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Theorem chebbnd2 23383
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 6300 . . . . . 6  |-  ( 2 [,) +oo )  e. 
_V
21a1i 11 . . . . 5  |-  ( T. 
->  ( 2 [,) +oo )  e.  _V )
3 ovex 6300 . . . . . 6  |-  ( (
theta `  x )  /  x )  e.  _V
43a1i 11 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  x )  e.  _V )
5 ovex 6300 . . . . . 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 2461 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  x ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (
theta `  x )  /  x ) ) )
8 simpr 461 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  ( 2 [,) +oo ) )
9 2re 10594 . . . . . . . . . . 11  |-  2  e.  RR
10 elicopnf 11609 . . . . . . . . . . 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 23170 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
1412, 13syl 16 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( theta `  x )  e.  RR+ )
1514rpcnne0d 11254 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  e.  CC  /\  ( theta `  x )  =/=  0
) )
16 ppinncl 23169 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
1712, 16syl 16 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (π `  x )  e.  NN )
1817nnrpd 11244 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (π `  x )  e.  RR+ )
1912simpld 459 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  RR )
20 1red 9600 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  e.  RR )
219a1i 11 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  2  e.  RR )
22 1lt2 10691 . . . . . . . . . . . 12  |-  1  <  2
2322a1i 11 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <  2 )
2412simprd 463 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  2  <_  x )
2520, 21, 19, 23, 24ltletrd 9730 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  <  x )
2619, 25rplogcld 22735 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  ( log `  x )  e.  RR+ )
2718, 26rpmulcld 11261 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  RR+ )
2827rpcnne0d 11254 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( (π `  x )  x.  ( log `  x
) )  e.  CC  /\  ( (π `  x )  x.  ( log `  x
) )  =/=  0
) )
29 recdiv 10239 . . . . . . 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 661 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
1  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )
3130mpteq2dva 4526 . . . . 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 6531 . . . 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 9586 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  0  e.  RR )
34 2pos 10616 . . . . . . . . . . 11  |-  0  <  2
3534a1i 11 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  0  <  2 )
3633, 21, 19, 35, 24ltletrd 9730 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  0  <  x )
3719, 36elrpd 11243 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  x  e.  RR+ )
3837rpcnne0d 11254 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
3927rpcnd 11247 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  CC )
40 dmdcan 10243 . . . . . . 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 1223 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( theta `  x
)  /  x )  x.  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  =  ( ( (π `  x
)  x.  ( log `  x ) )  /  x ) )
4218rpcnd 11247 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (π `  x )  e.  CC )
4326rpcnne0d 11254 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( log `  x
)  e.  CC  /\  ( log `  x )  =/=  0 ) )
44 divdiv2 10245 . . . . . . 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 1223 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
4641, 45eqtr4d 2504 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( ( theta `  x
)  /  x )  x.  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )  =  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )
4746mpteq2dva 4526 . . . 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 2501 . . 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 434 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
)
5049ssrdv 3503 . . . . 5  |-  ( T. 
->  ( 2 [,) +oo )  C_  RR+ )
51 chto1ub 23382 . . . . . 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 13335 . . . 4  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  x ) )  e.  O(1) )
54 ax-1cn 9539 . . . . . . 7  |-  1  e.  CC
5554a1i 11 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  e.  CC )
5614, 27rpdivcld 11262 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
5756rpcnd 11247 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  CC )
58 pnfxr 11310 . . . . . . . . 9  |- +oo  e.  RR*
59 icossre 11594 . . . . . . . . 9  |-  ( ( 2  e.  RR  /\ +oo  e.  RR* )  ->  (
2 [,) +oo )  C_  RR )
609, 58, 59mp2an 672 . . . . . . . 8  |-  ( 2 [,) +oo )  C_  RR
61 rlimconst 13316 . . . . . . . 8  |-  ( ( ( 2 [,) +oo )  C_  RR  /\  1  e.  CC )  ->  (
x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1 )
6260, 54, 61mp2an 672 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1
6362a1i 11 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1 )
64 chtppilim 23381 . . . . . . 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 9550 . . . . . . 7  |-  1  =/=  0
6766a1i 11 . . . . . 6  |-  ( T. 
->  1  =/=  0
)
6856rpne0d 11250 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
6955, 57, 63, 65, 67, 68rlimdiv 13417 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  ~~> r  ( 1  /  1 ) )
70 rlimo1 13388 . . . . 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 16 . . . 4  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  e.  O(1) )
72 o1mul 13386 . . . 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 661 . . 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 2549 . 2  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  e.  O(1) )
7574trud 1383 1  |-  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  e.  O(1)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374   T. wtru 1375    e. wcel 1762    =/= wne 2655   _Vcvv 3106    C_ wss 3469   class class class wbr 4440    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275    oFcof 6513   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    x. cmul 9486   +oocpnf 9614   RR*cxr 9616    < clt 9617    <_ cle 9618    / cdiv 10195   NNcn 10525   2c2 10574   RR+crp 11209   [,)cico 11520    ~~> r crli 13257   O(1)co1 13258   logclog 22663   thetaccht 23085  πcppi 23088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-ioc 11523  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-fac 12309  df-bc 12336  df-hash 12361  df-shft 12850  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-limsup 13243  df-clim 13260  df-rlim 13261  df-o1 13262  df-lo1 13263  df-sum 13458  df-ef 13654  df-e 13655  df-sin 13656  df-cos 13657  df-pi 13659  df-dvds 13837  df-gcd 13993  df-prm 14066  df-pc 14209  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cn 19487  df-cnp 19488  df-haus 19575  df-tx 19791  df-hmeo 19984  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-xms 20551  df-ms 20552  df-tms 20553  df-cncf 21110  df-limc 21998  df-dv 21999  df-log 22665  df-cxp 22666  df-cht 23091  df-ppi 23094
This theorem is referenced by: (None)
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