Theorem rngansg 20248

Description: Every additive subgroup of a non-unital ring is normal. (Contributed by AV, 25-Feb-2025.)

Assertion

Ref	Expression
rngansg	⊢ (𝑅 ∈ Rng → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))

Proof of Theorem rngansg

Step	Hyp	Ref	Expression
1		rngabl 20233	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2		ablnsg 19917	. 2 ⊢ (𝑅 ∈ Abel → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
3	1, 2	syl 18	1 ⊢ (𝑅 ∈ Rng → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ‘cfv 6537  SubGrpcsubg 19186  NrmSGrpcnsg 19187  Abelcabl 19851  Rngcrng 20230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-subg 19189  df-nsg 19190  df-cmn 19852  df-abl 19853  df-rng 20231

This theorem is referenced by: (None)