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

Theorem bnnv 21500
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 2316 . . 3  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 eqid 2316 . . 3  |-  ( IndMet `  U )  =  (
IndMet `  U )
31, 2iscbn 21498 . 2  |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec 
/\  ( IndMet `  U
)  e.  ( CMet `  ( BaseSet `  U )
) ) )
43simplbi 446 1  |-  ( U  e.  CBan  ->  U  e.  NrmCVec )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1701   ` cfv 5292   CMetcms 18733   NrmCVeccnv 21195   BaseSetcba 21197   IndMetcims 21202   CBanccbn 21496
This theorem is referenced by:  bnrel  21501  bnsscmcl  21502  ubthlem1  21504  ubthlem2  21505  ubthlem3  21506  minvecolem1  21508  hlnv  21525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-cbn 21497
  Copyright terms: Public domain W3C validator