Theorem rnggrp 20105

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 20102	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19727	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114  Grpcgrp 18875  Rngcrng 20099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-abl 19724  df-rng 20100

This theorem is referenced by:  rngacl  20109  rng0cl  20110  rngrz  20113  rngmneg1  20114  rngmneg2  20115  rngm2neg  20116  rngsubdi  20118  rngsubdir  20119  prdsrngd  20123  subrngsubg  20497  cntzsubrng  20512  rnglidlmcl  21183  rnglidl0  21196  rnglidl1  21199  2idlcpblrng  21238  rngqiprngimfolem  21257  rngqiprngimf1lem  21261  rngqiprngghm  21266  rngqiprngimf1  21267  rngqiprngimfo  21268  rngqiprngfulem3  21280  rngqiprngfulem4  21281  rngqiprngfulem5  21282  pzriprnglem4  21451  pzriprnglem10  21457