Theorem rngmgp 20121

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 20119	. 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 1146	1 ⊢ (𝑅 ∈ Rng → 𝐺 ∈ Smgrp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  +_gcplusg 17276  ._rcmulr 17277  Smgrpcsgrp 18701  Abelcabl 19767  mulGrpcmgp 20105  Rngcrng 20117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-nul 5281

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413  df-rng 20118

This theorem is referenced by:  rngmgpf  20122  rngass  20124  rngcl  20129  isringrng  20252  isrnghmmul  20407  idrnghm  20423  c0rnghm  20500  cntzsubrng  20532  rnglidlmsgrp  21212