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Theorem iscau 18704
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 16961. (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 18703 . . 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 2352 . 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 4951 . . . . . . 7  |-  ( f  =  F  ->  (
f  |`  ( ZZ>= `  k
) )  =  ( F  |`  ( ZZ>= `  k ) ) )
43feq1d 5381 . . . . . 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 2286 . . . . . . 7  |-  ( f  =  F  ->  ( ZZ>=
`  k )  =  ( ZZ>= `  k )
)
6 fveq1 5526 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  k )  =  ( F `  k ) )
76oveq1d 5875 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  k
) ( ball `  D
) x )  =  ( ( F `  k ) ( ball `  D ) x ) )
85, 7feq23d 5388 . . . . . 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 2566 . . . 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 2565 . . 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 2925 . 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 1625    e. wcel 1686   A.wral 2545   E.wrex 2546   {crab 2549    |` cres 4693   -->wf 5253   ` cfv 5257  (class class class)co 5860    ^pm cpm 6775   CCcc 8737   ZZcz 10026   ZZ>=cuz 10232   RR+crp 10356   * Metcxmt 16371   ballcbl 16373   Caucca 18681
This theorem is referenced by:  iscau2  18705  caufpm  18710  lmcau  18740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-map 6776  df-xr 8873  df-xmet 16375  df-cau 18684
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