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Theorem pnt 20765
Description: The Prime Number Theorem: the number of prime numbers less than  x tends asymptotically to  x  /  log (
x ) as  x goes to infinity. (Contributed by Mario Carneiro, 1-Jun-2016.)
Assertion
Ref Expression
pnt  |-  ( x  e.  ( 1 (,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1

Proof of Theorem pnt
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1re 8839 . . . . . . 7  |-  1  e.  RR
2 rexr 8879 . . . . . . 7  |-  ( 1  e.  RR  ->  1  e.  RR* )
31, 2ax-mp 8 . . . . . 6  |-  1  e.  RR*
4 1lt2 9888 . . . . . 6  |-  1  <  2
5 df-ioo 10662 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
6 df-ico 10664 . . . . . . 7  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
7 xrltletr 10490 . . . . . . 7  |-  ( ( 1  e.  RR*  /\  2  e.  RR*  /\  w  e. 
RR* )  ->  (
( 1  <  2  /\  2  <_  w )  ->  1  <  w
) )
85, 6, 7ixxss1 10676 . . . . . 6  |-  ( ( 1  e.  RR*  /\  1  <  2 )  ->  (
2 [,)  +oo )  C_  ( 1 (,)  +oo ) )
93, 4, 8mp2an 653 . . . . 5  |-  ( 2 [,)  +oo )  C_  (
1 (,)  +oo )
10 resmpt 5002 . . . . 5  |-  ( ( 2 [,)  +oo )  C_  ( 1 (,)  +oo )  ->  ( ( x  e.  ( 1 (,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  |`  ( 2 [,)  +oo ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) ) )
119, 10mp1i 11 . . . 4  |-  (  T. 
->  ( ( x  e.  ( 1 (,)  +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,)  +oo ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) ) )
129sseli 3178 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  ( 1 (,)  +oo ) )
13 ioossre 10714 . . . . . . . . . . 11  |-  ( 1 (,)  +oo )  C_  RR
1413sseli 3178 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  x  e.  RR )
1512, 14syl 15 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR )
16 2re 9817 . . . . . . . . . . 11  |-  2  e.  RR
17 pnfxr 10457 . . . . . . . . . . 11  |-  +oo  e.  RR*
18 elico2 10716 . . . . . . . . . . 11  |-  ( ( 2  e.  RR  /\  +oo 
e.  RR* )  ->  (
x  e.  ( 2 [,)  +oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  <  +oo ) ) )
1916, 17, 18mp2an 653 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  <  +oo ) )
2019simp2bi 971 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  <_  x )
21 chtrpcl 20415 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
2215, 20, 21syl2anc 642 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  e.  RR+ )
23 0re 8840 . . . . . . . . . . . 12  |-  0  e.  RR
2423a1i 10 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  0  e.  RR )
251a1i 10 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  1  e.  RR )
26 0lt1 9298 . . . . . . . . . . . 12  |-  0  <  1
2726a1i 10 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  0  <  1 )
28 eliooord 10712 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
1  <  x  /\  x  <  +oo ) )
2928simpld 445 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  1  <  x )
3024, 25, 14, 27, 29lttrd 8979 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  0  <  x )
3114, 30elrpd 10390 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) 
+oo )  ->  x  e.  RR+ )
3212, 31syl 15 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
3322, 32rpdivcld 10409 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( theta `  x )  /  x )  e.  RR+ )
3433adantl 452 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  x )  e.  RR+ )
35 ppinncl 20414 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
3615, 20, 35syl2anc 642 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  NN )
3736nnrpd 10391 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  RR+ )
3814, 29rplogcld 19982 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  ( log `  x )  e.  RR+ )
3912, 38syl 15 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  RR+ )
4037, 39rpmulcld 10408 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  RR+ )
4122, 40rpdivcld 10409 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
4241adantl 452 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
4332ssriv 3186 . . . . . . . 8  |-  ( 2 [,)  +oo )  C_  RR+
44 resmpt 5002 . . . . . . . 8  |-  ( ( 2 [,)  +oo )  C_  RR+  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,)  +oo ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( (
theta `  x )  /  x ) ) )
4543, 44ax-mp 8 . . . . . . 7  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,)  +oo ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( (
theta `  x )  /  x ) )
46 pnt2 20764 . . . . . . . 8  |-  ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1
47 rlimres 12034 . . . . . . . 8  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,)  +oo ) )  ~~> r  1 )
4846, 47mp1i 11 . . . . . . 7  |-  (  T. 
->  ( ( x  e.  RR+  |->  ( ( theta `  x )  /  x
) )  |`  (
2 [,)  +oo ) )  ~~> r  1 )
4945, 48syl5eqbrr 4059 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  x ) )  ~~> r  1 )
50 chtppilim 20626 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  ~~> r  1
5150a1i 10 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1 )
52 ax-1ne0 8808 . . . . . . 7  |-  1  =/=  0
5352a1i 10 . . . . . 6  |-  (  T. 
