Theorem issubrng2 13943

Description: Characterize the subrings of a ring by closure properties. (Contributed by AV, 15-Feb-2025.)

Hypotheses

Ref	Expression
issubrng2.b	⊢ 𝐵 = (Base‘𝑅)
issubrng2.t	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
issubrng2	⊢ (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦   𝑥, · ,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem issubrng2

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrngsubg 13937	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2		issubrng2.t	. . . . . 6 ⊢ · = (.r‘𝑅)
3	2	subrngmcl 13942	. . . . 5 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴)
4	3	3expb 1206	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 · 𝑦) ∈ 𝐴)
5	4	ralrimivva 2587	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)
6	1, 5	jca 306	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴))
7		simpl 109	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝑅 ∈ Rng)
8		simprl 529	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubGrp‘𝑅))
9		eqid 2204	. . . . . . 7 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴)
10	9	subgbas 13485	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 = (Base‘(𝑅 ↾s 𝐴)))
11	8, 10	syl 14	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 = (Base‘(𝑅 ↾s 𝐴)))
12		eqidd 2205	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴))
13		eqidd 2205	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (+g‘𝑅) = (+g‘𝑅))
14		id 19	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
15		subgrcl 13486	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝑅 ∈ Grp)
16	12, 13, 14, 15	ressplusgd 12932	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (+g‘𝑅) = (+g‘(𝑅 ↾s 𝐴)))
17	8, 16	syl 14	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (+g‘𝑅) = (+g‘(𝑅 ↾s 𝐴)))
18	9, 2	ressmulrg 12948	. . . . . 6 ⊢ ((𝐴 ∈ (SubGrp‘𝑅) ∧ 𝑅 ∈ Grp) → · = (.r‘(𝑅 ↾s 𝐴)))
19	8, 15, 18	syl2anc2 412	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → · = (.r‘(𝑅 ↾s 𝐴)))
20		rngabl 13668	. . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
21	9	subgabl 13639	. . . . . 6 ⊢ ((𝑅 ∈ Abel ∧ 𝐴 ∈ (SubGrp‘𝑅)) → (𝑅 ↾s 𝐴) ∈ Abel)
22	20, 8, 21	syl2an2r 595	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾s 𝐴) ∈ Abel)
23		simprr 531	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)
24		oveq1 5950	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
25	24	eleq1d 2273	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑦) ∈ 𝐴))
26		oveq2 5951	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
27	26	eleq1d 2273	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑢 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑣) ∈ 𝐴))
28	25, 27	rspc2v 2889	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 → (𝑢 · 𝑣) ∈ 𝐴))
29	23, 28	syl5com 29	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 · 𝑣) ∈ 𝐴))
30	29	3impib 1203	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 · 𝑣) ∈ 𝐴)
31		issubrng2.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
32	31	subgss 13481	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ⊆ 𝐵)
33	8, 32	syl 14	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ⊆ 𝐵)
34	33	sseld 3191	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝐵))
35	33	sseld 3191	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑣 ∈ 𝐴 → 𝑣 ∈ 𝐵))
36	33	sseld 3191	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵))
37	34, 35, 36	3anim123d 1331	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
38	37	imp 124	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))
39	31, 2	rngass 13672	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
40	39	adantlr 477	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
41	38, 40	syldan 282	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
42		eqid 2204	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅)
43	31, 42, 2	rngdi 13673	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢 · (𝑣(+g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+g‘𝑅)(𝑢 · 𝑤)))
44	43	adantlr 477	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢 · (𝑣(+g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+g‘𝑅)(𝑢 · 𝑤)))
45	38, 44	syldan 282	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑢 · (𝑣(+g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+g‘𝑅)(𝑢 · 𝑤)))
46	31, 42, 2	rngdir 13674	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g‘𝑅)(𝑣 · 𝑤)))
47	46	adantlr 477	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g‘𝑅)(𝑣 · 𝑤)))
48	38, 47	syldan 282	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑢(+g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g‘𝑅)(𝑣 · 𝑤)))
49	11, 17, 19, 22, 30, 41, 45, 48	isrngd 13686	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾s 𝐴) ∈ Rng)
50	31	issubrng 13932	. . . 4 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))
51	7, 49, 33, 50	syl3anbrc 1183	. . 3 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubRng‘𝑅))
52	51	ex 115	. 2 ⊢ (𝑅 ∈ Rng → ((𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴) → 𝐴 ∈ (SubRng‘𝑅)))
53	6, 52	impbid2 143	1 ⊢ (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1372   ∈ wcel 2175  ∀wral 2483   ⊆ wss 3165  ‘cfv 5270  (class class class)co 5943  Basecbs 12803   ↾_s cress 12804  +_gcplusg 12880  ._rcmulr 12881  Grpcgrp 13303  SubGrpcsubg 13474  Abelcabl 13592  Rngcrng 13665  SubRngcsubrng 13930

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-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-ndx 12806  df-slot 12807  df-base 12809  df-sets 12810  df-iress 12811  df-plusg 12893  df-mulr 12894  df-mgm 13159  df-sgrp 13205  df-grp 13306  df-subg 13477  df-cmn 13593  df-abl 13594  df-mgp 13654  df-rng 13666  df-subrng 13931

This theorem is referenced by:  opprsubrngg  13944  subrngintm  13945