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Theorem iscau 18629
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 16886. (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
StepHypRef Expression
1 caufval 18628 . . 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 2323 . 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 4902 . . . . . . 7  |-  ( f  =  F  ->  (
f  |`  ( ZZ>= `  k
) )  =  ( F  |`  ( ZZ>= `  k ) ) )
43feq1d 5282 . . . . . 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 2257 . . . . . . 7  |-  ( f  =  F  ->  ( ZZ>=
`  k )  =  ( ZZ>= `  k )
)
6 fveq1 5422 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  k )  =  ( F `  k ) )
76oveq1d 5772 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  k
) ( ball `  D
) x )  =  ( ( F `  k ) ( ball `  D ) x ) )
85, 7feq23d 5289 . . . . . 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 246 . . . . 5  |-  ( f  =  F  ->  (
( f  |`  ( ZZ>=
`  k ) ) : ( ZZ>= `  k
) --> ( ( f `
 k ) (
ball `  D )
x )  <->  ( F  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( F `  k ) ( ball `  D
) x ) ) )
109rexbidv 2535 . . . 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 2534 . . 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 2874 . 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 254 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   E.wrex 2517   {crab 2519    |` cres 4628   -->wf 4634   ` cfv 4638  (class class class)co 5757    ^pm cpm 6706   CCcc 8668   ZZcz 9956   ZZ>=cuz 10162   RR+crp 10286   * Metcxmt 16296   ballcbl 16298   Caucca 18606
This theorem is referenced by:  iscau2  18630  caufpm  18635  lmcau  18665
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-map 6707  df-xr 8804  df-xmet 16300  df-cau 18609
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