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Theorem nvvcop 21150
Description: A normed complex vector space is a vector space. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
Assertion
Ref Expression
nvvcop  |-  ( <. W ,  N >.  e.  NrmCVec 
->  W  e.  CVec OLD )

Proof of Theorem nvvcop
StepHypRef Expression
1 nvss 21149 . . 3  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
21sseli 3176 . 2  |-  ( <. W ,  N >.  e.  NrmCVec 
->  <. W ,  N >.  e.  ( CVec OLD  X. 
_V ) )
3 opelxp1 4722 . 2  |-  ( <. W ,  N >.  e.  ( CVec OLD  X.  _V )  ->  W  e. 
CVec OLD )
42, 3syl 15 1  |-  ( <. W ,  N >.  e.  NrmCVec 
->  W  e.  CVec OLD )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   _Vcvv 2788   <.cop 3643    X. cxp 4687   CVec OLDcvc 21101   NrmCVeccnv 21140
This theorem is referenced by:  nvex  21167  nvoprne  21244
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-oprab 5862  df-nv 21148
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