Theorem rnggrp 20227

Description: A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)

Assertion

Ref	Expression
rnggrp	⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Proof of Theorem rnggrp

Step	Hyp	Ref	Expression
1		rngabl 20224	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19847	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2145  Grpcgrp 18990  Rngcrng 20221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-ov 7403  df-abl 19844  df-rng 20222

This theorem is referenced by:  rngacl  20231  rng0cl  20232  rngrz  20235  rngmneg1  20236  rngmneg2  20237  rngm2neg  20238  rngsubdi  20240  rngsubdir  20241  prdsrngd  20245  rng1zr  20251  subrngsubg  20628  cntzsubrng  20643  rnglidlmcl  21310  rnglidl0  21324  rnglidl1  21327  2idlcpblrng  21372  rngqiprngimfolem  21392  rngqiprngimf1lem  21396  rngqiprngghm  21401  rngqiprngimf1  21402  rngqiprngimfo  21403  rngqiprngfulem3  21415  rngqiprngfulem4  21416  rngqiprngfulem5  21417  pzriprnglem4  21594  pzriprnglem10  21600