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Theorem nvvcop 21166
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 21165 . . 3  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
21sseli 3189 . 2  |-  ( <. W ,  N >.  e.  NrmCVec 
->  <. W ,  N >.  e.  ( CVec OLD  X. 
_V ) )
3 opelxp1 4738 . 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 1696   _Vcvv 2801   <.cop 3656    X. cxp 4703   CVec OLDcvc 21117   NrmCVeccnv 21156
This theorem is referenced by:  nvex  21183  nvoprne  21260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-oprab 5878  df-nv 21164
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