Theorem rnggrp 20061

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 20058	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19704	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2098  Grpcgrp 18861  Rngcrng 20055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-abl 19701  df-rng 20056

This theorem is referenced by:  rngacl  20065  rng0cl  20066  rngrz  20069  rngmneg1  20070  rngmneg2  20071  rngm2neg  20072  rngsubdi  20074  rngsubdir  20075  prdsrngd  20079  subrngsubg  20450  cntzsubrng  20465  rnglidlmcl  21073  rnglidl0  21086  rnglidl1  21089  2idlcpblrng  21126  rngqiprngimfolem  21141  rngqiprngimf1lem  21145  rngqiprngghm  21150  rngqiprngimf1  21151  rngqiprngimfo  21152  rngqiprngfulem3  21164  rngqiprngfulem4  21165  rngqiprngfulem5  21166  pzriprnglem4  21367  pzriprnglem10  21373