Theorem subrngringnsg 13837

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 13836	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2		subrngrcl 13835	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
3		rngabl 13567	. . . . . . . . 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 2196	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2196	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
9	7, 8	ablcom 13509	. . . . . 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 2265	. . . 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 2579	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) ∈ 𝐴 → (𝑦(+g‘𝑅)𝑥) ∈ 𝐴))
14	7, 8	isnsg2 13409	. 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 2167  ∀wral 2475  ‘cfv 5259  (class class class)co 5925  Basecbs 12703  +_gcplusg 12780  SubGrpcsubg 13373  NrmSGrpcnsg 13374  Abelcabl 13491  Rngcrng 13564  SubRngcsubrng 13829

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5928  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-mulr 12794  df-subg 13376  df-nsg 13377  df-cmn 13492  df-abl 13493  df-rng 13565  df-subrng 13830

This theorem is referenced by:  rng2idlnsg  14150  rng2idlsubgnsg  14153