Theorem rnggrp 20176

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 20173	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19819	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2106  Grpcgrp 18964  Rngcrng 20170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-abl 19816  df-rng 20171

This theorem is referenced by:  rngacl  20180  rng0cl  20181  rngrz  20184  rngmneg1  20185  rngmneg2  20186  rngm2neg  20187  rngsubdi  20189  rngsubdir  20190  prdsrngd  20194  subrngsubg  20569  cntzsubrng  20584  rnglidlmcl  21244  rnglidl0  21257  rnglidl1  21260  2idlcpblrng  21299  rngqiprngimfolem  21318  rngqiprngimf1lem  21322  rngqiprngghm  21327  rngqiprngimf1  21328  rngqiprngimfo  21329  rngqiprngfulem3  21341  rngqiprngfulem4  21342  rngqiprngfulem5  21343  pzriprnglem4  21513  pzriprnglem10  21519