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Theorem chto1lb 20623
Description: The  theta function is lower bounded by a linear term. Corollary of chebbnd1 20617. (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 5845 . . . . . 6  |-  ( 2 [,)  +oo )  e.  _V
21a1i 10 . . . . 5  |-  (  T. 
->  ( 2 [,)  +oo )  e.  _V )
3 2re 9811 . . . . . . . . . . . 12  |-  2  e.  RR
4 elicopnf 10735 . . . . . . . . . . . 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 186 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  e.  RR  /\  2  <_  x ) )
76simpld 445 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR )
8 0re 8834 . . . . . . . . . . 11  |-  0  e.  RR
98a1i 10 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  e.  RR )
103a1i 10 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  e.  RR )
11 2pos 9824 . . . . . . . . . . 11  |-  0  <  2
1211a1i 10 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  2 )
136simprd 449 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  <_  x )
149, 10, 7, 12, 13ltletrd 8972 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  x )
157, 14elrpd 10384 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
16 ppinncl 20408 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
1716nnrpd 10385 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  RR+ )
186, 17syl 15 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  RR+ )
19 1re 8833 . . . . . . . . . . . 12  |-  1  e.  RR
2019a1i 10 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  e.  RR )
21 1lt2 9882 . . . . . . . . . . . 12  |-  1  <  2
2221a1i 10 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  <  2 )
2320, 10, 7, 22, 13ltletrd 8972 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  <  x )
247, 23rplogcld 19976 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  RR+ )
2518, 24rpmulcld 10402 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  RR+ )
2615, 25rpdivcld 10403 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( (π `  x )  x.  ( log `  x ) ) )  e.  RR+ )
2726rpcnd 10388 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( (π `  x )  x.  ( log `  x ) ) )  e.  CC )
2827adantl 452 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
x  /  ( (π `  x )  x.  ( log `  x ) ) )  e.  CC )
29 chtrpcl 20409 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
306, 29syl 15 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  e.  RR+ )
3125, 30rpdivcld 10403 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  RR+ )
3231rpcnd 10388 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  CC )
3332adantl 452 . . . . 5  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  CC )
347recnd 8857 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  CC )
3524rpcnd 10388 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  CC )
3618rpcnd 10388 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  CC )
3724rpne0d 10391 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  =/=  0 )
3818rpne0d 10391 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  =/=  0
)
3934, 35, 36, 37, 38divdiv1d 9563 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  =  ( x  / 
( ( log `  x
)  x.  (π `  x
) ) ) )
4035, 36mulcomd 8852 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( log `  x
)  x.  (π `  x
) )  =  ( (π `  x )  x.  ( log `  x
) ) )
4140oveq2d 5836 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( ( log `  x )  x.  (π `  x ) ) )  =  ( x  /  ( (π `  x
)  x.  ( log `  x ) ) ) )
4239, 41eqtrd 2316 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  =  ( x  / 
( (π `  x )  x.  ( log `  x
) ) ) )
4342mpteq2ia 4103 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  =  ( x  e.  ( 2 [,) 
+oo )  |->  ( x  /  ( (π `  x
)  x.  ( log `  x ) ) ) )
4443a1i 10 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( x  / 
( (π `  x )  x.  ( log `  x
) ) ) ) )
4530rpcnd 10388 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  e.  CC )
4625rpcnd 10388 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  CC )
4730rpne0d 10391 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  =/=  0 )
4825rpne0d 10391 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  =/=  0
)
4945, 46, 47, 48recdivd 9549 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
1  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )
5049mpteq2ia 4103 . . . . . 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 10 . . . . 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 6057 . . . 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 ) ) ) ) )
5327, 32mulcomd 8852 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  (
(π `  x )  x.  ( log `  x
) ) )  x.  ( ( (π `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) )  =  ( ( ( (π `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
)  x.  ( x  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )
5434, 46, 45, 48, 47dmdcand 9561 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( (π `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
)  x.  ( x  /  ( (π `  x
)  x.  ( log `  x ) ) ) )  =  ( x  /  ( theta `  x
) ) )
5553, 54eqtrd 2316 . . . . 5  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  (
(π `  x )  x.  ( log `  x
) ) )  x.  ( ( (π `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) )  =  ( x  /  ( theta `  x ) ) )
5655mpteq2ia 4103 . . . 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
) ) )
5752, 56syl6eq 2332 . . 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
) ) ) )
58 chebbnd1 20617 . . . . 5  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( ( x  /  ( log `  x ) )  / 
(π `  x ) ) )  e.  O ( 1 )
5958a1i 10 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( x  / 
( log `  x
) )  /  (π `  x ) ) )  e.  O ( 1 ) )
60 ax-1cn 8791 . . . . . . 7  |-  1  e.  CC
6160a1i 10 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  CC )
6230, 25rpdivcld 10403 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
6362adantl 452 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
6463rpcnd 10388 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  CC )
657ssriv 3185 . . . . . . . 8  |-  ( 2 [,)  +oo )  C_  RR
66 rlimconst 12014 . . . . . . . 8  |-  ( ( ( 2 [,)  +oo )  C_  RR  /\  1  e.  CC )  ->  (
x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
6765, 60, 66mp2an 653 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  1 )  ~~> r  1
6867a1i 10 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  1 )  ~~> r  1 )
69 chtppilim 20620 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  ~~> r  1
7069a1i 10 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1 )
71 ax-1ne0 8802 . . . . . . 7  |-  1  =/=  0
7271a1i 10 . . . . . 6  |-  (  T. 
