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Theorem chto1lb 21160
Description: The  theta function is lower bounded by a linear term. Corollary of chebbnd1 21154. (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 6097 . . . . . 6  |-  ( 2 [,)  +oo )  e.  _V
21a1i 11 . . . . 5  |-  (  T. 
->  ( 2 [,)  +oo )  e.  _V )
3 2re 10058 . . . . . . . . . . . 12  |-  2  e.  RR
4 elicopnf 10989 . . . . . . . . . . . 12  |-  ( 2  e.  RR  ->  (
x  e.  ( 2 [,)  +oo )  <->  ( x  e.  RR  /\  2  <_  x ) ) )
53, 4ax-mp 8 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  <->  ( x  e.  RR  /\  2  <_  x ) )
65biimpi 187 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  e.  RR  /\  2  <_  x ) )
76simpld 446 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR )
8 0re 9080 . . . . . . . . . . 11  |-  0  e.  RR
98a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  e.  RR )
103a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  e.  RR )
11 2pos 10071 . . . . . . . . . . 11  |-  0  <  2
1211a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  2 )
136simprd 450 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  <_  x )
149, 10, 7, 12, 13ltletrd 9219 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  x )
157, 14elrpd 10635 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
16 ppinncl 20945 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
1716nnrpd 10636 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  RR+ )
186, 17syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  RR+ )
19 1re 9079 . . . . . . . . . . . 12  |-  1  e.  RR
2019a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  e.  RR )
21 1lt2 10131 . . . . . . . . . . . 12  |-  1  <  2
2221a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  <  2 )
2320, 10, 7, 22, 13ltletrd 9219 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  <  x )
247, 23rplogcld 20512 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  RR+ )
2518, 24rpmulcld 10653 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  RR+ )
2615, 25rpdivcld 10654 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( (π `  x )  x.  ( log `  x ) ) )  e.  RR+ )
2726rpcnd 10639 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( (π `  x )  x.  ( log `  x ) ) )  e.  CC )
2827adantl 453 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  /  ( (π `  x )  x.  ( log `  x ) ) )  e.  CC )
29 chtrpcl 20946 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
306, 29syl 16 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  e.  RR+ )
3125, 30rpdivcld 10654 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  RR+ )
3231rpcnd 10639 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  CC )
3332adantl 453 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  CC )
347recnd 9103 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  CC )
3524rpcnd 10639 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  CC )
3618rpcnd 10639 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  CC )
3724rpne0d 10642 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  =/=  0 )
3818rpne0d 10642 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  =/=  0
)
3934, 35, 36, 37, 38divdiv1d 9810 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  =  ( x  / 
( ( log `  x
)  x.  (π `  x
) ) ) )
4035, 36mulcomd 9098 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( log `  x
)  x.  (π `  x
) )  =  ( (π `  x )  x.  ( log `  x
) ) )
4140oveq2d 6088 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( ( log `  x )  x.  (π `  x ) ) )  =  ( x  /  ( (π `  x
)  x.  ( log `  x ) ) ) )
4239, 41eqtrd 2467 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  =  ( x  / 
( (π `  x )  x.  ( log `  x
) ) ) )
4342mpteq2ia 4283 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  =  ( x  e.  ( 2 [,) 
+oo )  |->  ( x  /  ( (π `  x
)  x.  ( log `  x ) ) ) )
4443a1i 11 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( x  / 
( (π `  x )  x.  ( log `  x
) ) ) ) )
4530rpcnd 10639 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  e.  CC )
4625rpcnd 10639 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  CC )
4730rpne0d 10642 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  =/=  0 )
4825rpne0d 10642 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  =/=  0
)
4945, 46, 47, 48recdivd 9796 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
1  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )
5049mpteq2ia 4283 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( 1  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) ) )  =  ( x  e.  ( 2 [,) 
+oo )  |->  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) )
5150a1i 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 ) ) ) )
522, 28, 33, 44, 51offval2 6313 . . . 4  |-  (  T. 
->  ( ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  o F  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 ) ) ) ) )
5334, 46, 45, 48, 47dmdcan2d 9809 . . . . 5  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  (
(π `  x )  x.  ( log `  x
) ) )  x.  ( ( (π `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) )  =  ( x  /  ( theta `  x ) ) )
5453mpteq2ia 4283 . . . 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
) ) )
5552, 54syl6eq 2483 . . 3  |-  (  T. 