->  1  =/=  0
)
5441rpne0d 10397 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
5554adantl 452 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
5634, 42, 49, 51, 53, 55rlimdiv 12121 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( ( theta `  x )  /  x
)  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) ) )  ~~> r  ( 1  /  1 ) )
5715recnd 8863 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  CC )
58 chtcl 20349 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  ( theta `  x )  e.  RR )
5914, 58syl 15 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) 
+oo )  ->  ( theta `  x )  e.  RR )
6059recnd 8863 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  ( theta `  x )  e.  CC )
6112, 60syl 15 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  e.  CC )
6257, 61mulcomd 8858 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  x.  ( theta `  x ) )  =  ( ( theta `  x
)  x.  x ) )
6362oveq2d 5876 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( theta `  x
)  x.  ( (π `  x )  x.  ( log `  x ) ) )  /  ( x  x.  ( theta `  x
) ) )  =  ( ( ( theta `  x )  x.  (
(π `  x )  x.  ( log `  x
) ) )  / 
( ( theta `  x
)  x.  x ) ) )
6440rpcnd 10394 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  CC )
6532rpne0d 10397 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  =/=  0 )
6622rpne0d 10397 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  =/=  0 )
6764, 57, 61, 65, 66divcan5d 9564 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( theta `  x
)  x.  ( (π `  x )  x.  ( log `  x ) ) )  /  ( (
theta `  x )  x.  x ) )  =  ( ( (π `  x
)  x.  ( log `  x ) )  /  x ) )
6863, 67eqtrd 2317 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( theta `  x
)  x.  ( (π `  x )  x.  ( log `  x ) ) )  /  ( x  x.  ( theta `  x
) ) )  =  ( ( (π `  x
)  x.  ( log `  x ) )  /  x ) )
6940rpne0d 10397 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  =/=  0
)
7061, 57, 61, 64, 65, 69, 66divdivdivd 9585 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( theta `  x
)  /  x )  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) )  =  ( ( (
theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) ) )
7136nncnd 9764 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  CC )
7239rpcnd 10394 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  CC )
7339rpne0d 10397 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  =/=  0 )
7471, 57, 72, 65, 73divdiv2d 9570 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
7568, 70, 743eqtr4d 2327 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( theta `  x
)  /  x )  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) )  =  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )
7675mpteq2ia 4104 . . . . 5  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
77 ax-1cn 8797 . . . . . 6  |-  1  e.  CC
7877div1i 9490 . . . . 5  |-  ( 1  /  1 )  =  1
7956, 76, 783brtr3g 4056 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
8011, 79eqbrtrd 4045 . . 3  |-  (  T. 
->  ( ( x  e.  ( 1 (,)  +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,)  +oo ) )  ~~> r  1 )
81 ppicl 20371 . . . . . . . . . 10  |-  ( x  e.  RR  ->  (π `  x )  e.  NN0 )
8214, 81syl 15 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (π `  x )  e.  NN0 )
8382nn0red 10021 . . . . . . . 8  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (π `  x )  e.  RR )
8431, 38rpdivcld 10409 . . . . . . . 8  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
8583, 84rerpdivcld 10419 . . . . . . 7  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  RR )
8685recnd 8863 . . . . . 6  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  CC )
8786adantl 452 . . . . 5  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  CC )
88 eqid 2285 . . . . 5  |-  ( x  e.  ( 1 (,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  =  ( x  e.  ( 1 (,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
8987, 88fmptd 5686 . . . 4  |-  (  T. 
->  ( x  e.  ( 1 (,)  +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) ) : ( 1 (,)  +oo ) --> CC )
9013a1i 10 . . . 4  |-  (  T. 
->  ( 1 (,)  +oo )  C_  RR )
9116a1i 10 . . . 4  |-  (  T. 
->  2  e.  RR )
9289, 90, 91rlimresb 12041 . . 3  |-  (  T. 
->  ( ( x  e.  ( 1 (,)  +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  ~~> r  1  <->  ( (
x  e.  ( 1 (,)  +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  |`  ( 2 [,)  +oo ) )  ~~> r  1 ) )
9380, 92mpbird 223 . 2  |-  (  T. 
->  ( x  e.  ( 1 (,)  +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
9493trud 1314 1  |-  ( x  e.  ( 1 (,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ w3a 934    T. wtru 1307    = wceq 1625    e. wcel 1686    =/= wne 2448    C_ wss 3154   class class class wbr 4025    e. cmpt 4079    |` cres 4693   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    x. cmul 8744    +oocpnf 8866   RR*cxr 8868    < clt 8869    <_ cle 8870    / cdiv 9425   NNcn 9748   2c2 9797   NN0cn0 9967   RR+crp 10356   (,)cioo 10658   [,)cico 10660    ~~> r crli 11961   logclog 19914   thetaccht 20330  πcppi 20333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-disj 3996  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-shft 11564  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-o1 11966  df-lo1 11967  df-sum 12161  df-ef 12351  df-e 12352  df-sin 12353  df-cos 12354  df-pi 12356  df-dvds 12534  df-gcd 12688  df-prm 12761  df-pc 12892  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-cmp 17116  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-limc 19218  df-dv 19219  df-log 19916  df-cxp 19917  df-em 20289  df-cht 20336  df-vma 20337  df-chp 20338  df-ppi 20339  df-mu 20340
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