Theorem zarcls 32122

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 2736	. . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
4		zarcls.1	. . . 4 ⊢ 𝑃 = (PrmIdeal‘𝑅)
5		eqid 2736	. . . 4 ⊢ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
6	2, 3, 4, 5	rspectopn 32115	. . 3 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopOpen‘𝑆))
7	1, 6	eqtr4id 2795	. 2 ⊢ (𝑅 ∈ Ring → 𝐽 = ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
8		nfv 1916	. . 3 ⊢ Ⅎ𝑠 𝑅 ∈ Ring
9		nfcv 2904	. . 3 ⊢ Ⅎ𝑠ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
10		nfrab1 3422	. . 3 ⊢ Ⅎ𝑠{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}
11		notrab 4258	. . . . . . . . . 10 ⊢ (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}
12	11	eqeq2i 2749	. . . . . . . . 9 ⊢ (𝑠 = (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ↔ 𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
13		ssrab2 4025	. . . . . . . . . . . 12 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑃 → {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃)
15		elpwi 4554	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑃 → 𝑠 ⊆ 𝑃)
16		ssdifsym 4210	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝒫 𝑃 → (𝑠 = (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ↔ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = (𝑃 ∖ 𝑠)))
18		eqcom 2743	. . . . . . . . . 10 ⊢ ({𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = (𝑃 ∖ 𝑠) ↔ (𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})
19	17, 18	bitrdi 286	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝒫 𝑃 → (𝑠 = (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ↔ (𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}))
20	12, 19	bitr3id 284	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝑃 → (𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ↔ (𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}))
21	20	ad2antlr 724	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑖 ∈ (LIdeal‘𝑅)) → (𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ↔ (𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}))
22	21	rexbidva 3169	. . . . . 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 6839	. . . . . . . 8 ⊢ 𝑃 ∈ V
25	24	rabex 5276	. . . . . . 7 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ V
26	23, 25	elrnmpti 5901	. . . . . 6 ⊢ ((𝑃 ∖ 𝑠) ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅)(𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})
27	22, 26	bitr4di 288	. . . . 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 4025	. . . . . . . . . 10 ⊢ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ⊆ 𝑃
30	24	elpw2 5289	. . . . . . . . . 10 ⊢ ({𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 ↔ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ⊆ 𝑃)
31	29, 30	mpbir 230	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃
32	31	rgenw 3065	. . . . . . . 8 ⊢ ∀𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃
33		eqid 2736	. . . . . . . . 9 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
34	33	rnmptss 7052	. . . . . . . 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 3928	. . . . . 6 ⊢ (𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) → 𝑠 ∈ 𝒫 𝑃)
37	36	pm4.71ri 561	. . . . 5 ⊢ (𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ 𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})))
38		vex 3445	. . . . . . 7 ⊢ 𝑠 ∈ V
39	33	elrnmpt 5897	. . . . . . 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 623	. . . . 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 274	. . . 4 ⊢ (𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ ∃𝑖 ∈ (LIdeal‘𝑅)𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
43		rabid 3423	. . . 4 ⊢ (𝑠 ∈ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} ↔ (𝑠 ∈ 𝒫 𝑃 ∧ (𝑃 ∖ 𝑠) ∈ ran 𝑉))
44	28, 42, 43	3bitr4g 313	. . 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 3951	. 2 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})
46	7, 45	eqtrd 2776	1 ⊢ (𝑅 ∈ Ring → 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3061  ∃wrex 3070  {crab 3403  Vcvv 3441   ∖ cdif 3895   ⊆ wss 3898  𝒫 cpw 4547   ↦ cmpt 5175  ran crn 5621  ‘cfv 6479  TopOpenctopn 17229  Ringcrg 19878  LIdealclidl 20538  PrmIdealcprmidl 31907  Speccrspec 32110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-nn 12075  df-2 12137  df-3 12138  df-4 12139  df-5 12140  df-6 12141  df-7 12142  df-8 12143  df-9 12144  df-n0 12335  df-z 12421  df-dec 12539  df-uz 12684  df-fz 13341  df-struct 16945  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-mulr 17073  df-tset 17078  df-ple 17079  df-rest 17230  df-topn 17231  df-prmidl 31908  df-idlsrg 31943  df-rspec 32111

This theorem is referenced by:  zartopn  32123