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Theorem nvvcop 21914
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 21913 . . 3  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
21sseli 3280 . 2  |-  ( <. W ,  N >.  e.  NrmCVec 
->  <. W ,  N >.  e.  ( CVec OLD  X. 
_V ) )
3 opelxp1 4844 . 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 1717   _Vcvv 2892   <.cop 3753    X. cxp 4809   CVec OLDcvc 21865   NrmCVeccnv 21904
This theorem is referenced by:  nvex  21931  nvoprne  22008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-opab 4201  df-xp 4817  df-oprab 6017  df-nv 21912
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