Theorem rngabl 20127

Description: A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)

Assertion

Ref	Expression
rngabl	⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)

Proof of Theorem rngabl

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2739	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2739	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
3		eqid 2739	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2739	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isrng 20126	. 2 ⊢ (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ (mulGrp‘𝑅) ∈ Smgrp ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)(𝑦(+g‘𝑅)𝑧)) = ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)(𝑥(.r‘𝑅)𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧)))))
6	5	simp1bi 1151	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  +_gcplusg 17211  ._rcmulr 17212  Smgrpcsgrp 18677  Abelcabl 19747  mulGrpcmgp 20112  Rngcrng 20124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-rng 20125

This theorem is referenced by:  rnggrp  20130  rnglz  20137  rngansg  20142  prdsrngd  20148  imasrng  20149  isringrng  20259  opprrng  20316  isrnghm  20412  isrnghmd  20422  idrnghm  20429  c0rnghm  20507  zrrnghm  20508  subrngringnsg  20525  issubrng2  20530  rnglidlrng  21240  2idlcpblrng  21264  qus2idrng  21266  rngqiprngimf1lem  21287  rngqiprngimfo  21294  rngqiprngfulem2  21305  rngqiprngfulem4  21307