Theorem rngansg 20243

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 20228	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2		ablnsg 19912	. 2 ⊢ (𝑅 ∈ Abel → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
3	1, 2	syl 18	1 ⊢ (𝑅 ∈ Rng → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ‘cfv 6536  SubGrpcsubg 19181  NrmSGrpcnsg 19182  Abelcabl 19846  Rngcrng 20225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-subg 19184  df-nsg 19185  df-cmn 19847  df-abl 19848  df-rng 20226

This theorem is referenced by: (None)