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Theorem isablo 21863
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 5087 . . . . 5  |-  ( g  =  G  ->  ran  g  =  ran  G )
2 isabl.1 . . . . 5  |-  X  =  ran  G
31, 2syl6eqr 2485 . . . 4  |-  ( g  =  G  ->  ran  g  =  X )
4 raleq 2896 . . . . 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 2904 . . . 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 16 . . 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 6079 . . . . 5  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
8 oveq 6079 . . . . 5  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
97, 8eqeq12d 2449 . . . 4  |-  ( g  =  G  ->  (
( x g y )  =  ( y g x )  <->  ( x G y )  =  ( y G x ) ) )
1092ralbidv 2739 . . 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 245 . 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 21862 . 2  |-  AbelOp  =  {
g  e.  GrpOp  |  A. x  e.  ran  g A. y  e.  ran  g ( x g y )  =  ( y g x ) }
1311, 12elrab2 3086 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 set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   ran crn 4871  (class class class)co 6073   GrpOpcgr 21766   AbelOpcablo 21861
This theorem is referenced by:  ablogrpo  21864  ablocom  21865  isabloi  21868  isabloda  21879  subgoablo  21891  ghablo  21949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-cnv 4878  df-dm 4880  df-rn 4881  df-iota 5410  df-fv 5454  df-ov 6076  df-ablo 21862
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