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Theorem isnvc 18205
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 18110 . 2  |- NrmVec  =  (NrmMod 
i^i  LVec )
21elin2 3359 1  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  W  e.  LVec ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   LVecclvec 15855  NrmModcnlm 18103  NrmVeccnvc 18104
This theorem is referenced by:  nvcnlm  18206  nvclvec  18207  isnvc2  18209  rlmnvc  18213  cphnvc  18612
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-nvc 18110
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