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

Theorem ablogrpo 26785
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 2610 . . 3 ran 𝐺 = ran 𝐺
21isablo 26784 . 2 (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
32simplbi 475 1 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  wral 2896  ran crn 5039  (class class class)co 6549  GrpOpcgr 26727  AbelOpcablo 26782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-iota 5768  df-fv 5812  df-ov 6552  df-ablo 26783
This theorem is referenced by:  ablo32  26787  ablo4  26788  ablomuldiv  26790  ablodivdiv  26791  ablodivdiv4  26792  ablonnncan  26794  ablonncan  26795  ablonnncan1  26796  vcgrp  26809  isvcOLD  26818  isvciOLD  26819  cnidOLD  26821  nvgrp  26856  cnnv  26916  cnnvba  26918  cncph  27058  hilid  27402  hhnv  27406  hhba  27408  hhph  27419  hhssabloilem  27502  hhssnv  27505  ablo4pnp  32849  rngogrpo  32879  iscringd  32967
  Copyright terms: Public domain W3C validator