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Theorem rngmgp 20096
Description: A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
Hypothesis
Ref Expression
rngmgp.g 𝐺 = (mulGrp‘𝑅)
Assertion
Ref Expression
rngmgp (𝑅 ∈ Rng → 𝐺 ∈ Smgrp)

Proof of Theorem rngmgp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2728 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 rngmgp.g . . 3 𝐺 = (mulGrp‘𝑅)
3 eqid 2728 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2728 . . 3 (.r𝑅) = (.r𝑅)
51, 2, 3, 4isrng 20094 . 2 (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ 𝐺 ∈ 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)))))
65simp2bi 1144 1 (𝑅 ∈ Rng → 𝐺 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3058  cfv 6548  (class class class)co 7420  Basecbs 17180  +gcplusg 17233  .rcmulr 17234  Smgrpcsgrp 18678  Abelcabl 19736  mulGrpcmgp 20074  Rngcrng 20092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-rng 20093
This theorem is referenced by:  rngmgpf  20097  rngass  20099  rngcl  20104  isringrng  20223  isrnghmmul  20381  idrnghm  20397  c0rnghm  20472  cntzsubrng  20504  rnglidlmsgrp  21141
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