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

Theorem flimcfil 18792
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 2316 . . . . 5  |-  U. J  =  U. J
21flimfil 17716 . . . 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 18039 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
65adantr 451 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  X  =  U. J )
76fveq2d 5567 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  ( Fil `  X )  =  ( Fil `  U. J
) )
83, 7eleqtrrd 2393 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  ( J  fLim  F )
)  ->  F  e.  ( Fil `  X ) )
91flimelbas 17715 . . . . . 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 2393 . . . 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 18038 . . . . . . 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 10408 . . . . . . . 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 18098 . . . . . . 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 18016 . . . . . . 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 16912 . . . . . 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 17714 . . . . 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 5907 . . . . . 6  |-  ( y  =  A  ->  (
y ( ball `  D
) x )  =  ( A ( ball `  D ) x ) )
2928eleq1d 2382 . . . . 5  |-  ( y  =  A  ->  (
( y ( ball `  D ) x )  e.  F  <->  ( A
( ball `  D )
x )  e.  F
) )
3029rspcev 2918 . . . 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 2660 . 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 18752 . . 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 1633    e. wcel 1701   A.wral 2577   E.wrex 2578   {csn 3674   U.cuni 3864   ` cfv 5292  (class class class)co 5900   RR*cxr 8911   RR+crp 10401   * Metcxmt 16418   ballcbl 16420   MetOpencmopn 16423   Topctop 16687   neicnei 16890   Filcfil 17592    fLim cflim 17681  CauFilccfil 18731
This theorem is referenced by:  cmetss  18793  fmcncfil  23386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-n0 10013  df-z 10072  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ico 10709  df-topgen 13393  df-xmet 16425  df-bl 16427  df-mopn 16428  df-fbas 16429  df-top 16692  df-bases 16694  df-topon 16695  df-nei 16891  df-fil 17593  df-flim 17686  df-cfil 18734
  Copyright terms: Public domain W3C validator