Theorem rngabl 20428

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 2763	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2763	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
3		eqid 2763	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2763	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isrng 20427	. 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 1163	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  +_gcplusg 17472  ._rcmulr 17473  Smgrpcsgrp 18957  Abelcabl 20046  mulGrpcmgp 20411  Rngcrng 20425

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5266

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-ov 7454  df-rng 20426

This theorem is used by:  rnggrp  20431  rnglz  20438  rngansg  20443  prdsrngd  20449  imasrng  20450  isringrng  20567  opprrng  20626  isrnghm  20722  isrnghmd  20732  idrnghm  20739  c0rnghm  20838  zrrnghm  20839  subrngringnsg  20856  issubrng2  20861  rnglidlrng  21586  2idlcpblrng  21616  qus2idrng  21618  rngqiprngimf1lem  21641  rngqiprngimfo  21648  rngqiprngfulem2  21659  rngqiprngfulem4  21661