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Theorem bnnvc 22227
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 2420 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
21isbn 22225 . 2  |-  ( W  e. Ban 
<->  ( W  e. NrmVec  /\  W  e. CMetSp  /\  (Scalar `  W
)  e. CMetSp ) )
32simp1bi 1020 1  |-  ( W  e. Ban  ->  W  e. NrmVec )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1867   ` cfv 5592  Scalarcsca 15153  NrmVeccnvc 21533  CMetSpccms 22219  Bancbn 22220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-iota 5556  df-fv 5600  df-bn 22223
This theorem is referenced by:  bnnlm  22228  lssbn  22238
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