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Theorem xrge0pluscn 24279
Description: The addition operation of the extended non-negative real numbers monoid is continuous. (Contributed by Thierry Arnoux, 24-Mar-2017.)
Hypotheses
Ref Expression
xrge0iifhmeo.1  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 ,  +oo ,  -u ( log `  x
) ) )
xrge0iifhmeo.k  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo )
)
xrge0pluscn.1  |-  .+  =  ( + e  |`  (
( 0 [,]  +oo )  X.  ( 0 [,] 
+oo ) ) )
Assertion
Ref Expression
xrge0pluscn  |-  .+  e.  ( ( J  tX  J )  Cn  J
)
Distinct variable group:    x, F
Allowed substitution hints:    .+ ( x)    J( x)

Proof of Theorem xrge0pluscn
Dummy variables  y  u  v  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrge0iifhmeo.1 . . 3  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 ,  +oo ,  -u ( log `  x
) ) )
2 xrge0iifhmeo.k . . 3  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo )
)
31, 2xrge0iifhmeo 24275 . 2  |-  F  e.  ( II  Homeo  J )
4 unitsscn 24247 . . . . 5  |-  ( 0 [,] 1 )  C_  CC
5 xpss12 4940 . . . . 5  |-  ( ( ( 0 [,] 1
)  C_  CC  /\  (
0 [,] 1 ) 
C_  CC )  -> 
( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  C_  ( CC  X.  CC ) )
64, 4, 5mp2an 654 . . . 4  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  C_  ( CC  X.  CC )
7 ax-mulf 9026 . . . . 5  |-  x.  :
( CC  X.  CC )
--> CC
8 ffn 5550 . . . . 5  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
9 fnssresb 5516 . . . . 5  |-  (  x.  Fn  ( CC  X.  CC )  ->  ( (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  <->  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) )  C_  ( CC  X.  CC ) ) )
107, 8, 9mp2b 10 . . . 4  |-  ( (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  <->  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) )  C_  ( CC  X.  CC ) )
116, 10mpbir 201 . . 3  |-  (  x.  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  Fn  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) )
12 ovres 6172 . . . . 5  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) v )  =  ( u  x.  v ) )
13 iimulcl 18915 . . . . 5  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u  x.  v )  e.  ( 0 [,] 1 ) )
1412, 13eqeltrd 2478 . . . 4  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) v )  e.  ( 0 [,] 1 ) )
1514rgen2a 2732 . . 3  |-  A. u  e.  ( 0 [,] 1
) A. v  e.  ( 0 [,] 1
) ( u (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) v )  e.  ( 0 [,] 1 )
16 ffnov 6133 . . 3  |-  ( (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> ( 0 [,] 1 )  <->  ( (  x.  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )  Fn  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  /\  A. u  e.  ( 0 [,] 1
) A. v  e.  ( 0 [,] 1
) ( u (  x.  |`  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) v )  e.  ( 0 [,] 1 ) ) )
1711, 15, 16mpbir2an 887 . 2  |-  (  x.  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) : ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) --> ( 0 [,] 1
)
18 iccssxr 10949 . . . . . 6  |-  ( 0 [,]  +oo )  C_  RR*
19 xpss12 4940 . . . . . 6  |-  ( ( ( 0 [,]  +oo )  C_  RR*  /\  (
0 [,]  +oo )  C_  RR* )  ->  ( (
0 [,]  +oo )  X.  ( 0 [,]  +oo ) )  C_  ( RR*  X.  RR* ) )
2018, 18, 19mp2an 654 . . . . 5  |-  ( ( 0 [,]  +oo )  X.  ( 0 [,]  +oo ) )  C_  ( RR*  X.  RR* )
21 xaddf 10766 . . . . . 6  |-  + e : ( RR*  X.  RR* )
--> RR*
22 ffn 5550 . . . . . 6  |-  ( + e : ( RR*  X. 
RR* ) --> RR*  ->  + e  Fn  ( RR*  X. 
RR* ) )
23 fnssresb 5516 . . . . . 6  |-  ( + e  Fn  ( RR*  X. 
