MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caufpm Structured version   Unicode version

Theorem caufpm 19227
Description: Inclusion of a Cauchy sequence, under our definition. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)
Assertion
Ref Expression
caufpm  |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  ( Cau `  D ) )  ->  F  e.  ( X  ^pm  CC ) )

Proof of Theorem caufpm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscau 19221 . 2  |-  ( D  e.  ( * Met `  X )  ->  ( F  e.  ( Cau `  D )  <->  ( F  e.  ( X  ^pm  CC )  /\  A. x  e.  RR+  E. y  e.  ZZ  ( F  |`  ( ZZ>= `  y ) ) : ( ZZ>= `  y ) --> ( ( F `  y ) ( ball `  D ) x ) ) ) )
21simprbda 607 1  |-  ( ( D  e.  ( * Met `  X )  /\  F  e.  ( Cau `  D ) )  ->  F  e.  ( X  ^pm  CC ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2697   E.wrex 2698    |` cres 4872   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^pm cpm 7011   CCcc 8980   ZZcz 10274   ZZ>=cuz 10480   RR+crp 10604   * Metcxmt 16678   ballcbl 16680   Caucca 19198
This theorem is referenced by:  cmetcaulem  19233  causs  19243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-xr 9116  df-xmet 16687  df-cau 19201
  Copyright terms: Public domain W3C validator