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Theorem cmetcusp 19261
Description: The uniform space generated by a complete metric is a complete uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
Assertion
Ref Expression
cmetcusp  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (toUnifSp `  (metUnif `  D )
)  e. CUnifSp )

Proof of Theorem cmetcusp
Dummy variables  x  c  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmetmet 19192 . . . . . 6  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
2 metxmet 18317 . . . . . 6  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
3 xmetpsmet 18331 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  D  e.  (PsMet `  X )
)
41, 2, 33syl 19 . . . . 5  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  (PsMet `  X ) )
54anim2i 553 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
) )
6 metuust 18555 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  e.  (UnifOn `  X ) )
7 eqid 2404 . . . . 5  |-  (toUnifSp `  (metUnif `  D ) )  =  (toUnifSp `  (metUnif `  D
) )
87tususp 18255 . . . 4  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (toUnifSp `  (metUnif `  D )
)  e. UnifSp )
95, 6, 83syl 19 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (toUnifSp `  (metUnif `  D )
)  e. UnifSp )
10 simpll 731 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )  ->  ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
) )
1110simprd 450 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )  ->  D  e.  (
CMet `  X )
)
121, 2syl 16 . . . . . . . . 9  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( * Met `  X
) )
1312ad3antlr 712 . . . . . . . 8  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )  ->  D  e.  ( * Met `  X
) )
147tusbas 18251 . . . . . . . . . . . . . 14  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  X  =  ( Base `  (toUnifSp `  (metUnif `  D )
) ) )
1514fveq2d 5691 . . . . . . . . . . . . 13  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  ( Fil `  X )  =  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D ) ) ) ) )
1615eleq2d 2471 . . . . . . . . . . . 12  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (
c  e.  ( Fil `  X )  <->  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) ) )
175, 6, 163syl 19 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
c  e.  ( Fil `  X )  <->  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) ) )
1817biimpar 472 . . . . . . . . . 10  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D ) ) ) ) )  ->  c  e.  ( Fil `  X
) )
1918adantr 452 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )  ->  c  e.  ( Fil `  X ) )
20 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )  ->  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D ) ) ) ) )
217tusunif 18252 . . . . . . . . . . . . . . . 16  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (metUnif `  D )  =  (
UnifSet `  (toUnifSp `  (metUnif `  D ) ) ) )
22 ustuni 18209 . . . . . . . . . . . . . . . . . 18  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  U. (metUnif `  D )  =  ( X  X.  X ) )
2321unieqd 3986 . . . . . . . . . . . . . . . . . 18  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  U. (metUnif `  D )  =  U. ( UnifSet `  (toUnifSp `  (metUnif `  D ) ) ) )
2414, 14xpeq12d 4862 . . . . . . . . . . . . . . . . . 18  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  ( X  X.  X )  =  ( ( Base `  (toUnifSp `  (metUnif `  D )
) )  X.  ( Base `  (toUnifSp `  (metUnif `  D ) ) ) ) )
2522, 23, 243eqtr3rd 2445 . . . . . . . . . . . . . . . . 17  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (
( Base `  (toUnifSp `  (metUnif `  D ) ) )  X.  ( Base `  (toUnifSp `  (metUnif `  D )
) ) )  = 
U. ( UnifSet `  (toUnifSp `  (metUnif `  D )
) ) )
26 eqid 2404 . . . . . . . . . . . . . . . . . 18  |-  ( Base `  (toUnifSp `  (metUnif `  D
) ) )  =  ( Base `  (toUnifSp `  (metUnif `  D )
) )
27 eqid 2404 . . . . . . . . . . . . . . . . . 18  |-  ( UnifSet `  (toUnifSp `  (metUnif `  D
) ) )  =  ( UnifSet `  (toUnifSp `  (metUnif `  D ) ) )
