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Theorem pnt 21175
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 9023 . . . . . . 7  |-  1  e.  RR
21rexri 9070 . . . . . 6  |-  1  e.  RR*
3 1lt2 10074 . . . . . 6  |-  1  <  2
4 df-ioo 10852 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
5 df-ico 10854 . . . . . . 7  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
6 xrltletr 10679 . . . . . . 7  |-  ( ( 1  e.  RR*  /\  2  e.  RR*  /\  w  e. 
RR* )  ->  (
( 1  <  2  /\  2  <_  w )  ->  1  <  w
) )
74, 5, 6ixxss1 10866 . . . . . 6  |-  ( ( 1  e.  RR*  /\  1  <  2 )  ->  (
2 [,)  +oo )  C_  ( 1 (,)  +oo ) )
82, 3, 7mp2an 654 . . . . 5  |-  ( 2 [,)  +oo )  C_  (
1 (,)  +oo )
9 resmpt 5131 . . . . 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 3287 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  ( 1 (,)  +oo ) )
12 ioossre 10904 . . . . . . . . . . 11  |-  ( 1 (,)  +oo )  C_  RR
1312sseli 3287 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  x  e.  RR )
1411, 13syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR )
15 2re 10001 . . . . . . . . . . 11  |-  2  e.  RR
16 pnfxr 10645 . . . . . . . . . . 11  |-  +oo  e.  RR*
17 elico2 10906 . . . . . . . . . . 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 20825 . . . . . . . . 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 9024 . . . . . . . . . . . 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 9482 . . . . . . . . . . . 12  |-  0  <  1
2625a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  0  <  1 )
27 eliooord 10902 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
1  <  x  /\  x  <  +oo ) )
2827simpld 446 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  1  <  x )
2923, 24, 13, 26, 28lttrd 9163 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  0  <  x )
3013, 29elrpd 10578 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) 
+oo )  ->  x  e.  RR+ )
3111, 30syl 16 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  RR+ )
3221, 31rpdivcld 10597 . . . . . . 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 20824 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
3514, 19, 34syl2anc 643 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  NN )
3635nnrpd 10579 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  RR+ )
3713, 28rplogcld 20391 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) 
+oo )  ->  ( log `  x )  e.  RR+ )
3811, 37syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  RR+ )
3936, 38rpmulcld 10596 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  RR+ )
4021, 39rpdivcld 10597 . . . . . . 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 3295 . . . . . . . 8  |-  ( 2 [,)  +oo )  C_  RR+
43 resmpt 5131 . . . . . . . 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 21174 . . . . . . . 8  |-  ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1
46 rlimres 12279 . . . . . . . 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 4187 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( theta `  x
)  /  x ) )  ~~> r  1 )
49 chtppilim 21036 . . . . . . 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 8992 . . . . . . 7  |-  1  =/=  0
5251a1i 11 . . . . . 6  |-  (  T. 
->  1  =/=  0
)
5340rpne0d 10585 . . . . . . 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 12366 . . . . 5  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( ( ( theta `  x )  /  x
)  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) ) )  ~~> r  ( 1  /  1 ) )
5614recnd 9047 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  e.  CC )
57 chtcl 20759 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  ( theta `  x )  e.  RR )
5813, 57syl 16 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) 
+oo )  ->  ( theta `  x )  e.  RR )
5958recnd 9047 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) 
+oo )  ->  ( theta `  x )  e.  CC )
6011, 59syl 16 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  e.  CC )
6156, 60mulcomd 9042 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
x  x.  ( theta `  x ) )  =  ( ( theta `  x
)  x.  x ) )
6261oveq2d 6036 . . . . . . . 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 10582 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  e.  CC )
6431rpne0d 10585 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  x  =/=  0 )
6521rpne0d 10585 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( theta `  x )  =/=  0 )
6663, 56, 60, 64, 65divcan5d 9748 . . . . . . . 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 2419 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( theta `  x
)  x.  ( (π `  x )  x.  ( log `  x ) ) )  /  ( x  x.  ( theta `  x
) ) )  =  ( ( (π `  x
)  x.  ( log `  x ) )  /  x ) )
6839rpne0d 10585 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  x.  ( log `  x
) )  =/=  0
)
6960, 56, 60, 63, 64, 68, 65divdivdivd 9769 . . . . . . 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 9948 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (π `  x )  e.  CC )
7138rpcnd 10582 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  e.  CC )
7238rpne0d 10585 . . . . . . . 8  |-  ( x  e.  ( 2 [,) 
+oo )  ->  ( log `  x )  =/=  0 )
7370, 56, 71, 64, 72divdiv2d 9754 . . . . . . 7  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
7467, 69, 733eqtr4d 2429 . . . . . 6  |-  ( x  e.  ( 2 [,) 
+oo )  ->  (
( ( theta `  x
)  /  x )  /  ( ( theta `  x )  /  (
(π `  x )  x.  ( log `  x
) ) ) )  =  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )
7574mpteq2ia 4232 . . . . 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 8981 . . . . . 6  |-  1  e.  CC
7776div1i 9674 . . . . 5  |-  ( 1  /  1 )  =  1
7855, 75, 773brtr3g 4184 . . . 4  |-  (  T. 
