Theorem rnggrp 20299

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 20296	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19919	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Grpcgrp 19063  Rngcrng 20293

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-nul 5267

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7420  df-abl 19916  df-rng 20294

This theorem is used by:  rngacl  20303  rng0cl  20304  rngrz  20307  rngmneg1  20308  rngmneg2  20309  rngm2neg  20310  rngsubdi  20312  rngsubdir  20313  prdsrngd  20317  rng1zr  20323  subrngsubg  20720  cntzsubrng  20735  rnglidlmcl  21410  rnglidl0  21424  rnglidl1  21427  2idlcpblrng  21479  rngqiprngimfolem  21499  rngqiprngimf1lem  21503  rngqiprngghm  21508  rngqiprngimf1  21509  rngqiprngimfo  21510  rngqiprngfulem3  21522  rngqiprngfulem4  21523  rngqiprngfulem5  21524  pzriprnglem4  21703  pzriprnglem10  21709