Theorem subrngringnsg 13577

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 13576	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2		subrngrcl 13575	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
3		rngabl 13314	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
4	2, 3	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Abel)
5	4	3anim1i 1187	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)))
6	5	3expb 1206	. . . . . 6 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)))
7		eqid 2189	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2189	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
9	7, 8	ablcom 13267	. . . . . 6 ⊢ ((𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = (𝑦(+g‘𝑅)𝑥))
10	6, 9	syl 14	. . . . 5 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑦(+g‘𝑅)𝑥))
11	10	eleq1d 2258	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(+g‘𝑅)𝑦) ∈ 𝐴 ↔ (𝑦(+g‘𝑅)𝑥) ∈ 𝐴))
12	11	biimpd 144	. . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(+g‘𝑅)𝑦) ∈ 𝐴 → (𝑦(+g‘𝑅)𝑥) ∈ 𝐴))
13	12	ralrimivva 2572	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) ∈ 𝐴 → (𝑦(+g‘𝑅)𝑥) ∈ 𝐴))
14	7, 8	isnsg2 13167	. 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 417	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2160  ∀wral 2468  ‘cfv 5238  (class class class)co 5900  Basecbs 12523  +_gcplusg 12600  SubGrpcsubg 13131  NrmSGrpcnsg 13132  Abelcabl 13249  Rngcrng 13311  SubRngcsubrng 13569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-cnex 7937  ax-resscn 7938  ax-1re 7940  ax-addrcl 7943

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-fv 5246  df-ov 5903  df-inn 8955  df-2 9013  df-3 9014  df-ndx 12526  df-slot 12527  df-base 12529  df-plusg 12613  df-mulr 12614  df-subg 13134  df-nsg 13135  df-cmn 13250  df-abl 13251  df-rng 13312  df-subrng 13570

This theorem is referenced by:  rng2idlnsg  13858  rng2idlsubgnsg  13861