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Theorem bnnv 22368
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 2436 . . 3  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 eqid 2436 . . 3  |-  ( IndMet `  U )  =  (
IndMet `  U )
31, 2iscbn 22366 . 2  |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec 
/\  ( IndMet `  U
)  e.  ( CMet `  ( BaseSet `  U )
) ) )
43simplbi 447 1  |-  ( U  e.  CBan  ->  U  e.  NrmCVec )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   ` cfv 5454   CMetcms 19207   NrmCVeccnv 22063   BaseSetcba 22065   IndMetcims 22070   CBanccbn 22364
This theorem is referenced by:  bnrel  22369  bnsscmcl  22370  ubthlem1  22372  ubthlem2  22373  ubthlem3  22374  minvecolem1  22376  hlnv  22393
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-cbn 22365
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