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Theorem bnnv 24420
Description: Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Assertion
Ref Expression
bnnv  |-  ( U  e.  CBan  ->  U  e.  NrmCVec )

Proof of Theorem bnnv
StepHypRef Expression
1 eqid 2454 . . 3  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 eqid 2454 . . 3  |-  ( IndMet `  U )  =  (
IndMet `  U )
31, 2iscbn 24418 . 2  |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec 
/\  ( IndMet `  U
)  e.  ( CMet `  ( BaseSet `  U )
) ) )
43simplbi 460 1  |-  ( U  e.  CBan  ->  U  e.  NrmCVec )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   ` cfv 5527   CMetcms 20898   NrmCVeccnv 24115   BaseSetcba 24117   IndMetcims 24122   CBanccbn 24416
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 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-iota 5490  df-fv 5535  df-cbn 24417
This theorem is referenced by:  bnrel  24421  bnsscmcl  24422  ubthlem1  24424  ubthlem2  24425  ubthlem3  24426  minvecolem1  24428  hlnv  24445
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