Theorem rnggrp 20156

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 20153	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19805	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2107  Grpcgrp 18952  Rngcrng 20150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-abl 19802  df-rng 20151

This theorem is referenced by:  rngacl  20160  rng0cl  20161  rngrz  20164  rngmneg1  20165  rngmneg2  20166  rngm2neg  20167  rngsubdi  20169  rngsubdir  20170  prdsrngd  20174  subrngsubg  20553  cntzsubrng  20568  rnglidlmcl  21227  rnglidl0  21240  rnglidl1  21243  2idlcpblrng  21282  rngqiprngimfolem  21301  rngqiprngimf1lem  21305  rngqiprngghm  21310  rngqiprngimf1  21311  rngqiprngimfo  21312  rngqiprngfulem3  21324  rngqiprngfulem4  21325  rngqiprngfulem5  21326  pzriprnglem4  21496  pzriprnglem10  21502