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Theorem cmetcusp 19269
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 19200 . . . . . 6  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
2 metxmet 18325 . . . . . 6  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
3 xmetpsmet 18339 . . . . . 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 18563 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  e.  (UnifOn `  X ) )
7 eqid 2412 . . . . 5  |-  (toUnifSp `  (metUnif `  D ) )  =  (toUnifSp `  (metUnif `  D
) )
87tususp 18263 . . . 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 18259 . . . . . . . . . . . . . 14  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  X  =  ( Base `  (toUnifSp `  (metUnif `  D )
) ) )
1514fveq2d 5699 . . . . . . . . . . . . 13  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  ( Fil `  X )  =  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D ) ) ) ) )
1615eleq2d 2479 . . . . . . . . . . . 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 18260 . . . . . . . . . . . . . . . 16  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (metUnif `  D )  =  (
UnifSet `  (toUnifSp `  (metUnif `  D ) ) ) )
22 ustuni 18217 . . . . . . . . . . . . . . . . . 18  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  U. (metUnif `  D )  =  ( X  X.  X ) )
2321unieqd 3994 . . . . . . . . . . . . . . . . . 18  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  U. (metUnif `  D )  =  U. ( UnifSet `  (toUnifSp `  (metUnif `  D ) ) ) )
2414, 14xpeq12d 4870 . . . . . . . . . . . . . . . . . 18  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  ( X  X.  X )  =  ( ( Base `  (toUnifSp `  (metUnif `  D )
) )  X.  ( Base `  (toUnifSp `  (metUnif `  D ) ) ) ) )
2522, 23, 243eqtr3rd 2453 . . . . . . . . . . . . . . . . 17  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (
( Base `  (toUnifSp `  (metUnif `  D ) ) )  X.  ( Base `  (toUnifSp `  (metUnif `  D )
) ) )  = 
U. ( UnifSet `  (toUnifSp `  (metUnif `  D )
) ) )
26 eqid 2412 . . . . . . . . . . . . . . . . . 18  |-  ( Base `  (toUnifSp `  (metUnif `  D
) ) )  =  ( Base `  (toUnifSp `  (metUnif `  D )
) )
27 eqid 2412 . . . . . . . . . . . . . . . . . 18  |-  ( UnifSet `  (toUnifSp `  (metUnif `  D
) ) )  =  ( UnifSet `  (toUnifSp `  (metUnif `  D ) ) )
2826, 27ussid 18251 . . . . . . . . . . . . . . . . 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 2444 . . . . . . . . . . . . . . 15  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (metUnif `  D )  =  (UnifSt `  (toUnifSp `  (metUnif `  D
) ) ) )
3130fveq2d 5699 . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . 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 18565 . . . . . . . . . . . . 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 19179 . . . . . . . . 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 2412 . . . . . . . 8  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4645cmetcvg 19199 . . . . . . 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 2412 . . . . . . . . . . . 12  |-  (unifTop `  (metUnif `  D ) )  =  (unifTop `  (metUnif `  D
) )
497, 48tustopn 18262 . . . . . . . . . . 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 18575 . . . . . . . . . . 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 2446 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  ( TopOpen
`  (toUnifSp `  (metUnif `  D
) ) )  =  ( MetOpen `  D )
)
5554oveq1d 6063 . . . . . . . 8  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
( TopOpen `  (toUnifSp `  (metUnif `  D ) ) ) 
fLim  c )  =  ( ( MetOpen `  D
)  fLim  c )
)
5655neeq1d 2588 . . . . . . 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 2757 . . 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 18290 . 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 2575   A.wral 2674   E.wrex 2675    C_ wss 3288   (/)c0 3596   U.cuni 3983    X. cxp 4843   "cima 4848   ` cfv 5421  (class class class)co 6048   0cc0 8954   RR+crp 10576   [,)cico 10882   Basecbs 13432   UnifSetcunif 13502   TopOpenctopn 13612  PsMetcpsmet 16648   * Metcxmt 16649   Metcme 16650   fBascfbas 16652   MetOpencmopn 16654  metUnifcmetu 16656   Filcfil 17838    fLim cflim 17927  UnifOncust 18190  unifTopcutop 18221  UnifStcuss 18244  UnifSpcusp 18245  toUnifSpctus 18246  CauFiluccfilu 18277  CUnifSpccusp 18288  CauFilccfil 19166   CMetcms 19168
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ico 10886  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-tset 13511  df-unif 13515  df-rest 13613  df-topn 13614  df-topgen 13630  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-fbas 16662  df-fg 16663  df-metu 16665  df-fil 17839  df-ust 18191  df-utop 18222  df-uss 18247  df-usp 18248  df-tus 18249  df-cfilu 18278  df-cusp 18289  df-cfil 19169  df-cmet 19171
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