Theorem subrngringnsg 14222

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 14221	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2		subrngrcl 14220	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
3		rngabl 13951	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
4	2, 3	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Abel)
5	4	3anim1i 1211	. . . . . . 7 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)))
6	5	3expb 1230	. . . . . 6 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)))
7		eqid 2231	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2231	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
9	7, 8	ablcom 13892	. . . . . 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 2300	. . . 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 2614	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) ∈ 𝐴 → (𝑦(+g‘𝑅)𝑥) ∈ 𝐴))
14	7, 8	isnsg2 13792	. 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 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ‘cfv 5326  (class class class)co 6018  Basecbs 13084  +_gcplusg 13162  SubGrpcsubg 13756  NrmSGrpcnsg 13757  Abelcabl 13874  Rngcrng 13948  SubRngcsubrng 14214

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6021  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13087  df-slot 13088  df-base 13090  df-plusg 13175  df-mulr 13176  df-subg 13759  df-nsg 13760  df-cmn 13875  df-abl 13876  df-rng 13949  df-subrng 14215

This theorem is referenced by:  rng2idlnsg  14535  rng2idlsubgnsg  14538