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Theorem ablogrpo 28251
Description: An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
Assertion
Ref Expression
ablogrpo (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

Proof of Theorem ablogrpo
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2818 . . 3 ran 𝐺 = ran 𝐺
21isablo 28250 . 2 (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
32simplbi 498 1 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wral 3135  ran crn 5549  (class class class)co 7145  GrpOpcgr 28193  AbelOpcablo 28248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-cnv 5556  df-dm 5558  df-rn 5559  df-iota 6307  df-fv 6356  df-ov 7148  df-ablo 28249
This theorem is referenced by:  ablo32  28253  ablo4  28254  ablomuldiv  28256  ablodivdiv  28257  ablodivdiv4  28258  ablonncan  28260  ablonnncan1  28261  vcgrp  28274  isvcOLD  28283  isvciOLD  28284  cnidOLD  28286  nvgrp  28321  cnnv  28381  cnnvba  28383  cncph  28523  hilid  28865  hhnv  28869  hhba  28871  hhph  28882  hhssabloilem  28965  hhssnv  28968  ablo4pnp  35039  rngogrpo  35069  iscringd  35157
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