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Theorem nvex 22082
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 22065 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  <. G ,  S >.  e.  CVec OLD )
2 vcex 22051 . . 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 22064 . . . 4  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
54sseli 3336 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  <. <. G ,  S >. ,  N >.  e.  ( CVec OLD  X.  _V ) )
6 opelxp2 4904 . . 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 1725   _Vcvv 2948   <.cop 3809    X. cxp 4868   CVec OLDcvc 22016   NrmCVeccnv 22055
This theorem is referenced by:  isnv  22083  h2hva  22469  h2hsm  22470  h2hnm  22471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-oprab 6077  df-vc 22017  df-nv 22063
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