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Theorem isablo 20966
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 4920 . . . . 5  |-  ( g  =  G  ->  ran  g  =  ran  G )
2 isabl.1 . . . . 5  |-  X  =  ran  G
31, 2syl6eqr 2346 . . . 4  |-  ( g  =  G  ->  ran  g  =  X )
4 raleq 2749 . . . . 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 2757 . . . 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 15 . . 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 5880 . . . . 5  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
8 oveq 5880 . . . . 5  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
97, 8eqeq12d 2310 . . . 4  |-  ( g  =  G  ->  (
( x g y )  =  ( y g x )  <->  ( x G y )  =  ( y G x ) ) )
1092ralbidv 2598 . . 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 244 . 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 20965 . 2  |-  AbelOp  =  {
g  e.  GrpOp  |  A. x  e.  ran  g A. y  e.  ran  g ( x g y )  =  ( y g x ) }
1311, 12elrab2 2938 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   ran crn 4706  (class class class)co 5874   GrpOpcgr 20869   AbelOpcablo 20964
This theorem is referenced by:  ablogrpo  20967  ablocom  20968  isabloi  20971  isabloda  20982  subgoablo  20994  ghablo  21052  ablocomgrp  25445  tcnvec  25793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716  df-iota 5235  df-fv 5279  df-ov 5877  df-ablo 20965
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