495 . . . . . . . . . 10  |-  ( ( log `  2 )  -  ( 1  / 
( 2  x.  _e ) ) )  =/=  0
99 recdiv 9652 . . . . . . . . . 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 10582 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  CC )
10325nncnd 9948 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  CC )
10423rpne0d 10585 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( log `  x ) )  =/=  0 )
10525nnne0d 9976 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  =/=  0
)
106102, 103, 104, 105recdivd 9739 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
1  /  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  =  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
107103, 102, 104divrecd 9725 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  =  ( (π `  x )  x.  (
1  /  ( x  /  ( log `  x
) ) ) ) )
10821rpcnne0d 10589 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
10922rpcnne0d 10589 . . . . . . . . . . . 12  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
110 recdiv 9652 . . . . . . . . . . . 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 6036 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( 1  /  (
x  /  ( log `  x ) ) ) )  =  ( (π `  x )  x.  (
( log `  x
)  /  x ) ) )
113106, 107, 1123eqtrd 2423 . . . . . . . . 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 4183 . . . . . . 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 10546 . . . . . . . . 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 9860 . . . . . . . . . 10  |-  0  <  ( 2  /  (
( log `  2
)  -  ( 1  /  ( 2  x.  _e ) ) ) )
119 ltrec 9823 . . . . . . . . . 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 10580 . . . . . . 7  |-  ( ( x  e.  ( 2 [,)  +oo )  /\  8  <_  x )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  e.  RR )
125 ltle 9096 . . . . . . 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 4173 . . . 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 12257 . 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 1717    =/= wne 2550    C_ wss 3263   class class class wbr 4153    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    x. cmul 8928    +oocpnf 9050   RR*cxr 9052    < clt 9053    <_ cle 9054    - cmin 9223    / cdiv 9609   NNcn 9932   2c2 9981   3c3 9982   4c4 9983   8c8 9987   ZZcz 10214   RR+crp 10544   [,)cico 10850   |_cfl 11128   ^cexp 11309   abscabs 11966   O ( 1 )co1 12207   _eceu 12592   logclog 20319  πcppi 20743
This theorem is referenced by:  chtppilimlem2  21035  chto1lb  21039
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ioc 10853  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-fac 11494  df-bc 11521  df-hash 11546  df-shft 11809  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-limsup 12192  df-clim 12209  df-rlim 12210  df-o1 12211  df-lo1 12212  df-sum 12407  df-ef 12597  df-e 12598  df-sin 12599  df-cos 12600  df-pi 12602  df-dvds 12780  df-gcd 12934  df-prm 13007  df-pc 13138  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-hom 13480  df-cco 13481  df-rest 13577  df-topn 13578  df-topgen 13594  df-pt 13595  df-prds 13598  df-xrs 13653  df-0g 13654  df-gsum 13655  df-qtop 13660  df-imas 13661  df-xps 13663  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-mulg 14742  df-cntz 15043  df-cmn 15341  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-fbas 16623  df-fg 16624  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cld 17006  df-ntr 17007  df-cls 17008  df-nei 17085  df-lp 17123  df-perf 17124  df-cn 17213  df-cnp 17214  df-haus 17301  df-tx 17515  df-hmeo 17708  df-fil 17799  df-fm 17891  df-flim 17892  df-flf 17893  df-xms 18259  df-ms 18260  df-tms 18261  df-cncf 18779  df-limc 19620  df-dv 19621  df-log 20321  df-ppi 20749
  Copyright terms: Public domain W3C validator