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Theorem rngmgp 41649
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 ∈ SGrp)

Proof of Theorem rngmgp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2609 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 rngmgp.g . . 3 𝐺 = (mulGrp‘𝑅)
3 eqid 2609 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2609 . . 3 (.r𝑅) = (.r𝑅)
51, 2, 3, 4isrng 41647 . 2 (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ 𝐺 ∈ SGrp ∧ ∀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 1069 1 (𝑅 ∈ Rng → 𝐺 ∈ SGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  cfv 5789  (class class class)co 6526  Basecbs 15643  +gcplusg 15716  .rcmulr 15717  SGrpcsgrp 17054  Abelcabl 17965  mulGrpcmgp 18260  Rngcrng 41645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4711
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5753  df-fv 5797  df-ov 6529  df-rng0 41646
This theorem is referenced by:  isringrng  41652  rngcl  41654  isrnghmmul  41664  idrnghm  41679  c0rnghm  41684
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