Theorem rngmgp 20202

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 2762	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		rngmgp.g	. . 3 ⊢ 𝐺 = (mulGrp‘𝑅)
3		eqid 2762	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2762	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isrng 20200	. 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 1159	1 ⊢ (𝑅 ∈ Rng → 𝐺 ∈ Smgrp)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  +_gcplusg 17286  ._rcmulr 17287  Smgrpcsgrp 18752  Abelcabl 19821  mulGrpcmgp 20186  Rngcrng 20198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5256

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529  df-ov 7399  df-rng 20199

This theorem is referenced by:  rngmgpf  20203  rngass  20205  rngcl  20210  isringrng  20333  isrnghmmul  20487  idrnghm  20503  c0rnghm  20581  cntzsubrng  20613  rnglidlmsgrp  21313