Theorem ablogrpo 30618

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 2736	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 30617	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
3	2	simplbi 496	1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ran crn 5632  (class class class)co 7367  GrpOpcgr 30560  AbelOpcablo 30615

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-cnv 5639  df-dm 5641  df-rn 5642  df-iota 6454  df-fv 6506  df-ov 7370  df-ablo 30616

This theorem is referenced by:  ablo32  30620  ablo4  30621  ablomuldiv  30623  ablodivdiv  30624  ablodivdiv4  30625  ablonncan  30627  ablonnncan1  30628  vcgrp  30641  isvcOLD  30650  isvciOLD  30651  cnidOLD  30653  nvgrp  30688  cnnv  30748  cnnvba  30750  cncph  30890  hilid  31232  hhnv  31236  hhba  31238  hhph  31249  hhssabloilem  31332  hhssnv  31335  ablo4pnp  38201  rngogrpo  38231  iscringd  38319