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Theorem vcrel 25638
Description: The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Assertion
Ref Expression
vcrel  |-  Rel  CVecOLD

Proof of Theorem vcrel
Dummy variables  g 
s  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-vc 25637 . 2  |-  CVecOLD  =  { <. g ,  s
>.  |  ( g  e.  AbelOp  /\  s :
( CC  X.  ran  g ) --> ran  g  /\  A. x  e.  ran  g ( ( 1 s x )  =  x  /\  A. y  e.  CC  ( A. z  e.  ran  g ( y s ( x g z ) )  =  ( ( y s x ) g ( y s z ) )  /\  A. z  e.  CC  ( ( ( y  +  z ) s x )  =  ( ( y s x ) g ( z s x ) )  /\  ( ( y  x.  z ) s x )  =  ( y s ( z s x ) ) ) ) ) ) }
21relopabi 5116 1  |-  Rel  CVecOLD
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804    X. cxp 4986   ran crn 4989   Rel wrel 4993   -->wf 5566  (class class class)co 6270   CCcc 9479   1c1 9482    + caddc 9484    x. cmul 9486   AbelOpcablo 25481   CVecOLDcvc 25636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-opab 4498  df-xp 4994  df-rel 4995  df-vc 25637
This theorem is referenced by:  vcoprnelem  25669  vcex  25671  nvvop  25700  phop  25931
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