Theorem rspectopn 34288

Description: The topology component of the spectrum of a ring. (Contributed by Thierry Arnoux, 4-Jun-2024.)

Hypotheses

Ref	Expression
rspecbas.1	⊢ 𝑆 = (Spec‘𝑅)
rspectopn.1	⊢ 𝐼 = (LIdeal‘𝑅)
rspectopn.2	⊢ 𝑃 = (PrmIdeal‘𝑅)
rspectopn.3	⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})

Assertion

Ref	Expression
rspectopn	⊢ (𝑅 ∈ Ring → 𝐽 = (TopOpen‘𝑆))

Distinct variable groups:   𝑖,𝐼,𝑗   𝑃,𝑖,𝑗   𝑅,𝑖,𝑗

Allowed substitution hints:   𝑆(𝑖, 𝑗)   𝐽(𝑖, 𝑗)

Proof of Theorem rspectopn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rspecval 34285	. . . . 5 ⊢ (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))
2		rspecbas.1	. . . . 5 ⊢ 𝑆 = (Spec‘𝑅)
3		rspectopn.2	. . . . . 6 ⊢ 𝑃 = (PrmIdeal‘𝑅)
4	3	oveq2i 7434	. . . . 5 ⊢ ((IDLsrg‘𝑅) ↾s 𝑃) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅))
5	1, 2, 4	3eqtr4g 2826	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑆 = ((IDLsrg‘𝑅) ↾s 𝑃))
6	5	fveq2d 6892	. . 3 ⊢ (𝑅 ∈ Ring → (TopOpen‘𝑆) = (TopOpen‘((IDLsrg‘𝑅) ↾s 𝑃)))
7		eqid 2766	. . . 4 ⊢ ((IDLsrg‘𝑅) ↾s 𝑃) = ((IDLsrg‘𝑅) ↾s 𝑃)
8		eqid 2766	. . . 4 ⊢ (TopOpen‘(IDLsrg‘𝑅)) = (TopOpen‘(IDLsrg‘𝑅))
9	7, 8	resstopn 23380	. . 3 ⊢ ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃) = (TopOpen‘((IDLsrg‘𝑅) ↾s 𝑃))
10	6, 9	eqtr4di 2819	. 2 ⊢ (𝑅 ∈ Ring → (TopOpen‘𝑆) = ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃))
11		eqid 2766	. . . . 5 ⊢ (IDLsrg‘𝑅) = (IDLsrg‘𝑅)
12		rspectopn.1	. . . . 5 ⊢ 𝐼 = (LIdeal‘𝑅)
13		eqid 2766	. . . . 5 ⊢ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})
14	11, 12, 13	idlsrgtset 33829	. . . 4 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopSet‘(IDLsrg‘𝑅)))
15	12	fvexi 6902	. . . . . . . . . . 11 ⊢ 𝐼 ∈ V
16	15	rabex 5314	. . . . . . . . . 10 ⊢ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V
17	16	a1i 11	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V)
18		simp2 1155	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼 ∧ ¬ 𝑖 ⊆ 𝑗) → 𝑗 ∈ 𝐼)
19	11, 12	idlsrgbas 33825	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝐼 = (Base‘(IDLsrg‘𝑅)))
20	19	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
21	20	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼 ∧ ¬ 𝑖 ⊆ 𝑗) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
22	18, 21	eleqtrd 2868	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼 ∧ ¬ 𝑖 ⊆ 𝑗) → 𝑗 ∈ (Base‘(IDLsrg‘𝑅)))
23	22	rabssdv 4031	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ⊆ (Base‘(IDLsrg‘𝑅)))
24	17, 23	elpwd 4573	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
25	24	ralrimiva 3160	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ∀𝑖 ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
26		eqid 2766	. . . . . . . 8 ⊢ (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})
27	26	rnmptss 7125	. . . . . . 7 ⊢ (∀𝑖 ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)) → ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)))
28	25, 27	syl 18	. . . . . 6 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)))
29	14, 28	eqsstrrd 3975	. . . . 5 ⊢ (𝑅 ∈ Ring → (TopSet‘(IDLsrg‘𝑅)) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)))
30		eqid 2766	. . . . . 6 ⊢ (Base‘(IDLsrg‘𝑅)) = (Base‘(IDLsrg‘𝑅))
31		eqid 2766	. . . . . 6 ⊢ (TopSet‘(IDLsrg‘𝑅)) = (TopSet‘(IDLsrg‘𝑅))
32	30, 31	topnid 17513	. . . . 5 ⊢ ((TopSet‘(IDLsrg‘𝑅)) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)) → (TopSet‘(IDLsrg‘𝑅)) = (TopOpen‘(IDLsrg‘𝑅)))
33	29, 32	syl 18	. . . 4 ⊢ (𝑅 ∈ Ring → (TopSet‘(IDLsrg‘𝑅)) = (TopOpen‘(IDLsrg‘𝑅)))
34	14, 33	eqtrd 2801	. . 3 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopOpen‘(IDLsrg‘𝑅)))
35	34	oveq1d 7438	. 2 ⊢ (𝑅 ∈ Ring → (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾t 𝑃) = ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃))
36	15	mptex 7228	. . . . . . 7 ⊢ (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ∈ V
37	36	rnex 7916	. . . . . 6 ⊢ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ∈ V
38	3	fvexi 6902	. . . . . 6 ⊢ 𝑃 ∈ V
39		elrest 17505	. . . . . 6 ⊢ ((ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ∈ V ∧ 𝑃 ∈ V) → (𝑥 ∈ (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾t 𝑃) ↔ ∃𝑦 ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})𝑥 = (𝑦 ∩ 𝑃)))
