Theorem rnggrp 20061

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 20058	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19683	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18830  Rngcrng 20055

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 5248

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 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-ov 7356  df-abl 19680  df-rng 20056

This theorem is referenced by:  rngacl  20065  rng0cl  20066  rngrz  20069  rngmneg1  20070  rngmneg2  20071  rngm2neg  20072  rngsubdi  20074  rngsubdir  20075  prdsrngd  20079  subrngsubg  20455  cntzsubrng  20470  rnglidlmcl  21141  rnglidl0  21154  rnglidl1  21157  2idlcpblrng  21196  rngqiprngimfolem  21215  rngqiprngimf1lem  21219  rngqiprngghm  21224  rngqiprngimf1  21225  rngqiprngimfo  21226  rngqiprngfulem3  21238  rngqiprngfulem4  21239  rngqiprngfulem5  21240  pzriprnglem4  21409  pzriprnglem10  21415