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Theorem nvex 21167
Description: The components of a normed complex vector space are sets. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
Assertion
Ref Expression
nvex  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V ) )

Proof of Theorem nvex
StepHypRef Expression
1 nvvcop 21150 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  <. G ,  S >.  e.  CVec OLD )
2 vcex 21136 . . 3  |-  ( <. G ,  S >.  e. 
CVec OLD  ->  ( G  e.  _V  /\  S  e. 
_V ) )
31, 2syl 15 . 2  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  ( G  e.  _V  /\  S  e.  _V ) )
4 nvss 21149 . . . 4  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
54sseli 3176 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  <. <. G ,  S >. ,  N >.  e.  ( CVec OLD  X.  _V ) )
6 opelxp2 4723 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  ( CVec OLD  X.  _V )  ->  N  e.  _V )
75, 6syl 15 . 2  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  N  e.  _V )
8 df-3an 936 . 2  |-  ( ( G  e.  _V  /\  S  e.  _V  /\  N  e.  _V )  <->  ( ( G  e.  _V  /\  S  e.  _V )  /\  N  e.  _V ) )
93, 7, 8sylanbrc 645 1  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684   _Vcvv 2788   <.cop 3643    X. cxp 4687   CVec OLDcvc 21101   NrmCVeccnv 21140
This theorem is referenced by:  isnv  21168  h2hva  21554  h2hsm  21555  h2hnm  21556
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-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-oprab 5862  df-vc 21102  df-nv 21148
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