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Theorem nvvcop 22066
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 22065 . . 3  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
21sseli 3337 . 2  |-  ( <. W ,  N >.  e.  NrmCVec 
->  <. W ,  N >.  e.  ( CVec OLD  X. 
_V ) )
3 opelxp1 4904 . 2  |-  ( <. W ,  N >.  e.  ( CVec OLD  X.  _V )  ->  W  e. 
CVec OLD )
42, 3syl 16 1  |-  ( <. W ,  N >.  e.  NrmCVec 
->  W  e.  CVec OLD )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   _Vcvv 2949   <.cop 3810    X. cxp 4869   CVec OLDcvc 22017   NrmCVeccnv 22056
This theorem is referenced by:  nvex  22083  nvoprne  22160
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  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-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-opab 4260  df-xp 4877  df-oprab 6078  df-nv 22064
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