->  ( x  e.  ( 2 [,)  +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
7910, 78eqbrtrd 4173 . . 3  |-  (  T. 
->  ( ( x  e.  ( 1 (,)  +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,)  +oo ) )  ~~> r  1 )
80 ppicl 20781 . . . . . . . . . 10  |-  ( x  e.  RR  ->  (π `  x )  e.  NN0 )
8113, 80syl 16 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (π `  x )  e.  NN0 )
8281nn0red 10207 . . . . . . . 8  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (π `  x )  e.  RR )
8330, 37rpdivcld 10597 . . . . . . . 8  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
x  /  ( log `  x ) )  e.  RR+ )
8482, 83rerpdivcld 10607 . . . . . . 7  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  RR )
8584recnd 9047 . . . . . 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 2387 . . . . 5  |-  ( x  e.  ( 1 (,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  =  ( x  e.  ( 1 (,) 
+oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
8886, 87fmptd 5832 . . . 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 12286 . . 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 1717    =/= wne 2550    C_ wss 3263   class class class wbr 4153    e. cmpt 4207    |` cres 4820   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    x. cmul 8928    +oocpnf 9050   RR*cxr 9052    < clt 9053    <_ cle 9054    / cdiv 9609   NNcn 9932   2c2 9981   NN0cn0 10153   RR+crp 10544   (,)cioo 10848   [,)cico 10850    ~~> r crli 12206   logclog 20319   thetaccht 20740  πcppi 20743
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-disj 4124  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ioc 10853  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-fac 11494  df-bc 11521  df-hash 11546  df-shft 11809  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-limsup 12192  df-clim 12209  df-rlim 12210  df-o1 12211  df-lo1 12212  df-sum 12407  df-ef 12597  df-e 12598  df-sin 12599  df-cos 12600  df-pi 12602  df-dvds 12780  df-gcd 12934  df-prm 13007  df-pc 13138  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-hom 13480  df-cco 13481  df-rest 13577  df-topn 13578  df-topgen 13594  df-pt 13595  df-prds 13598  df-xrs 13653  df-0g 13654  df-gsum 13655  df-qtop 13660  df-imas 13661  df-xps 13663  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-mulg 14742  df-cntz 15043  df-cmn 15341  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-fbas 16623  df-fg 16624  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cld 17006  df-ntr 17007  df-cls 17008  df-nei 17085  df-lp 17123  df-perf 17124  df-cn 17213  df-cnp 17214  df-haus 17301  df-cmp 17372  df-tx 17515  df-hmeo 17708  df-fil 17799  df-fm 17891  df-flim 17892  df-flf 17893  df-xms 18259  df-ms 18260  df-tms 18261  df-cncf 18779  df-limc 19620  df-dv 19621  df-log 20321  df-cxp 20322  df-em 20698  df-cht 20746  df-vma 20747  df-chp 20748  df-ppi 20749  df-mu 20750
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