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Theorem nvss 21149
Description: Structure of the class of all normed complex vectors spaces. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
Assertion
Ref Expression
nvss  |-  NrmCVec  C_  ( CVec OLD  X.  _V )

Proof of Theorem nvss
Dummy variables  g 
s  n  w  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2343 . . . . . . 7  |-  ( w  =  <. g ,  s
>.  ->  ( w  e. 
CVec OLD  <->  <. g ,  s
>.  e.  CVec OLD ) )
21biimpar 471 . . . . . 6  |-  ( ( w  =  <. g ,  s >.  /\  <. g ,  s >.  e.  CVec OLD )  ->  w  e.  CVec
OLD )
323ad2antr1 1120 . . . . 5  |-  ( ( w  =  <. g ,  s >.  /\  ( <. g ,  s >.  e.  CVec OLD  /\  n : ran  g --> RR  /\  A. x  e.  ran  g
( ( ( n `
 x )  =  0  ->  x  =  (GId `  g ) )  /\  A. y  e.  CC  ( n `  ( y s x ) )  =  ( ( abs `  y
)  x.  ( n `
 x ) )  /\  A. y  e. 
ran  g ( n `
 ( x g y ) )  <_ 
( ( n `  x )  +  ( n `  y ) ) ) ) )  ->  w  e.  CVec OLD )
43exlimivv 1667 . . . 4  |-  ( E. g E. s ( w  =  <. g ,  s >.  /\  ( <. g ,  s >.  e.  CVec OLD  /\  n : ran  g --> RR  /\  A. x  e.  ran  g
( ( ( n `
 x )  =  0  ->  x  =  (GId `  g ) )  /\  A. y  e.  CC  ( n `  ( y s x ) )  =  ( ( abs `  y
)  x.  ( n `
 x ) )  /\  A. y  e. 
ran  g ( n `
 ( x g y ) )  <_ 
( ( n `  x )  +  ( n `  y ) ) ) ) )  ->  w  e.  CVec OLD )
5 vex 2791 . . . 4  |-  n  e. 
_V
64, 5jctir 524 . . 3  |-  ( E. g E. s ( w  =  <. g ,  s >.  /\  ( <. g ,  s >.  e.  CVec OLD  /\  n : ran  g --> RR  /\  A. x  e.  ran  g
( ( ( n `
 x )  =  0  ->  x  =  (GId `  g ) )  /\  A. y  e.  CC  ( n `  ( y s x ) )  =  ( ( abs `  y
)  x.  ( n `
 x ) )  /\  A. y  e. 
ran  g ( n `
 ( x g y ) )  <_ 
( ( n `  x )  +  ( n `  y ) ) ) ) )  ->  ( w  e. 
CVec OLD  /\  n  e. 
_V ) )
76ssopab2i 4292 . 2  |-  { <. w ,  n >.  |  E. g E. s ( w  =  <. g ,  s
>.  /\  ( <. g ,  s >.  e.  CVec OLD 
/\  n : ran  g
--> RR  /\  A. x  e.  ran  g ( ( ( n `  x
)  =  0  ->  x  =  (GId `  g
) )  /\  A. y  e.  CC  (
n `  ( y
s x ) )  =  ( ( abs `  y )  x.  (
n `  x )
)  /\  A. y  e.  ran  g ( n `
 ( x g y ) )  <_ 
( ( n `  x )  +  ( n `  y ) ) ) ) ) }  C_  { <. w ,  n >.  |  (
w  e.  CVec OLD  /\  n  e.  _V ) }
8 df-nv 21148 . . 3  |-  NrmCVec  =  { <. <. g ,  s
>. ,  n >.  |  ( <. g ,  s
>.  e.  CVec OLD  /\  n : ran  g --> RR  /\  A. x  e.  ran  g
( ( ( n `
 x )  =  0  ->  x  =  (GId `  g ) )  /\  A. y  e.  CC  ( n `  ( y s x ) )  =  ( ( abs `  y
)  x.  ( n `
 x ) )  /\  A. y  e. 
ran  g ( n `
 ( x g y ) )  <_ 
( ( n `  x )  +  ( n `  y ) ) ) ) }
9 dfoprab2 5895 . . 3  |-  { <. <.
g ,  s >. ,  n >.  |  ( <. g ,  s >.  e.  CVec OLD  /\  n : ran  g --> RR  /\  A. x  e.  ran  g
( ( ( n `
 x )  =  0  ->  x  =  (GId `  g ) )  /\  A. y  e.  CC  ( n `  ( y s x ) )  =  ( ( abs `  y
)  x.  ( n `
 x ) )  /\  A. y  e. 
ran  g ( n `
 ( x g y ) )  <_ 
( ( n `  x )  +  ( n `  y ) ) ) ) }  =  { <. w ,  n >.  |  E. g E. s ( w  =  <. g ,  s
>.  /\  ( <. g ,  s >.  e.  CVec OLD 
/\  n : ran  g
--> RR  /\  A. x  e.  ran  g ( ( ( n `  x
)  =  0  ->  x  =  (GId `  g
) )  /\  A. y  e.  CC  (
n `  ( y
s x ) )  =  ( ( abs `  y )  x.  (
n `  x )
)  /\  A. y  e.  ran  g ( n `
 ( x g y ) )  <_ 
( ( n `  x )  +  ( n `  y ) ) ) ) ) }
108, 9eqtri 2303 . 2  |-  NrmCVec  =  { <. w ,  n >.  |  E. g E. s
( w  =  <. g ,  s >.  /\  ( <. g ,  s >.  e.  CVec OLD  /\  n : ran  g --> RR  /\  A. x  e.  ran  g
( ( ( n `
 x )  =  0  ->  x  =  (GId `  g ) )  /\  A. y  e.  CC  ( n `  ( y s x ) )  =  ( ( abs `  y
)  x.  ( n `
 x ) )  /\  A. y  e. 
ran  g ( n `
 ( x g y ) )  <_ 
( ( n `  x )  +  ( n `  y ) ) ) ) ) }
11 df-xp 4695 . 2  |-  ( CVec
OLD  X.  _V )  =  { <. w ,  n >.  |  ( w  e. 
CVec OLD  /\  n  e. 
_V ) }
127, 10, 113sstr4i 3217 1  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   <.cop 3643   class class class wbr 4023   {copab 4076    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   {coprab 5859   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    <_ cle 8868   abscabs 11719  GIdcgi 20854   CVec OLDcvc 21101   NrmCVeccnv 21140
This theorem is referenced by:  nvvcop  21150  nvrel  21158  nvvop  21165  nvex  21167
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-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-opab 4078  df-xp 4695  df-oprab 5862  df-nv 21148
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