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Theorem pnt 21261
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 9046 . . . . . . 7  |-  1  e.  RR
21rexri 9093 . . . . . 6  |-  1  e.  RR*
3 1lt2 10098 . . . . . 6  |-  1  <  2
4 df-ioo 10876 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
5 df-ico 10878 . . . . . . 7  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
6 xrltletr 10703 . . . . . . 7  |-  ( ( 1  e.  RR*  /\  2  e.  RR*  /\  w  e. 
RR* )  ->  (
( 1  <  2  /\  2  <_  w )  ->  1  <  w
) )
74, 5, 6ixxss1 10890 . . . . . 6  |-  ( ( 1  e.  RR*  /\  1  <  2 )  ->  (
2 [,)  +oo )  C_  ( 1 (,)  +oo ) )
82, 3, 7mp2an 654 . . . . 5  |-  ( 2 [,)  +oo )  C_  (
1 (,)  +oo )
9 resmpt 5150 . . . . 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 ) ) ) ) )
108, 9mp1i 12 . . . 4  |-  (  T. 
->  ( ( x  e.  ( 1 (,)  +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,)  +oo ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) ) )
118sseli 3304 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  ( 1 (,)  +oo ) )
12 ioossre 10928 . . . . . . . . . . 11  |-  ( 1 (,)  +oo )  C_  RR
1312sseli 3304 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  x  e.  RR )
1411, 13syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR )
15 2re 10025 . . . . . . . . . . 11  |-  2  e.  RR
16 pnfxr 10669 . . . . . . . . . . 11  |-  +oo  e.  RR*
17 elico2 10930 . . . . . . . . . . 11  |-  ( ( 2  e.  RR  /\  +oo 
e.  RR* )  ->  (
x  e.  ( 2 [,)  +oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  <  +oo ) ) )
1815, 16, 17mp2an 654 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  <  +oo ) )
1918simp2bi 973 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  <_  x )
20 chtrpcl 20911 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
2114, 19, 20syl2anc 643 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  e.  RR+ )
22 0re 9047 . . . . . . . . . . . 12  |-  0  e.  RR
2322a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  0  e.  RR )
241a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  1  e.  RR )
25 0lt1 9506 . . . . . . . . . . . 12  |-  0  <  1
2625a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  0  <  1 )
27 eliooord 10926 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
1  <  x  /\  x  <  +oo ) )
2827simpld 446 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  1  <  x )
2923, 24, 13, 26, 28lttrd 9187 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  0  <  x )
3013, 29elrpd 10602 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) 
+oo )  ->  x  e.  RR+ )
3111, 30syl 16 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
3221, 31rpdivcld 10621 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( theta `  x )  /  x )  e.  RR+ )
3332adantl 453 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  x )  e.  RR+ )
34 ppinncl 20910 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
3514, 19, 34syl2anc 643 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  NN )
3635nnrpd 10603 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  RR+ )
3713, 28rplogcld 20477 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  ( log `  x )  e.  RR+ )
3811, 37syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  RR+ )
3936, 38rpmulcld 10620 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  RR+ )
4021, 39rpdivcld 10621 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
4140adantl 453 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
4231ssriv 3312 . . . . . . . 8  |-  ( 2 [,)  +oo )  C_  RR+
43 resmpt 5150 . . . . . . . 8  |-  ( ( 2 [,)  +oo )  C_  RR+  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,)  +oo ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( (
theta `  x )  /  x ) ) )
4442, 43ax-mp 8 . . . . . . 7  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,)  +oo ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( (
theta `  x )  /  x ) )
45 pnt2 21260 . . . . . . . 8  |-  ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1
46 rlimres 12307 . . . . . . . 8  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,)  +oo ) )  ~~> r  1 )
4745, 46mp1i 12 . . . . . . 7  |-  (  T. 
->  ( ( x  e.  RR+  |->  ( ( theta `  x )  /  x
) )  |`  (
2 [,)  +oo ) )  ~~> r  1 )
4844, 47syl5eqbrr 4206 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  x ) )  ~~> r  1 )
49 chtppilim 21122 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  ~~> r  1
5049a1i 11 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1 )
51 ax-1ne0 9015 . . . . . . 7  |-  1  =/=  0
5251a1i 11 . . . . . 6  |-  (  T. 
