Theorem ablogrpo 30635

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 2737	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 30634	. 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 3052  ran crn 5633  (class class class)co 7368  GrpOpcgr 30577  AbelOpcablo 30632

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 2709

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-cnv 5640  df-dm 5642  df-rn 5643  df-iota 6456  df-fv 6508  df-ov 7371  df-ablo 30633

This theorem is referenced by:  ablo32  30637  ablo4  30638  ablomuldiv  30640  ablodivdiv  30641  ablodivdiv4  30642  ablonncan  30644  ablonnncan1  30645  vcgrp  30658  isvcOLD  30667  isvciOLD  30668  cnidOLD  30670  nvgrp  30705  cnnv  30765  cnnvba  30767  cncph  30907  hilid  31249  hhnv  31253  hhba  31255  hhph  31266  hhssabloilem  31349  hhssnv  31352  ablo4pnp  38131  rngogrpo  38161  iscringd  38249