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Theorem ablogrpo 20967
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 2296 . . 3  |-  ran  G  =  ran  G
21isablo 20966 . 2  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  ran  G A. y  e.  ran  G ( x G y )  =  ( y G x ) ) )
32simplbi 446 1  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   ran crn 4706  (class class class)co 5874   GrpOpcgr 20869   AbelOpcablo 20964
This theorem is referenced by:  ablo32  20969  ablo4  20970  ablomuldiv  20972  ablodivdiv  20973  ablodivdiv4  20974  ablonnncan  20976  ablonncan  20977  ablonnncan1  20978  gxdi  20979  cnid  21034  addinv  21035  readdsubgo  21036  zaddsubgo  21037  mulid  21039  ghablo  21052  efghgrp  21056  rngogrpo  21073  cnrngo  21086  rngosn  21087  vcgrp  21130  vcoprnelem  21150  isvc  21153  isvci  21154  nvgrp  21189  cnnv  21261  cnnvba  21263  cncph  21413  hilid  21756  hhnv  21760  hhba  21762  hhph  21773  hhssabloi  21855  hhssnv  21857  abloinvop  25456  fprodneg  25481  fprodsub  25482  clfsebs5  25485  rngoinvcl  25524  zintdom  25541  claddinvvec  25563  vec2inv  25564  sum2vv  25565  addnull1  25566  addnull2  25567  addvecass  25568  invaddvec  25570  vecsrcan  25572  vecslcan  25573  vwit  25574  sub2vec  25575  mvecrtol  25576  vecrcan  25578  veclcan  25579  mvecrtol2  25580  mulinvsca  25583  muldisc  25584  svli2  25587  svs2  25590  ablo4pnp  26673  iscringd  26727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716  df-iota 5235  df-fv 5279  df-ov 5877  df-ablo 20965
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