->  1  =/=  0
)
5340rpne0d 10609 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
5453adantl 453 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
5533, 41, 48, 50, 52, 54rlimdiv 12394 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( ( theta `  x )  /  x
)  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) ) )  ~~> r  ( 1  /  1 ) )
5614recnd 9070 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  CC )
57 chtcl 20845 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  ( theta `  x )  e.  RR )
5813, 57syl 16 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) 
+oo )  ->  ( theta `  x )  e.  RR )
5958recnd 9070 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  ( theta `  x )  e.  CC )
6011, 59syl 16 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  e.  CC )
6156, 60mulcomd 9065 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  x.  ( theta `  x ) )  =  ( ( theta `  x
)  x.  x ) )
6261oveq2d 6056 . . . . . . . 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 ) ) )
6339rpcnd 10606 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  CC )
6431rpne0d 10609 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  =/=  0 )
6521rpne0d 10609 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  =/=  0 )
6663, 56, 60, 64, 65divcan5d 9772 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( theta `  x
)  x.  ( (π `  x )  x.  ( log `  x ) ) )  /  ( (
theta `  x )  x.  x ) )  =  ( ( (π `  x
)  x.  ( log `  x ) )  /  x ) )
6762, 66eqtrd 2436 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( theta `  x
)  x.  ( (π `  x )  x.  ( log `  x ) ) )  /  ( x  x.  ( theta `  x
) ) )  =  ( ( (π `  x
)  x.  ( log `  x ) )  /  x ) )
6839rpne0d 10609 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  =/=  0
)
6960, 56, 60, 63, 64, 68, 65divdivdivd 9793 . . . . . . 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 ) ) ) )
7035nncnd 9972 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  CC )
7138rpcnd 10606 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  CC )
7238rpne0d 10609 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  =/=  0 )
7370, 56, 71, 64, 72divdiv2d 9778 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
7467, 69, 733eqtr4d 2446 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( theta `  x
)  /  x )  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) )  =  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )
7574mpteq2ia 4251 . . . . 5  |-  ( x  e.  ( 2 [,) 
+oo )  |->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  =  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
76 ax-1cn 9004 . . . . . 6  |-  1  e.  CC
7776div1i 9698 . . . . 5  |-  ( 1  /  1 )  =  1
7855, 75, 773brtr3g 4203 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
7910, 78eqbrtrd 4192 . . 3  |-  (  T. 
->  ( ( x  e.  ( 1 (,)  +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,)  +oo ) )  ~~> r  1 )
80 ppicl 20867 . . . . . . . . . 10  |-  ( x  e.  RR  ->  (π `  x )  e.  NN0 )
8113, 80syl 16 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (π `  x )  e.  NN0 )
8281nn0red 10231 . . . . . . . 8  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (π `  x )  e.  RR )
8330, 37rpdivcld 10621 . . . . . . . 8  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
8482, 83rerpdivcld 10631 . . . . . . 7  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  RR )
8584recnd 9070 . . . . . 6  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  CC )
8685adantl 453 . . . . 5  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  CC )
87 eqid 2404 . . . . 5  |-  ( x  e.  ( 1 (,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  =  ( x  e.  ( 1 (,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
8886, 87fmptd 5852 . . . 4  |-  (  T. 
->  ( x  e.  ( 1 (,)  +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) ) : ( 1 (,)  +oo ) --> CC )
8912a1i 11 . . . 4  |-  (  T. 
->  ( 1 (,)  +oo )  C_  RR )
9015a1i 11 . . . 4  |-  (  T. 
->  2  e.  RR )
9188, 89, 90rlimresb 12314 . . 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 ) )
9279, 91mpbird 224 . 2  |-  (  T. 
->  ( x  e.  ( 1 (,)  +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
9392trud 1329 1  |-  ( x  e.  ( 1 (,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ w3a 936    T. wtru 1322    = wceq 1649    e. wcel 1721    =/= wne 2567    C_ wss 3280   class class class wbr 4172    e. cmpt 4226    |` cres 4839   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    +oocpnf 9073   RR*cxr 9075    < clt 9076    <_ cle 9077    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   RR+crp 10568   (,)cioo 10872   [,)cico 10874    ~~> r crli 12234   logclog 20405   thetaccht 20826  πcppi 20829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-o1 12239  df-lo1 12240  df-sum 12435  df-ef 12625  df-e 12626  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-em 20784  df-cht 20832  df-vma 20833  df-chp 20834  df-ppi 20835  df-mu 20836
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