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Theorem chebbnd2 24041
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 6305 . . . . . 6  |-  ( 2 [,) +oo )  e. 
_V
21a1i 11 . . . . 5  |-  ( T. 
->  ( 2 [,) +oo )  e.  _V )
3 ovex 6305 . . . . . 6  |-  ( (
theta `  x )  /  x )  e.  _V
43a1i 11 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  x )  e.  _V )
5 ovex 6305 . . . . . 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 2403 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  x ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (
theta `  x )  /  x ) ) )
8 simpr 459 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  ( 2 [,) +oo ) )
9 2re 10645 . . . . . . . . . . 11  |-  2  e.  RR
10 elicopnf 11672 . . . . . . . . . . 11  |-  ( 2  e.  RR  ->  (
x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x ) ) )
119, 10ax-mp 5 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x ) )
128, 11sylib 196 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  e.  RR  /\  2  <_  x ) )
13 chtrpcl 23828 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
1412, 13syl 17 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( theta `  x )  e.  RR+ )
1514rpcnne0d 11312 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0
) )
16 ppinncl 23827 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
1712, 16syl 17 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (π `  x )  e.  NN )
1817nnrpd 11301 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (π `  x )  e.  RR+ )
1912simpld 457 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  RR )
20 1red 9640 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  e.  RR )
219a1i 11 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  2  e.  RR )
22 1lt2 10742 . . . . . . . . . . . 12  |-  1  <  2
2322a1i 11 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <  2 )
2412simprd 461 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  2  <_  x )
2520, 21, 19, 23, 24ltletrd 9775 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <  x )
2619, 25rplogcld 23306 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( log `  x )  e.  RR+ )
2718, 26rpmulcld 11319 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  RR+ )
2827rpcnne0d 11312 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( (π `  x )  x.  ( log `  x
) )  e.  CC  /\  ( (π `  x )  x.  ( log `  x
) )  =/=  0
) )
29 recdiv 10290 . . . . . . 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
) ) )
3015, 28, 29syl2anc 659 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )
3130mpteq2dva 4480 . . . . 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 ) ) ) )
322, 4, 6, 7, 31offval2 6537 . . . 4  |-  ( T. 
->  ( ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x )  /  x
) )  oF  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 ) ) ) ) )
33 0red 9626 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  0  e.  RR )
34 2pos 10667 . . . . . . . . . . 11  |-  0  <  2
3534a1i 11 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  0  <  2 )
3633, 21, 19, 35, 24ltletrd 9775 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  0  <  x )
3719, 36elrpd 11300 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  RR+ )
3837rpcnne0d 11312 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
3927rpcnd 11305 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  CC )
40 dmdcan 10294 . . . . . . 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
) )
4115, 38, 39, 40syl3anc 1230 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( theta `  x
)  /  x )  x.  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  =  ( ( (π `  x
)  x.  ( log `  x ) )  /  x ) )
4218rpcnd 11305 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (π `  x )  e.  CC )
4326rpcnne0d 11312 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
44 divdiv2 10296 . . . . . . 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 ) )
4542, 38, 43, 44syl3anc 1230 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
4641, 45eqtr4d 2446 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( theta `  x
)  /  x )  x.  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  =  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )
4746mpteq2dva 4480 . . . 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 ) ) ) ) )
4832, 47eqtrd 2443 . . 3  |-  ( T. 
->  ( ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x )  /  x
) )  oF  x.  ( x  e.  ( 2 [,) +oo )  |->  ( 1  / 
( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) ) )
4937ex 432 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
)
5049ssrdv 3447 . . . . 5  |-  ( T. 
