Theorem ablogrpo 30476

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 2729	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 30475	. 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 1540   ∈ wcel 2109  ∀wral 3044  ran crn 5639  (class class class)co 7387  GrpOpcgr 30418  AbelOpcablo 30473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-cnv 5646  df-dm 5648  df-rn 5649  df-iota 6464  df-fv 6519  df-ov 7390  df-ablo 30474

This theorem is referenced by:  ablo32  30478  ablo4  30479  ablomuldiv  30481  ablodivdiv  30482  ablodivdiv4  30483  ablonncan  30485  ablonnncan1  30486  vcgrp  30499  isvcOLD  30508  isvciOLD  30509  cnidOLD  30511  nvgrp  30546  cnnv  30606  cnnvba  30608  cncph  30748  hilid  31090  hhnv  31094  hhba  31096  hhph  31107  hhssabloilem  31190  hhssnv  31193  ablo4pnp  37874  rngogrpo  37904  iscringd  37992