Theorem ablogrpo 30571

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.

Step	Hyp	Ref	Expression
1		eqid 2734	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 30570	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
3	2	simplbi 497	1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ran crn 5623  (class class class)co 7356  GrpOpcgr 30513  AbelOpcablo 30568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-cnv 5630  df-dm 5632  df-rn 5633  df-iota 6446  df-fv 6498  df-ov 7359  df-ablo 30569

This theorem is referenced by:  ablo32  30573  ablo4  30574  ablomuldiv  30576  ablodivdiv  30577  ablodivdiv4  30578  ablonncan  30580  ablonnncan1  30581  vcgrp  30594  isvcOLD  30603  isvciOLD  30604  cnidOLD  30606  nvgrp  30641  cnnv  30701  cnnvba  30703  cncph  30843  hilid  31185  hhnv  31189  hhba  31191  hhph  31202  hhssabloilem  31285  hhssnv  31288  ablo4pnp  38020  rngogrpo  38050  iscringd  38138