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Theorem nvss 24150
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_  ( CVecOLD  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 2526 . . . . . . 7  |-  ( w  =  <. g ,  s
>.  ->  ( w  e. 
CVecOLD  <->  <. g ,  s
>.  e.  CVecOLD ) )
21biimpar 485 . . . . . 6  |-  ( ( w  =  <. g ,  s >.  /\  <. g ,  s >.  e.  CVecOLD )  ->  w  e.  CVecOLD )
323ad2antr1 1153 . . . . 5  |-  ( ( w  =  <. g ,  s >.  /\  ( <. g ,  s >.  e.  CVecOLD  /\  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.  CVecOLD )
43exlimivv 1690 . . . 4  |-  ( E. g E. s ( w  =  <. g ,  s >.  /\  ( <. g ,  s >.  e.  CVecOLD  /\  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.  CVecOLD )
5 vex 3081 . . . 4  |-  n  e. 
_V
64, 5jctir 538 . . 3  |-  ( E. g E. s ( w  =  <. g ,  s >.  /\  ( <. g ,  s >.  e.  CVecOLD  /\  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. 
CVecOLD  /\  n  e. 
_V ) )
76ssopab2i 4727 . 2  |-  { <. w ,  n >.  |  E. g E. s ( w  =  <. g ,  s
>.  /\  ( <. g ,  s >.  e.  CVecOLD 
/\  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.  CVecOLD  /\  n  e.  _V ) }
8 df-nv 24149 . . 3  |-  NrmCVec  =  { <. <. g ,  s
>. ,  n >.  |  ( <. g ,  s
>.  e.  CVecOLD  /\  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 6245 . . 3  |-  { <. <.
g ,  s >. ,  n >.  |  ( <. g ,  s >.  e.  CVecOLD  /\  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.  CVecOLD 
/\  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 2483 . 2  |-  NrmCVec  =  { <. w ,  n >.  |  E. g E. s
( w  =  <. g ,  s >.  /\  ( <. g ,  s >.  e.  CVecOLD  /\  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 4957 . 2  |-  ( CVecOLD  X.  _V )  =  { <. w ,  n >.  |  ( w  e. 
CVecOLD  /\  n  e. 
_V ) }
127, 10, 113sstr4i 3506 1  |-  NrmCVec  C_  ( CVecOLD  X.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   A.wral 2799   _Vcvv 3078    C_ wss 3439   <.cop 3994   class class class wbr 4403   {copab 4460    X. cxp 4949   ran crn 4952   -->wf 5525   ` cfv 5529  (class class class)co 6203   {coprab 6204   CCcc 9395   RRcr 9396   0cc0 9397    + caddc 9400    x. cmul 9402    <_ cle 9534   abscabs 12845  GIdcgi 23853   CVecOLDcvc 24102   NrmCVeccnv 24141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-opab 4462  df-xp 4957  df-oprab 6207  df-nv 24149
This theorem is referenced by:  nvvcop  24151  nvrel  24159  nvvop  24166  nvex  24168
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