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Theorem rngabl 45435
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 45434 . 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 1144 1 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wral 3064  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  .rcmulr 16963  Smgrpcsgrp 18374  Abelcabl 19387  mulGrpcmgp 19720  Rngcrng 45432
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-rng0 45433
This theorem is referenced by:  isringrng  45439  rnglz  45442  isrnghm  45450  isrnghmd  45460  idrnghm  45466  c0rnghm  45471  zrrnghm  45475
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