Theorem zarcls 33864

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 2729	. . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
4		zarcls.1	. . . 4 ⊢ 𝑃 = (PrmIdeal‘𝑅)
5		eqid 2729	. . . 4 ⊢ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
6	2, 3, 4, 5	rspectopn 33857	. . 3 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopOpen‘𝑆))
7	1, 6	eqtr4id 2783	. 2 ⊢ (𝑅 ∈ Ring → 𝐽 = ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
8		nfv 1914	. . 3 ⊢ Ⅎ𝑠 𝑅 ∈ Ring
9		nfcv 2891	. . 3 ⊢ Ⅎ𝑠ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
10		nfrab1 3426	. . 3 ⊢ Ⅎ𝑠{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}
11		notrab 4285	. . . . . . . . . 10 ⊢ (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}
12	11	eqeq2i 2742	. . . . . . . . 9 ⊢ (𝑠 = (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ↔ 𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
13		ssrab2 4043	. . . . . . . . . . . 12 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑃 → {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃)
15		elpwi 4570	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑃 → 𝑠 ⊆ 𝑃)
16		ssdifsym 4237	. . . . . . . . . . 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 2736	. . . . . . . . . 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 727	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑖 ∈ (LIdeal‘𝑅)) → (𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ↔ (𝑃 ∖ 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}))
22	21	rexbidva 3155	. . . . . 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 6872	. . . . . . . 8 ⊢ 𝑃 ∈ V
25	24	rabex 5294	. . . . . . 7 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ V
26	23, 25	elrnmpti 5926	. . . . . 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 4043	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃
32	31	rgenw 3048	. . . . . . . 8 ⊢ ∀𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃
33		eqid 2729	. . . . . . . . 9 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
34	33	rnmptss 7095	. . . . . . . 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 3942	. . . . . 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 3451	. . . . . . 7 ⊢ 𝑠 ∈ V
39	33	elrnmpt 5922	. . . . . . 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 275	. . . 4 ⊢ (𝑠 ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ ∃𝑖 ∈ (LIdeal‘𝑅)𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
43		rabid 3427	. . . 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 3966	. 2 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})
46	7, 45	eqtrd 2764	1 ⊢ (𝑅 ∈ Ring → 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3405  Vcvv 3447   ∖ cdif 3911   ⊆ wss 3914  𝒫 cpw 4563   ↦ cmpt 5188  ran crn 5639  ‘cfv 6511  TopOpenctopn 17384  Ringcrg 20142  LIdealclidl 21116  PrmIdealcprmidl 33406  Speccrspec 33852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-tset 17239  df-ple 17240  df-rest 17385  df-topn 17386  df-prmidl 33407  df-idlsrg 33472  df-rspec 33853

This theorem is referenced by:  zartopn  33865