Theorem zarcls 34617

Description: The open sets of the Zariski topology are the complements of the closed sets. (Contributed by Thierry Arnoux, 16-Jun-2024.)

Hypotheses

Ref	Expression
zartop.1	⊢ 𝑆 = (Spec‘𝑅)
zartop.2	⊢ 𝐽 = (TopOpen‘𝑆)
zarcls.1	⊢ 𝑃 = (PrmIdeal‘𝑅)
zarcls.2	⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})

Assertion

Ref	Expression
zarcls	⊢ (𝑅 ∈ Ring → 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})

Distinct variable groups:   𝑃,𝑖,𝑗,𝑠   𝑅,𝑖,𝑗,𝑠   𝑉,𝑠

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

Proof of Theorem zarcls

Step	Hyp	Ref	Expression
1		zartop.2	. . 3 ⊢ 𝐽 = (TopOpen‘𝑆)
2		zartop.1	. . . 4 ⊢ 𝑆 = (Spec‘𝑅)
3		eqid 2763	. . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
4		zarcls.1	. . . 4 ⊢ 𝑃 = (PrmIdeal‘𝑅)
5		eqid 2763	. . . 4 ⊢ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
6	2, 3, 4, 5	rspectopn 34610	. . 3 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopOpen‘𝑆))
7	1, 6	eqtr4id 2817	. 2 ⊢ (𝑅 ∈ Ring → 𝐽 = ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
8		nfv 1949	. . 3 ⊢ Ⅎ𝑠 𝑅 ∈ Ring
9		nfcv 2926	. . 3 ⊢ Ⅎ𝑠ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
10		nfrab1 3435	. . 3 ⊢ Ⅎ𝑠{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}
11		notrab 4270	. . . . . . . . . 10 ⊢ (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}
12	11	eqeq2i 2776	. . . . . . . . 9 ⊢ (𝑠 = (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ↔ 𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
13		ssrab2 4008	. . . . . . . . . . . 12 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑃 → {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃)
15		elpwi 4567	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑃 → 𝑠 ⊆ 𝑃)
16		ssdifsym 4222	. . . . . . . . . . 11 ⊢ (({𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃 ∧ 𝑠 ⊆ 𝑃) → (𝑠 = (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ↔ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = (𝑃 ∖ 𝑠)))
17	14, 15, 16	syl2anc 596	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝒫 𝑃 → (𝑠 = (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ↔ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = (𝑃 ∖ 𝑠)))
18		eqcom 2770	. . . . . . . . . 10 ⊢ ({𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = (𝑃 ∖ 𝑠) ↔ (𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})
19	17, 18	bitrdi 290	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝒫 𝑃 → (𝑠 = (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ↔ (𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}))
20	12, 19	bitr3id 288	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝑃 → (𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ↔ (𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}))
21	20	ad2antlr 740	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑖 ∈ (LIdeal‘𝑅)) → (𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ↔ (𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}))
22	21	rexbidva 3188	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝒫 𝑃) → (∃𝑖 ∈ (LIdeal‘𝑅)𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ↔ ∃𝑖 ∈ (LIdeal‘𝑅)(𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}))
23		zarcls.2	. . . . . . 7 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})
24	4	fvexi 6930	. . . . . . . 8 ⊢ 𝑃 ∈ V
25	24	rabex 5307	. . . . . . 7 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ V
26	23, 25	elrnmpti 5956	. . . . . 6 ⊢ ((𝑃 ∖ 𝑠) ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)(𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})
27	22, 26	bitr4di 292	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝒫 𝑃) → (∃𝑖 ∈ (LIdeal‘𝑅)𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ↔ (𝑃 ∖ 𝑠) ∈ ran 𝑉))
28	27	pm5.32da 590	. . . 4 ⊢ (𝑅 ∈ Ring → ((𝑠 ∈ 𝒫 𝑃 ∧ ∃𝑖 ∈ (LIdeal‘𝑅)𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ (𝑃 ∖ 𝑠) ∈ ran 𝑉)))
29		ssrab2 4008	. . . . . . . . . 10 ⊢ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ⊆ 𝑃
30	24	elpw2 5303	. . . . . . . . . 10 ⊢ ({𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 ↔ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ⊆ 𝑃)
31	29, 30	mpbir 234	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃
32	31	rgenw 3084	. . . . . . . 8 ⊢ ∀𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃
33		eqid 2763	. . . . . . . . 9 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
34	33	rnmptss 7155	. . . . . . . 8 ⊢ (∀𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ⊆ 𝒫 𝑃)
35	32, 34	ax-mp 5	. . . . . . 7 ⊢ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ⊆ 𝒫 𝑃
36	35	sseli 3909	. . . . . 6 ⊢ (𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) → 𝑠 ∈ 𝒫 𝑃)
37	36	pm4.71ri 570	. . . . 5 ⊢ (𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ 𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})))
38		vex 3458	. . . . . . 7 ⊢ 𝑠 ∈ V
39	33	elrnmpt 5952	. . . . . . 7 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ ∃𝑖 ∈ (LIdeal‘𝑅)𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
40	38, 39	ax-mp 5	. . . . . 6 ⊢ (𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ ∃𝑖 ∈ (LIdeal‘𝑅)𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
41	40	anbi2i 635	. . . . 5 ⊢ ((𝑠 ∈ 𝒫 𝑃 ∧ 𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ ∃𝑖 ∈ (LIdeal‘𝑅)𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
42	37, 41	bitri 278	. . . 4 ⊢ (𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ ∃𝑖 ∈ (LIdeal‘𝑅)𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
43		rabid 3436	. . . 4 ⊢ (𝑠 ∈ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} ↔ (𝑠 ∈ 𝒫 𝑃 ∧ (𝑃 ∖ 𝑠) ∈ ran 𝑉))
44	28, 42, 43	3bitr4g 317	. . 3 ⊢ (𝑅 ∈ Ring → (𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ 𝑠 ∈ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}))
45	8, 9, 10, 44	eqrd 3932	. 2 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})
46	7, 45	eqtrd 2798	1 ⊢ (𝑅 ∈ Ring → 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  𝒫 cpw 4560   ↦ cmpt 5192  ran crn 5659  ‘cfv 6566  TopOpenctopn 17642  Ringcrg 20510  LIdealclidl 21535  PrmIdealcprmidl 21667  Speccrspec 34605

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-fz 13680  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-mulr 17487  df-tset 17492  df-ple 17493  df-rest 17643  df-topn 17644  df-prmidl 21668  df-idlsrg 34137  df-rspec 34606

This theorem is used by:  zartopn  34618