Theorem rngmgp 20096

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 2728	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		rngmgp.g	. . 3 ⊢ 𝐺 = (mulGrp‘𝑅)
3		eqid 2728	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2728	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isrng 20094	. 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 1144	1 ⊢ (𝑅 ∈ Rng → 𝐺 ∈ Smgrp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3058  ‘cfv 6548  (class class class)co 7420  Basecbs 17180  +_gcplusg 17233  ._rcmulr 17234  Smgrpcsgrp 18678  Abelcabl 19736  mulGrpcmgp 20074  Rngcrng 20092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-rng 20093

This theorem is referenced by:  rngmgpf  20097  rngass  20099  rngcl  20104  isringrng  20223  isrnghmmul  20381  idrnghm  20397  c0rnghm  20472  cntzsubrng  20504  rnglidlmsgrp  21141