Theorem rngabl 20291

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 2760	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2760	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
3		eqid 2760	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2760	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isrng 20290	. 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 2145  ∀wral 3076  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  +_gcplusg 17343  ._rcmulr 17344  Smgrpcsgrp 18821  Abelcabl 19909  mulGrpcmgp 20274  Rngcrng 20288

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 2732  ax-nul 5263

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 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7417  df-rng 20289

This theorem is used by:  rnggrp  20294  rnglz  20301  rngansg  20306  prdsrngd  20312  imasrng  20313  isringrng  20429  opprrng  20487  isrnghm  20583  isrnghmd  20593  idrnghm  20600  c0rnghm  20698  zrrnghm  20699  subrngringnsg  20716  issubrng2  20721  rnglidlrng  21445  2idlcpblrng  21474  qus2idrng  21476  rngqiprngimf1lem  21498  rngqiprngimfo  21505  rngqiprngfulem2  21516  rngqiprngfulem4  21518