Theorem rnggrp 20386

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 20383	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 20006	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  Grpcgrp 19150  Rngcrng 20380

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6547  df-ov 7424  df-abl 20003  df-rng 20381

This theorem is used by:  rngacl  20390  rng0cl  20391  rngrz  20394  rngmneg1  20395  rngmneg2  20396  rngm2neg  20397  rngsubdi  20399  rngsubdir  20400  prdsrngd  20404  rng1zr  20410  subrngsubg  20810  cntzsubrng  20825  rnglidlmcl  21501  rnglidl0  21515  rnglidl1  21518  2idlcpblrng  21571  rngqiprngimfolem  21592  rngqiprngimf1lem  21596  rngqiprngghm  21601  rngqiprngimf1  21602  rngqiprngimfo  21603  rngqiprngfulem3  21615  rngqiprngfulem4  21616  rngqiprngfulem5  21617  pzriprnglem4  21796  pzriprnglem10  21802