Theorem rnggrp 20067

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 20064	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19716	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18865  Rngcrng 20061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-abl 19713  df-rng 20062

This theorem is referenced by:  rngacl  20071  rng0cl  20072  rngrz  20075  rngmneg1  20076  rngmneg2  20077  rngm2neg  20078  rngsubdi  20080  rngsubdir  20081  prdsrngd  20085  subrngsubg  20461  cntzsubrng  20476  rnglidlmcl  21126  rnglidl0  21139  rnglidl1  21142  2idlcpblrng  21181  rngqiprngimfolem  21200  rngqiprngimf1lem  21204  rngqiprngghm  21209  rngqiprngimf1  21210  rngqiprngimfo  21211  rngqiprngfulem3  21223  rngqiprngfulem4  21224  rngqiprngfulem5  21225  pzriprnglem4  21394  pzriprnglem10  21400