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Theorem cbncms 24445
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 24444 . 2  |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec 
/\  D  e.  (
CMet `  X )
) )
43simprbi 464 1  |-  ( U  e.  CBan  ->  D  e.  ( CMet `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   ` cfv 5529   CMetcms 20907   NrmCVeccnv 24141   BaseSetcba 24143   IndMetcims 24148   CBanccbn 24442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-cbn 24443
This theorem is referenced by:  bnsscmcl  24448  ubthlem1  24450  ubthlem2  24451  minvecolem4a  24457  hlcmet  24474
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