Theorem rngmgp 20137

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.

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		rngmgp.g	. . 3 ⊢ 𝐺 = (mulGrp‘𝑅)
3		eqid 2736	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2736	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isrng 20135	. 2 ⊢ (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ 𝐺 ∈ Smgrp ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)(𝑦(+g‘𝑅)𝑧)) = ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)(𝑥(.r‘𝑅)𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧)))))
6	5	simp2bi 1147	1 ⊢ (𝑅 ∈ Rng → 𝐺 ∈ Smgrp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  Smgrpcsgrp 18686  Abelcabl 19756  mulGrpcmgp 20121  Rngcrng 20133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-rng 20134

This theorem is referenced by:  rngmgpf  20138  rngass  20140  rngcl  20145  isringrng  20268  isrnghmmul  20422  idrnghm  20438  c0rnghm  20512  cntzsubrng  20544  rnglidlmsgrp  21244