Theorem ablogrpo 27974

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 2778	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	isablo 27973	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
3	2	simplbi 493	1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ran crn 5356  (class class class)co 6922  GrpOpcgr 27916  AbelOpcablo 27971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-cnv 5363  df-dm 5365  df-rn 5366  df-iota 6099  df-fv 6143  df-ov 6925  df-ablo 27972

This theorem is referenced by:  ablo32  27976  ablo4  27977  ablomuldiv  27979  ablodivdiv  27980  ablodivdiv4  27981  ablonncan  27983  ablonnncan1  27984  vcgrp  27997  isvcOLD  28006  isvciOLD  28007  cnidOLD  28009  nvgrp  28044  cnnv  28104  cnnvba  28106  cncph  28246  hilid  28590  hhnv  28594  hhba  28596  hhph  28607  hhssabloilem  28690  hhssnv  28693  ablo4pnp  34303  rngogrpo  34333  iscringd  34421