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Theorem rngmgp 42396
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 2770 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 rngmgp.g . . 3 𝐺 = (mulGrp‘𝑅)
3 eqid 2770 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2770 . . 3 (.r𝑅) = (.r𝑅)
51, 2, 3, 4isrng 42394 . 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 1139 1 (𝑅 ∈ Rng → 𝐺 ∈ SGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wral 3060  cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  .rcmulr 16149  SGrpcsgrp 17490  Abelcabl 18400  mulGrpcmgp 18696  Rngcrng 42392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-ov 6795  df-rng0 42393
This theorem is referenced by:  isringrng  42399  rngcl  42401  isrnghmmul  42411  idrnghm  42426  c0rnghm  42431
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