->  ( ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  o F  x.  (
x  e.  ( 2 [,)  +oo )  |->  ( 1  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( x  /  ( theta `  x
) ) ) )
56 chebbnd1 21154 . . . 4  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  e.  O ( 1 )
57 ax-1cn 9037 . . . . . . 7  |-  1  e.  CC
5857a1i 11 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  CC )
5930, 25rpdivcld 10654 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
6059adantl 453 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
6160rpcnd 10639 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  CC )
627ssriv 3344 . . . . . . . 8  |-  ( 2 [,)  +oo )  C_  RR
63 rlimconst 12326 . . . . . . . 8  |-  ( ( ( 2 [,)  +oo )  C_  RR  /\  1  e.  CC )  ->  (
x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
6462, 57, 63mp2an 654 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  1 )  ~~> r  1
6564a1i 11 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
66 chtppilim 21157 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  ~~> r  1
6766a1i 11 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1 )
68 ax-1ne0 9048 . . . . . . 7  |-  1  =/=  0
6968a1i 11 . . . . . 6  |-  (  T. 
->  1  =/=  0
)
7059rpne0d 10642 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
7170adantl 453 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
7258, 61, 65, 67, 69, 71rlimdiv 12427 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  ~~> r  ( 1  /  1 ) )
73 rlimo1 12398 . . . . 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 ) )
7472, 73syl 16 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  e.  O
( 1 ) )
75 o1mul 12396 . . . 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 ) ) )  o F  x.  (
x  e.  ( 2 [,)  +oo )  |->  ( 1  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) ) ) )  e.  O
( 1 ) )
7656, 74, 75sylancr 645 . . 3  |-  (  T. 
->  ( ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  /  ( log `  x
) )  /  (π `  x ) ) )  o F  x.  (
x  e.  ( 2 [,)  +oo )  |->  ( 1  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) ) ) )  e.  O
( 1 ) )
7755, 76eqeltrrd 2510 . 2  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( x  /  ( theta `  x ) ) )  e.  O ( 1 ) )
7877trud 1332 1  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( x  /  ( theta `  x
) ) )  e.  O ( 1 )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    T. wtru 1325    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   ` cfv 5445  (class class class)co 6072    o Fcof 6294   CCcc 8977   RRcr 8978   0cc0 8979   1c1 8980    x. cmul 8984    +oocpnf 9106    < clt 9109    <_ cle 9110    / cdiv 9666   2c2 10038   RR+crp 10601   [,)cico 10907    ~~> r crli 12267   O (
1 )co1 12268   logclog 20440   thetaccht 20861  πcppi 20864
This theorem is referenced by:  chpchtlim  21161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058  ax-mulf 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-sup 7437  df-oi 7468  df-card 7815  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ioo 10909  df-ioc 10910  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-fl 11190  df-mod 11239  df-seq 11312  df-exp 11371  df-fac 11555  df-bc 11582  df-hash 11607  df-shft 11870  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-limsup 12253  df-clim 12270  df-rlim 12271  df-o1 12272  df-lo1 12273  df-sum 12468  df-ef 12658  df-e 12659  df-sin 12660  df-cos 12661  df-pi 12663  df-dvds 12841  df-gcd 12995  df-prm 13068  df-pc 13199  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-starv 13532  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-unif 13540  df-hom 13541  df-cco 13542  df-rest 13638  df-topn 13639  df-topgen 13655  df-pt 13656  df-prds 13659  df-xrs 13714  df-0g 13715  df-gsum 13716  df-qtop 13721  df-imas 13722  df-xps 13724  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-cntz 15104  df-cmn 15402  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-cnfld 16692  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-lp 17188  df-perf 17189  df-cn 17279  df-cnp 17280  df-haus 17367  df-tx 17582  df-hmeo 17775  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-xms 18338  df-ms 18339  df-tms 18340  df-cncf 18896  df-limc 19741  df-dv 19742  df-log 20442  df-cxp 20443  df-cht 20867  df-ppi 20870
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