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Theorem isablo 25058
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 5228 . . . . 5  |-  ( g  =  G  ->  ran  g  =  ran  G )
2 isabl.1 . . . . 5  |-  X  =  ran  G
31, 2syl6eqr 2526 . . . 4  |-  ( g  =  G  ->  ran  g  =  X )
4 raleq 3058 . . . . 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 3066 . . . 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 6291 . . . . 5  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
8 oveq 6291 . . . . 5  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
97, 8eqeq12d 2489 . . . 4  |-  ( g  =  G  ->  (
( x g y )  =  ( y g x )  <->  ( x G y )  =  ( y G x ) ) )
1092ralbidv 2908 . . 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 253 . 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 25057 . 2  |-  AbelOp  =  {
g  e.  GrpOp  |  A. x  e.  ran  g A. y  e.  ran  g ( x g y )  =  ( y g x ) }
1311, 12elrab2 3263 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   ran crn 5000  (class class class)co 6285   GrpOpcgr 24961   AbelOpcablo 25056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-cnv 5007  df-dm 5009  df-rn 5010  df-iota 5551  df-fv 5596  df-ov 6288  df-ablo 25057
This theorem is referenced by:  ablogrpo  25059  ablocom  25060  isabloi  25063  isabloda  25074  subgoablo  25086  ghablo  25144
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