2826, 27ussid 18243 . . . . . . . . . . . . . . . . 17  |-  ( ( ( Base `  (toUnifSp `  (metUnif `  D )
) )  X.  ( Base `  (toUnifSp `  (metUnif `  D ) ) ) )  =  U. ( UnifSet
`  (toUnifSp `  (metUnif `  D
) ) )  -> 
( UnifSet `  (toUnifSp `  (metUnif `  D ) ) )  =  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) )
2925, 28syl 16 . . . . . . . . . . . . . . . 16  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  ( UnifSet
`  (toUnifSp `  (metUnif `  D
) ) )  =  (UnifSt `  (toUnifSp `  (metUnif `  D ) ) ) )
3021, 29eqtrd 2436 . . . . . . . . . . . . . . 15  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (metUnif `  D )  =  (UnifSt `  (toUnifSp `  (metUnif `  D
) ) ) )
3130fveq2d 5691 . . . . . . . . . . . . . 14  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (CauFilu `  (metUnif `  D ) )  =  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )
325, 6, 313syl 19 . . . . . . . . . . . . 13  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (CauFilu `  (metUnif `  D ) )  =  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )
3332eleq2d 2471 . . . . . . . . . . . 12  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
c  e.  (CauFilu `  (metUnif `  D ) )  <->  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D ) ) ) ) ) )
3433biimpar 472 . . . . . . . . . . 11  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )  ->  c  e.  (CauFilu `  (metUnif `  D )
) )
3510, 20, 34syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )  ->  c  e.  (CauFilu `  (metUnif `  D )
) )
36 cfilucfil2 18557 . . . . . . . . . . . . 13  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( c  e.  (CauFilu `  (metUnif `  D
) )  <->  ( c  e.  ( fBas `  X
)  /\  A. x  e.  RR+  E. y  e.  c  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
375, 36syl 16 . . . . . . . . . . . 12  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
c  e.  (CauFilu `  (metUnif `  D ) )  <->  ( c  e.  ( fBas `  X
)  /\  A. x  e.  RR+  E. y  e.  c  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
3837biimpa 471 . . . . . . . . . . 11  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  c  e.  (CauFilu `  (metUnif `  D
) ) )  -> 
( c  e.  (
fBas `  X )  /\  A. x  e.  RR+  E. y  e.  c  ( D " ( y  X.  y ) ) 
C_  ( 0 [,) x ) ) )
3938simprd 450 . . . . . . . . . 10  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  c  e.  (CauFilu `  (metUnif `  D
) ) )  ->  A. x  e.  RR+  E. y  e.  c  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) )
4010, 35, 39syl2anc 643 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )  ->  A. x  e.  RR+  E. y  e.  c  ( D " ( y  X.  y ) ) 
C_  ( 0 [,) x ) )
4119, 40jca 519 . . . . . . . 8  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )  ->  ( c  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  c  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) )
42 iscfil 19171 . . . . . . . . 9  |-  ( D  e.  ( * Met `  X )  ->  (
c  e.  (CauFil `  D )  <->  ( c  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  c  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
4342biimpar 472 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  ( c  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  c  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) )  ->  c  e.  (CauFil `  D )
)
4413, 41, 43syl2anc 643 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )  ->  c  e.  (CauFil `  D ) )
45 eqid 2404 . . . . . . . 8  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4645cmetcvg 19191 . . . . . . 7  |-  ( ( D  e.  ( CMet `  X )  /\  c  e.  (CauFil `  D )
)  ->  ( ( MetOpen
`  D )  fLim  c )  =/=  (/) )
4711, 44, 46syl2anc 643 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )  ->  ( ( MetOpen `  D )  fLim  c
)  =/=  (/) )
48 eqid 2404 . . . . . . . . . . . 12  |-  (unifTop `  (metUnif `  D ) )  =  (unifTop `  (metUnif `  D
) )
497, 48tustopn 18254 . . . . . . . . . . 11  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (unifTop `  (metUnif `  D )
)  =  ( TopOpen `  (toUnifSp `  (metUnif `  D
) ) ) )
