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Theorem flimcfil 19266
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 2436 . . . . 5  |-  U. J  =  U. J
21flimfil 18001 . . . 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 18471 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
65adantr 452 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  X  =  U. J )
76fveq2d 5732 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  ( Fil `  X )  =  ( Fil `  U. J
) )
83, 7eleqtrrd 2513 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  F  e.  ( Fil `  X ) )
91flimelbas 18000 . . . . . 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 2513 . . . 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 18470 . . . . . . 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 10619 . . . . . . . 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 18530 . . . . . . 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 18443 . . . . . . 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 17183 . . . . . 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 17999 . . . . 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 6088 . . . . . 6  |-  ( y  =  A  ->  (
y ( ball `  D
) x )  =  ( A ( ball `  D ) x ) )
2928eleq1d 2502 . . . . 5  |-  ( y  =  A  ->  (
( y ( ball `  D ) x )  e.  F  <->  ( A
( ball `  D )
x )  e.  F
) )
3029rspcev 3052 . . . 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 2789 . 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 19226 . . 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 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   {csn 3814   U.cuni 4015   ` cfv 5454  (class class class)co 6081   RR*cxr 9119   RR+crp 10612   * Metcxmt 16686   ballcbl 16688   MetOpencmopn 16691   Topctop 16958   neicnei 17161   Filcfil 17877    fLim cflim 17966  CauFilccfil 19205
This theorem is referenced by:  cmetss  19267  fmcncfil  24317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ico 10922  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-bl 16697  df-mopn 16698  df-fbas 16699  df-top 16963  df-bases 16965  df-topon 16966  df-nei 17162  df-fil 17878  df-flim 17971  df-cfil 19208
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