RR* )  ->  (
( + e  |`  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) )  Fn  ( ( 0 [,]  +oo )  X.  ( 0 [,]  +oo ) )  <->  ( (
0 [,]  +oo )  X.  ( 0 [,]  +oo ) )  C_  ( RR*  X.  RR* ) ) )
2421, 22, 23mp2b 10 . . . . 5  |-  ( ( + e  |`  (
( 0 [,]  +oo )  X.  ( 0 [,] 
+oo ) ) )  Fn  ( ( 0 [,]  +oo )  X.  (
0 [,]  +oo ) )  <-> 
( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) 
C_  ( RR*  X.  RR* ) )
2520, 24mpbir 201 . . . 4  |-  ( + e  |`  ( (
0 [,]  +oo )  X.  ( 0 [,]  +oo ) ) )  Fn  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) )
26 xrge0pluscn.1 . . . . 5  |-  .+  =  ( + e  |`  (
( 0 [,]  +oo )  X.  ( 0 [,] 
+oo ) ) )
2726fneq1i 5498 . . . 4  |-  (  .+  Fn  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) )  <-> 
( + e  |`  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) )  Fn  ( ( 0 [,]  +oo )  X.  ( 0 [,]  +oo ) ) )
2825, 27mpbir 201 . . 3  |-  .+  Fn  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) )
2926oveqi 6053 . . . . 5  |-  ( a 
.+  b )  =  ( a ( + e  |`  ( (
0 [,]  +oo )  X.  ( 0 [,]  +oo ) ) ) b )
30 ovres 6172 . . . . . 6  |-  ( ( a  e.  ( 0 [,]  +oo )  /\  b  e.  ( 0 [,]  +oo ) )  ->  (
a ( + e  |`  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) ) b )  =  ( a + e
b ) )
31 ge0xaddcl 10967 . . . . . 6  |-  ( ( a  e.  ( 0 [,]  +oo )  /\  b  e.  ( 0 [,]  +oo ) )  ->  (
a + e b )  e.  ( 0 [,]  +oo ) )
3230, 31eqeltrd 2478 . . . . 5  |-  ( ( a  e.  ( 0 [,]  +oo )  /\  b  e.  ( 0 [,]  +oo ) )  ->  (
a ( + e  |`  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) ) b )  e.  ( 0 [,]  +oo ) )
3329, 32syl5eqel 2488 . . . 4  |-  ( ( a  e.  ( 0 [,]  +oo )  /\  b  e.  ( 0 [,]  +oo ) )  ->  (
a  .+  b )  e.  ( 0 [,]  +oo ) )
3433rgen2a 2732 . . 3  |-  A. a  e.  ( 0 [,]  +oo ) A. b  e.  ( 0 [,]  +oo )
( a  .+  b
)  e.  ( 0 [,]  +oo )
35 ffnov 6133 . . 3  |-  (  .+  : ( ( 0 [,]  +oo )  X.  (
0 [,]  +oo ) ) --> ( 0 [,]  +oo ) 
<->  (  .+  Fn  (
( 0 [,]  +oo )  X.  ( 0 [,] 
+oo ) )  /\  A. a  e.  ( 0 [,]  +oo ) A. b  e.  ( 0 [,]  +oo ) ( a  .+  b )  e.  ( 0 [,]  +oo )
) )
3628, 34, 35mpbir2an 887 . 2  |-  .+  :
( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) --> ( 0 [,]  +oo )
37 iitopon 18862 . 2  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
38 letopon 17223 . . . 4  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
39 resttopon 17179 . . . 4  |-  ( ( (ordTop `  <_  )  e.  (TopOn `  RR* )  /\  ( 0 [,]  +oo )  C_  RR* )  ->  (
(ordTop `  <_  )t  ( 0 [,]  +oo ) )  e.  (TopOn `  ( 0 [,]  +oo ) ) )
4038, 18, 39mp2an 654 . . 3  |-  ( (ordTop `  <_  )t  ( 0 [,] 
+oo ) )  e.  (TopOn `  ( 0 [,]  +oo ) )
412, 40eqeltri 2474 . 2  |-  J  e.  (TopOn `  ( 0 [,]  +oo ) )
4226oveqi 6053 . . . 4  |-  ( ( F `  u ) 
.+  ( F `  v ) )  =  ( ( F `  u ) ( + e  |`  ( (
0 [,]  +oo )  X.  ( 0 [,]  +oo ) ) ) ( F `  v ) )
431xrge0iifcnv 24272 . . . . . . . 8  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  /\  `' F  =  ( y  e.  ( 0 [,]  +oo )  |->  if ( y  = 
+oo ,  0 , 
( exp `  -u y
) ) ) )
4443simpli 445 . . . . . . 7  |-  F :
( 0 [,] 1
)
-1-1-onto-> ( 0 [,]  +oo )
45 f1of 5633 . . . . . . 7  |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  ->  F : ( 0 [,] 1 ) --> ( 0 [,]  +oo ) )
4644, 45ax-mp 8 . . . . . 6  |-  F :
( 0 [,] 1
) --> ( 0 [,] 
+oo )
4746ffvelrni 5828 . . . . 5  |-  ( u  e.  ( 0 [,] 1 )  ->  ( F `  u )  e.  ( 0 [,]  +oo ) )
4846ffvelrni 5828 . . . . 5  |-  ( v  e.  ( 0 [,] 1 )  ->  ( F `  v )  e.  ( 0 [,]  +oo ) )
49 ovres 6172 . . . . 5  |-  ( ( ( F `  u
)  e.  ( 0 [,]  +oo )  /\  ( F `  v )  e.  ( 0 [,]  +oo ) )  ->  (
( F `  u
) ( + e  |`  ( ( 0 [,] 
+oo )  X.  (
0 [,]  +oo ) ) ) ( F `  v ) )  =  ( ( F `  u ) + e
( F `  v
) ) )
5047, 48, 49syl2an 464 . . . 4  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( ( F `
 u ) ( + e  |`  (
( 0 [,]  +oo )  X.  ( 0 [,] 
+oo ) ) ) ( F `  v
) )  =  ( ( F `  u
) + e ( F `  v ) ) )
5142, 50syl5eq 2448 . . 3  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( ( F `
 u )  .+  ( F `  v ) )  =  ( ( F `  u ) + e ( F `
 v ) ) )
521, 2xrge0iifhom 24276 . . 3  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( F `  ( u  x.  v
) )  =  ( ( F `  u
) + e ( F `  v ) ) )
5312eqcomd 2409 . . . 4  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u  x.  v )  =  ( u (  x.  |`  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) ) v ) )
5453fveq2d 5691 . . 3  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( F `  ( u  x.  v
) )  =  ( F `  ( u (  x.  |`  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) ) v ) ) )
5551, 52, 543eqtr2rd 2443 . 2  |-  ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( F `  ( u (  x.  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) v ) )  =  ( ( F `  u ) 
.+  ( F `  v ) ) )
56 eqid 2404 . . . 4  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  =  ( (mulGrp ` fld )s  ( 0 [,] 1 ) )
5756iistmd 24253 . . 3  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  e. TopMnd
58 cnfldex 16661 . . . . . 6  |-fld  e.  _V
59 ovex 6065 . . . . . 6  |-  ( 0 [,] 1 )  e. 
