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Theorem rngabl 38692
 Description: A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
Assertion
Ref Expression
rngabl Rng

Proof of Theorem rngabl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2420 . . 3
2 eqid 2420 . . 3 mulGrp mulGrp
3 eqid 2420 . . 3
4 eqid 2420 . . 3
51, 2, 3, 4isrng 38691 . 2 Rng mulGrp SGrp
65simp1bi 1020 1 Rng
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   wceq 1437   wcel 1867  wral 2773  cfv 5592  (class class class)co 6296  cbs 15081   cplusg 15150  cmulr 15151  SGrpcsgrp 16478  cabl 17372  mulGrpcmgp 17664  Rngcrng 38689 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-nul 4547 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-iota 5556  df-fv 5600  df-ov 6299  df-rng0 38690 This theorem is referenced by:  isringrng  38696  rnglz  38699  isrnghm  38707  isrnghmd  38717  idrnghm  38723  c0rnghm  38728  zrrnghm  38732
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