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Theorem nvex 25631
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 25614 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  <. G ,  S >.  e.  CVecOLD )
2 vcex 25600 . . 3  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  ( G  e.  _V  /\  S  e. 
_V ) )
31, 2syl 16 . 2  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  ( G  e.  _V  /\  S  e.  _V ) )
4 nvss 25613 . . . 4  |-  NrmCVec  C_  ( CVecOLD  X.  _V )
54sseli 3495 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  <. <. G ,  S >. ,  N >.  e.  ( CVecOLD  X.  _V ) )
6 opelxp2 5042 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  ( CVecOLD  X.  _V )  ->  N  e.  _V )
75, 6syl 16 . 2  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  N  e.  _V )
8 df-3an 975 . 2  |-  ( ( G  e.  _V  /\  S  e.  _V  /\  N  e.  _V )  <->  ( ( G  e.  _V  /\  S  e.  _V )  /\  N  e.  _V ) )
93, 7, 8sylanbrc 664 1  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1819   _Vcvv 3109   <.cop 4038    X. cxp 5006   CVecOLDcvc 25565   NrmCVeccnv 25604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-oprab 6300  df-vc 25566  df-nv 25612
This theorem is referenced by:  isnv  25632  h2hva  26018  h2hsm  26019  h2hnm  26020
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