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Theorem iscau 18698
Description: Express the property " F is a Cauchy sequence of metric  D." Part of Definition 1.4-3 of [Kreyszig] p. 28. The condition  F  C_  ( CC  X.  X ) allows us to use objects more general than sequences when convenient; see the comment in df-lm 16955. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
Assertion
Ref Expression
iscau  |-  ( D  e.  ( * Met `  X )  ->  ( F  e.  ( Cau `  D )  <->  ( F  e.  ( X  ^pm  CC )  /\  A. x  e.  RR+  E. k  e.  ZZ  ( F  |`  ( ZZ>= `  k ) ) : ( ZZ>= `  k ) --> ( ( F `  k ) ( ball `  D ) x ) ) ) )
Distinct variable groups:    x, k, D    k, F, x    k, X, x

Proof of Theorem iscau
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 caufval 18697 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ( Cau `  D )  =  { f  e.  ( X  ^pm  CC )  |  A. x  e.  RR+  E. k  e.  ZZ  (
f  |`  ( ZZ>= `  k
) ) : (
ZZ>= `  k ) --> ( ( f `  k
) ( ball `  D
) x ) } )
21eleq2d 2351 . 2  |-  ( D  e.  ( * Met `  X )  ->  ( F  e.  ( Cau `  D )  <->  F  e.  { f  e.  ( X 
^pm  CC )  |  A. x  e.  RR+  E. k  e.  ZZ  ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  D
) x ) } ) )
3 reseq1 4948 . . . . . . 7  |-  ( f  =  F  ->  (
f  |`  ( ZZ>= `  k
) )  =  ( F  |`  ( ZZ>= `  k ) ) )
43feq1d 5345 . . . . . 6  |-  ( f  =  F  ->  (
( f  |`  ( ZZ>=
`  k ) ) : ( ZZ>= `  k
) --> ( ( f `
 k ) (
ball `  D )
x )  <->  ( F  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  D
) x ) ) )
5 eqidd 2285 . . . . . . 7  |-  ( f  =  F  ->  ( ZZ>=
`  k )  =  ( ZZ>= `  k )
)
6 fveq1 5485 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  k )  =  ( F `  k ) )
76oveq1d 5835 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  k
) ( ball `  D
) x )  =  ( ( F `  k ) ( ball `  D ) x ) )
85, 7feq23d 5352 . . . . . 6  |-  ( f  =  F  ->  (
( F  |`  ( ZZ>=
`  k ) ) : ( ZZ>= `  k
) --> ( ( f `
 k ) (
ball `  D )
x )  <->  ( F  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( F `  k ) ( ball `  D
) x ) ) )
94, 8bitrd 244 . . . . 5  |-  ( f  =  F  ->  (
( f  |`  ( ZZ>=
`  k ) ) : ( ZZ>= `  k
) --> ( ( f `
 k ) (
ball `  D )
x )  <->  ( F  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( F `  k ) ( ball `  D
) x ) ) )
109rexbidv 2565 . . . 4  |-  ( f  =  F  ->  ( E. k  e.  ZZ  ( f  |`  ( ZZ>=
`  k ) ) : ( ZZ>= `  k
) --> ( ( f `
 k ) (
ball `  D )
x )  <->  E. k  e.  ZZ  ( F  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( F `  k ) ( ball `  D
) x ) ) )
1110ralbidv 2564 . . 3  |-  ( f  =  F  ->  ( A. x  e.  RR+  E. k  e.  ZZ  ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  D
) x )  <->  A. x  e.  RR+  E. k  e.  ZZ  ( F  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( F `  k ) ( ball `  D
) x ) ) )
1211elrab 2924 . 2  |-  ( F  e.  { f  e.  ( X  ^pm  CC )  |  A. x  e.  RR+  E. k  e.  ZZ  ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  D
) x ) }  <-> 
( F  e.  ( X  ^pm  CC )  /\  A. x  e.  RR+  E. k  e.  ZZ  ( F  |`  ( ZZ>= `  k
) ) : (
ZZ>= `  k ) --> ( ( F `  k
) ( ball `  D
) x ) ) )
132, 12syl6bb 252 1  |-  ( D  e.  ( * Met `  X )  ->  ( F  e.  ( Cau `  D )  <->  ( F  e.  ( X  ^pm  CC )  /\  A. x  e.  RR+  E. k  e.  ZZ  ( F  |`  ( ZZ>= `  k ) ) : ( ZZ>= `  k ) --> ( ( F `  k ) ( ball `  D ) x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685   A.wral 2544   E.wrex 2545   {crab 2548    |` cres 4690   -->wf 5217   ` cfv 5221  (class class class)co 5820    ^pm cpm 6769   CCcc 8731   ZZcz 10020   ZZ>=cuz 10226   RR+crp 10350   * Metcxmt 16365   ballcbl 16367   Caucca 18675
This theorem is referenced by:  iscau2  18699  caufpm  18704  lmcau  18734
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-map 6770  df-xr 8867  df-xmet 16369  df-cau 18678
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