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Theorem chto1lb 22686
Description: The  theta function is lower bounded by a linear term. Corollary of chebbnd1 22680. (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 6115 . . . . . 6  |-  ( 2 [,) +oo )  e. 
_V
21a1i 11 . . . . 5  |-  ( T. 
->  ( 2 [,) +oo )  e.  _V )
3 2re 10387 . . . . . . . . . . . 12  |-  2  e.  RR
4 elicopnf 11381 . . . . . . . . . . . 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 456 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR )
8 0red 9383 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  0  e.  RR )
93a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  2  e.  RR )
10 2pos 10409 . . . . . . . . . . 11  |-  0  <  2
1110a1i 11 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  0  <  2 )
126simprd 460 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  2  <_  x )
138, 9, 7, 11, 12ltletrd 9527 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  0  < 
x )
147, 13elrpd 11021 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
15 ppinncl 22471 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
1615nnrpd 11022 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  RR+ )
176, 16syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  RR+ )
18 1red 9397 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  1  e.  RR )
19 1lt2 10484 . . . . . . . . . . . 12  |-  1  <  2
2019a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) +oo )  ->  1  <  2 )
2118, 9, 7, 20, 12ltletrd 9527 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  1  < 
x )
227, 21rplogcld 22037 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  RR+ )
2317, 22rpmulcld 11039 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  RR+ )
2414, 23rpdivcld 11040 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
2524rpcnd 11025 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  CC )
2625adantl 463 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
x  /  ( (π `  x )  x.  ( log `  x ) ) )  e.  CC )
27 chtrpcl 22472 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
286, 27syl 16 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  RR+ )
2923, 28rpdivcld 11040 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  RR+ )
3029rpcnd 11025 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  CC )
3130adantl 463 . . . . 5  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  CC )
327recnd 9408 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  CC )
3322rpcnd 11025 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  CC )
3417rpcnd 11025 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  CC )
3522rpne0d 11028 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  =/=  0
)
3617rpne0d 11028 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  =/=  0 )
3732, 33, 34, 35, 36divdiv1d 10134 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  =  ( x  / 
( ( log `  x
)  x.  (π `  x
) ) ) )
3833, 34mulcomd 9403 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( log `  x )  x.  (π `  x ) )  =  ( (π `  x
)  x.  ( log `  x ) ) )
3938oveq2d 6106 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  /  ( ( log `  x )  x.  (π `  x ) ) )  =  ( x  / 
( (π `  x )  x.  ( log `  x
) ) ) )
4037, 39eqtrd 2473 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) )  =  ( x  / 
( (π `  x )  x.  ( log `  x
) ) ) )
4140mpteq2ia 4371 . . . . . 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 11025 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  CC )
4423rpcnd 11025 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  CC )
4528rpne0d 11028 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  =/=  0
)
4623rpne0d 11028 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  =/=  0 )
4743, 44, 45, 46recdivd 10120 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( 1  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )
4847mpteq2ia 4371 . . . . . 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 6335 . . . 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 10133 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( x  /  ( (π `  x )  x.  ( log `  x ) ) )  x.  ( ( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) ) )  =  ( x  /  ( theta `  x
) ) )
5251mpteq2ia 4371 . . . 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 2489 . . 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 22680 . . . 4  |-  ( x  e.  ( 2 [,) +oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  e.  O(1)
55 ax-1cn 9336 . . . . . . 7  |-  1  e.  CC
5655a1i 11 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  1  e.  CC )
5728, 23rpdivcld 11040 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  e.  RR+ )
5857adantl 463 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
5958rpcnd 11025 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  CC )
607ssriv 3357 . . . . . . . 8  |-  ( 2 [,) +oo )  C_  RR
61 rlimconst 13018 . . . . . . . 8  |-  ( ( ( 2 [,) +oo )  C_  RR  /\  1  e.  CC )  ->  (
x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1 )
6260, 55, 61mp2an 667 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1
6362a1i 11 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  1 )  ~~> r  1 )
64 chtppilim 22683 . . . . . . 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 9347 . . . . . . 7  |-  1  =/=  0
6766a1i 11 . . . . . 6  |-  ( T. 
->  1  =/=  0
)
6857rpne0d 11028 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  =/=  0 )
6968adantl 463 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
7056, 59, 63, 65, 67, 69rlimdiv 13119 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  ~~> r  ( 1  /  1 ) )
71 rlimo1 13090 . . . . 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 13088 . . . 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 658 . . 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 2516 . 2  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x ) ) )  e.  O(1) )
7675trud 1373 1  |-  ( x  e.  ( 2 [,) +oo )  |->  ( x  /  ( theta `  x
) ) )  e.  O(1)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1364   T. wtru 1365    e. wcel 1761    =/= wne 2604   _Vcvv 2970    C_ wss 3325   class class class wbr 4289    e. cmpt 4347   ` cfv 5415  (class class class)co 6090    oFcof 6317   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    x. cmul 9283   +oocpnf 9411    < clt 9414    <_ cle 9415    / cdiv 9989   2c2 10367   RR+crp 10987   [,)cico 11298    ~~> r crli 12959   O(1)co1 12960   logclog 21965   thetaccht 22387  πcppi 22390
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-o1 12964  df-lo1 12965  df-sum 13160  df-ef 13349  df-e 13350  df-sin 13351  df-cos 13352  df-pi 13354  df-dvds 13532  df-gcd 13687  df-prm 13760  df-pc 13900  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967  df-cxp 21968  df-cht 22393  df-ppi 22396
This theorem is referenced by:  chpchtlim  22687
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