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Theorem rngabl 44998
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 2738 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2738 . . 3 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 eqid 2738 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2738 . . 3 (.r𝑅) = (.r𝑅)
51, 2, 3, 4isrng 44997 . 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 399   = wceq 1542  wcel 2114  wral 3053  cfv 6339  (class class class)co 7172  Basecbs 16588  +gcplusg 16670  .rcmulr 16671  Smgrpcsgrp 18018  Abelcabl 19027  mulGrpcmgp 19360  Rngcrng 44995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-nul 5174
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-iota 6297  df-fv 6347  df-ov 7175  df-rng0 44996
This theorem is referenced by:  isringrng  45002  rnglz  45005  isrnghm  45013  isrnghmd  45023  idrnghm  45029  c0rnghm  45034  zrrnghm  45038
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