rspectopn - Mathbox for Thierry Arnoux

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Theorem rspectopn 33404

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 33401	. . . . 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 7425	. . . . 5 ⊢ ((IDLsrg‘𝑅) ↾_s 𝑃) = ((IDLsrg‘𝑅) ↾_s (PrmIdeal‘𝑅))
5	1, 2, 4	3eqtr4g 2792	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑆 = ((IDLsrg‘𝑅) ↾_s 𝑃))
6	5	fveq2d 6895	. . 3 ⊢ (𝑅 ∈ Ring → (TopOpen‘𝑆) = (TopOpen‘((IDLsrg‘𝑅) ↾_s 𝑃)))
7		eqid 2727	. . . 4 ⊢ ((IDLsrg‘𝑅) ↾_s 𝑃) = ((IDLsrg‘𝑅) ↾_s 𝑃)
8		eqid 2727	. . . 4 ⊢ (TopOpen‘(IDLsrg‘𝑅)) = (TopOpen‘(IDLsrg‘𝑅))
9	7, 8	resstopn 23077	. . 3 ⊢ ((TopOpen‘(IDLsrg‘𝑅)) ↾_t 𝑃) = (TopOpen‘((IDLsrg‘𝑅) ↾_s 𝑃))
10	6, 9	eqtr4di 2785	. 2 ⊢ (𝑅 ∈ Ring → (TopOpen‘𝑆) = ((TopOpen‘(IDLsrg‘𝑅)) ↾_t 𝑃))
11		eqid 2727	. . . . 5 ⊢ (IDLsrg‘𝑅) = (IDLsrg‘𝑅)
12		rspectopn.1	. . . . 5 ⊢ 𝐼 = (LIdeal‘𝑅)
13		eqid 2727	. . . . 5 ⊢ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})
14	11, 12, 13	idlsrgtset 33155	. . . 4 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopSet‘(IDLsrg‘𝑅)))
15	12	fvexi 6905	. . . . . . . . . . 11 ⊢ 𝐼 ∈ V
16	15	rabex 5328	. . . . . . . . . 10 ⊢ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V
17	16	a1i 11	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V)
18		simp2 1135	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼 ∧ ¬ 𝑖 ⊆ 𝑗) → 𝑗 ∈ 𝐼)
19	11, 12	idlsrgbas 33151	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝐼 = (Base‘(IDLsrg‘𝑅)))
20	19	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
21	20	3ad2ant1 1131	. . . . . . . . . . 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 4068	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ⊆ (Base‘(IDLsrg‘𝑅)))
24	17, 23	elpwd 4604	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) → {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
25	24	ralrimiva 3141	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ∀𝑖 ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
26		eqid 2727	. . . . . . . 8 ⊢ (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})
27	26	rnmptss 7127	. . . . . . 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 4017	. . . . 5 ⊢ (𝑅 ∈ Ring → (TopSet‘(IDLsrg‘𝑅)) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)))
30		eqid 2727	. . . . . 6 ⊢ (Base‘(IDLsrg‘𝑅)) = (Base‘(IDLsrg‘𝑅))
31		eqid 2727	. . . . . 6 ⊢ (TopSet‘(IDLsrg‘𝑅)) = (TopSet‘(IDLsrg‘𝑅))
32	30, 31	topnid 17408	. . . . 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 2767	. . 3 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopOpen‘(IDLsrg‘𝑅)))
35	34	oveq1d 7429	. 2 ⊢ (𝑅 ∈ Ring → (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾_t 𝑃) = ((TopOpen‘(IDLsrg‘𝑅)) ↾_t 𝑃))
36	15	mptex 7229	. . . . . . 7 ⊢ (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ∈ V
37	36	rnex 7912	. . . . . 6 ⊢ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ∈ V
38	3	fvexi 6905	. . . . . 6 ⊢ 𝑃 ∈ V
39		elrest 17400	. . . . . 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 691	. . . . 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 3060	. . . . . . 7 ⊢ ∀𝑖 ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V
42		ineq1 4201	. . . . . . . . 9 ⊢ (𝑦 = {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} → (𝑦 ∩ 𝑃) = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃))
43	42	eqeq2d 2738	. . . . . . . 8 ⊢ (𝑦 = {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} → (𝑥 = (𝑦 ∩ 𝑃) ↔ 𝑥 = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃)))
44	26, 43	rexrnmptw 7099	. . . . . . 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 4303	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) = {𝑗 ∈ (𝐼 ∩ 𝑃) ∣ ¬ 𝑖 ⊆ 𝑗}
47		prmidlssidl 33096	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
48	47, 3, 12	3sstr4g 4023	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑃 ⊆ 𝐼)
49		sseqin2 4211	. . . . . . . . . . 11 ⊢ (𝑃 ⊆ 𝐼 ↔ (𝐼 ∩ 𝑃) = 𝑃)
50	48, 49	sylib 217	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (𝐼 ∩ 𝑃) = 𝑃)
51	50	rabeqdv 3442	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → {𝑗 ∈ (𝐼 ∩ 𝑃) ∣ ¬ 𝑖 ⊆ 𝑗} = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
52	46, 51	eqtrid 2779	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
53	52	eqeq2d 2738	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑥 = ({𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗} ∩ 𝑃) ↔ 𝑥 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
54	53	rexbidv 3173	. . . . . 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 2727	. . . . . 6 ⊢ (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
60	38	rabex 5328	. . . . . 6 ⊢ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V
61	59, 60	elrnmpti 5956	. . . . 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 2725	. 2 ⊢ (𝑅 ∈ Ring → (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ↾_t 𝑃) = 𝐽)
65	10, 35, 64	3eqtr2rd 2774	1 ⊢ (𝑅 ∈ Ring → 𝐽 = (TopOpen‘𝑆))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 {crab 3427 Vcvv 3469 ∩ cin 3943 ⊆ wss 3944 𝒫 cpw 4598 ↦ cmpt 5225 ran crn 5673 ‘cfv 6542 (class class class)co 7414 Basecbs 17171 ↾_s cress 17200 TopSetcts 17230 ↾_t crest 17393 TopOpenctopn 17394 Ringcrg 20164 LIdealclidl 21091 PrmIdealcprmidl 33086 IDLsrgcidlsrg 33147 Speccrspec 33399

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-tset 17243 df-ple 17244 df-rest 17395 df-topn 17396 df-prmidl 33087 df-idlsrg 33148 df-rspec 33400

This theorem is referenced by: zarcls 33411 zar0ring 33415

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