Theorem rngabl 20088

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 2734	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2734	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
3		eqid 2734	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2734	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isrng 20087	. 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 1145	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  +_gcplusg 17175  ._rcmulr 17176  Smgrpcsgrp 18641  Abelcabl 19708  mulGrpcmgp 20073  Rngcrng 20085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-nul 5249

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-iota 6446  df-fv 6498  df-ov 7359  df-rng 20086

This theorem is referenced by:  rnggrp  20091  rnglz  20098  rngansg  20103  prdsrngd  20109  imasrng  20110  isringrng  20220  opprrng  20279  isrnghm  20375  isrnghmd  20385  idrnghm  20392  c0rnghm  20466  zrrnghm  20467  subrngringnsg  20484  issubrng2  20489  rnglidlrng  21200  2idlcpblrng  21224  qus2idrng  21226  rngqiprngimf1lem  21247  rngqiprngimfo  21254  rngqiprngfulem2  21265  rngqiprngfulem4  21267