Theorem ablogrpo 31036

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 2762	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 31035	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
3	2	simplbi 502	1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ran crn 5660  (class class class)co 7417  GrpOpcgr 30978  AbelOpcablo 31033

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-cnv 5667  df-dm 5669  df-rn 5670  df-iota 6493  df-fv 6545  df-ov 7420  df-ablo 31034

This theorem is used by:  ablo32  31038  ablo4  31039  ablomuldiv  31041  ablodivdiv  31042  ablodivdiv4  31043  ablonncan  31045  ablonnncan1  31046  vcgrp  31059  isvcOLD  31068  isvciOLD  31069  cnidOLD  31071  nvgrp  31106  cnnv  31166  cnnvba  31168  cncph  31308  hilid  31650  hhnv  31654  hhba  31656  hhph  31667  hhssabloilem  31750  hhssnv  31753  ablo4pnp  38638  rngogrpo  38668  iscringd  38756