MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  chebbnd2 Unicode version

Theorem chebbnd2 21039
Description: The Chebyshev bound, part 2: The function π ( x ) is eventually upper bounded by a positive constant times  x  /  log ( x ). Alternatively stated, the function π ( x )  /  (
x  /  log (
x ) ) is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.)
Assertion
Ref Expression
chebbnd2  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  e.  O ( 1 )

Proof of Theorem chebbnd2
StepHypRef Expression
1 ovex 6046 . . . . . 6  |-  ( 2 [,)  +oo )  e.  _V
21a1i 11 . . . . 5  |-  (  T. 
->  ( 2 [,)  +oo )  e.  _V )
3 ovex 6046 . . . . . 6  |-  ( (
theta `  x )  /  x )  e.  _V
43a1i 11 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  x )  e.  _V )
5 ovex 6046 . . . . . 6  |-  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  _V
65a1i 11 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  _V )
7 eqidd 2389 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  x ) )  =  ( x  e.  ( 2 [,) 
+oo )  |->  ( (
theta `  x )  /  x ) ) )
8 simpr 448 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  ( 2 [,)  +oo ) )
9 2re 10002 . . . . . . . . . . 11  |-  2  e.  RR
10 elicopnf 10933 . . . . . . . . . . 11  |-  ( 2  e.  RR  ->  (
x  e.  ( 2 [,)  +oo )  <->  ( x  e.  RR  /\  2  <_  x ) ) )
119, 10ax-mp 8 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  <->  ( x  e.  RR  /\  2  <_  x ) )
128, 11sylib 189 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  e.  RR  /\  2  <_  x ) )
13 chtrpcl 20826 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
1412, 13syl 16 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( theta `  x )  e.  RR+ )
1514rpcnne0d 10590 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0
) )
16 ppinncl 20825 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
1712, 16syl 16 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  NN )
1817nnrpd 10580 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  RR+ )
1912simpld 446 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR )
20 1re 9024 . . . . . . . . . . . 12  |-  1  e.  RR
2120a1i 11 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  RR )
229a1i 11 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  e.  RR )
23 1lt2 10075 . . . . . . . . . . . 12  |-  1  <  2
2423a1i 11 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <  2 )
2512simprd 450 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  <_  x )
2621, 22, 19, 24, 25ltletrd 9163 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <  x )
2719, 26rplogcld 20392 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( log `  x )  e.  RR+ )
2818, 27rpmulcld 10597 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  RR+ )
2928rpcnne0d 10590 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( (π `  x )  x.  ( log `  x
) )  e.  CC  /\  ( (π `  x )  x.  ( log `  x
) )  =/=  0
) )
30 recdiv 9653 . . . . . . 7  |-  ( ( ( ( theta `  x
)  e.  CC  /\  ( theta `  x )  =/=  0 )  /\  (
( (π `  x )  x.  ( log `  x
) )  e.  CC  /\  ( (π `  x )  x.  ( log `  x
) )  =/=  0
) )  ->  (
1  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )
3115, 29, 30syl2anc 643 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )
3231mpteq2dva 4237 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) ) )
332, 4, 6, 7, 32offval2 6262 . . . 4  |-  (  T. 
->  ( ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x )  /  x
) )  o F  x.  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  / 
( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( ( (
theta `  x )  /  x )  x.  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) ) ) )
34 0re 9025 . . . . . . . . . . 11  |-  0  e.  RR
3534a1i 11 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  e.  RR )
36 2pos 10015 . . . . . . . . . . 11  |-  0  <  2
3736a1i 11 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <  2 )
3835, 22, 19, 37, 25ltletrd 9163 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <  x )
3919, 38elrpd 10579 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR+ )
4039rpcnne0d 10590 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
4128rpcnd 10583 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  CC )
42 dmdcan 9657 . . . . . . 7  |-  ( ( ( ( theta `  x
)  e.  CC  /\  ( theta `  x )  =/=  0 )  /\  (
x  e.  CC  /\  x  =/=  0 )  /\  ( (π `  x )  x.  ( log `  x
) )  e.  CC )  ->  ( ( (
theta `  x )  /  x )  x.  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
4315, 40, 41, 42syl3anc 1184 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( theta `  x
)  /  x )  x.  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  =  ( ( (π `  x
)  x.  ( log `  x ) )  /  x ) )
4418rpcnd 10583 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  CC )
4527rpcnne0d 10590 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
46 divdiv2 9659 . . . . . . 7  |-  ( ( (π `  x )  e.  CC  /\  ( x  e.  CC  /\  x  =/=  0 )  /\  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )  ->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) )  =  ( ( (π `  x
)  x.  ( log `  x ) )  /  x ) )
4744, 40, 45, 46syl3anc 1184 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
4843, 47eqtr4d 2423 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( theta `  x
)  /  x )  x.  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  =  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )
4948mpteq2dva 4237 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( ( theta `  x )  /  x
)  x.  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) ) )
5033, 49eqtrd 2420 . . 3  |-  (  T. 
