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Theorem isabloi 21837
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 2740 . 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 21761 . . 3  |-  X  =  ran  G
65isablo 21832 . 2  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
71, 3, 6mpbir2an 887 1  |-  G  e. 
AbelOp
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674    X. cxp 4843   dom cdm 4845  (class class class)co 6048   GrpOpcgr 21735   AbelOpcablo 21830
This theorem is referenced by:  ablosn  21896  cnaddablo  21899  ablomul  21904  hilablo  22623  hhssabloi  22723
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
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-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fo 5427  df-fv 5429  df-ov 6051  df-grpo 21740  df-ablo 21831
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