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

Theorem flimcfil 19219
Description: Every convergent filter in a metric space is a Cauchy filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
Hypothesis
Ref Expression
lmcau.1  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
flimcfil  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  F  e.  (CauFil `  D ) )

Proof of Theorem flimcfil
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . . . 5  |-  U. J  =  U. J
21flimfil 17954 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. J
) )
32adantl 453 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  F  e.  ( Fil `  U. J
) )
4 lmcau.1 . . . . . 6  |-  J  =  ( MetOpen `  D )
54mopnuni 18424 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
65adantr 452 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  X  =  U. J )
76fveq2d 5691 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  ( Fil `  X )  =  ( Fil `  U. J
) )
83, 7eleqtrrd 2481 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  F  e.  ( Fil `  X ) )
91flimelbas 17953 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  U. J )
109ad2antlr 708 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  A  e. 
U. J )
115ad2antrr 707 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  X  = 
U. J )
1210, 11eleqtrrd 2481 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  A  e.  X )
13 simplr 732 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  A  e.  ( J  fLim  F
) )
144mopntop 18423 . . . . . . 7  |-  ( D  e.  ( * Met `  X )  ->  J  e.  Top )
1514ad2antrr 707 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  J  e. 
Top )
16 simpll 731 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  D  e.  ( * Met `  X
) )
17 rpxr 10575 . . . . . . . 8  |-  ( x  e.  RR+  ->  x  e. 
RR* )
1817adantl 453 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  x  e. 
RR* )
194blopn 18483 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  x  e.  RR* )  ->  ( A ( ball `  D ) x )  e.  J )
2016, 12, 18, 19syl3anc 1184 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  ( A ( ball `  D
) x )  e.  J )
21 simpr 448 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
22 blcntr 18396 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  x  e.  RR+ )  ->  A  e.  ( A ( ball `  D
) x ) )
2316, 12, 21, 22syl3anc 1184 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  A  e.  ( A ( ball `  D ) x ) )
24 opnneip 17138 . . . . . 6  |-  ( ( J  e.  Top  /\  ( A ( ball `  D
) x )  e.  J  /\  A  e.  ( A ( ball `  D ) x ) )  ->  ( A
( ball `  D )
x )  e.  ( ( nei `  J
) `  { A } ) )
2515, 20, 23, 24syl3anc 1184 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  ( A ( ball `  D
) x )  e.  ( ( nei `  J
) `  { A } ) )
26 flimnei 17952 . . . . 5  |-  ( ( A  e.  ( J 
fLim  F )  /\  ( A ( ball `  D
) x )  e.  ( ( nei `  J
) `  { A } ) )  -> 
( A ( ball `  D ) x )  e.  F )
2713, 25, 26syl2anc 643 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  ( A ( ball `  D
) x )  e.  F )
28 oveq1 6047 . . . . . 6  |-  ( y  =  A  ->  (
y ( ball `  D
) x )  =  ( A ( ball `  D ) x ) )
2928eleq1d 2470 . . . . 5  |-  ( y  =  A  ->  (
( y ( ball `  D ) x )  e.  F  <->  ( A
( ball `  D )
x )  e.  F
) )
3029rspcev 3012 . . . 4  |-  ( ( A  e.  X  /\  ( A ( ball `  D
) x )  e.  F )  ->  E. y  e.  X  ( y
( ball `  D )
x )  e.  F
)
3112, 27, 30syl2anc 643 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  E. y  e.  X  ( y
( ball `  D )
x )  e.  F
)
3231ralrimiva 2749 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  A. x  e.  RR+  E. y  e.  X  ( y (
ball `  D )
x )  e.  F
)
33 iscfil3 19179 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ( F  e.  (CauFil `  D
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  RR+  E. y  e.  X  ( y (
ball `  D )
x )  e.  F
) ) )
3433adantr 452 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  ( F  e.  (CauFil `  D )  <->  ( F  e.  ( Fil `  X )  /\  A. x  e.  RR+  E. y  e.  X  ( y
( ball `  D )
x )  e.  F
) ) )
358, 32, 34mpbir2and 889 1  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  F  e.  (CauFil `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   {csn 3774   U.cuni 3975   ` cfv 5413  (class class class)co 6040   RR*cxr 9075   RR+crp 10568   * Metcxmt 16641   ballcbl 16643   MetOpencmopn 16646   Topctop 16913   neicnei 17116   Filcfil 17830    fLim cflim 17919  CauFilccfil 19158
This theorem is referenced by:  cmetss  19220  fmcncfil  24270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ico 10878  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-bl 16652  df-mopn 16653  df-fbas 16654  df-top 16918  df-bases 16920  df-topon 16921  df-nei 17117  df-fil 17831  df-flim 17924  df-cfil 19161
  Copyright terms: Public domain W3C validator