->  1  =/=  0
)
7362rpne0d 10391 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
7473adantl 452 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
7561, 64, 68, 70, 72, 74rlimdiv 12115 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  ~~> r  ( 1  /  1 ) )
76 rlimo1 12086 . . . . 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 ) )
7775, 76syl 15 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  e.  O
( 1 ) )
78 o1mul 12084 . . . 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 ) )
7959, 77, 78syl2anc 642 . . 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 ) )
8057, 79eqeltrrd 2359 . 2  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( x  /  ( theta `  x ) ) )  e.  O ( 1 ) )
8180trud 1314 1  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( x  /  ( theta `  x
) ) )  e.  O ( 1 )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    T. wtru 1307    = wceq 1623    e. wcel 1685    =/= wne 2447   _Vcvv 2789    C_ wss 3153   class class class wbr 4024    e. cmpt 4078   ` cfv 5221  (class class class)co 5820    o Fcof 6038   CCcc 8731   RRcr 8732   0cc0 8733   1c1 8734    x. cmul 8738    +oocpnf 8860    < clt 8863    <_ cle 8864    / cdiv 9419   2c2 9791   RR+crp 10350   [,)cico 10654    ~~> r crli 11955   O (
1 )co1 11956   logclog 19908   thetaccht 20324  πcppi 20327
This theorem is referenced by:  chpchtlim  20624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-er 6656  df-map 6770  df-pm 6771  df-ixp 6814  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-fi 7161  df-sup 7190  df-oi 7221  df-card 7568  df-cda 7790  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-q 10313  df-rp 10351  df-xneg 10448  df-xadd 10449  df-xmul 10450  df-ioo 10656  df-ioc 10657  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101  df-fac 11285  df-bc 11312  df-hash 11334  df-shft 11558  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-limsup 11941  df-clim 11958  df-rlim 11959  df-o1 11960  df-lo1 11961  df-sum 12155  df-ef 12345  df-e 12346  df-sin 12347  df-cos 12348  df-pi 12350  df-dvds 12528  df-gcd 12682  df-prm 12755  df-pc 12886  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-starv 13219  df-sca 13220  df-vsca 13221  df-tset 13223  df-ple 13224  df-ds 13226  df-hom 13228  df-cco 13229  df-rest 13323  df-topn 13324  df-topgen 13340  df-pt 13341  df-prds 13344  df-xrs 13399  df-0g 13400  df-gsum 13401  df-qtop 13406  df-imas 13407  df-xps 13409  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-submnd 14412  df-mulg 14488  df-cntz 14789  df-cmn 15087  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cld 16752  df-ntr 16753  df-cls 16754  df-nei 16831  df-lp 16864  df-perf 16865  df-cn 16953  df-cnp 16954  df-haus 17039  df-tx 17253  df-hmeo 17442  df-fbas 17516  df-fg 17517  df-fil 17537  df-fm 17629  df-flim 17630  df-flf 17631  df-xms 17881  df-ms 17882  df-tms 17883  df-cncf 18378  df-limc 19212  df-dv 19213  df-log 19910  df-cxp 19911  df-cht 20330  df-ppi 20333
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