Theorem ablogrpo 29800

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 2733	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 29799	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
3	2	simplbi 499	1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ran crn 5678  (class class class)co 7409  GrpOpcgr 29742  AbelOpcablo 29797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-cnv 5685  df-dm 5687  df-rn 5688  df-iota 6496  df-fv 6552  df-ov 7412  df-ablo 29798

This theorem is referenced by:  ablo32  29802  ablo4  29803  ablomuldiv  29805  ablodivdiv  29806  ablodivdiv4  29807  ablonncan  29809  ablonnncan1  29810  vcgrp  29823  isvcOLD  29832  isvciOLD  29833  cnidOLD  29835  nvgrp  29870  cnnv  29930  cnnvba  29932  cncph  30072  hilid  30414  hhnv  30418  hhba  30420  hhph  30431  hhssabloilem  30514  hhssnv  30517  ablo4pnp  36748  rngogrpo  36778  iscringd  36866