505, 6, 493syl 19 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (unifTop `  (metUnif `  D )
)  =  ( TopOpen `  (toUnifSp `  (metUnif `  D
) ) ) )
5112anim2i 553 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) ) )
52 xmetutop 18567 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  (unifTop `  (metUnif `  D )
)  =  ( MetOpen `  D ) )
5351, 52syl 16 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (unifTop `  (metUnif `  D )
)  =  ( MetOpen `  D ) )
5450, 53eqtr3d 2438 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  ( TopOpen
`  (toUnifSp `  (metUnif `  D
) ) )  =  ( MetOpen `  D )
)
5554oveq1d 6055 . . . . . . . 8  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
( TopOpen `  (toUnifSp `  (metUnif `  D ) ) ) 
fLim  c )  =  ( ( MetOpen `  D
)  fLim  c )
)
5655neeq1d 2580 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
( ( TopOpen `  (toUnifSp `  (metUnif `  D )
) )  fLim  c
)  =/=  (/)  <->  ( ( MetOpen
`  D )  fLim  c )  =/=  (/) ) )
5756biimpar 472 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  (
( MetOpen `  D )  fLim  c )  =/=  (/) )  -> 
( ( TopOpen `  (toUnifSp `  (metUnif `  D )
) )  fLim  c
)  =/=  (/) )
5810, 47, 57syl2anc 643 . . . . 5  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (
CMet `  X )
)  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )  ->  ( ( TopOpen `  (toUnifSp `  (metUnif `  D
) ) )  fLim  c )  =/=  (/) )
5958ex 424 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D ) ) ) ) )  ->  (
c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D
) ) ) )  ->  ( ( TopOpen `  (toUnifSp `  (metUnif `  D
) ) )  fLim  c )  =/=  (/) ) )
6059ralrimiva 2749 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  A. c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D ) ) ) ) ( c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) )  -> 
( ( TopOpen `  (toUnifSp `  (metUnif `  D )
) )  fLim  c
)  =/=  (/) ) )
619, 60jca 519 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
(toUnifSp `  (metUnif `  D
) )  e. UnifSp  /\  A. c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D )
) ) ) ( c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D
) ) ) )  ->  ( ( TopOpen `  (toUnifSp `  (metUnif `  D
) ) )  fLim  c )  =/=  (/) ) ) )
62 iscusp 18282 . 2  |-  ( (toUnifSp `  (metUnif `  D )
)  e. CUnifSp  <->  ( (toUnifSp `  (metUnif `  D ) )  e. UnifSp  /\  A. c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) ( c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D ) ) ) )  ->  ( ( TopOpen
`  (toUnifSp `  (metUnif `  D
) ) )  fLim  c )  =/=  (/) ) ) )
6361, 62sylibr 204 1  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (toUnifSp `  (metUnif `  D )
)  e. CUnifSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667    C_ wss 3280   (/)c0 3588   U.cuni 3975    X. cxp 4835   "cima 4840   ` cfv 5413  (class class class)co 6040   0cc0 8946   RR+crp 10568   [,)cico 10874   Basecbs 13424   UnifSetcunif 13494   TopOpenctopn 13604  PsMetcpsmet 16640   * Metcxmt 16641   Metcme 16642   fBascfbas 16644   MetOpencmopn 16646  metUnifcmetu 16648   Filcfil 17830    fLim cflim 17919  UnifOncust 18182  unifTopcutop 18213  UnifStcuss 18236  UnifSpcusp 18237  toUnifSpctus 18238  CauFiluccfilu 18269  CUnifSpccusp 18280  CauFilccfil 19158   CMetcms 19160
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-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
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-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-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-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  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-ico 10878  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-tset 13503  df-unif 13507  df-rest 13605  df-topn 13606  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-metu 16657  df-fil 17831  df-ust 18183  df-utop 18214  df-uss 18239  df-usp 18240  df-tus 18241  df-cfilu 18270  df-cusp 18281  df-cfil 19161  df-cmet 19163
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