Theorem zarcls 34034

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 2737	. . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
4		zarcls.1	. . . 4 ⊢ 𝑃 = (PrmIdeal‘𝑅)
5		eqid 2737	. . . 4 ⊢ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
6	2, 3, 4, 5	rspectopn 34027	. . 3 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopOpen‘𝑆))
7	1, 6	eqtr4id 2791	. 2 ⊢ (𝑅 ∈ Ring → 𝐽 = ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
8		nfv 1916	. . 3 ⊢ Ⅎ𝑠 𝑅 ∈ Ring
9		nfcv 2899	. . 3 ⊢ Ⅎ𝑠ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
10		nfrab1 3410	. . 3 ⊢ Ⅎ𝑠{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}
11		notrab 4263	. . . . . . . . . 10 ⊢ (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}
12	11	eqeq2i 2750	. . . . . . . . 9 ⊢ (𝑠 = (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ↔ 𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
13		ssrab2 4021	. . . . . . . . . . . 12 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑃 → {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃)
15		elpwi 4549	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑃 → 𝑠 ⊆ 𝑃)
16		ssdifsym 4215	. . . . . . . . . . 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 585	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝒫 𝑃 → (𝑠 = (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ↔ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = (𝑃 ∖ 𝑠)))
18		eqcom 2744	. . . . . . . . . 10 ⊢ ({𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = (𝑃 ∖ 𝑠) ↔ (𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})
19	17, 18	bitrdi 287	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝒫 𝑃 → (𝑠 = (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ↔ (𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}))
20	12, 19	bitr3id 285	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝑃 → (𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ↔ (𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}))
21	20	ad2antlr 728	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑖 ∈ (LIdeal‘𝑅)) → (𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ↔ (𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}))
22	21	rexbidva 3160	. . . . . 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 6848	. . . . . . . 8 ⊢ 𝑃 ∈ V
25	24	rabex 5276	. . . . . . 7 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ V
26	23, 25	elrnmpti 5911	. . . . . 6 ⊢ ((𝑃 ∖ 𝑠) ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)(𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})
27	22, 26	bitr4di 289	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝒫 𝑃) → (∃𝑖 ∈ (LIdeal‘𝑅)𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ↔ (𝑃 ∖ 𝑠) ∈ ran 𝑉))
28	27	pm5.32da 579	. . . 4 ⊢ (𝑅 ∈ Ring → ((𝑠 ∈ 𝒫 𝑃 ∧ ∃𝑖 ∈ (LIdeal‘𝑅)𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ (𝑃 ∖ 𝑠) ∈ ran 𝑉)))
29		ssrab2 4021	. . . . . . . . . 10 ⊢ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ⊆ 𝑃
30	24	elpw2 5271	. . . . . . . . . 10 ⊢ ({𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 ↔ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ⊆ 𝑃)
31	29, 30	mpbir 231	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃
32	31	rgenw 3056	. . . . . . . 8 ⊢ ∀𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃
33		eqid 2737	. . . . . . . . 9 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
34	33	rnmptss 7069	. . . . . . . 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 3918	. . . . . 6 ⊢ (𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) → 𝑠 ∈ 𝒫 𝑃)
37	36	pm4.71ri 560	. . . . 5 ⊢ (𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ 𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})))
38		vex 3434	. . . . . . 7 ⊢ 𝑠 ∈ V
39	33	elrnmpt 5907	. . . . . . 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 624	. . . . 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 275	. . . 4 ⊢ (𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ ∃𝑖 ∈ (LIdeal‘𝑅)𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
43		rabid 3411	. . . 4 ⊢ (𝑠 ∈ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} ↔ (𝑠 ∈ 𝒫 𝑃 ∧ (𝑃 ∖ 𝑠) ∈ ran 𝑉))
44	28, 42, 43	3bitr4g 314	. . 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 3942	. 2 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})
46	7, 45	eqtrd 2772	1 ⊢ (𝑅 ∈ Ring → 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  𝒫 cpw 4542   ↦ cmpt 5167  ran crn 5625  ‘cfv 6492  TopOpenctopn 17375  Ringcrg 20205  LIdealclidl 21196  PrmIdealcprmidl 33510  Speccrspec 34022

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

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 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-tset 17230  df-ple 17231  df-rest 17376  df-topn 17377  df-prmidl 33511  df-idlsrg 33576  df-rspec 34023

This theorem is referenced by:  zartopn  34035