Theorem rnggrp 20413

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 20410	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 20033	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Grpcgrp 19176  Rngcrng 20407

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5264

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-ov 7449  df-abl 20030  df-rng 20408

This theorem is used by:  rngacl  20417  rng0cl  20418  rngrz  20421  rngmneg1  20422  rngmneg2  20423  rngm2neg  20424  rngsubdi  20426  rngsubdir  20427  prdsrngd  20431  rng1zr  20437  subrngsubg  20837  cntzsubrng  20852  rnglidlmcl  21528  rnglidl0  21542  rnglidl1  21545  2idlcpblrng  21598  rngqiprngimfolem  21619  rngqiprngimf1lem  21623  rngqiprngghm  21628  rngqiprngimf1  21629  rngqiprngimfo  21630  rngqiprngfulem3  21642  rngqiprngfulem4  21643  rngqiprngfulem5  21644  pzriprnglem4  21823  pzriprnglem10  21829