Theorem rspectopn 34044

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 34041	. . . . 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 7379	. . . . 5 ⊢ ((IDLsrg‘𝑅) ↾s 𝑃) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅))
5	1, 2, 4	3eqtr4g 2797	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑆 = ((IDLsrg‘𝑅) ↾s 𝑃))
6	5	fveq2d 6846	. . 3 ⊢ (𝑅 ∈ Ring → (TopOpen‘𝑆) = (TopOpen‘((IDLsrg‘𝑅) ↾s 𝑃)))
7		eqid 2737	. . . 4 ⊢ ((IDLsrg‘𝑅) ↾s 𝑃) = ((IDLsrg‘𝑅) ↾s 𝑃)
8		eqid 2737	. . . 4 ⊢ (TopOpen‘(IDLsrg‘𝑅)) = (TopOpen‘(IDLsrg‘𝑅))
9	7, 8	resstopn 23142	. . 3 ⊢ ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃) = (TopOpen‘((IDLsrg‘𝑅) ↾s 𝑃))
10	6, 9	eqtr4di 2790	. 2 ⊢ (𝑅 ∈ Ring → (TopOpen‘𝑆) = ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃))
11		eqid 2737	. . . . 5 ⊢ (IDLsrg‘𝑅) = (IDLsrg‘𝑅)
12		rspectopn.1	. . . . 5 ⊢ 𝐼 = (LIdeal‘𝑅)
13		eqid 2737	. . . . 5 ⊢ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})
14	11, 12, 13	idlsrgtset 33600	. . . 4 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopSet‘(IDLsrg‘𝑅)))
15	12	fvexi 6856	. . . . . . . . . . 11 ⊢ 𝐼 ∈ V
16	15	rabex 5286	. . . . . . . . . 10 ⊢ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V
17	16	a1i 11	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V)
18		simp2 1138	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼 ∧ ¬ 𝑖 ⊆ 𝑗) → 𝑗 ∈ 𝐼)
19	11, 12	idlsrgbas 33596	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝐼 = (Base‘(IDLsrg‘𝑅)))
20	19	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
21	20	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼 ∧ ¬ 𝑖 ⊆ 𝑗) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
22	18, 21	eleqtrd 2839	. . . . . . . . . 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 4562	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
25	24	ralrimiva 3130	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ∀𝑖 ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
26		eqid 2737	. . . . . . . 8 ⊢ (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})
27	26	rnmptss 7077	. . . . . . 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 3971	. . . . 5 ⊢ (𝑅 ∈ Ring → (TopSet‘(IDLsrg‘𝑅)) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)))
30		eqid 2737	. . . . . 6 ⊢ (Base‘(IDLsrg‘𝑅)) = (Base‘(IDLsrg‘𝑅))
31		eqid 2737	. . . . . 6 ⊢ (TopSet‘(IDLsrg‘𝑅)) = (TopSet‘(IDLsrg‘𝑅))
32	30, 31	topnid 17367	. . . . 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 2772	. . 3 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopOpen‘(IDLsrg‘𝑅)))
35	34	oveq1d 7383	. 2 ⊢ (𝑅 ∈ Ring → (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾t 𝑃) = ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃))
36	15	mptex 7179	. . . . . . 7 ⊢ (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ∈ V
37	36	rnex 7862	. . . . . 6 ⊢ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ∈ V
38	3	fvexi 6856	. . . . . 6 ⊢ 𝑃 ∈ V
39		elrest 17359	. . . . . 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 693	. . . . 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 3056	. . . . . . 7 ⊢ ∀𝑖 ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V
42		ineq1 4167	. . . . . . . . 9 ⊢ (𝑦 = {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} → (𝑦 ∩ 𝑃) = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃))
43	42	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑦 = {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} → (𝑥 = (𝑦 ∩ 𝑃) ↔ 𝑥 = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃)))
44	26, 43	rexrnmptw 7049	. . . . . . 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 4271	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) = {𝑗 ∈ (𝐼 ∩ 𝑃) ∣ ¬ 𝑖 ⊆ 𝑗}
47		prmidlssidl 33537	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
48	47, 3, 12	3sstr4g 3989	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ⊆ 𝐼)
49		sseqin2 4177	. . . . . . . . . . 11 ⊢ (𝑃 ⊆ 𝐼 ↔ (𝐼 ∩ 𝑃) = 𝑃)
50	48, 49	sylib 218	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (𝐼 ∩ 𝑃) = 𝑃)
51	50	rabeqdv 3416	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → {𝑗 ∈ (𝐼 ∩ 𝑃) ∣ ¬ 𝑖 ⊆ 𝑗} = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
52	46, 51	eqtrid 2784	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
53	52	eqeq2d 2748	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑥 = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) ↔ 𝑥 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
54	53	rexbidv 3162	. . . . . 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 2829	. . . . 5 ⊢ (𝑥 ∈ 𝐽 ↔ 𝑥 ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
59		eqid 2737	. . . . . 6 ⊢ (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
60	38	rabex 5286	. . . . . 6 ⊢ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V
61	59, 60	elrnmpti 5919	. . . . 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 2735	. 2 ⊢ (𝑅 ∈ Ring → (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾t 𝑃) = 𝐽)
65	10, 35, 64	3eqtr2rd 2779	1 ⊢ (𝑅 ∈ Ring → 𝐽 = (TopOpen‘𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556   ↦ cmpt 5181  ran crn 5633  ‘cfv 6500  (class class class)co 7368  Basecbs 17148   ↾_s cress 17169  TopSetcts 17195   ↾_t crest 17352  TopOpenctopn 17353  Ringcrg 20180  LIdealclidl 21173  PrmIdealcprmidl 33527  IDLsrgcidlsrg 33592  Speccrspec 34039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-tset 17208  df-ple 17209  df-rest 17354  df-topn 17355  df-prmidl 33528  df-idlsrg 33593  df-rspec 34040

This theorem is referenced by:  zarcls  34051  zar0ring  34055