Theorem rspectopn 33833

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 33830	. . . . 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 7364	. . . . 5 ⊢ ((IDLsrg‘𝑅) ↾s 𝑃) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅))
5	1, 2, 4	3eqtr4g 2789	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑆 = ((IDLsrg‘𝑅) ↾s 𝑃))
6	5	fveq2d 6830	. . 3 ⊢ (𝑅 ∈ Ring → (TopOpen‘𝑆) = (TopOpen‘((IDLsrg‘𝑅) ↾s 𝑃)))
7		eqid 2729	. . . 4 ⊢ ((IDLsrg‘𝑅) ↾s 𝑃) = ((IDLsrg‘𝑅) ↾s 𝑃)
8		eqid 2729	. . . 4 ⊢ (TopOpen‘(IDLsrg‘𝑅)) = (TopOpen‘(IDLsrg‘𝑅))
9	7, 8	resstopn 23089	. . 3 ⊢ ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃) = (TopOpen‘((IDLsrg‘𝑅) ↾s 𝑃))
10	6, 9	eqtr4di 2782	. 2 ⊢ (𝑅 ∈ Ring → (TopOpen‘𝑆) = ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃))
11		eqid 2729	. . . . 5 ⊢ (IDLsrg‘𝑅) = (IDLsrg‘𝑅)
12		rspectopn.1	. . . . 5 ⊢ 𝐼 = (LIdeal‘𝑅)
13		eqid 2729	. . . . 5 ⊢ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})
14	11, 12, 13	idlsrgtset 33455	. . . 4 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopSet‘(IDLsrg‘𝑅)))
15	12	fvexi 6840	. . . . . . . . . . 11 ⊢ 𝐼 ∈ V
16	15	rabex 5281	. . . . . . . . . 10 ⊢ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V
17	16	a1i 11	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V)
18		simp2 1137	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼 ∧ ¬ 𝑖 ⊆ 𝑗) → 𝑗 ∈ 𝐼)
19	11, 12	idlsrgbas 33451	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝐼 = (Base‘(IDLsrg‘𝑅)))
20	19	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
21	20	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼 ∧ ¬ 𝑖 ⊆ 𝑗) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
22	18, 21	eleqtrd 2830	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼 ∧ ¬ 𝑖 ⊆ 𝑗) → 𝑗 ∈ (Base‘(IDLsrg‘𝑅)))
23	22	rabssdv 4028	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ⊆ (Base‘(IDLsrg‘𝑅)))
24	17, 23	elpwd 4559	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
25	24	ralrimiva 3121	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ∀𝑖 ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
26		eqid 2729	. . . . . . . 8 ⊢ (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})
27	26	rnmptss 7061	. . . . . . 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 17	. . . . . 6 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)))
29	14, 28	eqsstrrd 3973	. . . . 5 ⊢ (𝑅 ∈ Ring → (TopSet‘(IDLsrg‘𝑅)) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)))
30		eqid 2729	. . . . . 6 ⊢ (Base‘(IDLsrg‘𝑅)) = (Base‘(IDLsrg‘𝑅))
31		eqid 2729	. . . . . 6 ⊢ (TopSet‘(IDLsrg‘𝑅)) = (TopSet‘(IDLsrg‘𝑅))
32	30, 31	topnid 17357	. . . . 5 ⊢ ((TopSet‘(IDLsrg‘𝑅)) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)) → (TopSet‘(IDLsrg‘𝑅)) = (TopOpen‘(IDLsrg‘𝑅)))
33	29, 32	syl 17	. . . 4 ⊢ (𝑅 ∈ Ring → (TopSet‘(IDLsrg‘𝑅)) = (TopOpen‘(IDLsrg‘𝑅)))
34	14, 33	eqtrd 2764	. . 3 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopOpen‘(IDLsrg‘𝑅)))
35	34	oveq1d 7368	. 2 ⊢ (𝑅 ∈ Ring → (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾t 𝑃) = ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃))
36	15	mptex 7163	. . . . . . 7 ⊢ (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ∈ V
37	36	rnex 7850	. . . . . 6 ⊢ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ∈ V
38	3	fvexi 6840	. . . . . 6 ⊢ 𝑃 ∈ V
39		elrest 17349	. . . . . 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 692	. . . . 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 3048	. . . . . . 7 ⊢ ∀𝑖 ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V
42		ineq1 4166	. . . . . . . . 9 ⊢ (𝑦 = {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} → (𝑦 ∩ 𝑃) = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃))
43	42	eqeq2d 2740	. . . . . . . 8 ⊢ (𝑦 = {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} → (𝑥 = (𝑦 ∩ 𝑃) ↔ 𝑥 = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃)))
44	26, 43	rexrnmptw 7033	. . . . . . 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 4270	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) = {𝑗 ∈ (𝐼 ∩ 𝑃) ∣ ¬ 𝑖 ⊆ 𝑗}
47		prmidlssidl 33392	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
48	47, 3, 12	3sstr4g 3991	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ⊆ 𝐼)
49		sseqin2 4176	. . . . . . . . . . 11 ⊢ (𝑃 ⊆ 𝐼 ↔ (𝐼 ∩ 𝑃) = 𝑃)
50	48, 49	sylib 218	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (𝐼 ∩ 𝑃) = 𝑃)
51	50	rabeqdv 3412	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → {𝑗 ∈ (𝐼 ∩ 𝑃) ∣ ¬ 𝑖 ⊆ 𝑗} = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
52	46, 51	eqtrid 2776	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
53	52	eqeq2d 2740	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑥 = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) ↔ 𝑥 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
54	53	rexbidv 3153	. . . . . 6 ⊢ (𝑅 ∈ Ring → (∃𝑖 ∈ 𝐼 𝑥 = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) ↔ ∃𝑖 ∈ 𝐼 𝑥 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
55	45, 54	bitrid 283	. . . . 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 283	. . . 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 2820	. . . . 5 ⊢ (𝑥 ∈ 𝐽 ↔ 𝑥 ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
59		eqid 2729	. . . . . 6 ⊢ (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
60	38	rabex 5281	. . . . . 6 ⊢ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V
61	59, 60	elrnmpti 5908	. . . . 5 ⊢ (𝑥 ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ ∃𝑖 ∈ 𝐼 𝑥 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
62	58, 61	bitri 275	. . . 4 ⊢ (𝑥 ∈ 𝐽 ↔ ∃𝑖 ∈ 𝐼 𝑥 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
63	56, 62	bitr4di 289	. . 3 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾t 𝑃) ↔ 𝑥 ∈ 𝐽))
64	63	eqrdv 2727	. 2 ⊢ (𝑅 ∈ Ring → (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾t 𝑃) = 𝐽)
65	10, 35, 64	3eqtr2rd 2771	1 ⊢ (𝑅 ∈ Ring → 𝐽 = (TopOpen‘𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3396  Vcvv 3438   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4553   ↦ cmpt 5176  ran crn 5624  ‘cfv 6486  (class class class)co 7353  Basecbs 17138   ↾_s cress 17159  TopSetcts 17185   ↾_t crest 17342  TopOpenctopn 17343  Ringcrg 20136  LIdealclidl 21131  PrmIdealcprmidl 33382  IDLsrgcidlsrg 33447  Speccrspec 33828

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 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-z 12490  df-dec 12610  df-uz 12754  df-fz 13429  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-mulr 17193  df-tset 17198  df-ple 17199  df-rest 17344  df-topn 17345  df-prmidl 33383  df-idlsrg 33448  df-rspec 33829

This theorem is referenced by:  zarcls  33840  zar0ring  33844