MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isablo Unicode version

Theorem isablo 21720
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 5036 . . . . 5  |-  ( g  =  G  ->  ran  g  =  ran  G )
2 isabl.1 . . . . 5  |-  X  =  ran  G
31, 2syl6eqr 2438 . . . 4  |-  ( g  =  G  ->  ran  g  =  X )
4 raleq 2848 . . . . 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 2856 . . . 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 6027 . . . . 5  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
8 oveq 6027 . . . . 5  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
97, 8eqeq12d 2402 . . . 4  |-  ( g  =  G  ->  (
( x g y )  =  ( y g x )  <->  ( x G y )  =  ( y G x ) ) )
1092ralbidv 2692 . . 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 21719 . 2  |-  AbelOp  =  {
g  e.  GrpOp  |  A. x  e.  ran  g A. y  e.  ran  g ( x g y )  =  ( y g x ) }
1311, 12elrab2 3038 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 1649    e. wcel 1717   A.wral 2650   ran crn 4820  (class class class)co 6021   GrpOpcgr 21623   AbelOpcablo 21718
This theorem is referenced by:  ablogrpo  21721  ablocom  21722  isabloi  21725  isabloda  21736  subgoablo  21748  ghablo  21806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-cnv 4827  df-dm 4829  df-rn 4830  df-iota 5359  df-fv 5403  df-ov 6024  df-ablo 21719
  Copyright terms: Public domain W3C validator