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

Theorem cbncms 22359
 Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
iscbn.x
iscbn.8
Assertion
Ref Expression
cbncms

Proof of Theorem cbncms
StepHypRef Expression
1 iscbn.x . . 3
2 iscbn.8 . . 3
31, 2iscbn 22358 . 2
43simprbi 451 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  cfv 5446  cms 19199  cnv 22055  cba 22057  cims 22062  ccbn 22356 This theorem is referenced by:  bnsscmcl  22362  ubthlem1  22364  ubthlem2  22365  minvecolem4a  22371  hlcmet  22388 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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-cbn 22357
 Copyright terms: Public domain W3C validator