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Theorem iscau 18702
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 16959. (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 18701 . . 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 2350 . 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 4949 . . . . . . 7  |-  ( f  =  F  ->  (
f  |`  ( ZZ>= `  k
) )  =  ( F  |`  ( ZZ>= `  k ) ) )
43feq1d 5379 . . . . . 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 2284 . . . . . . 7  |-  ( f  =  F  ->  ( ZZ>=
`  k )  =  ( ZZ>= `  k )
)
6 fveq1 5524 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  k )  =  ( F `  k ) )
76oveq1d 5873 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  k
) ( ball `  D
) x )  =  ( ( F `  k ) ( ball `  D ) x ) )
85, 7feq23d 5386 . . . . . 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 2564 . . . 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 2563 . . 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 2923 . 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 1684   A.wral 2543   E.wrex 2544   {crab 2547    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   CCcc 8735   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   * Metcxmt 16369   ballcbl 16371   Caucca 18679
This theorem is referenced by:  iscau2  18703  caufpm  18708  lmcau  18738
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-xr 8871  df-xmet 16373  df-cau 18682
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