Theorem rspectopn 33857

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 33854	. . . . 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 7398	. . . . 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 6862	. . 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 23073	. . 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 33479	. . . 4 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopSet‘(IDLsrg‘𝑅)))
15	12	fvexi 6872	. . . . . . . . . . 11 ⊢ 𝐼 ∈ V
16	15	rabex 5294	. . . . . . . . . 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 33475	. . . . . . . . . . . . 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 4038	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ⊆ (Base‘(IDLsrg‘𝑅)))
24	17, 23	elpwd 4569	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
25	24	ralrimiva 3125	. . . . . . 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 7095	. . . . . . 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 3982	. . . . 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 17398	. . . . 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 7402	. 2 ⊢ (𝑅 ∈ Ring → (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾t 𝑃) = ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃))
36	15	mptex 7197	. . . . . . 7 ⊢ (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ∈ V
37	36	rnex 7886	. . . . . 6 ⊢ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ∈ V
38	3	fvexi 6872	. . . . . 6 ⊢ 𝑃 ∈ V
39		elrest 17390	. . . . . 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 4176	. . . . . . . . 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 7067	. . . . . . 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 4280	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) = {𝑗 ∈ (𝐼 ∩ 𝑃) ∣ ¬ 𝑖 ⊆ 𝑗}
47		prmidlssidl 33416	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
48	47, 3, 12	3sstr4g 4000	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ⊆ 𝐼)
49		sseqin2 4186	. . . . . . . . . . 11 ⊢ (𝑃 ⊆ 𝐼 ↔ (𝐼 ∩ 𝑃) = 𝑃)
50	48, 49	sylib 218	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (𝐼 ∩ 𝑃) = 𝑃)
51	50	rabeqdv 3421	. . . . . . . . 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 3157	. . . . . 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 5294	. . . . . 6 ⊢ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V
61	59, 60	elrnmpti 5926	. . . . 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 3405  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  𝒫 cpw 4563   ↦ cmpt 5188  ran crn 5639  ‘cfv 6511  (class class class)co 7387  Basecbs 17179   ↾_s cress 17200  TopSetcts 17226   ↾_t crest 17383  TopOpenctopn 17384  Ringcrg 20142  LIdealclidl 21116  PrmIdealcprmidl 33406  IDLsrgcidlsrg 33471  Speccrspec 33852

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-tset 17239  df-ple 17240  df-rest 17385  df-topn 17386  df-prmidl 33407  df-idlsrg 33472  df-rspec 33853

This theorem is referenced by:  zarcls  33864  zar0ring  33868