MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablogrpo Structured version   Visualization version   GIF version

Theorem ablogrpo 29838
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 2732 . . 3 ran 𝐺 = ran 𝐺
21isablo 29837 . 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 1541  wcel 2106  wral 3061  ran crn 5677  (class class class)co 7411  GrpOpcgr 29780  AbelOpcablo 29835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-iota 6495  df-fv 6551  df-ov 7414  df-ablo 29836
This theorem is referenced by:  ablo32  29840  ablo4  29841  ablomuldiv  29843  ablodivdiv  29844  ablodivdiv4  29845  ablonncan  29847  ablonnncan1  29848  vcgrp  29861  isvcOLD  29870  isvciOLD  29871  cnidOLD  29873  nvgrp  29908  cnnv  29968  cnnvba  29970  cncph  30110  hilid  30452  hhnv  30456  hhba  30458  hhph  30469  hhssabloilem  30552  hhssnv  30555  ablo4pnp  36834  rngogrpo  36864  iscringd  36952
  Copyright terms: Public domain W3C validator