Theorem rnggrp 20137

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 20134	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19759	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2119  Grpcgrp 18907  Rngcrng 20131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-abl 19756  df-rng 20132

This theorem is referenced by:  rngacl  20141  rng0cl  20142  rngrz  20145  rngmneg1  20146  rngmneg2  20147  rngm2neg  20148  rngsubdi  20150  rngsubdir  20151  prdsrngd  20155  subrngsubg  20531  cntzsubrng  20546  rnglidlmcl  21216  rnglidl0  21229  rnglidl1  21232  2idlcpblrng  21271  rngqiprngimfolem  21290  rngqiprngimf1lem  21294  rngqiprngghm  21299  rngqiprngimf1  21300  rngqiprngimfo  21301  rngqiprngfulem3  21313  rngqiprngfulem4  21314  rngqiprngfulem5  21315  pzriprnglem4  21466  pzriprnglem10  21472