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Theorem isnvc 18730
Description: A normed vector space is just a normed module which is algebraically a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
isnvc  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  W  e.  LVec ) )

Proof of Theorem isnvc
StepHypRef Expression
1 df-nvc 18635 . 2  |- NrmVec  =  (NrmMod 
i^i  LVec )
21elin2 3531 1  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  W  e.  LVec ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1725   LVecclvec 16174  NrmModcnlm 18628  NrmVeccnvc 18629
This theorem is referenced by:  nvcnlm  18731  nvclvec  18732  isnvc2  18734  rlmnvc  18738  cphnvc  19139
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-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-in 3327  df-nvc 18635
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