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Theorem cbncms 21444
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  |-  X  =  ( BaseSet `  U )
iscbn.8  |-  D  =  ( IndMet `  U )
Assertion
Ref Expression
cbncms  |-  ( U  e.  CBan  ->  D  e.  ( CMet `  X
) )

Proof of Theorem cbncms
StepHypRef Expression
1 iscbn.x . . 3  |-  X  =  ( BaseSet `  U )
2 iscbn.8 . . 3  |-  D  =  ( IndMet `  U )
31, 2iscbn 21443 . 2  |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec 
/\  D  e.  (
CMet `  X )
) )
43simprbi 450 1  |-  ( U  e.  CBan  ->  D  e.  ( CMet `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255   CMetcms 18680   NrmCVeccnv 21140   BaseSetcba 21142   IndMetcims 21147   CBanccbn 21441
This theorem is referenced by:  bnsscmcl  21447  ubthlem1  21449  ubthlem2  21450  minvecolem4a  21456  hlcmet  21473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-cbn 21442
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