Theorem rnggrp 20267

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 20264	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19887	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Grpcgrp 19031  Rngcrng 20261

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-ov 7426  df-abl 19884  df-rng 20262

This theorem is used by:  rngacl  20271  rng0cl  20272  rngrz  20275  rngmneg1  20276  rngmneg2  20277  rngm2neg  20278  rngsubdi  20280  rngsubdir  20281  prdsrngd  20285  rng1zr  20291  subrngsubg  20688  cntzsubrng  20703  rnglidlmcl  21378  rnglidl0  21392  rnglidl1  21395  2idlcpblrng  21447  rngqiprngimfolem  21467  rngqiprngimf1lem  21471  rngqiprngghm  21476  rngqiprngimf1  21477  rngqiprngimfo  21478  rngqiprngfulem3  21490  rngqiprngfulem4  21491  rngqiprngfulem5  21492  pzriprnglem4  21671  pzriprnglem10  21677