Theorem ablogrpo 30474

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 2735	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 30473	. 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 2108  ∀wral 3051  ran crn 5655  (class class class)co 7403  GrpOpcgr 30416  AbelOpcablo 30471

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-cnv 5662  df-dm 5664  df-rn 5665  df-iota 6483  df-fv 6538  df-ov 7406  df-ablo 30472

This theorem is referenced by:  ablo32  30476  ablo4  30477  ablomuldiv  30479  ablodivdiv  30480  ablodivdiv4  30481  ablonncan  30483  ablonnncan1  30484  vcgrp  30497  isvcOLD  30506  isvciOLD  30507  cnidOLD  30509  nvgrp  30544  cnnv  30604  cnnvba  30606  cncph  30746  hilid  31088  hhnv  31092  hhba  31094  hhph  31105  hhssabloilem  31188  hhssnv  31191  ablo4pnp  37850  rngogrpo  37880  iscringd  37968