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Theorem chto1lb 20739
Description: The  theta function is lower bounded by a linear term. Corollary of chebbnd1 20733. (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 5970 . . . . . 6  |-  ( 2 [,)  +oo )  e.  _V
21a1i 10 . . . . 5  |-  (  T. 
->  ( 2 [,)  +oo )  e.  _V )
3 2re 9905 . . . . . . . . . . . 12  |-  2  e.  RR
4 elicopnf 10831 . . . . . . . . . . . 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 8928 . . . . . . . . . . 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 9918 . . . . . . . . . . 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 9066 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  0  <  x )
157, 14elrpd 10480 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
16 ppinncl 20524 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
1716nnrpd 10481 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  RR+ )
186, 17syl 15 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  RR+ )
19 1re 8927 . . . . . . . . . . . 12  |-  1  e.  RR
2019a1i 10 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  e.  RR )
21 1lt2 9978 . . . . . . . . . . . 12  |-  1  <  2
2221a1i 10 . . . . . . . . . . 11  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  <  2 )
2320, 10, 7, 22, 13ltletrd 9066 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  1  <  x )
247, 23rplogcld 20091 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  RR+ )
2518, 24rpmulcld 10498 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  RR+ )
2615, 25rpdivcld 10499 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( (π `  x )  x.  ( log `  x ) ) )  e.  RR+ )
2726rpcnd 10484 . . . . . 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 20525 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
306, 29syl 15 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  e.  RR+ )
3125, 30rpdivcld 10499 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( (π `  x )  x.  ( log `  x
) )  /  ( theta `  x ) )  e.  RR+ )
3231rpcnd 10484 . . . . . 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 8951 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  CC )
3524rpcnd 10484 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  CC )
3618rpcnd 10484 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  CC )
3724rpne0d 10487 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  =/=  0 )
3818rpne0d 10487 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  =/=  0
)
3934, 35, 36, 37, 38divdiv1d 9657 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  =  ( x  / 
( ( log `  x
)  x.  (π `  x
) ) ) )
4035, 36mulcomd 8946 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( log `  x
)  x.  (π `  x
) )  =  ( (π `  x )  x.  ( log `  x
) ) )
4140oveq2d 5961 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  /  ( ( log `  x )  x.  (π `  x ) ) )  =  ( x  /  ( (π `  x
)  x.  ( log `  x ) ) ) )
4239, 41eqtrd 2390 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  ( log `  x ) )  /  (π `  x ) )  =  ( x  / 
( (π `  x )  x.  ( log `  x
) ) ) )
4342mpteq2ia 4183 . . . . . 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 10484 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  e.  CC )
4625rpcnd 10484 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  CC )
4730rpne0d 10487 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  =/=  0 )
4825rpne0d 10487 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  =/=  0
)
4945, 46, 47, 48recdivd 9643 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
1  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  ( theta `  x
) ) )
5049mpteq2ia 4183 . . . . . 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 6182 . . . 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 8946 . . . . . 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 9655 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( (π `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
)  x.  ( x  /  ( (π `  x
)  x.  ( log `  x ) ) ) )  =  ( x  /  ( theta `  x
) ) )
5553, 54eqtrd 2390 . . . . 5  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( x  /  (
(π `  x )  x.  ( log `  x
) ) )  x.  ( ( (π `  x
)  x.  ( log `  x ) )  / 
( theta `  x )
) )  =  ( x  /  ( theta `  x ) ) )
5655mpteq2ia 4183 . . . 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 2406 . . 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 20733 . . . . 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 8885 . . . . . . 7  |-  1  e.  CC
6160a1i 10 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  1  e.  CC )
6230, 25rpdivcld 10499 . . . . . . . 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 10484 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  CC )
657ssriv 3260 . . . . . . . 8  |-  ( 2 [,)  +oo )  C_  RR
66 rlimconst 12114 . . . . . . . 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 20736 . . . . . . 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 8896 . . . . . . 7  |-  1  =/=  0
7271a1i 10 . . . . . 6  |-  (  T. 
->  1  =/=  0
)
7362rpne0d 10487 . . . . . . 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 12215 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( 1  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  ~~> r  ( 1  /  1 ) )
76 rlimo1 12186 . . . . 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 12184 . . . 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 2433 . 2  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( x  /  ( theta `  x ) ) )  e.  O ( 1 ) )
8180trud 1323 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 1316    = wceq 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864    C_ wss 3228   class class class wbr 4104    e. cmpt 4158   ` cfv 5337  (class class class)co 5945    o Fcof 6163   CCcc 8825   RRcr 8826   0cc0 8827   1c1 8828    x. cmul 8832    +oocpnf 8954    < clt 8957    <_ cle 8958    / cdiv 9513   2c2 9885   RR+crp 10446   [,)cico 10750    ~~> r crli 12055   O (
1 )co1 12056   logclog 20019   thetaccht 20440  πcppi 20443
This theorem is referenced by:  chpchtlim  20740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-fi 7255  df-sup 7284  df-oi 7315  df-card 7662  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-q 10409  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-ioo 10752  df-ioc 10753  df-ico 10754  df-icc 10755  df-fz 10875  df-fzo 10963  df-fl 11017  df-mod 11066  df-seq 11139  df-exp 11198  df-fac 11382  df-bc 11409  df-hash 11431  df-shft 11658  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-limsup 12041  df-clim 12058  df-rlim 12059  df-o1 12060  df-lo1 12061  df-sum 12256  df-ef 12446  df-e 12447  df-sin 12448  df-cos 12449  df-pi 12451  df-dvds 12629  df-gcd 12783  df-prm 12856  df-pc 12987  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-hom 13329  df-cco 13330  df-rest 13426  df-topn 13427  df-topgen 13443  df-pt 13444  df-prds 13447  df-xrs 13502  df-0g 13503  df-gsum 13504  df-qtop 13509  df-imas 13510  df-xps 13512  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-submnd 14515  df-mulg 14591  df-cntz 14892  df-cmn 15190  df-xmet 16475  df-met 16476  df-bl 16477  df-mopn 16478  df-fbas 16479  df-fg 16480  df-cnfld 16483  df-top 16742  df-bases 16744  df-topon 16745  df-topsp 16746  df-cld 16862  df-ntr 16863  df-cls 16864  df-nei 16941  df-lp 16974  df-perf 16975  df-cn 17063  df-cnp 17064  df-haus 17149  df-tx 17363  df-hmeo 17552  df-fil 17643  df-fm 17735  df-flim 17736  df-flf 17737  df-xms 17987  df-ms 17988  df-tms 17989  df-cncf 18485  df-limc 19320  df-dv 19321  df-log 20021  df-cxp 20022  df-cht 20446  df-ppi 20449
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