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Theorem cfilucfil3OLD 20964
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.)
Assertion
Ref Expression
cfilucfil3OLD  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( ( C  e.  ( Fil `  X
)  /\  C  e.  (CauFilu `  (metUnifOLD
`  D ) ) )  <->  C  e.  (CauFil `  D ) ) )

Proof of Theorem cfilucfil3OLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfilucfil2OLD 20283 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( C  e.  (CauFilu `  (metUnifOLD
`  D ) )  <-> 
( C  e.  (
fBas `  X )  /\  A. x  e.  RR+  E. y  e.  C  ( D " ( y  X.  y ) ) 
C_  ( 0 [,) x ) ) ) )
21anbi2d 703 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( ( C  e.  ( Fil `  X
)  /\  C  e.  (CauFilu `  (metUnifOLD
`  D ) ) )  <->  ( C  e.  ( Fil `  X
)  /\  ( C  e.  ( fBas `  X
)  /\  A. x  e.  RR+  E. y  e.  C  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) ) )
3 filfbas 19556 . . . . . 6  |-  ( C  e.  ( Fil `  X
)  ->  C  e.  ( fBas `  X )
)
43pm4.71i 632 . . . . 5  |-  ( C  e.  ( Fil `  X
)  <->  ( C  e.  ( Fil `  X
)  /\  C  e.  ( fBas `  X )
) )
54anbi1i 695 . . . 4  |-  ( ( 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 ) ) )
6 anass 649 . . . 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
)  /\  ( C  e.  ( fBas `  X
)  /\  A. x  e.  RR+  E. y  e.  C  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
75, 6bitr2i 250 . . 3  |-  ( ( 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 ) ) )
82, 7syl6bb 261 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( ( C  e.  ( Fil `  X
)  /\  C  e.  (CauFilu `  (metUnifOLD
`  D ) ) )  <->  ( C  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  C  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
9 iscfil 20911 . . 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 ) ) ) )
109adantl 466 . 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 ) ) ) )
118, 10bitr4d 256 1  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( ( C  e.  ( Fil `  X
)  /\  C  e.  (CauFilu `  (metUnifOLD
`  D ) ) )  <->  C  e.  (CauFil `  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800    C_ wss 3439   (/)c0 3748    X. cxp 4949   "cima 4954   ` cfv 5529  (class class class)co 6203   0cc0 9396   RR+crp 11105   [,)cico 11416   *Metcxmt 17929   fBascfbas 17932  metUnifOLDcmetuOLD 17935   Filcfil 19553  CauFiluccfilu 19996  CauFilccfil 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-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
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 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-2 10494  df-rp 11106  df-xneg 11203  df-xadd 11204  df-xmul 11205  df-ico 11420  df-xmet 17938  df-fbas 17942  df-fg 17943  df-metuOLD 17944  df-fil 19554  df-ust 19910  df-cfilu 19997  df-cfil 20901
This theorem is referenced by:  cfilucfil4OLD  20966
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