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Theorem isabloi 23598
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 2772 . 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 23522 . . 3  |-  X  =  ran  G
65isablo 23593 . 2  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
71, 3, 6mpbir2an 904 1  |-  G  e. 
AbelOp
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   A.wral 2705    X. cxp 4825   dom cdm 4827  (class class class)co 6080   GrpOpcgr 23496   AbelOpcablo 23591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-fo 5412  df-fv 5414  df-ov 6083  df-grpo 23501  df-ablo 23592
This theorem is referenced by:  ablosn  23657  cnaddablo  23660  ablomul  23665  hilablo  24385  hhssabloi  24486
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