Theorem rnggrp 20093

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 20090	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19715	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2113  Grpcgrp 18863  Rngcrng 20087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-nul 5251

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-abl 19712  df-rng 20088

This theorem is referenced by:  rngacl  20097  rng0cl  20098  rngrz  20101  rngmneg1  20102  rngmneg2  20103  rngm2neg  20104  rngsubdi  20106  rngsubdir  20107  prdsrngd  20111  subrngsubg  20485  cntzsubrng  20500  rnglidlmcl  21171  rnglidl0  21184  rnglidl1  21187  2idlcpblrng  21226  rngqiprngimfolem  21245  rngqiprngimf1lem  21249  rngqiprngghm  21254  rngqiprngimf1  21255  rngqiprngimfo  21256  rngqiprngfulem3  21268  rngqiprngfulem4  21269  rngqiprngfulem5  21270  pzriprnglem4  21439  pzriprnglem10  21445