MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cfilucfil3 Structured version   Unicode version

Theorem cfilucfil3 20948
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 20041 . . 3  |-  ( D  e.  ( *Met `  X )  ->  D  e.  (PsMet `  X )
)
2 cfilucfil2 20267 . . . . 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 703 . . . 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 19539 . . . . . . 7  |-  ( C  e.  ( Fil `  X
)  ->  C  e.  ( fBas `  X )
)
54pm4.71i 632 . . . . . 6  |-  ( C  e.  ( Fil `  X
)  <->  ( C  e.  ( Fil `  X
)  /\  C  e.  ( fBas `  X )
) )
65anbi1i 695 . . . . 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 649 . . . . 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 474 . 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 20894 . . 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 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 ) ) ) )
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 369    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796    C_ wss 3428   (/)c0 3737    X. cxp 4938   "cima 4943   ` cfv 5518  (class class class)co 6192   0cc0 9385   RR+crp 11094   [,)cico 11405  PsMetcpsmet 17911   *Metcxmt 17912   fBascfbas 17915  metUnifcmetu 17919   Filcfil 19536  CauFiluccfilu 19979  CauFilccfil 20881
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 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-2 10483  df-rp 11095  df-xneg 11192  df-xadd 11193  df-xmul 11194  df-ico 11409  df-psmet 17920  df-xmet 17921  df-fbas 17925  df-fg 17926  df-metu 17928  df-fil 19537  df-ust 19893  df-cfilu 19980  df-cfil 20884
This theorem is referenced by:  cfilucfil4  20950
  Copyright terms: Public domain W3C validator