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Theorem ablogrpo 25686
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 2402 . . 3  |-  ran  G  =  ran  G
21isablo 25685 . 2  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  ran  G A. y  e.  ran  G ( x G y )  =  ( y G x ) ) )
32simplbi 458 1  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   A.wral 2753   ran crn 4823  (class class class)co 6277   GrpOpcgr 25588   AbelOpcablo 25683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-cnv 4830  df-dm 4832  df-rn 4833  df-iota 5532  df-fv 5576  df-ov 6280  df-ablo 25684
This theorem is referenced by:  ablo32  25688  ablo4  25689  ablomuldiv  25691  ablodivdiv  25692  ablodivdiv4  25693  ablonnncan  25695  ablonncan  25696  ablonnncan1  25697  gxdi  25698  cnid  25753  addinv  25754  readdsubgo  25755  zaddsubgo  25756  mulid  25758  ghabloOLD  25771  efghgrpOLD  25775  rngogrpo  25792  cnrngo  25805  rngosn  25806  vcgrp  25851  vcoprnelem  25871  isvc  25874  isvci  25875  nvgrp  25910  cnnv  25982  cnnvba  25984  cncph  26134  hilid  26478  hhnv  26482  hhba  26484  hhph  26495  hhssabloi  26578  hhssnv  26580  ablo4pnp  31604  iscringd  31658
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