MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnvc Unicode version

Theorem isnvc 18301
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 18206 . 2  |- NrmVec  =  (NrmMod 
i^i  LVec )
21elin2 3435 1  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  W  e.  LVec ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1710   LVecclvec 15948  NrmModcnlm 18199  NrmVeccnvc 18200
This theorem is referenced by:  nvcnlm  18302  nvclvec  18303  isnvc2  18305  rlmnvc  18309  cphnvc  18710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866  df-in 3235  df-nvc 18206
  Copyright terms: Public domain W3C validator