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Theorem chebbnd2 20626
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 5883 . . . . . 6  |-  ( 2 [,)  +oo )  e.  _V
21a1i 10 . . . . 5  |-  (  T. 
->  ( 2 [,)  +oo )  e.  _V )
3 ovex 5883 . . . . . 6  |-  ( (
theta `  x )  /  x )  e.  _V
43a1i 10 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  x )  e.  _V )
5 ovex 5883 . . . . . 6  |-  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  _V
65a1i 10 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  _V )
7 eqidd 2284 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  x ) )  =  ( x  e.  ( 2 [,) 
+oo )  |->  ( (
theta `  x )  /  x ) ) )
8 simpr 447 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  ( 2 [,)  +oo ) )
9 2re 9815 . . . . . . . . . . 11  |-  2  e.  RR
10 elicopnf 10739 . . . . . . . . . . 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 188 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  e.  RR  /\  2  <_  x ) )
13 chtrpcl 20413 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
1412, 13syl 15 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( theta `  x )  e.  RR+ )
1514rpcnne0d 10399 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0
) )
16 ppinncl 20412 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
1712, 16syl 15 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  NN )
1817nnrpd 10389 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  RR+ )
1912simpld 445 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR )
20 1re 8837 . . . . . . . . . . . 12  |-  1  e.  RR
2120a1i 10 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  RR )
229a1i 10 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  e.  RR )
23 1lt2 9886 . . . . . . . . . . . 12  |-  1  <  2
2423a1i 10 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <  2 )
2512simprd 449 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  2  <_  x )
2621, 22, 19, 24, 25ltletrd 8976 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  <  x )
2719, 26rplogcld 19980 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  ( log `  x )  e.  RR+ )
2818, 27rpmulcld 10406 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  RR+ )
2928rpcnne0d 10399 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( (π `  x )  x.  ( log `  x
) )  e.  CC  /\  ( (π `  x )  x.  ( log `  x
) )  =/=  0
) )
30 recdiv 9466 . . . . . . 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 642 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
1  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )
3231mpteq2dva 4106 . . . . 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 6095 . . . 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 8838 . . . . . . . . . . 11  |-  0  e.  RR
3534a1i 10 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  e.  RR )
36 2pos 9828 . . . . . . . . . . 11  |-  0  <  2
3736a1i 10 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <  2 )
3835, 22, 19, 37, 25ltletrd 8976 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  0  <  x )
3919, 38elrpd 10388 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  x  e.  RR+ )
4039rpcnne0d 10399 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
4128rpcnd 10392 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  CC )
42 dmdcan 9470 . . . . . . 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 1182 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( theta `  x
)  /  x )  x.  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  =  ( ( (π `  x
)  x.  ( log `  x ) )  /  x ) )
4418rpcnd 10392 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (π `  x )  e.  CC )
4527rpcnne0d 10399 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
46 divdiv2 9472 . . . . . . 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 1182 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
4843, 47eqtr4d 2318 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( ( theta `  x
)  /  x )  x.  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  =  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )
4948mpteq2dva 4106 . . . 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 2315 . . 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 423 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  ->  x  e.  RR+ )
)
5251ssrdv 3185 . . . . 5  |-  (  T. 
->  ( 2 [,)  +oo )  C_  RR+ )
53 chto1ub 20625 . . . . . 6  |-  ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  e.  O ( 1 )
5453a1i 10 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( theta `  x
)  /  x ) )  e.  O ( 1 ) )
5552, 54o1res2 12037 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  x ) )  e.  O ( 1 ) )
56 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
5756a1i 10 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  CC )
5814, 28rpdivcld 10407 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
5958rpcnd 10392 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  CC )
60 pnfxr 10455 . . . . . . . . 9  |-  +oo  e.  RR*
61 icossre 10730 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\  +oo 
e.  RR* )  ->  (
2 [,)  +oo )  C_  RR )
629, 60, 61mp2an 653 . . . . . . . 8  |-  ( 2 [,)  +oo )  C_  RR
63 rlimconst 12018 . . . . . . . 8  |-  ( ( ( 2 [,)  +oo )  C_  RR  /\  1  e.  CC )  ->  (
x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
6462, 56, 63mp2an 653 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  1 )  ~~> r  1
6564a1i 10 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
66 chtppilim 20624 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  ~~> r  1
6766a1i 10 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1 )
68 ax-1ne0 8806 . . . . . . 7  |-  1  =/=  0
6968a1i 10 . . . . . 6  |-  (  T. 
->  1  =/=  0
)
7058rpne0d 10395 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
7157, 59, 65, 67, 69, 70rlimdiv 12119 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  ~~> r  ( 1  /  1 ) )
72 rlimo1 12090 . . . . 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 15 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  e.  O
( 1 ) )
74 o1mul 12088 . . . 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 642 . . 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 2358 . 2  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  e.  O
( 1 ) )
7776trud 1314 1  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  e.  O ( 1 )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    T. wtru 1307    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   2c2 9795   RR+crp 10354   [,)cico 10658    ~~> r crli 11959   O (
1 )co1 11960   logclog 19912   thetaccht 20328  πcppi 20331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-o1 11964  df-lo1 11965  df-sum 12159  df-ef 12349  df-e 12350  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915  df-cht 20334  df-ppi 20337
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