40	37, 38, 39	mp2an 705	. . . . 5 ⊢ (𝑥 ∈ (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾t 𝑃) ↔ ∃𝑦 ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})𝑥 = (𝑦 ∩ 𝑃))
41	16	rgenw 3086	. . . . . . 7 ⊢ ∀𝑖 ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V
42		ineq1 4169	. . . . . . . . 9 ⊢ (𝑦 = {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} → (𝑦 ∩ 𝑃) = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃))
43	42	eqeq2d 2777	. . . . . . . 8 ⊢ (𝑦 = {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} → (𝑥 = (𝑦 ∩ 𝑃) ↔ 𝑥 = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃)))
44	26, 43	rexrnmptw 7097	. . . . . . 7 ⊢ (∀𝑖 ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V → (∃𝑦 ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})𝑥 = (𝑦 ∩ 𝑃) ↔ ∃𝑖 ∈ 𝐼 𝑥 = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃)))
45	41, 44	ax-mp 5	. . . . . 6 ⊢ (∃𝑦 ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})𝑥 = (𝑦 ∩ 𝑃) ↔ ∃𝑖 ∈ 𝐼 𝑥 = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃))
46		inrab2 4273	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) = {𝑗 ∈ (𝐼 ∩ 𝑃) ∣ ¬ 𝑖 ⊆ 𝑗}
47		prmidlssidl 21507	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
48	47, 3, 12	3sstr4g 3993	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ⊆ 𝐼)
49		sseqin2 4179	. . . . . . . . . . 11 ⊢ (𝑃 ⊆ 𝐼 ↔ (𝐼 ∩ 𝑃) = 𝑃)
50	48, 49	sylib 221	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (𝐼 ∩ 𝑃) = 𝑃)
51	50	rabeqdv 3434	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → {𝑗 ∈ (𝐼 ∩ 𝑃) ∣ ¬ 𝑖 ⊆ 𝑗} = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
52	46, 51	eqtrid 2813	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
53	52	eqeq2d 2777	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑥 = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) ↔ 𝑥 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
54	53	rexbidv 3192	. . . . . 6 ⊢ (𝑅 ∈ Ring → (∃𝑖 ∈ 𝐼 𝑥 = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) ↔ ∃𝑖 ∈ 𝐼 𝑥 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
55	45, 54	bitrid 286	. . . . 5 ⊢ (𝑅 ∈ Ring → (∃𝑦 ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})𝑥 = (𝑦 ∩ 𝑃) ↔ ∃𝑖 ∈ 𝐼 𝑥 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
56	40, 55	bitrid 286	. . . 4 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾t 𝑃) ↔ ∃𝑖 ∈ 𝐼 𝑥 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
57		rspectopn.3	. . . . . 6 ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
58	57	eleq2i 2858	. . . . 5 ⊢ (𝑥 ∈ 𝐽 ↔ 𝑥 ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
59		eqid 2766	. . . . . 6 ⊢ (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
60	38	rabex 5314	. . . . . 6 ⊢ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V
61	59, 60	elrnmpti 5957	. . . . 5 ⊢ (𝑥 ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ ∃𝑖 ∈ 𝐼 𝑥 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
62	58, 61	bitri 278	. . . 4 ⊢ (𝑥 ∈ 𝐽 ↔ ∃𝑖 ∈ 𝐼 𝑥 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
63	56, 62	bitr4di 292	. . 3 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾t 𝑃) ↔ 𝑥 ∈ 𝐽))
64	63	eqrdv 2764	. 2 ⊢ (𝑅 ∈ Ring → (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾t 𝑃) = 𝐽)
65	10, 35, 64	3eqtr2rd 2808	1 ⊢ (𝑅 ∈ Ring → 𝐽 = (TopOpen‘𝑆))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092  {crab 3419  Vcvv 3458   ∩ cin 3907   ⊆ wss 3908  𝒫 cpw 4567   ↦ cmpt 5197  ran crn 5667  ‘cfv 6543  (class class class)co 7423  Basecbs 17294   ↾_s cress 17315  TopSetcts 17341   ↾_t crest 17498  TopOpenctopn 17499  Ringcrg 20346  LIdealclidl 21367  PrmIdealcprmidl 21497  IDLsrgcidlsrg 33821  Speccrspec 34283

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12523  df-z 12610  df-dec 12730  df-uz 12881  df-fz 13554  df-struct 17232  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-plusg 17348  df-mulr 17349  df-tset 17354  df-ple 17355  df-rest 17500  df-topn 17501  df-prmidl 21498  df-idlsrg 33822  df-rspec 34284

This theorem is used by:  zarcls  34295  zar0ring  34299