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Theorem chto1lb 23527
Description: The  theta function is lower bounded by a linear term. Corollary of chebbnd1 23521. (Contributed by Mario Carneiro, 8-Apr-2016.)
Assertion
Ref Expression
chto1lb  |-  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x
) ) )  e.  O(1)

Proof of Theorem chto1lb
StepHypRef Expression
1 ovex 6320 . . . . . 6  |-  ( 2 [,) +oo )  e. 
_V
21a1i 11 . . . . 5  |-  ( T. 
->  ( 2 [,) +oo )  e.  _V )
3 2re 10617 . . . . . . . . . . . 12  |-  2  e.  RR
4 elicopnf 11632 . . . . . . . . . . . 12  |-  ( 2  e.  RR  ->  (
x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x ) ) )
53, 4ax-mp 5 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x ) )
65biimpi 194 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  e.  RR  /\  2  <_  x ) )
76simpld 459 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR )
8 0red 9609 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  0  e.  RR )
93a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  2  e.  RR )
10 2pos 10639 . . . . . . . . . . 11  |-  0  <  2
1110a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  0  <  2 )
126simprd 463 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  2  <_  x )
138, 9, 7, 11, 12ltletrd 9753 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  0  < 
x )
147, 13elrpd 11266 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
15 ppinncl 23312 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
1615nnrpd 11267 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  RR+ )
176, 16syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  RR+ )
18 1red 9623 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  1  e.  RR )
19 1lt2 10714 . . . . . . . . . . . 12  |-  1  <  2
2019a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  1  <  2 )
2118, 9, 7, 20, 12ltletrd 9753 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  1  < 
x )
227, 21rplogcld 22878 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  RR+ )
2317, 22rpmulcld 11284 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  RR+ )
2414, 23rpdivcld 11285 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
2524rpcnd 11270 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  CC )
2625adantl 466 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  /  ( (π `  x )  x.  ( log `  x ) ) )  e.  CC )
27 chtrpcl 23313 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
286, 27syl 16 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  RR+ )
2923, 28rpdivcld 11285 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  RR+ )
3029rpcnd 11270 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  CC )
3130adantl 466 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  CC )
327recnd 9634 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  CC )
3322rpcnd 11270 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  CC )
3417rpcnd 11270 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  CC )
3522rpne0d 11273 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  =/=  0
)
3617rpne0d 11273 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  =/=  0 )
3732, 33, 34, 35, 36divdiv1d 10363 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  =  ( x  / 
( ( log `  x
)  x.  (π `  x
) ) ) )
3833, 34mulcomd 9629 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( log `  x )  x.  (π `  x ) )  =  ( (π `  x
)  x.  ( log `  x ) ) )
3938oveq2d 6311 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( ( log `  x )  x.  (π `  x ) ) )  =  ( x  / 
( (π `  x )  x.  ( log `  x
) ) ) )
4037, 39eqtrd 2508 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  =  ( x  / 
( (π `  x )  x.  ( log `  x
) ) ) )
4140mpteq2ia 4535 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( (π `  x
)  x.  ( log `  x ) ) ) )
4241a1i 11 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( x  / 
( (π `  x )  x.  ( log `  x
) ) ) ) )
4328rpcnd 11270 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  CC )
4423rpcnd 11270 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  CC )
4528rpne0d 11273 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  =/=  0
)
4623rpne0d 11273 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  =/=  0 )
4743, 44, 45, 46recdivd 10349 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( 1  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )
4847mpteq2ia 4535 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  |->  ( 1  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) )
4948a1i 11 . . . . 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 ) ) ) )
502, 26, 31, 42, 49offval2 6551 . . . 4  |-  ( T. 