_V
60 eqid 2404 . . . . . . 7  |-  (flds  ( 0 [,] 1 ) )  =  (flds  ( 0 [,] 1 ) )
61 eqid 2404 . . . . . . 7  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
6260, 61mgpress 15614 . . . . . 6  |-  ( (fld  e. 
_V  /\  ( 0 [,] 1 )  e. 
_V )  ->  (
(mulGrp ` fld )s  ( 0 [,] 1 ) )  =  (mulGrp `  (flds  ( 0 [,] 1 ) ) ) )
6358, 59, 62mp2an 654 . . . . 5  |-  ( (mulGrp ` fld )s  ( 0 [,] 1
) )  =  (mulGrp `  (flds  ( 0 [,] 1 ) ) )
6460dfii4 18867 . . . . 5  |-  II  =  ( TopOpen `  (flds  ( 0 [,] 1 ) ) )
6563, 64mgptopn 15612 . . . 4  |-  II  =  ( TopOpen `  ( (mulGrp ` fld )s  ( 0 [,] 1 ) ) )
66 cnfldbas 16662 . . . . . . 7  |-  CC  =  ( Base ` fld )
6761, 66mgpbas 15609 . . . . . 6  |-  CC  =  ( Base `  (mulGrp ` fld ) )
68 cnfldmul 16664 . . . . . . 7  |-  x.  =  ( .r ` fld )
6961, 68mgpplusg 15607 . . . . . 6  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
707, 8ax-mp 8 . . . . . 6  |-  x.  Fn  ( CC  X.  CC )
7167, 56, 69, 70, 4ressplusf 24136 . . . . 5  |-  ( + f `  ( (mulGrp ` fld )s  ( 0 [,] 1
) ) )  =  (  x.  |`  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
7271eqcomi 2408 . . . 4  |-  (  x.  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  =  ( + f `  (
(mulGrp ` fld )s  ( 0 [,] 1 ) ) )
7365, 72tmdcn 18066 . . 3  |-  ( ( (mulGrp ` fld )s  ( 0 [,] 1 ) )  e. TopMnd  ->  (  x.  |`  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  e.  ( ( II  tX  II )  Cn  II ) )
7457, 73ax-mp 8 . 2  |-  (  x.  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  e.  ( ( II  tX  II )  Cn  II )
753, 17, 36, 37, 41, 55, 74mndpluscn 24265 1  |-  .+  e.  ( ( J  tX  J )  Cn  J
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    C_ wss 3280   ifcif 3699    e. cmpt 4226    X. cxp 4835   `'ccnv 4836    |` cres 4839    Fn wfn 5408   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    x. cmul 8951    +oocpnf 9073   RR*cxr 9075    <_ cle 9077   -ucneg 9248   + ecxad 10664   [,]cicc 10875   expce 12619   ↾s cress 13425   ↾t crest 13603  ordTopcordt 13676   + fcplusf 14642  mulGrpcmgp 15603  ℂfldccnfld 16658  TopOnctopon 16914    Cn ccn 17242    tX ctx 17545  TopMndctmd 18053   IIcii 18858   logclog 20405
This theorem is referenced by:  xrge0tmdOLD  24284
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-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-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  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-ordt 13680  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-ps 14584  df-tsr 14585  df-mnd 14645  df-plusf 14646  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-subrg 15821  df-abv 15860  df-lmod 15907  df-scaf 15908  df-sra 16199  df-rgmod 16200  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-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-tmd 18055  df-tgp 18056  df-trg 18142  df-xms 18303  df-ms 18304  df-tms 18305  df-nm 18583  df-ngp 18584  df-nrg 18586  df-nlm 18587  df-ii 18860  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407
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