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Theorem flimcfil 20829
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 2443 . . . . 5  |-  U. J  =  U. J
21flimfil 19547 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. J
) )
32adantl 466 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  F  e.  ( Fil `  U. J
) )
4 lmcau.1 . . . . . 6  |-  J  =  ( MetOpen `  D )
54mopnuni 20021 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  X  =  U. J )
65adantr 465 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  X  =  U. J )
76fveq2d 5700 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  ( Fil `  X )  =  ( Fil `  U. J
) )
83, 7eleqtrrd 2520 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  F  e.  ( Fil `  X ) )
91flimelbas 19546 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  U. J )
109ad2antlr 726 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  A  e. 
U. J )
115ad2antrr 725 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  X  = 
U. J )
1210, 11eleqtrrd 2520 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  A  e.  X )
13 simplr 754 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  A  e.  ( J  fLim  F
) )
144mopntop 20020 . . . . . . 7  |-  ( D  e.  ( *Met `  X )  ->  J  e.  Top )
1514ad2antrr 725 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  J  e. 
Top )
16 simpll 753 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  D  e.  ( *Met `  X ) )
17 rpxr 11003 . . . . . . . 8  |-  ( x  e.  RR+  ->  x  e. 
RR* )
1817adantl 466 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  x  e. 
RR* )
194blopn 20080 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  x  e.  RR* )  ->  ( A ( ball `  D ) x )  e.  J )
2016, 12, 18, 19syl3anc 1218 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  ( A ( ball `  D
) x )  e.  J )
21 simpr 461 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
22 blcntr 19993 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  x  e.  RR+ )  ->  A  e.  ( A ( ball `  D
) x ) )
2316, 12, 21, 22syl3anc 1218 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  A  e.  ( A ( ball `  D ) x ) )
24 opnneip 18728 . . . . . 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 1218 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  ( A ( ball `  D
) x )  e.  ( ( nei `  J
) `  { A } ) )
26 flimnei 19545 . . . . 5  |-  ( ( A  e.  ( J 
fLim  F )  /\  ( A ( ball `  D
) x )  e.  ( ( nei `  J
) `  { A } ) )  -> 
( A ( ball `  D ) x )  e.  F )
2713, 25, 26syl2anc 661 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  ( A ( ball `  D
) x )  e.  F )
28 oveq1 6103 . . . . . 6  |-  ( y  =  A  ->  (
y ( ball `  D
) x )  =  ( A ( ball `  D ) x ) )
2928eleq1d 2509 . . . . 5  |-  ( y  =  A  ->  (
( y ( ball `  D ) x )  e.  F  <->  ( A
( ball `  D )
x )  e.  F
) )
3029rspcev 3078 . . . 4  |-  ( ( A  e.  X  /\  ( A ( ball `  D
) x )  e.  F )  ->  E. y  e.  X  ( y
( ball `  D )
x )  e.  F
)
3112, 27, 30syl2anc 661 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  E. y  e.  X  ( y
( ball `  D )
x )  e.  F
)
3231ralrimiva 2804 . 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 20789 . . 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 465 . 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 913 1  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  F  e.  (CauFil `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   {csn 3882   U.cuni 4096   ` cfv 5423  (class class class)co 6096   RR*cxr 9422   RR+crp 10996   *Metcxmt 17806   ballcbl 17808   MetOpencmopn 17811   Topctop 18503   neicnei 18706   Filcfil 19423    fLim cflim 19512  CauFilccfil 20768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ico 11311  df-topgen 14387  df-psmet 17814  df-xmet 17815  df-bl 17817  df-mopn 17818  df-fbas 17819  df-top 18508  df-bases 18510  df-topon 18511  df-nei 18707  df-fil 19424  df-flim 19517  df-cfil 20771
This theorem is referenced by:  cmetss  20830  fmcncfil  26366
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