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Theorem ablogrpo 24962
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 2467 . . 3  |-  ran  G  =  ran  G
21isablo 24961 . 2  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  ran  G A. y  e.  ran  G ( x G y )  =  ( y G x ) ) )
32simplbi 460 1  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2814   ran crn 5000  (class class class)co 6282   GrpOpcgr 24864   AbelOpcablo 24959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-cnv 5007  df-dm 5009  df-rn 5010  df-iota 5549  df-fv 5594  df-ov 6285  df-ablo 24960
This theorem is referenced by:  ablo32  24964  ablo4  24965  ablomuldiv  24967  ablodivdiv  24968  ablodivdiv4  24969  ablonnncan  24971  ablonncan  24972  ablonnncan1  24973  gxdi  24974  cnid  25029  addinv  25030  readdsubgo  25031  zaddsubgo  25032  mulid  25034  ghablo  25047  efghgrp  25051  rngogrpo  25068  cnrngo  25081  rngosn  25082  vcgrp  25127  vcoprnelem  25147  isvc  25150  isvci  25151  nvgrp  25186  cnnv  25258  cnnvba  25260  cncph  25410  hilid  25754  hhnv  25758  hhba  25760  hhph  25771  hhssabloi  25854  hhssnv  25856  ablo4pnp  29945  iscringd  29999
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