->  ( ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  oF  x.  (
x  e.  ( 2 [,) +oo )  |->  ( 1  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  (
(π `  x )  x.  ( log `  x
) ) )  x.  ( ( (π `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) ) ) )
5132, 44, 43, 46, 45dmdcan2d 10362 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( (π `  x )  x.  ( log `  x ) ) )  x.  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) )  =  ( x  /  ( theta `  x
) ) )
5251mpteq2ia 4535 . . . 4  |-  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( (π `  x )  x.  ( log `  x ) ) )  x.  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x ) ) )
5350, 52syl6eq 2524 . . 3  |-  ( T. 
->  ( ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  oF  x.  (
x  e.  ( 2 [,) +oo )  |->  ( 1  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x ) ) ) )
54 chebbnd1 23521 . . . 4  |-  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  e.  O(1)
55 ax-1cn 9562 . . . . . . 7  |-  1  e.  CC
5655a1i 11 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  e.  CC )
5728, 23rpdivcld 11285 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  e.  RR+ )
5857adantl 466 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
5958rpcnd 11270 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  CC )
607ssriv 3513 . . . . . . . 8  |-  ( 2 [,) +oo )  C_  RR
61 rlimconst 13346 . . . . . . . 8  |-  ( ( ( 2 [,) +oo )  C_  RR  /\  1  e.  CC )  ->  (
x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1 )
6260, 55, 61mp2an 672 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1
6362a1i 11 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1 )
64 chtppilim 23524 . . . . . . 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 9573 . . . . . . 7  |-  1  =/=  0
6766a1i 11 . . . . . 6  |-  ( T. 
->  1  =/=  0
)
6857rpne0d 11273 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  =/=  0 )
6968adantl 466 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
7056, 59, 63, 65, 67, 69rlimdiv 13447 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  ~~> r  ( 1  /  1 ) )
71 rlimo1 13418 . . . . 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) )
7270, 71syl 16 . . . 4  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  e.  O(1) )
73 o1mul 13416 . . . 4  |-  ( ( ( x  e.  ( 2 [,) +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  e.  O(1)  /\  ( x  e.  ( 2 [,) +oo )  |->  ( 1  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) ) )  e.  O(1) )  -> 
( ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  oF  x.  (
x  e.  ( 2 [,) +oo )  |->  ( 1  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) ) ) )  e.  O(1) )
7454, 72, 73sylancr 663 . . 3  |-  ( T. 
->  ( ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  oF  x.  (
x  e.  ( 2 [,) +oo )  |->  ( 1  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) ) ) )  e.  O(1) )
7553, 74eqeltrrd 2556 . 2  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x ) ) )  e.  O(1) )
7675trud 1388 1  |-  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x
) ) )  e.  O(1)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379   T. wtru 1380    e. wcel 1767    =/= wne 2662   _Vcvv 3118    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295    oFcof 6533   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    x. cmul 9509   +oocpnf 9637    < clt 9640    <_ cle 9641    / cdiv 10218   2c2 10597   RR+crp 11232   [,)cico 11543    ~~> r crli 13287   O(1)co1 13288   logclog 22806   thetaccht 23228  πcppi 23231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-o1 13292  df-lo1 13293  df-sum 13488  df-ef 13681  df-e 13682  df-sin 13683  df-cos 13684  df-pi 13686  df-dvds 13864  df-gcd 14020  df-prm 14093  df-pc 14236  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-hom 14595  df-cco 14596  df-rest 14694  df-topn 14695  df-0g 14713  df-gsum 14714  df-topgen 14715  df-pt 14716  df-prds 14719  df-xrs 14773  df-qtop 14778  df-imas 14779  df-xps 14781  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-submnd 15839  df-mulg 15931  df-cntz 16226  df-cmn 16671  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-fbas 18284  df-fg 18285  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-lp 19503  df-perf 19504  df-cn 19594  df-cnp 19595  df-haus 19682  df-tx 19929  df-hmeo 20122  df-fil 20213  df-fm 20305  df-flim 20306  df-flf 20307  df-xms 20689  df-ms 20690  df-tms 20691  df-cncf 21248  df-limc 22136  df-dv 22137  df-log 22808  df-cxp 22809  df-cht 23234  df-ppi 23237
This theorem is referenced by:  chpchtlim  23528
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