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Theorem vcrel 24078
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 24077 . 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 5074 1  |-  Rel  CVecOLD
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799    X. cxp 4947   ran crn 4950   Rel wrel 4954   -->wf 5523  (class class class)co 6201   CCcc 9392   1c1 9395    + caddc 9397    x. cmul 9399   AbelOpcablo 23921   CVecOLDcvc 24076
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 4522  ax-nul 4530  ax-pr 4640
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-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-opab 4460  df-xp 4955  df-rel 4956  df-vc 24077
This theorem is referenced by:  vcoprnelem  24109  vcex  24111  nvvop  24140  phop  24371
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