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Theorem iscau 18917
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 17176. (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 18916 . . 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 2433 . 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 5052 . . . . . . 7  |-  ( f  =  F  ->  (
f  |`  ( ZZ>= `  k
) )  =  ( F  |`  ( ZZ>= `  k ) ) )
43feq1d 5484 . . . . . 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 2367 . . . . . . 7  |-  ( f  =  F  ->  ( ZZ>=
`  k )  =  ( ZZ>= `  k )
)
6 fveq1 5631 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  k )  =  ( F `  k ) )
76oveq1d 5996 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  k
) ( ball `  D
) x )  =  ( ( F `  k ) ( ball `  D ) x ) )
85, 7feq23d 5492 . . . . . 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 2649 . . . 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 2648 . . 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 3009 . 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 1647    e. wcel 1715   A.wral 2628   E.wrex 2629   {crab 2632    |` cres 4794   -->wf 5354   ` cfv 5358  (class class class)co 5981    ^pm cpm 6916   CCcc 8882   ZZcz 10175   ZZ>=cuz 10381   RR+crp 10505   * Metcxmt 16579   ballcbl 16581   Caucca 18894
This theorem is referenced by:  iscau2  18918  caufpm  18923  lmcau  18953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-map 6917  df-xr 9018  df-xmet 16586  df-cau 18897
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