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Theorem bnnvc 21606
Description: A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
bnnvc  |-  ( W  e. Ban  ->  W  e. NrmVec )

Proof of Theorem bnnvc
StepHypRef Expression
1 eqid 2467 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
21isbn 21604 . 2  |-  ( W  e. Ban 
<->  ( W  e. NrmVec  /\  W  e. CMetSp  /\  (Scalar `  W
)  e. CMetSp ) )
32simp1bi 1011 1  |-  ( W  e. Ban  ->  W  e. NrmVec )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   ` cfv 5588  Scalarcsca 14561  NrmVeccnvc 20929  CMetSpccms 21598  Bancbn 21599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-bn 21602
This theorem is referenced by:  bnnlm  21607  lssbn  21617
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