Theorem rnggrp 20139

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 20136	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19761	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114  Grpcgrp 18909  Rngcrng 20133

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 2708  ax-nul 5241

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 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-abl 19758  df-rng 20134

This theorem is referenced by:  rngacl  20143  rng0cl  20144  rngrz  20147  rngmneg1  20148  rngmneg2  20149  rngm2neg  20150  rngsubdi  20152  rngsubdir  20153  prdsrngd  20157  subrngsubg  20529  cntzsubrng  20544  rnglidlmcl  21214  rnglidl0  21227  rnglidl1  21230  2idlcpblrng  21269  rngqiprngimfolem  21288  rngqiprngimf1lem  21292  rngqiprngghm  21297  rngqiprngimf1  21298  rngqiprngimfo  21299  rngqiprngfulem3  21311  rngqiprngfulem4  21312  rngqiprngfulem5  21313  pzriprnglem4  21464  pzriprnglem10  21470