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Theorem nvvcop 26294
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.  CVecOLD )

Proof of Theorem nvvcop
StepHypRef Expression
1 nvss 26293 . . 3  |-  NrmCVec  C_  ( CVecOLD  X.  _V )
21sseli 3414 . 2  |-  ( <. W ,  N >.  e.  NrmCVec 
->  <. W ,  N >.  e.  ( CVecOLD  X. 
_V ) )
3 opelxp1 4872 . 2  |-  ( <. W ,  N >.  e.  ( CVecOLD  X.  _V )  ->  W  e. 
CVecOLD )
42, 3syl 17 1  |-  ( <. W ,  N >.  e.  NrmCVec 
->  W  e.  CVecOLD )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1904   _Vcvv 3031   <.cop 3965    X. cxp 4837   CVecOLDcvc 26245   NrmCVeccnv 26284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-xp 4845  df-oprab 6312  df-nv 26292
This theorem is referenced by:  nvex  26311  nvoprne  26388
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