Theorem subrngringnsg 20689

Description: A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)

Assertion

Ref	Expression
subrngringnsg	⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅))

Proof of Theorem subrngringnsg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrngsubg 20688	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2		subrngrcl 20687	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
3		rngabl 20264	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
4	2, 3	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Abel)
5	4	3anim1i 1170	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)))
6	5	3expb 1138	. . . . . 6 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)))
7		eqid 2766	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2766	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
9	7, 8	ablcom 19900	. . . . . 6 ⊢ ((𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = (𝑦(+g‘𝑅)𝑥))
10	6, 9	syl 18	. . . . 5 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑦(+g‘𝑅)𝑥))
11	10	eleq1d 2851	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(+g‘𝑅)𝑦) ∈ 𝐴 ↔ (𝑦(+g‘𝑅)𝑥) ∈ 𝐴))
12	11	biimpd 232	. . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(+g‘𝑅)𝑦) ∈ 𝐴 → (𝑦(+g‘𝑅)𝑥) ∈ 𝐴))
13	12	ralrimivva 3211	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) ∈ 𝐴 → (𝑦(+g‘𝑅)𝑥) ∈ 𝐴))
14	7, 8	isnsg2 19253	. 2 ⊢ (𝐴 ∈ (NrmSGrp‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) ∈ 𝐴 → (𝑦(+g‘𝑅)𝑥) ∈ 𝐴)))
15	1, 13, 14	sylanbrc 595	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  SubGrpcsubg 19217  NrmSGrpcnsg 19218  Abelcabl 19882  Rngcrng 20261  SubRngcsubrng 20681

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409

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-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-subg 19220  df-nsg 19221  df-cmn 19883  df-abl 19884  df-rng 20262  df-subrng 20682

This theorem is used by:  rng2idlnsg  21442  rng2idlsubgnsg  21445