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Theorem nvss 25781
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 2472 . . . . . . 7  |-  ( w  =  <. g ,  s
>.  ->  ( w  e. 
CVecOLD  <->  <. g ,  s
>.  e.  CVecOLD ) )
21biimpar 483 . . . . . 6  |-  ( ( w  =  <. g ,  s >.  /\  <. g ,  s >.  e.  CVecOLD )  ->  w  e.  CVecOLD )
323ad2antr1 1160 . . . . 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 1742 . . . 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 3059 . . . 4  |-  n  e. 
_V
64, 5jctir 536 . . 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 4715 . 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 25780 . . 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 6278 . . 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 2429 . 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 4946 . 2  |-  ( CVecOLD  X.  _V )  =  { <. w ,  n >.  |  ( w  e. 
CVecOLD  /\  n  e. 
_V ) }
127, 10, 113sstr4i 3478 1  |-  NrmCVec  C_  ( CVecOLD  X.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403   E.wex 1631    e. wcel 1840   A.wral 2751   _Vcvv 3056    C_ wss 3411   <.cop 3975   class class class wbr 4392   {copab 4449    X. cxp 4938   ran crn 4941   -->wf 5519   ` cfv 5523  (class class class)co 6232   {coprab 6233   CCcc 9438   RRcr 9439   0cc0 9440    + caddc 9443    x. cmul 9445    <_ cle 9577   abscabs 13121  GIdcgi 25484   CVecOLDcvc 25733   NrmCVeccnv 25772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-opab 4451  df-xp 4946  df-oprab 6236  df-nv 25780
This theorem is referenced by:  nvvcop  25782  nvrel  25790  nvvop  25797  nvex  25799
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