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Theorem isablo 25856
Description: The predicate "is an Abelian (commutative) group operation." (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
isabl.1  |-  X  =  ran  G
Assertion
Ref Expression
isablo  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
Distinct variable groups:    x, y, G    x, X, y

Proof of Theorem isablo
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 rneq 5080 . . . . 5  |-  ( g  =  G  ->  ran  g  =  ran  G )
2 isabl.1 . . . . 5  |-  X  =  ran  G
31, 2syl6eqr 2488 . . . 4  |-  ( g  =  G  ->  ran  g  =  X )
4 raleq 3032 . . . . 5  |-  ( ran  g  =  X  -> 
( A. y  e. 
ran  g ( x g y )  =  ( y g x )  <->  A. y  e.  X  ( x g y )  =  ( y g x ) ) )
54raleqbi1dv 3040 . . . 4  |-  ( ran  g  =  X  -> 
( A. x  e. 
ran  g A. y  e.  ran  g ( x g y )  =  ( y g x )  <->  A. x  e.  X  A. y  e.  X  ( x g y )  =  ( y g x ) ) )
63, 5syl 17 . . 3  |-  ( g  =  G  ->  ( A. x  e.  ran  g A. y  e.  ran  g ( x g y )  =  ( y g x )  <->  A. x  e.  X  A. y  e.  X  ( x g y )  =  ( y g x ) ) )
7 oveq 6311 . . . . 5  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
8 oveq 6311 . . . . 5  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
97, 8eqeq12d 2451 . . . 4  |-  ( g  =  G  ->  (
( x g y )  =  ( y g x )  <->  ( x G y )  =  ( y G x ) ) )
1092ralbidv 2876 . . 3  |-  ( g  =  G  ->  ( A. x  e.  X  A. y  e.  X  ( x g y )  =  ( y g x )  <->  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
116, 10bitrd 256 . 2  |-  ( g  =  G  ->  ( A. x  e.  ran  g A. y  e.  ran  g ( x g y )  =  ( y g x )  <->  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
12 df-ablo 25855 . 2  |-  AbelOp  =  {
g  e.  GrpOp  |  A. x  e.  ran  g A. y  e.  ran  g ( x g y )  =  ( y g x ) }
1311, 12elrab2 3237 1  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   ran crn 4855  (class class class)co 6305   GrpOpcgr 25759   AbelOpcablo 25854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-cnv 4862  df-dm 4864  df-rn 4865  df-iota 5565  df-fv 5609  df-ov 6308  df-ablo 25855
This theorem is referenced by:  ablogrpo  25857  ablocom  25858  isabloi  25861  isabloda  25872  subgoablo  25884  ghabloOLD  25942
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