Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngabl Structured version   Visualization version   GIF version

Theorem rngabl 46261
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 2733 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2733 . . 3 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 eqid 2733 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2733 . . 3 (.r𝑅) = (.r𝑅)
51, 2, 3, 4isrng 46260 . 2 (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ (mulGrp‘𝑅) ∈ Smgrp ∧ ∀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 1146 1 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  .rcmulr 17139  Smgrpcsgrp 18550  Abelcabl 19568  mulGrpcmgp 19901  Rngcrng 46258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-rng 46259
This theorem is referenced by:  isringrng  46265  rnglz  46268  isrnghm  46276  isrnghmd  46286  idrnghm  46292  c0rnghm  46297  zrrnghm  46301
  Copyright terms: Public domain W3C validator