Theorem rnggrp 20360

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 20357	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19980	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Grpcgrp 19124  Rngcrng 20354

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 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

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 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6487  df-fv 6539  df-ov 7415  df-abl 19977  df-rng 20355

This theorem is used by:  rngacl  20364  rng0cl  20365  rngrz  20368  rngmneg1  20369  rngmneg2  20370  rngm2neg  20371  rngsubdi  20373  rngsubdir  20374  prdsrngd  20378  rng1zr  20384  subrngsubg  20784  cntzsubrng  20799  rnglidlmcl  21475  rnglidl0  21489  rnglidl1  21492  2idlcpblrng  21545  rngqiprngimfolem  21566  rngqiprngimf1lem  21570  rngqiprngghm  21575  rngqiprngimf1  21576  rngqiprngimfo  21577  rngqiprngfulem3  21589  rngqiprngfulem4  21590  rngqiprngfulem5  21591  pzriprnglem4  21770  pzriprnglem10  21776