Theorem ablogrpo 29552

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 2731	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 29551	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
3	2	simplbi 498	1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ran crn 5639  (class class class)co 7362  GrpOpcgr 29494  AbelOpcablo 29549

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 2702

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 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-cnv 5646  df-dm 5648  df-rn 5649  df-iota 6453  df-fv 6509  df-ov 7365  df-ablo 29550

This theorem is referenced by:  ablo32  29554  ablo4  29555  ablomuldiv  29557  ablodivdiv  29558  ablodivdiv4  29559  ablonncan  29561  ablonnncan1  29562  vcgrp  29575  isvcOLD  29584  isvciOLD  29585  cnidOLD  29587  nvgrp  29622  cnnv  29682  cnnvba  29684  cncph  29824  hilid  30166  hhnv  30170  hhba  30172  hhph  30183  hhssabloilem  30266  hhssnv  30269  ablo4pnp  36412  rngogrpo  36442  iscringd  36530