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Theorem cmetcusp 21662
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 21593 . . . . . 6  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
2 metxmet 20705 . . . . . 6  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
3 xmetpsmet 20719 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  D  e.  (PsMet `  X )
)
41, 2, 33syl 20 . . . . 5  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  (PsMet `  X ) )
54anim2i 569 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
) )
6 metuust 20943 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  e.  (UnifOn `  X ) )
7 eqid 2467 . . . . 5  |-  (toUnifSp `  (metUnif `  D ) )  =  (toUnifSp `  (metUnif `  D
) )
87tususp 20643 . . . 4  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (toUnifSp `  (metUnif `  D )
)  e. UnifSp )
95, 6, 83syl 20 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (toUnifSp `  (metUnif `  D )
)  e. UnifSp )
10 simpll 753 . . . . . 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 463 . . . . . . 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 730 . . . . . . . 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 20639 . . . . . . . . . . . . . 14  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  X  =  ( Base `  (toUnifSp `  (metUnif `  D )
) ) )
1514fveq2d 5876 . . . . . . . . . . . . 13  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  ( Fil `  X )  =  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D ) ) ) ) )
1615eleq2d 2537 . . . . . . . . . . . 12  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (
c  e.  ( Fil `  X )  <->  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) ) )
175, 6, 163syl 20 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
c  e.  ( Fil `  X )  <->  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D
) ) ) ) ) )
1817biimpar 485 . . . . . . . . . 10  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  c  e.  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D ) ) ) ) )  ->  c  e.  ( Fil `  X
) )
1918adantr 465 . . . . . . . . 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 461 . . . . . . . . . . 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 20640 . . . . . . . . . . . . . . . 16  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (metUnif `  D )  =  (
UnifSet `  (toUnifSp `  (metUnif `  D ) ) ) )
22 ustuni 20597 . . . . . . . . . . . . . . . . . 18  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  U. (metUnif `  D )  =  ( X  X.  X ) )
2321unieqd 4261 . . . . . . . . . . . . . . . . . 18  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  U. (metUnif `  D )  =  U. ( UnifSet `  (toUnifSp `  (metUnif `  D ) ) ) )
2414, 14xpeq12d 5030 . . . . . . . . . . . . . . . . . 18  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  ( X  X.  X )  =  ( ( Base `  (toUnifSp `  (metUnif `  D )
) )  X.  ( Base `  (toUnifSp `  (metUnif `  D ) ) ) ) )
2522, 23, 243eqtr3rd 2517 . . . . . . . . . . . . . . . . 17  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (
( Base `  (toUnifSp `  (metUnif `  D ) ) )  X.  ( Base `  (toUnifSp `  (metUnif `  D )
) ) )  = 
U. ( UnifSet `  (toUnifSp `  (metUnif `  D )
) ) )
26 eqid 2467 . . . . . . . . . . . . . . . . . 18  |-  ( Base `  (toUnifSp `  (metUnif `  D
) ) )  =  ( Base `  (toUnifSp `  (metUnif `  D )
) )
27 eqid 2467 . . . . . . . . . . . . . . . . . 18  |-  ( UnifSet `  (toUnifSp `  (metUnif `  D
) ) )  =  ( UnifSet `  (toUnifSp `  (metUnif `  D ) ) )
2826, 27ussid 20631 . . . . . . . . . . . . . . . . 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 2508 . . . . . . . . . . . . . . 15  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (metUnif `  D )  =  (UnifSt `  (toUnifSp `  (metUnif `  D
) ) ) )
3130fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (CauFilu `  (metUnif `  D ) )  =  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )
325, 6, 313syl 20 . . . . . . . . . . . . 13  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (CauFilu `  (metUnif `  D ) )  =  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )
3332eleq2d 2537 . . . . . . . . . . . 12  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
c  e.  (CauFilu `  (metUnif `  D ) )  <->  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D ) ) ) ) ) )
3433biimpar 485 . . . . . . . . . . 11  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  c  e.  (CauFilu `  (UnifSt `  (toUnifSp `  (metUnif `  D )
) ) ) )  ->  c  e.  (CauFilu `  (metUnif `  D )
) )
3510, 20, 34syl2anc 661 . . . . . . . . . 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 20945 . . . . . . . . . . . . 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 484 . . . . . . . . . . 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 463 . . . . . . . . . 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 661 . . . . . . . . 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 532 . . . . . . . 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 21572 . . . . . . . . 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 485 . . . . . . . 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 661 . . . . . . 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 2467 . . . . . . . 8  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4645cmetcvg 21592 . . . . . . 7  |-  ( ( D  e.  ( CMet `  X )  /\  c  e.  (CauFil `  D )
)  ->  ( ( MetOpen
`  D )  fLim  c )  =/=  (/) )
4711, 44, 46syl2anc 661 . . . . . 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 2467 . . . . . . . . . . . 12  |-  (unifTop `  (metUnif `  D ) )  =  (unifTop `  (metUnif `  D
) )
497, 48tustopn 20642 . . . . . . . . . . 11  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (unifTop `  (metUnif `  D )
)  =  ( TopOpen `  (toUnifSp `  (metUnif `  D
) ) ) )
505, 6, 493syl 20 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (unifTop `  (metUnif `  D )
)  =  ( TopOpen `  (toUnifSp `  (metUnif `  D
) ) ) )
5112anim2i 569 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) ) )
52 xmetutop 20955 . . . . . . . . . . 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 2510 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  ( TopOpen
`  (toUnifSp `  (metUnif `  D
) ) )  =  ( MetOpen `  D )
)
5554oveq1d 6310 . . . . . . . 8  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
( TopOpen `  (toUnifSp `  (metUnif `  D ) ) ) 
fLim  c )  =  ( ( MetOpen `  D
)  fLim  c )
)
5655neeq1d 2744 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
( ( TopOpen `  (toUnifSp `  (metUnif `  D )
) )  fLim  c
)  =/=  (/)  <->  ( ( MetOpen
`  D )  fLim  c )  =/=  (/) ) )
5756biimpar 485 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  /\  (
( MetOpen `  D )  fLim  c )  =/=  (/) )  -> 
( ( TopOpen `  (toUnifSp `  (metUnif `  D )
) )  fLim  c
)  =/=  (/) )
5810, 47, 57syl2anc 661 . . . . 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 434 . . . 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 2881 . . 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 532 . 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 20670 . 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 212 1  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (toUnifSp `  (metUnif `  D )
)  e. CUnifSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    C_ wss 3481   (/)c0 3790   U.cuni 4251    X. cxp 5003   "cima 5008   ` cfv 5594  (class class class)co 6295   0cc0 9504   RR+crp 11232   [,)cico 11543   Basecbs 14507   UnifSetcunif 14582   TopOpenctopn 14694  PsMetcpsmet 18272   *Metcxmt 18273   Metcme 18274   fBascfbas 18276   MetOpencmopn 18278  metUnifcmetu 18280   Filcfil 20214    fLim cflim 20303  UnifOncust 20570  unifTopcutop 20601  UnifStcuss 20624  UnifSpcusp 20625  toUnifSpctus 20626  CauFiluccfilu 20657  CUnifSpccusp 20668  CauFilccfil 21559   CMetcms 21561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ico 11547  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-tset 14591  df-unif 14595  df-rest 14695  df-topn 14696  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-metu 18289  df-fil 20215  df-ust 20571  df-utop 20602  df-uss 20627  df-usp 20628  df-tus 20629  df-cfilu 20658  df-cusp 20669  df-cfil 21562  df-cmet 21564
This theorem is referenced by: (None)
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