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Theorem rngmgp 20103
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 20101 . 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 394   = wceq 1533  wcel 2098  wral 3058  cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  .rcmulr 17241  Smgrpcsgrp 18685  Abelcabl 19743  mulGrpcmgp 20081  Rngcrng 20099
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 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-rng 20100
This theorem is referenced by:  rngmgpf  20104  rngass  20106  rngcl  20111  isringrng  20230  isrnghmmul  20388  idrnghm  20404  c0rnghm  20479  cntzsubrng  20511  rnglidlmsgrp  21148
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