Theorem rngabl 20115

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 2735	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2735	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
3		eqid 2735	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2735	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isrng 20114	. 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 1540   ∈ wcel 2108  ∀wral 3051  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  +_gcplusg 17271  ._rcmulr 17272  Smgrpcsgrp 18696  Abelcabl 19762  mulGrpcmgp 20100  Rngcrng 20112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-ov 7408  df-rng 20113

This theorem is referenced by:  rnggrp  20118  rnglz  20125  rngansg  20130  prdsrngd  20136  imasrng  20137  isringrng  20247  opprrng  20305  isrnghm  20401  isrnghmd  20411  idrnghm  20418  c0rnghm  20495  zrrnghm  20496  subrngringnsg  20513  issubrng2  20518  rnglidlrng  21208  2idlcpblrng  21232  qus2idrng  21234  rngqiprngimf1lem  21255  rngqiprngimfo  21262  rngqiprngfulem2  21273  rngqiprngfulem4  21275