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Theorem cfilucfil3 22050
Description: Given a metric  D and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the Cauchy filters for the metric. (Contributed by Thierry Arnoux, 15-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
cfilucfil3  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( ( C  e.  ( Fil `  X
)  /\  C  e.  (CauFilu `  (metUnif `  D )
) )  <->  C  e.  (CauFil `  D ) ) )

Proof of Theorem cfilucfil3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xmetpsmet 21143 . . 3  |-  ( D  e.  ( *Met `  X )  ->  D  e.  (PsMet `  X )
)
2 cfilucfil2 21369 . . . . 5  |-  ( ( 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 ) ) ) )
32anbi2d 702 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( ( C  e.  ( Fil `  X )  /\  C  e.  (CauFilu `  (metUnif `  D
) ) )  <->  ( C  e.  ( Fil `  X
)  /\  ( C  e.  ( fBas `  X
)  /\  A. x  e.  RR+  E. y  e.  C  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) ) )
4 filfbas 20641 . . . . . . 7  |-  ( C  e.  ( Fil `  X
)  ->  C  e.  ( fBas `  X )
)
54pm4.71i 630 . . . . . 6  |-  ( C  e.  ( Fil `  X
)  <->  ( C  e.  ( Fil `  X
)  /\  C  e.  ( fBas `  X )
) )
65anbi1i 693 . . . . 5  |-  ( ( C  e.  ( Fil `  X )  /\  A. x  e.  RR+  E. y  e.  C  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) )  <->  ( ( C  e.  ( Fil `  X
)  /\  C  e.  ( fBas `  X )
)  /\  A. x  e.  RR+  E. y  e.  C  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) )
7 anass 647 . . . . 5  |-  ( ( ( C  e.  ( Fil `  X )  /\  C  e.  (
fBas `  X )
)  /\  A. x  e.  RR+  E. y  e.  C  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) )  <->  ( C  e.  ( Fil `  X
)  /\  ( C  e.  ( fBas `  X
)  /\  A. x  e.  RR+  E. y  e.  C  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
86, 7bitr2i 250 . . . 4  |-  ( ( C  e.  ( Fil `  X )  /\  ( C  e.  ( fBas `  X )  /\  A. x  e.  RR+  E. y  e.  C  ( D " ( y  X.  y
) )  C_  (
0 [,) x ) ) )  <->  ( C  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  C  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) )
93, 8syl6bb 261 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( ( C  e.  ( Fil `  X )  /\  C  e.  (CauFilu `  (metUnif `  D
) ) )  <->  ( C  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  C  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
101, 9sylan2 472 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( ( C  e.  ( Fil `  X
)  /\  C  e.  (CauFilu `  (metUnif `  D )
) )  <->  ( C  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  C  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
11 iscfil 21996 . . 3  |-  ( 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 ) ) ) )
1211adantl 464 . 2  |-  ( ( X  =/=  (/)  /\  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 ) ) ) )
1310, 12bitr4d 256 1  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( ( C  e.  ( Fil `  X
)  /\  C  e.  (CauFilu `  (metUnif `  D )
) )  <->  C  e.  (CauFil `  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755    C_ wss 3414   (/)c0 3738    X. cxp 4821   "cima 4826   ` cfv 5569  (class class class)co 6278   0cc0 9522   RR+crp 11265   [,)cico 11584  PsMetcpsmet 18722   *Metcxmt 18723   fBascfbas 18726  metUnifcmetu 18730   Filcfil 20638  CauFiluccfilu 21081  CauFilccfil 21983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-2 10635  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ico 11588  df-psmet 18731  df-xmet 18732  df-fbas 18736  df-fg 18737  df-metu 18739  df-fil 20639  df-ust 20995  df-cfilu 21082  df-cfil 21986
This theorem is referenced by:  cfilucfil4  22052
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