->  ( ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x )  /  x
) )  o F  x.  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  / 
( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) ) )
5139ex 424 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  ->  x  e.  RR+ )
)
5251ssrdv 3298 . . . . 5  |-  (  T. 
->  ( 2 [,)  +oo )  C_  RR+ )
53 chto1ub 21038 . . . . . 6  |-  ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  e.  O ( 1 )
5453a1i 11 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( theta `  x
)  /  x ) )  e.  O ( 1 ) )
5552, 54o1res2 12285 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  x ) )  e.  O ( 1 ) )
56 ax-1cn 8982 . . . . . . 7  |-  1  e.  CC
5756a1i 11 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  CC )
5814, 28rpdivcld 10598 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
5958rpcnd 10583 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  CC )
60 pnfxr 10646 . . . . . . . . 9  |-  +oo  e.  RR*
61 icossre 10924 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  +oo 
e.  RR* )  ->  (
2 [,)  +oo )  C_  RR )
629, 60, 61mp2an 654 . . . . . . . 8  |-  ( 2 [,)  +oo )  C_  RR
63 rlimconst 12266 . . . . . . . 8  |-  ( ( ( 2 [,)  +oo )  C_  RR  /\  1  e.  CC )  ->  (
x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
6462, 56, 63mp2an 654 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  1 )  ~~> r  1
6564a1i 11 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
66 chtppilim 21037 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  ~~> r  1
6766a1i 11 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1 )
68 ax-1ne0 8993 . . . . . . 7  |-  1  =/=  0
6968a1i 11 . . . . . 6  |-  (  T. 
->  1  =/=  0
)
7058rpne0d 10586 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
7157, 59, 65, 67, 69, 70rlimdiv 12367 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  ~~> r  ( 1  /  1 ) )
72 rlimo1 12338 . . . . 5  |-  ( ( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) ) )  ~~> r  ( 1  /  1 )  -> 
( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  e.  O
( 1 ) )
7371, 72syl 16 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  e.  O
( 1 ) )
74 o1mul 12336 . . . 4  |-  ( ( ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  x ) )  e.  O ( 1 )  /\  (
x  e.  ( 2 [,)  +oo )  |->  ( 1  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) ) )  e.  O ( 1 ) )  -> 
( ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x )  /  x
) )  o F  x.  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  / 
( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) ) ) )  e.  O ( 1 ) )
7555, 73, 74syl2anc 643 . . 3  |-  (  T. 
->  ( ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x )  /  x
) )  o F  x.  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  / 
( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) ) ) )  e.  O ( 1 ) )
7650, 75eqeltrrd 2463 . 2  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  e.  O
( 1 ) )
7776trud 1329 1  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  e.  O ( 1 )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    T. wtru 1322    = wceq 1649    e. wcel 1717    =/= wne 2551   _Vcvv 2900    C_ wss 3264   class class class wbr 4154    e. cmpt 4208   ` cfv 5395  (class class class)co 6021    o Fcof 6243   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925    x. cmul 8929    +oocpnf 9051   RR*cxr 9053    < clt 9054    <_ cle 9055    / cdiv 9610   NNcn 9933   2c2 9982   RR+crp 10545   [,)cico 10851    ~~> r crli 12207   O (
1 )co1 12208   logclog 20320   thetaccht 20741  πcppi 20744
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ioc 10854  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-shft 11810  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-limsup 12193  df-clim 12210  df-rlim 12211  df-o1 12212  df-lo1 12213  df-sum 12408  df-ef 12598  df-e 12599  df-sin 12600  df-cos 12601  df-pi 12603  df-dvds 12781  df-gcd 12935  df-prm 13008  df-pc 13139  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lp 17124  df-perf 17125  df-cn 17214  df-cnp 17215  df-haus 17302  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cncf 18780  df-limc 19621  df-dv 19622  df-log 20322  df-cxp 20323  df-cht 20747  df-ppi 20750
  Copyright terms: Public domain W3C validator