Theorem issubrng2 14288

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 14282	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2		issubrng2.t	. . . . . 6 ⊢ · = (.r‘𝑅)
3	2	subrngmcl 14287	. . . . 5 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴)
4	3	3expb 1231	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 · 𝑦) ∈ 𝐴)
5	4	ralrimivva 2615	. . 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 531	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubGrp‘𝑅))
9		eqid 2231	. . . . . . 7 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴)
10	9	subgbas 13828	. . . . . 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 2232	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴))
13		eqidd 2232	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (+g‘𝑅) = (+g‘𝑅))
14		id 19	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
15		subgrcl 13829	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝑅 ∈ Grp)
16	12, 13, 14, 15	ressplusgd 13275	. . . . . 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 13291	. . . . . 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 14012	. . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
21	9	subgabl 13982	. . . . . 6 ⊢ ((𝑅 ∈ Abel ∧ 𝐴 ∈ (SubGrp‘𝑅)) → (𝑅 ↾s 𝐴) ∈ Abel)
22	20, 8, 21	syl2an2r 599	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾s 𝐴) ∈ Abel)
23		simprr 533	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)
24		oveq1 6035	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
25	24	eleq1d 2300	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑦) ∈ 𝐴))
26		oveq2 6036	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
27	26	eleq1d 2300	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑢 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑣) ∈ 𝐴))
28	25, 27	rspc2v 2924	. . . . . . 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 1228	. . . . 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 13824	. . . . . . . . . 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 3227	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝐵))
35	33	sseld 3227	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑣 ∈ 𝐴 → 𝑣 ∈ 𝐵))
36	33	sseld 3227	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵))
37	34, 35, 36	3anim123d 1356	. . . . . . 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 14016	. . . . . . 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 2231	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅)
43	31, 42, 2	rngdi 14017	. . . . . . 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 14018	. . . . . . 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 14030	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾s 𝐴) ∈ Rng)
50	31	issubrng 14277	. . . 4 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))
51	7, 49, 33, 50	syl3anbrc 1208	. . 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 1005   = wceq 1398   ∈ wcel 2202  ∀wral 2511   ⊆ wss 3201  ‘cfv 5333  (class class class)co 6028  Basecbs 13145   ↾_s cress 13146  +_gcplusg 13223  ._rcmulr 13224  Grpcgrp 13646  SubGrpcsubg 13817  Abelcabl 13935  Rngcrng 14009  SubRngcsubrng 14275

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-3 9245  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-mulr 13237  df-mgm 13502  df-sgrp 13548  df-grp 13649  df-subg 13820  df-cmn 13936  df-abl 13937  df-mgp 13998  df-rng 14010  df-subrng 14276

This theorem is referenced by:  opprsubrngg  14289  subrngintm  14290