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Theorem rngmgp 32522
 Description: A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
Hypothesis
Ref Expression
rngmgp.g mulGrp
Assertion
Ref Expression
rngmgp Rng SGrp

Proof of Theorem rngmgp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . 3
2 rngmgp.g . . 3 mulGrp
3 eqid 2443 . . 3
4 eqid 2443 . . 3
51, 2, 3, 4isrng 32520 . 2 Rng SGrp
65simp2bi 1013 1 Rng SGrp
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1383   wcel 1804  wral 2793  cfv 5578  (class class class)co 6281  cbs 14614   cplusg 14679  cmulr 14680  SGrpcsgrp 15889  cabl 16778  mulGrpcmgp 17120  Rngcrng 32518 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-nul 4566 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284  df-rng0 32519 This theorem is referenced by:  isringrng  32525  isrnghmmul  32534  idrnghm  32549
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