Theorem rngabl 20389

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 20388	. 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 1569   ∈ wcel 2146  ∀wral 3079  ‘cfv 6546  (class class class)co 7427  Basecbs 17399  +_gcplusg 17440  ._rcmulr 17441  Smgrpcsgrp 18919  Abelcabl 20007  mulGrpcmgp 20372  Rngcrng 20386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-nul 5262

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3416  df-v 3455  df-sbc 3742  df-ss 3900  df-dif 4049  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-iota 6500  df-fv 6554  df-ov 7430  df-rng 20387

This theorem is used by:  rnggrp  20392  rnglz  20399  rngansg  20404  prdsrngd  20410  imasrng  20411  isringrng  20528  opprrng  20587  isrnghm  20683  isrnghmd  20693  idrnghm  20700  c0rnghm  20799  zrrnghm  20800  subrngringnsg  20817  issubrng2  20822  rnglidlrng  21547  2idlcpblrng  21577  qus2idrng  21579  rngqiprngimf1lem  21602  rngqiprngimfo  21609  rngqiprngfulem2  21620  rngqiprngfulem4  21622