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Theorem cmetcusp 20999
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 20930 . . . . . 6  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
2 metxmet 20042 . . . . . 6  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
3 xmetpsmet 20056 . . . . . 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 20280 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  e.  (UnifOn `  X ) )
7 eqid 2454 . . . . 5  |-  (toUnifSp `  (metUnif `  D ) )  =  (toUnifSp `  (metUnif `  D
) )
87tususp 19980 . . . 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 19976 . . . . . . . . . . . . . 14  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  X  =  ( Base `  (toUnifSp `  (metUnif `  D )
) ) )
1514fveq2d 5804 . . . . . . . . . . . . 13  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  ( Fil `  X )  =  ( Fil `  ( Base `  (toUnifSp `  (metUnif `  D ) ) ) ) )
1615eleq2d 2524 . . . . . . . . . . . 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 19977 . . . . . . . . . . . . . . . 16  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (metUnif `  D )  =  (
UnifSet `  (toUnifSp `  (metUnif `  D ) ) ) )
22 ustuni 19934 . . . . . . . . . . . . . . . . . 18  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  U. (metUnif `  D )  =  ( X  X.  X ) )
2321unieqd 4210 . . . . . . . . . . . . . . . . . 18  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  U. (metUnif `  D )  =  U. ( UnifSet `  (toUnifSp `  (metUnif `  D ) ) ) )
2414, 14xpeq12d 4974 . . . . . . . . . . . . . . . . . 18  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  ( X  X.  X )  =  ( ( Base `  (toUnifSp `  (metUnif `  D )
) )  X.  ( Base `  (toUnifSp `  (metUnif `  D ) ) ) ) )
2522, 23, 243eqtr3rd 2504 . . . . . . . . . . . . . . . . 17  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (
( Base `  (toUnifSp `  (metUnif `  D ) ) )  X.  ( Base `  (toUnifSp `  (metUnif `  D )
) ) )  = 
U. ( UnifSet `  (toUnifSp `  (metUnif `  D )
) ) )
26 eqid 2454 . . . . . . . . . . . . . . . . . 18  |-  ( Base `  (toUnifSp `  (metUnif `  D
) ) )  =  ( Base `  (toUnifSp `  (metUnif `  D )
) )
27 eqid 2454 . . . . . . . . . . . . . . . . . 18  |-  ( UnifSet `  (toUnifSp `  (metUnif `  D
) ) )  =  ( UnifSet `  (toUnifSp `  (metUnif `  D ) ) )
2826, 27ussid 19968 . . . . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . . . 15  |-  ( (metUnif `  D )  e.  (UnifOn `  X )  ->  (metUnif `  D )  =  (UnifSt `  (toUnifSp `  (metUnif `  D
) ) ) )
3130fveq2d 5804 . . . . . . . . . . . . . 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 2524 . . . . . . . . . . . 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 20282 . . . . . . . . . . . . 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 20909 . . . . . . . . 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 2454 . . . . . . . 8  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4645cmetcvg 20929 . . . . . . 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 2454 . . . . . . . . . . . 12  |-  (unifTop `  (metUnif `  D ) )  =  (unifTop `  (metUnif `  D
) )
497, 48tustopn 19979 . . . . . . . . . . 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 20292 . . . . . . . . . . 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 2497 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  ( TopOpen
`  (toUnifSp `  (metUnif `  D
) ) )  =  ( MetOpen `  D )
)
5554oveq1d 6216 . . . . . . . 8  |-  ( ( X  =/=  (/)  /\  D  e.  ( CMet `  X
) )  ->  (
( TopOpen `  (toUnifSp `  (metUnif `  D ) ) ) 
fLim  c )  =  ( ( MetOpen `  D
)  fLim  c )
)
5655neeq1d 2729 . . . . . . 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 2830 . . 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 20007 . 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 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800    C_ wss 3437   (/)c0 3746   U.cuni 4200    X. cxp 4947   "cima 4952   ` cfv 5527  (class class class)co 6201   0cc0 9394   RR+crp 11103   [,)cico 11414   Basecbs 14293   UnifSetcunif 14368   TopOpenctopn 14480  PsMetcpsmet 17926   *Metcxmt 17927   Metcme 17928   fBascfbas 17930   MetOpencmopn 17932  metUnifcmetu 17934   Filcfil 19551    fLim cflim 19640  UnifOncust 19907  unifTopcutop 19938  UnifStcuss 19961  UnifSpcusp 19962  toUnifSpctus 19963  CauFiluccfilu 19994  CUnifSpccusp 20005  CauFilccfil 20896   CMetcms 20898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-sup 7803  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-q 11066  df-rp 11104  df-xneg 11201  df-xadd 11202  df-xmul 11203  df-ico 11418  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-tset 14377  df-unif 14381  df-rest 14481  df-topn 14482  df-topgen 14502  df-psmet 17935  df-xmet 17936  df-met 17937  df-bl 17938  df-mopn 17939  df-fbas 17940  df-fg 17941  df-metu 17943  df-fil 19552  df-ust 19908  df-utop 19939  df-uss 19964  df-usp 19965  df-tus 19966  df-cfilu 19995  df-cusp 20006  df-cfil 20899  df-cmet 20901
This theorem is referenced by: (None)
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