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Theorem isabloi 23726
Description: Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
isabli.1  |-  G  e. 
GrpOp
isabli.2  |-  dom  G  =  ( X  X.  X )
isabli.3  |-  ( ( x  e.  X  /\  y  e.  X )  ->  ( x G y )  =  ( y G x ) )
Assertion
Ref Expression
isabloi  |-  G  e. 
AbelOp
Distinct variable groups:    x, y, G    x, X, y

Proof of Theorem isabloi
StepHypRef Expression
1 isabli.1 . 2  |-  G  e. 
GrpOp
2 isabli.3 . . 3  |-  ( ( x  e.  X  /\  y  e.  X )  ->  ( x G y )  =  ( y G x ) )
32rgen2a 2777 . 2  |-  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x )
4 isabli.2 . . . 4  |-  dom  G  =  ( X  X.  X )
51, 4grporn 23650 . . 3  |-  X  =  ran  G
65isablo 23721 . 2  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
71, 3, 6mpbir2an 911 1  |-  G  e. 
AbelOp
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710    X. cxp 4833   dom cdm 4835  (class class class)co 6086   GrpOpcgr 23624   AbelOpcablo 23719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421  df-ov 6089  df-grpo 23629  df-ablo 23720
This theorem is referenced by:  ablosn  23785  cnaddablo  23788  ablomul  23793  hilablo  24513  hhssabloi  24614
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