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Theorem flimcfil 18739
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 2283 . . . . 5  |-  U. J  =  U. J
21flimfil 17664 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. J
) )
32adantl 452 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  F  e.  ( Fil `  U. J
) )
4 lmcau.1 . . . . . 6  |-  J  =  ( MetOpen `  D )
54mopnuni 17987 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
65adantr 451 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  X  =  U. J )
76fveq2d 5529 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  ( Fil `  X )  =  ( Fil `  U. J
) )
83, 7eleqtrrd 2360 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  F  e.  ( Fil `  X ) )
91flimelbas 17663 . . . . . 6  |-  ( A  e.  ( J  fLim  F )  ->  A  e.  U. J )
109ad2antlr 707 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  A  e. 
U. J )
115ad2antrr 706 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  X  = 
U. J )
1210, 11eleqtrrd 2360 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  A  e.  X )
13 simplr 731 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  A  e.  ( J  fLim  F
) )
144mopntop 17986 . . . . . . 7  |-  ( D  e.  ( * Met `  X )  ->  J  e.  Top )
1514ad2antrr 706 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  J  e. 
Top )
16 simpll 730 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  D  e.  ( * Met `  X
) )
17 rpxr 10361 . . . . . . . 8  |-  ( x  e.  RR+  ->  x  e. 
RR* )
1817adantl 452 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  x  e. 
RR* )
194blopn 18046 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  x  e.  RR* )  ->  ( A ( ball `  D ) x )  e.  J )
2016, 12, 18, 19syl3anc 1182 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  ( A ( ball `  D
) x )  e.  J )
21 simpr 447 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
22 blcntr 17964 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  x  e.  RR+ )  ->  A  e.  ( A ( ball `  D
) x ) )
2316, 12, 21, 22syl3anc 1182 . . . . . 6  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  A  e.  ( A ( ball `  D ) x ) )
24 opnneip 16856 . . . . . 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 1182 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  ( A ( ball `  D
) x )  e.  ( ( nei `  J
) `  { A } ) )
26 flimnei 17662 . . . . 5  |-  ( ( A  e.  ( J 
fLim  F )  /\  ( A ( ball `  D
) x )  e.  ( ( nei `  J
) `  { A } ) )  -> 
( A ( ball `  D ) x )  e.  F )
2713, 25, 26syl2anc 642 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  ( A ( ball `  D
) x )  e.  F )
28 oveq1 5865 . . . . . 6  |-  ( y  =  A  ->  (
y ( ball `  D
) x )  =  ( A ( ball `  D ) x ) )
2928eleq1d 2349 . . . . 5  |-  ( y  =  A  ->  (
( y ( ball `  D ) x )  e.  F  <->  ( A
( ball `  D )
x )  e.  F
) )
3029rspcev 2884 . . . 4  |-  ( ( A  e.  X  /\  ( A ( ball `  D
) x )  e.  F )  ->  E. y  e.  X  ( y
( ball `  D )
x )  e.  F
)
3112, 27, 30syl2anc 642 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  A  e.  ( J  fLim  F ) )  /\  x  e.  RR+ )  ->  E. y  e.  X  ( y
( ball `  D )
x )  e.  F
)
3231ralrimiva 2626 . 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 18699 . . 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 451 . 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 888 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {csn 3640   U.cuni 3827   ` cfv 5255  (class class class)co 5858   RR*cxr 8866   RR+crp 10354   * Metcxmt 16369   ballcbl 16371   MetOpencmopn 16372   Topctop 16631   neicnei 16834   Filcfil 17540    fLim cflim 17629  CauFilccfil 18678
This theorem is referenced by:  cmetss  18740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ico 10662  df-topgen 13344  df-xmet 16373  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-nei 16835  df-fbas 17520  df-fil 17541  df-flim 17634  df-cfil 18681
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