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

Theorem chebbnd2 20620
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 5844 . . . . . 6  |-  ( 2 [,)  +oo )  e.  _V
21a1i 12 . . . . 5  |-  (  T. 
->  ( 2 [,)  +oo )  e.  _V )
3 ovex 5844 . . . . . 6  |-  ( (
theta `  x )  /  x )  e.  _V
43a1i 12 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  x )  e.  _V )
5 ovex 5844 . . . . . 6  |-  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  _V
65a1i 12 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  _V )
7 eqidd 2285 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  x ) )  =  ( x  e.  ( 2 [,) 
+oo )  |->  ( (
theta `  x )  /  x ) ) )
8 simpr 449 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  ( 2 [,)  +oo ) )
9 2re 9810 . . . . . . . . . . 11  |-  2  e.  RR
10 elicopnf 10733 . . . . . . . . . . 11  |-  ( 2  e.  RR  ->  (
x  e.  ( 2 [,)  +oo )  <->  ( x  e.  RR  /\  2  <_  x ) ) )
119, 10ax-mp 10 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  <->  ( x  e.  RR  /\  2  <_  x ) )
128, 11sylib 190 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  e.  RR  /\  2  <_  x ) )
13 chtrpcl 20407 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
1412, 13syl 17 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( theta `  x )  e.  RR+ )
1514rpcnne0d 10394 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0
) )
16 ppinncl 20406 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
1712, 16syl 17 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  NN )
1817nnrpd 10384 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  RR+ )
1912simpld 447 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR )
20 1re 8832 . . . . . . . . . . . 12  |-  1  e.  RR
2120a1i 12 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  RR )
229a1i 12 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  e.  RR )
23 1lt2 9881 . . . . . . . . . . . 12  |-  1  <  2
2423a1i 12 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <  2 )
2512simprd 451 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  <_  x )
2621, 22, 19, 24, 25ltletrd 8971 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <  x )
2719, 26rplogcld 19974 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( log `  x )  e.  RR+ )
2818, 27rpmulcld 10401 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  RR+ )
2928rpcnne0d 10394 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( (π `  x )  x.  ( log `  x
) )  e.  CC  /\  ( (π `  x )  x.  ( log `  x
) )  =/=  0
) )
30 recdiv 9461 . . . . . . 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 644 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )
3231mpteq2dva 4107 . . . . 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 6056 . . . 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 8833 . . . . . . . . . . 11  |-  0  e.  RR
3534a1i 12 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  e.  RR )
36 2pos 9823 . . . . . . . . . . 11  |-  0  <  2
3736a1i 12 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <  2 )
3835, 22, 19, 37, 25ltletrd 8971 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <  x )
3919, 38elrpd 10383 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR+ )
4039rpcnne0d 10394 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
4128rpcnd 10387 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  CC )
42 dmdcan 9465 . . . . . . 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 10387 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  CC )
4527rpcnne0d 10394 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
46 divdiv2 9467 . . . . . . 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 2319 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( theta `  x
)  /  x )  x.  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  =  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )
4948mpteq2dva 4107 . . . 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 2316 . . 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 425 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  ->  x  e.  RR+ )
)
5251ssrdv 3186 . . . . 5  |-  (  T. 
->  ( 2 [,)  +oo )  C_  RR+ )
53 chto1ub 20619 . . . . . 6  |-  ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  e.  O ( 1 )
5453a1i 12 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( theta `  x
)  /  x ) )  e.  O ( 1 ) )
5552, 54o1res2 12031 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  x ) )  e.  O ( 1 ) )
56 ax-1cn 8790 . . . . . . 7  |-  1  e.  CC
5756a1i 12 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  CC )
5814, 28rpdivcld 10402 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
5958rpcnd 10387 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  CC )
60 pnfxr 10450 . . . . . . . . 9  |-  +oo  e.  RR*
61 icossre 10724 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  +oo 
e.  RR* )  ->  (
2 [,)  +oo )  C_  RR )
629, 60, 61mp2an 655 . . . . . . . 8  |-  ( 2 [,)  +oo )  C_  RR
63 rlimconst 12012 . . . . . . . 8  |-  ( ( ( 2 [,)  +oo )  C_  RR  /\  1  e.  CC )  ->  (
x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
6462, 56, 63mp2an 655 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  1 )  ~~> r  1
6564a1i 12 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
66 chtppilim 20618 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  ~~> r  1
6766a1i 12 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1 )
68 ax-1ne0 8801 . . . . . . 7  |-  1  =/=  0
6968a1i 12 . . . . . 6  |-  (  T. 
->  1  =/=  0
)
7058rpne0d 10390 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
7157, 59, 65, 67, 69, 70rlimdiv 12113 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  ~~> r  ( 1  /  1 ) )
72 rlimo1 12084 . . . . 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 17 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  e.  O
( 1 ) )
74 o1mul 12082 . . . 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 644 . . 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 2359 . 2  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  e.  O
( 1 ) )
7776trud 1316 1  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  e.  O ( 1 )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    T. wtru 1309    = wceq 1624    e. wcel 1685    =/= wne 2447   _Vcvv 2789    C_ wss 3153   class class class wbr 4024    e. cmpt 4078   ` cfv 5221  (class class class)co 5819    o Fcof 6037   CCcc 8730   RRcr 8731   0cc0 8732   1c1 8733    x. cmul 8737    +oocpnf 8859   RR*cxr 8861    < clt 8862    <_ cle 8863    / cdiv 9418   NNcn 9741   2c2 9790   RR+crp 10349   [,)cico 10652    ~~> r crli 11953   O (
1 )co1 11954   logclog 19906   thetaccht 20322  πcppi 20325
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-map 6769  df-pm 6770  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10654  df-ioc 10655  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-fl 10919  df-mod 10968  df-seq 11041  df-exp 11099  df-fac 11283  df-bc 11310  df-hash 11332  df-shft 11556  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-limsup 11939  df-clim 11956  df-rlim 11957  df-o1 11958  df-lo1 11959  df-sum 12153  df-ef 12343  df-e 12344  df-sin 12345  df-cos 12346  df-pi 12348  df-dvds 12526  df-gcd 12680  df-prm 12753  df-pc 12884  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-starv 13217  df-sca 13218  df-vsca 13219  df-tset 13221  df-ple 13222  df-ds 13224  df-hom 13226  df-cco 13227  df-rest 13321  df-topn 13322  df-topgen 13338  df-pt 13339  df-prds 13342  df-xrs 13397  df-0g 13398  df-gsum 13399  df-qtop 13404  df-imas 13405  df-xps 13407  df-mre 13482  df-mrc 13483  df-acs 13485  df-mnd 14361  df-submnd 14410  df-mulg 14486  df-cntz 14787  df-cmn 15085  df-xmet 16367  df-met 16368  df-bl 16369  df-mopn 16370  df-cnfld 16372  df-top 16630  df-bases 16632  df-topon 16633  df-topsp 16634  df-cld 16750  df-ntr 16751  df-cls 16752  df-nei 16829  df-lp 16862  df-perf 16863  df-cn 16951  df-cnp 16952  df-haus 17037  df-tx 17251  df-hmeo 17440  df-fbas 17514  df-fg 17515  df-fil 17535  df-fm 17627  df-flim 17628  df-flf 17629  df-xms 17879  df-ms 17880  df-tms 17881  df-cncf 18376  df-limc 19210  df-dv 19211  df-log 19908  df-cxp 19909  df-cht 20328  df-ppi 20331
  Copyright terms: Public domain W3C validator