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Theorem nvss 22033
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 2472 . . . . . . 7  |-  ( w  =  <. g ,  s
>.  ->  ( w  e. 
CVec OLD  <->  <. g ,  s
>.  e.  CVec OLD ) )
21biimpar 472 . . . . . 6  |-  ( ( w  =  <. g ,  s >.  /\  <. g ,  s >.  e.  CVec OLD )  ->  w  e.  CVec
OLD )
323ad2antr1 1122 . . . . 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 1642 . . . 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 2927 . . . 4  |-  n  e. 
_V
64, 5jctir 525 . . 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 4450 . 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 22032 . . 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 6088 . . 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 2432 . 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 4851 . 2  |-  ( CVec
OLD  X.  _V )  =  { <. w ,  n >.  |  ( w  e. 
CVec OLD  /\  n  e. 
_V ) }
127, 10, 113sstr4i 3355 1  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2674   _Vcvv 2924    C_ wss 3288   <.cop 3785   class class class wbr 4180   {copab 4233    X. cxp 4843   ran crn 4846   -->wf 5417   ` cfv 5421  (class class class)co 6048   {coprab 6049   CCcc 8952   RRcr 8953   0cc0 8954    + caddc 8957    x. cmul 8959    <_ cle 9085   abscabs 12002  GIdcgi 21736   CVec OLDcvc 21985   NrmCVeccnv 22024
This theorem is referenced by:  nvvcop  22034  nvrel  22042  nvvop  22049  nvex  22051
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
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 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-opab 4235  df-xp 4851  df-oprab 6052  df-nv 22032
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