Theorem rnggrp 20123

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 20120	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19772	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18921  Rngcrng 20117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-nul 5281

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413  df-abl 19769  df-rng 20118

This theorem is referenced by:  rngacl  20127  rng0cl  20128  rngrz  20131  rngmneg1  20132  rngmneg2  20133  rngm2neg  20134  rngsubdi  20136  rngsubdir  20137  prdsrngd  20141  subrngsubg  20517  cntzsubrng  20532  rnglidlmcl  21182  rnglidl0  21195  rnglidl1  21198  2idlcpblrng  21237  rngqiprngimfolem  21256  rngqiprngimf1lem  21260  rngqiprngghm  21265  rngqiprngimf1  21266  rngqiprngimfo  21267  rngqiprngfulem3  21279  rngqiprngfulem4  21280  rngqiprngfulem5  21281  pzriprnglem4  21450  pzriprnglem10  21456