Theorem ablogrpo 27732

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 2761	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 27731	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
3	2	simplbi 478	1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2140  ∀wral 3051  ran crn 5268  (class class class)co 6815  GrpOpcgr 27674  AbelOpcablo 27729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-cnv 5275  df-dm 5277  df-rn 5278  df-iota 6013  df-fv 6058  df-ov 6818  df-ablo 27730

This theorem is referenced by:  ablo32  27734  ablo4  27735  ablomuldiv  27737  ablodivdiv  27738  ablodivdiv4  27739  ablonnncan  27741  ablonncan  27742  ablonnncan1  27743  vcgrp  27756  isvcOLD  27765  isvciOLD  27766  cnidOLD  27768  nvgrp  27803  cnnv  27863  cnnvba  27865  cncph  28005  hilid  28349  hhnv  28353  hhba  28355  hhph  28366  hhssabloilem  28449  hhssnv  28452  ablo4pnp  34011  rngogrpo  34041  iscringd  34129