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Theorem nvss 22077
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 2498 . . . . . . 7  |-  ( w  =  <. g ,  s
>.  ->  ( w  e. 
CVec OLD  <->  <. g ,  s
>.  e.  CVec OLD ) )
21biimpar 473 . . . . . 6  |-  ( ( w  =  <. g ,  s >.  /\  <. g ,  s >.  e.  CVec OLD )  ->  w  e.  CVec
OLD )
323ad2antr1 1123 . . . . 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 1646 . . . 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 2961 . . . 4  |-  n  e. 
_V
64, 5jctir 526 . . 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 4485 . 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 22076 . . 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 6124 . . 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 2458 . 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 4887 . 2  |-  ( CVec
OLD  X.  _V )  =  { <. w ,  n >.  |  ( w  e. 
CVec OLD  /\  n  e. 
_V ) }
127, 10, 113sstr4i 3389 1  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    C_ wss 3322   <.cop 3819   class class class wbr 4215   {copab 4268    X. cxp 4879   ran crn 4882   -->wf 5453   ` cfv 5457  (class class class)co 6084   {coprab 6085   CCcc 8993   RRcr 8994   0cc0 8995    + caddc 8998    x. cmul 9000    <_ cle 9126   abscabs 12044  GIdcgi 21780   CVec OLDcvc 22029   NrmCVeccnv 22068
This theorem is referenced by:  nvvcop  22078  nvrel  22086  nvvop  22093  nvex  22095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4270  df-xp 4887  df-oprab 6088  df-nv 22076
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