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Theorem bnnv 25924
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 2396 . . 3  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 eqid 2396 . . 3  |-  ( IndMet `  U )  =  (
IndMet `  U )
31, 2iscbn 25922 . 2  |-  ( U  e.  CBan  <->  ( U  e.  NrmCVec 
/\  ( IndMet `  U
)  e.  ( CMet `  ( BaseSet `  U )
) ) )
43simplbi 458 1  |-  ( U  e.  CBan  ->  U  e.  NrmCVec )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1836   ` cfv 5513   CMetcms 21801   NrmCVeccnv 25619   BaseSetcba 25621   IndMetcims 25626   CBanccbn 25920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-rex 2752  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-iota 5477  df-fv 5521  df-cbn 25921
This theorem is referenced by:  bnrel  25925  bnsscmcl  25926  ubthlem1  25928  ubthlem2  25929  ubthlem3  25930  minvecolem1  25932  hlnv  25949
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