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Theorem rngabl 41667
 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.
StepHypRef Expression
1 eqid 2610 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2610 . . 3 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 eqid 2610 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2610 . . 3 (.r𝑅) = (.r𝑅)
51, 2, 3, 4isrng 41666 . 2 (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ (mulGrp‘𝑅) ∈ SGrp ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥(.r𝑅)(𝑦(+g𝑅)𝑧)) = ((𝑥(.r𝑅)𝑦)(+g𝑅)(𝑥(.r𝑅)𝑧)) ∧ ((𝑥(+g𝑅)𝑦)(.r𝑅)𝑧) = ((𝑥(.r𝑅)𝑧)(+g𝑅)(𝑦(.r𝑅)𝑧)))))
65simp1bi 1069 1 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  SGrpcsgrp 17106  Abelcabl 18017  mulGrpcmgp 18312  Rngcrng 41664 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-rng0 41665 This theorem is referenced by:  isringrng  41671  rnglz  41674  isrnghm  41682  isrnghmd  41692  idrnghm  41698  c0rnghm  41703  zrrnghm  41707
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