Theorem issubrng2 20523

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 20517	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2		issubrng2.t	. . . . . 6 ⊢ · = (.r‘𝑅)
3	2	subrngmcl 20522	. . . . 5 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴)
4	3	3expb 1120	. . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 · 𝑦) ∈ 𝐴)
5	4	ralrimivva 3188	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)
6	1, 5	jca 511	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴))
7		simpl 482	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝑅 ∈ Rng)
8		simprl 770	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubGrp‘𝑅))
9		eqid 2736	. . . . . . 7 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴)
10	9	subgbas 19118	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 = (Base‘(𝑅 ↾s 𝐴)))
11	8, 10	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 = (Base‘(𝑅 ↾s 𝐴)))
12		eqid 2736	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
13	9, 12	ressplusg 17310	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (+g‘𝑅) = (+g‘(𝑅 ↾s 𝐴)))
14	8, 13	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (+g‘𝑅) = (+g‘(𝑅 ↾s 𝐴)))
15	9, 2	ressmulr 17326	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → · = (.r‘(𝑅 ↾s 𝐴)))
16	8, 15	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → · = (.r‘(𝑅 ↾s 𝐴)))
17		rngabl 20120	. . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
18	9	subgabl 19822	. . . . . 6 ⊢ ((𝑅 ∈ Abel ∧ 𝐴 ∈ (SubGrp‘𝑅)) → (𝑅 ↾s 𝐴) ∈ Abel)
19	17, 8, 18	syl2an2r 685	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾s 𝐴) ∈ Abel)
20		simprr 772	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)
21		oveq1 7417	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
22	21	eleq1d 2820	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑦) ∈ 𝐴))
23		oveq2 7418	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
24	23	eleq1d 2820	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑢 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑣) ∈ 𝐴))
25	22, 24	rspc2v 3617	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 → (𝑢 · 𝑣) ∈ 𝐴))
26	20, 25	syl5com 31	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 · 𝑣) ∈ 𝐴))
27	26	3impib 1116	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 · 𝑣) ∈ 𝐴)
28		issubrng2.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
29	28	subgss 19115	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ⊆ 𝐵)
30	8, 29	syl 17	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ⊆ 𝐵)
31	30	sseld 3962	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝐵))
32	30	sseld 3962	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑣 ∈ 𝐴 → 𝑣 ∈ 𝐵))
33	30	sseld 3962	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵))
34	31, 32, 33	3anim123d 1445	. . . . . . 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)))
35	34	imp 406	. . . . . 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))
36	28, 2	rngass 20124	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
37	36	adantlr 715	. . . . . 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)))
38	35, 37	syldan 591	. . . . 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)))
39	28, 12, 2	rngdi 20125	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢 · (𝑣(+g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+g‘𝑅)(𝑢 · 𝑤)))
40	39	adantlr 715	. . . . . 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)))
41	35, 40	syldan 591	. . . . 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)))
42	28, 12, 2	rngdir 20126	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g‘𝑅)(𝑣 · 𝑤)))
43	42	adantlr 715	. . . . . 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)))
44	35, 43	syldan 591	. . . . 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)))
45	11, 14, 16, 19, 27, 38, 41, 44	isrngd 20138	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾s 𝐴) ∈ Rng)
46	28	issubrng 20512	. . . 4 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))
47	7, 45, 30, 46	syl3anbrc 1344	. . 3 ⊢ ((𝑅 ∈ Rng ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubRng‘𝑅))
48	47	ex 412	. 2 ⊢ (𝑅 ∈ Rng → ((𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴) → 𝐴 ∈ (SubRng‘𝑅)))
49	6, 48	impbid2 226	1 ⊢ (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052   ⊆ wss 3931  ‘cfv 6536  (class class class)co 7410  Basecbs 17233   ↾_s cress 17256  +_gcplusg 17276  ._rcmulr 17277  SubGrpcsubg 19108  Abelcabl 19767  Rngcrng 20117  SubRngcsubrng 20510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-mulr 17290  df-mgm 18623  df-sgrp 18702  df-grp 18924  df-subg 19111  df-cmn 19768  df-abl 19769  df-mgp 20106  df-rng 20118  df-subrng 20511

This theorem is referenced by:  opprsubrng  20524  subrngint  20525  rhmimasubrng  20531  cntzsubrng  20532  pzriprnglem5  21451