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Theorem nvex 21931
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 21914 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  <. G ,  S >.  e.  CVec OLD )
2 vcex 21900 . . 3  |-  ( <. G ,  S >.  e. 
CVec OLD  ->  ( G  e.  _V  /\  S  e. 
_V ) )
31, 2syl 16 . 2  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  ( G  e.  _V  /\  S  e.  _V ) )
4 nvss 21913 . . . 4  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
54sseli 3280 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  <. <. G ,  S >. ,  N >.  e.  ( CVec OLD  X.  _V ) )
6 opelxp2 4845 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  ( CVec OLD  X.  _V )  ->  N  e.  _V )
75, 6syl 16 . 2  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  N  e.  _V )
8 df-3an 938 . 2  |-  ( ( G  e.  _V  /\  S  e.  _V  /\  N  e.  _V )  <->  ( ( G  e.  _V  /\  S  e.  _V )  /\  N  e.  _V ) )
93, 7, 8sylanbrc 646 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 359    /\ w3a 936    e. wcel 1717   _Vcvv 2892   <.cop 3753    X. cxp 4809   CVec OLDcvc 21865   NrmCVeccnv 21904
This theorem is referenced by:  isnv  21932  h2hva  22318  h2hsm  22319  h2hnm  22320
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-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-oprab 6017  df-vc 21866  df-nv 21912
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