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Theorem rngmgp 20059
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 2726 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 rngmgp.g . . 3 𝐺 = (mulGrp‘𝑅)
3 eqid 2726 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2726 . . 3 (.r𝑅) = (.r𝑅)
51, 2, 3, 4isrng 20057 . 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 1143 1 (𝑅 ∈ Rng → 𝐺 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  cfv 6536  (class class class)co 7404  Basecbs 17151  +gcplusg 17204  .rcmulr 17205  Smgrpcsgrp 18649  Abelcabl 19699  mulGrpcmgp 20037  Rngcrng 20055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-rng 20056
This theorem is referenced by:  rngmgpf  20060  rngass  20062  rngcl  20067  isringrng  20184  isrnghmmul  20342  idrnghm  20358  c0rnghm  20433  cntzsubrng  20465  rnglidlmsgrp  21102
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