->  ( 2 [,) +oo )  C_  RR+ )
51 chto1ub 24040 . . . . . 6  |-  ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  e.  O(1)
5251a1i 11 . . . . 5  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( theta `  x
)  /  x ) )  e.  O(1) )
5350, 52o1res2 13533 . . . 4  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  x ) )  e.  O(1) )
54 ax-1cn 9579 . . . . . . 7  |-  1  e.  CC
5554a1i 11 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  e.  CC )
5614, 27rpdivcld 11320 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
5756rpcnd 11305 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  CC )
58 pnfxr 11373 . . . . . . . . 9  |- +oo  e.  RR*
59 icossre 11657 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\ +oo  e.  RR* )  ->  (
2 [,) +oo )  C_  RR )
609, 58, 59mp2an 670 . . . . . . . 8  |-  ( 2 [,) +oo )  C_  RR
61 rlimconst 13514 . . . . . . . 8  |-  ( ( ( 2 [,) +oo )  C_  RR  /\  1  e.  CC )  ->  (
x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1 )
6260, 54, 61mp2an 670 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1
6362a1i 11 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1 )
64 chtppilim 24039 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  |->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  ~~> r  1
6564a1i 11 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1 )
66 ax-1ne0 9590 . . . . . . 7  |-  1  =/=  0
6766a1i 11 . . . . . 6  |-  ( T. 
->  1  =/=  0
)
6856rpne0d 11308 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
6955, 57, 63, 65, 67, 68rlimdiv 13615 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  ~~> r  ( 1  /  1 ) )
70 rlimo1 13586 . . . . 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) )
7169, 70syl 17 . . . 4  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  e.  O(1) )
72 o1mul 13584 . . . 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
) )  oF  x.  ( x  e.  ( 2 [,) +oo )  |->  ( 1  / 
( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) ) ) )  e.  O(1) )
7353, 71, 72syl2anc 659 . . 3  |-  ( T. 
->  ( ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x )  /  x
) )  oF  x.  ( x  e.  ( 2 [,) +oo )  |->  ( 1  / 
( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) ) ) )  e.  O(1) )
7448, 73eqeltrrd 2491 . 2  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  e.  O(1) )
7574trud 1414 1  |-  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  e.  O(1)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405   T. wtru 1406    e. wcel 1842    =/= wne 2598   _Vcvv 3058    C_ wss 3413   class class class wbr 4394    |-> cmpt 4452   ` cfv 5568  (class class class)co 6277    oFcof 6518   CCcc 9519   RRcr 9520   0cc0 9521   1c1 9522    x. cmul 9526   +oocpnf 9654   RR*cxr 9656    < clt 9657    <_ cle 9658    / cdiv 10246   NNcn 10575   2c2 10625   RR+crp 11264   [,)cico 11583    ~~> r crli 13455   O(1)co1 13456   logclog 23232   thetaccht 23743  πcppi 23746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-addf 9600  ax-mulf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-fi 7904  df-sup 7934  df-oi 7968  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-q 11227  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-ioo 11585  df-ioc 11586  df-ico 11587  df-icc 11588  df-fz 11725  df-fzo 11853  df-fl 11964  df-mod 12033  df-seq 12150  df-exp 12209  df-fac 12396  df-bc 12423  df-hash 12451  df-shft 13047  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-limsup 13441  df-clim 13458  df-rlim 13459  df-o1 13460  df-lo1 13461  df-sum 13656  df-ef 14010  df-e 14011  df-sin 14012  df-cos 14013  df-pi 14015  df-dvds 14194  df-gcd 14352  df-prm 14425  df-pc 14568  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-starv 14922  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-hom 14931  df-cco 14932  df-rest 15035  df-topn 15036  df-0g 15054  df-gsum 15055  df-topgen 15056  df-pt 15057  df-prds 15060  df-xrs 15114  df-qtop 15119  df-imas 15120  df-xps 15122  df-mre 15198  df-mrc 15199  df-acs 15201  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-submnd 16289  df-mulg 16382  df-cntz 16677  df-cmn 17122  df-psmet 18729  df-xmet 18730  df-met 18731  df-bl 18732  df-mopn 18733  df-fbas 18734  df-fg 18735  df-cnfld 18739  df-top 19689  df-bases 19691  df-topon 19692  df-topsp 19693  df-cld 19810  df-ntr 19811  df-cls 19812  df-nei 19890  df-lp 19928  df-perf 19929  df-cn 20019  df-cnp 20020  df-haus 20107  df-tx 20353  df-hmeo 20546  df-fil 20637  df-fm 20729  df-flim 20730  df-flf 20731  df-xms 21113  df-ms 21114  df-tms 21115  df-cncf 21672  df-limc 22560  df-dv 22561  df-log 23234  df-cxp 23235  df-cht 23749  df-ppi 23752
This theorem is referenced by: (None)
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