Theorem ablogrpo 27401

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.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ran G = ran G
2	1	isablo 27400	. 2 |- ( G e. AbelOp <-> ( G e. GrpOp /\ A. x e. ran G A. y e. ran G ( x G y ) = ( y G x ) ) )
3	2	simplbi 476	1 |- ( G e. AbelOp -> G e. GrpOp )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   ran crn 5115  (class class class)co 6650   GrpOpcgr 27343   AbelOpcablo 27398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896  df-ov 6653  df-ablo 27399

This theorem is referenced by:  ablo32  27403  ablo4  27404  ablomuldiv  27406  ablodivdiv  27407  ablodivdiv4  27408  ablonnncan  27410  ablonncan  27411  ablonnncan1  27412  vcgrp  27425  isvcOLD  27434  isvciOLD  27435  cnidOLD  27437  nvgrp  27472  cnnv  27532  cnnvba  27534  cncph  27674  hilid  28018  hhnv  28022  hhba  28024  hhph  28035  hhssabloilem  28118  hhssnv  28121  ablo4pnp  33679  rngogrpo  33709  iscringd  33797