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Theorem isnvc 18683
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 18588 . 2  |- NrmVec  =  (NrmMod 
i^i  LVec )
21elin2 3491 1  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  W  e.  LVec ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1721   LVecclvec 16129  NrmModcnlm 18581  NrmVeccnvc 18582
This theorem is referenced by:  nvcnlm  18684  nvclvec  18685  isnvc2  18687  rlmnvc  18691  cphnvc  19092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-in 3287  df-nvc 18588
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