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Theorem pnt 21310
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 9092 . . . . . . 7  |-  1  e.  RR
21rexri 9139 . . . . . 6  |-  1  e.  RR*
3 1lt2 10144 . . . . . 6  |-  1  <  2
4 df-ioo 10922 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
5 df-ico 10924 . . . . . . 7  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
6 xrltletr 10749 . . . . . . 7  |-  ( ( 1  e.  RR*  /\  2  e.  RR*  /\  w  e. 
RR* )  ->  (
( 1  <  2  /\  2  <_  w )  ->  1  <  w
) )
74, 5, 6ixxss1 10936 . . . . . 6  |-  ( ( 1  e.  RR*  /\  1  <  2 )  ->  (
2 [,)  +oo )  C_  ( 1 (,)  +oo ) )
82, 3, 7mp2an 655 . . . . 5  |-  ( 2 [,)  +oo )  C_  (
1 (,)  +oo )
9 resmpt 5193 . . . . 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 3346 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  ( 1 (,)  +oo ) )
12 ioossre 10974 . . . . . . . . . . 11  |-  ( 1 (,)  +oo )  C_  RR
1312sseli 3346 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  x  e.  RR )
1411, 13syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR )
15 2re 10071 . . . . . . . . . . 11  |-  2  e.  RR
16 pnfxr 10715 . . . . . . . . . . 11  |-  +oo  e.  RR*
17 elico2 10976 . . . . . . . . . . 11  |-  ( ( 2  e.  RR  /\  +oo 
e.  RR* )  ->  (
x  e.  ( 2 [,)  +oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  <  +oo ) ) )
1815, 16, 17mp2an 655 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  <  +oo ) )
1918simp2bi 974 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  2  <_  x )
20 chtrpcl 20960 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
2114, 19, 20syl2anc 644 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  e.  RR+ )
22 0re 9093 . . . . . . . . . . . 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 9552 . . . . . . . . . . . 12  |-  0  <  1
2625a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  0  <  1 )
27 eliooord 10972 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
1  <  x  /\  x  <  +oo ) )
2827simpld 447 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  1  <  x )
2923, 24, 13, 26, 28lttrd 9233 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  0  <  x )
3013, 29elrpd 10648 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) 
+oo )  ->  x  e.  RR+ )
3111, 30syl 16 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
3221, 31rpdivcld 10667 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( theta `  x )  /  x )  e.  RR+ )
3332adantl 454 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  x )  e.  RR+ )
34 ppinncl 20959 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
3514, 19, 34syl2anc 644 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  NN )
3635nnrpd 10649 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  RR+ )
3713, 28rplogcld 20526 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  ( log `  x )  e.  RR+ )
3811, 37syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  RR+ )
3936, 38rpmulcld 10666 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  RR+ )
4021, 39rpdivcld 10667 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
4140adantl 454 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
4231ssriv 3354 . . . . . . . 8  |-  ( 2 [,)  +oo )  C_  RR+
43 resmpt 5193 . . . . . . . 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 21309 . . . . . . . 8  |-  ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1
46 rlimres 12354 . . . . . . . 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 4248 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  x ) )  ~~> r  1 )
49 chtppilim 21171 . . . . . . 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 9061 . . . . . . 7  |-  1  =/=  0
5251a1i 11 . . . . . 6  |-  (  T. 
->  1  =/=  0
)
5340rpne0d 10655 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
5453adantl 454 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 2 [,)  +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
5533, 41, 48, 50, 52, 54rlimdiv 12441 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( ( theta `  x )  /  x
)  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) ) )  ~~> r  ( 1  /  1 ) )
5614recnd 9116 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  CC )
57 chtcl 20894 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  ( theta `  x )  e.  RR )
5813, 57syl 16 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) 
+oo )  ->  ( theta `  x )  e.  RR )
5958recnd 9116 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  ( theta `  x )  e.  CC )
6011, 59syl 16 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  e.  CC )
6156, 60mulcomd 9111 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  x.  ( theta `  x ) )  =  ( ( theta `  x
)  x.  x ) )
6261oveq2d 6099 . . . . . . . 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 10652 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  CC )
6431rpne0d 10655 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  =/=  0 )
6521rpne0d 10655 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  =/=  0 )
6663, 56, 60, 64, 65divcan5d 9818 . . . . . . . 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 2470 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( theta `  x
)  x.  ( (π `  x )  x.  ( log `  x ) ) )  /  ( x  x.  ( theta `  x
) ) )  =  ( ( (π `  x
)  x.  ( log `  x ) )  /  x ) )
6839rpne0d 10655 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  =/=  0
)
6960, 56, 60, 63, 64, 68, 65divdivdivd 9839 . . . . . . 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 10018 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  CC )
7138rpcnd 10652 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  CC )
7238rpne0d 10655 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  =/=  0 )
7370, 56, 71, 64, 72divdiv2d 9824 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
7467, 69, 733eqtr4d 2480 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( theta `  x
)  /  x )  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) )  =  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )
7574mpteq2ia 4293 . . . . 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 9050 . . . . . 6  |-  1  e.  CC
7776div1i 9744 . . . . 5  |-  ( 1  /  1 )  =  1
7855, 75, 773brtr3g 4245 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
7910, 78eqbrtrd 4234 . . 3  |-  (  T. 
->  ( ( x  e.  ( 1 (,)  +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,)  +oo ) )  ~~> r  1 )
80 ppicl 20916 . . . . . . . . . 10  |-  ( x  e.  RR  ->  (π `  x )  e.  NN0 )
8113, 80syl 16 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (π `  x )  e.  NN0 )
8281nn0red 10277 . . . . . . . 8  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (π `  x )  e.  RR )
8330, 37rpdivcld 10667 . . . . . . . 8  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
8482, 83rerpdivcld 10677 . . . . . . 7  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  RR )
8584recnd 9116 . . . . . 6  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  CC )
8685adantl 454 . . . . 5  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  CC )
87 eqid 2438 . . . . 5  |-  ( x  e.  ( 1 (,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  =  ( x  e.  ( 1 (,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
8886, 87fmptd 5895 . . . 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 12361 . . 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 225 . 2  |-  (  T. 
->  ( x  e.  ( 1 (,)  +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
9392trud 1333 1  |-  ( x  e.  ( 1 (,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ w3a 937    T. wtru 1326    = wceq 1653    e. wcel 1726    =/= wne 2601    C_ wss 3322   class class class wbr 4214    e. cmpt 4268    |` cres 4882   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    x. cmul 8997    +oocpnf 9119   RR*cxr 9121    < clt 9122    <_ cle 9123    / cdiv 9679   NNcn 10002   2c2 10051   NN0cn0 10223   RR+crp 10614   (,)cioo 10918   [,)cico 10920    ~~> r crli 12281   logclog 20454   thetaccht 20875  πcppi 20878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-disj 4185  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ioc 10923  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-o1 12286  df-lo1 12287  df-sum 12482  df-ef 12672  df-e 12673  df-sin 12674  df-cos 12675  df-pi 12677  df-dvds 12855  df-gcd 13009  df-prm 13082  df-pc 13213  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cn 17293  df-cnp 17294  df-haus 17381  df-cmp 17452  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-limc 19755  df-dv 19756  df-log 20456  df-cxp 20457  df-em 20833  df-cht 20881  df-vma 20882  df-chp 20883  df-ppi 20884  df-mu 20885
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