Theorem rnggrp 20237

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 20234	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19857	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2143  Grpcgrp 19001  Rngcrng 20231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-ov 7415  df-abl 19854  df-rng 20232

This theorem is referenced by:  rngacl  20241  rng0cl  20242  rngrz  20245  rngmneg1  20246  rngmneg2  20247  rngm2neg  20248  rngsubdi  20250  rngsubdir  20251  prdsrngd  20255  rng1zr  20261  subrngsubg  20638  cntzsubrng  20653  rnglidlmcl  21322  rnglidl0  21336  rnglidl1  21339  2idlcpblrng  21391  rngqiprngimfolem  21411  rngqiprngimf1lem  21415  rngqiprngghm  21420  rngqiprngimf1  21421  rngqiprngimfo  21422  rngqiprngfulem3  21434  rngqiprngfulem4  21435  rngqiprngfulem5  21436  pzriprnglem4  21615  pzriprnglem10  21621