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Theorem ablogrpo 23722
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 2438 . . 3  |-  ran  G  =  ran  G
21isablo 23721 . 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 1369    e. wcel 1756   A.wral 2710   ran crn 4836  (class class class)co 6086   GrpOpcgr 23624   AbelOpcablo 23719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-cnv 4843  df-dm 4845  df-rn 4846  df-iota 5376  df-fv 5421  df-ov 6089  df-ablo 23720
This theorem is referenced by:  ablo32  23724  ablo4  23725  ablomuldiv  23727  ablodivdiv  23728  ablodivdiv4  23729  ablonnncan  23731  ablonncan  23732  ablonnncan1  23733  gxdi  23734  cnid  23789  addinv  23790  readdsubgo  23791  zaddsubgo  23792  mulid  23794  ghablo  23807  efghgrp  23811  rngogrpo  23828  cnrngo  23841  rngosn  23842  vcgrp  23887  vcoprnelem  23907  isvc  23910  isvci  23911  nvgrp  23946  cnnv  24018  cnnvba  24020  cncph  24170  hilid  24514  hhnv  24518  hhba  24520  hhph  24531  hhssabloi  24614  hhssnv  24616  ablo4pnp  28698  iscringd  28752
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