Theorem rnggrp 20074

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 20071	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19723	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18872  Rngcrng 20068

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 2702  ax-nul 5264

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 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393  df-abl 19720  df-rng 20069

This theorem is referenced by:  rngacl  20078  rng0cl  20079  rngrz  20082  rngmneg1  20083  rngmneg2  20084  rngm2neg  20085  rngsubdi  20087  rngsubdir  20088  prdsrngd  20092  subrngsubg  20468  cntzsubrng  20483  rnglidlmcl  21133  rnglidl0  21146  rnglidl1  21149  2idlcpblrng  21188  rngqiprngimfolem  21207  rngqiprngimf1lem  21211  rngqiprngghm  21216  rngqiprngimf1  21217  rngqiprngimfo  21218  rngqiprngfulem3  21230  rngqiprngfulem4  21231  rngqiprngfulem5  21232  pzriprnglem4  21401  pzriprnglem10  21407