Theorem rnggrp 46644

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 46641	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	ablgrpd 19653	1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2106  Grpcgrp 18818  Rngcrng 46638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-abl 19650  df-rng 46639

This theorem is referenced by:  rngacl  46651  rng0cl  46652  rngrz  46655  rngmneg1  46656  rngmneg2  46657  rngm2neg  46658  rngsubdi  46660  rngsubdir  46661  prdsrngd  46667  subrngsubg  46721  cntzsubrng  46736  rnglidlmcl  46738  rnglidl0  46742  rnglidl1  46743  2idlcpblrng  46756  rngqiprngimfolem  46765  rngqiprngimf1lem  46769  rngqiprngghm  46774  rngqiprngimf1  46775  rngqiprngimfo  46776  rngqiprngfulem3  46788  rngqiprngfulem4  46789  rngqiprngfulem5  46790  pzriprnglem4  46798  pzriprnglem10  46804