Theorem ablogrpo 30840

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 2769	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 30839	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
3	2	simplbi 501	1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ran crn 5663  (class class class)co 7411  GrpOpcgr 30782  AbelOpcablo 30837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-cnv 5670  df-dm 5672  df-rn 5673  df-iota 6493  df-fv 6545  df-ov 7414  df-ablo 30838

This theorem is referenced by:  ablo32  30842  ablo4  30843  ablomuldiv  30845  ablodivdiv  30846  ablodivdiv4  30847  ablonncan  30849  ablonnncan1  30850  vcgrp  30863  isvcOLD  30872  isvciOLD  30873  cnidOLD  30875  nvgrp  30910  cnnv  30970  cnnvba  30972  cncph  31112  hilid  31454  hhnv  31458  hhba  31460  hhph  31471  hhssabloilem  31554  hhssnv  31557  ablo4pnp  38419  rngogrpo  38449  iscringd  38537