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Theorem ablogrpo 21825
Description: An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
Assertion
Ref Expression
ablogrpo  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )

Proof of Theorem ablogrpo
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . 3  |-  ran  G  =  ran  G
21isablo 21824 . 2  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  ran  G A. y  e.  ran  G ( x G y )  =  ( y G x ) ) )
32simplbi 447 1  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   A.wral 2666   ran crn 4838  (class class class)co 6040   GrpOpcgr 21727   AbelOpcablo 21822
This theorem is referenced by:  ablo32  21827  ablo4  21828  ablomuldiv  21830  ablodivdiv  21831  ablodivdiv4  21832  ablonnncan  21834  ablonncan  21835  ablonnncan1  21836  gxdi  21837  cnid  21892  addinv  21893  readdsubgo  21894  zaddsubgo  21895  mulid  21897  ghablo  21910  efghgrp  21914  rngogrpo  21931  cnrngo  21944  rngosn  21945  vcgrp  21990  vcoprnelem  22010  isvc  22013  isvci  22014  nvgrp  22049  cnnv  22121  cnnvba  22123  cncph  22273  hilid  22616  hhnv  22620  hhba  22622  hhph  22633  hhssabloi  22715  hhssnv  22717  ablo4pnp  26445  iscringd  26499
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-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-cnv 4845  df-dm 4847  df-rn 4848  df-iota 5377  df-fv 5421  df-ov 6043  df-ablo 21823
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