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Theorem nvex 21183
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 21166 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  <. G ,  S >.  e.  CVec OLD )
2 vcex 21152 . . 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 21165 . . . 4  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
54sseli 3189 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  <. <. G ,  S >. ,  N >.  e.  ( CVec OLD  X.  _V ) )
6 opelxp2 4739 . . 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 1696   _Vcvv 2801   <.cop 3656    X. cxp 4703   CVec OLDcvc 21117   NrmCVeccnv 21156
This theorem is referenced by:  isnv  21184  h2hva  21570  h2hsm  21571  h2hnm  21572
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-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-oprab 5878  df-vc 21118  df-nv 21164
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