Theorem zarcls 34553

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 2760	. . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
4		zarcls.1	. . . 4 ⊢ 𝑃 = (PrmIdeal‘𝑅)
5		eqid 2760	. . . 4 ⊢ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
6	2, 3, 4, 5	rspectopn 34546	. . 3 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopOpen‘𝑆))
7	1, 6	eqtr4id 2814	. 2 ⊢ (𝑅 ∈ Ring → 𝐽 = ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}))
8		nfv 1946	. . 3 ⊢ Ⅎ𝑠 𝑅 ∈ Ring
9		nfcv 2922	. . 3 ⊢ Ⅎ𝑠ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
10		nfrab1 3431	. . 3 ⊢ Ⅎ𝑠{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}
11		notrab 4267	. . . . . . . . . 10 ⊢ (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}
12	11	eqeq2i 2773	. . . . . . . . 9 ⊢ (𝑠 = (𝑃 ∖ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ↔ 𝑠 = {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
13		ssrab2 4005	. . . . . . . . . . . 12 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑃 → {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃)
15		elpwi 4563	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 𝑃 → 𝑠 ⊆ 𝑃)
16		ssdifsym 4219	. . . . . . . . . . 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 2767	. . . . . . . . . 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 3184	. . . . . 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 6900	. . . . . . . 8 ⊢ 𝑃 ∈ V
25	24	rabex 5299	. . . . . . 7 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ V
26	23, 25	elrnmpti 5943	. . . . . 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 4005	. . . . . . . . . 10 ⊢ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ⊆ 𝑃
30	24	elpw2 5295	. . . . . . . . . 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 3080	. . . . . . . 8 ⊢ ∀𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃
33		eqid 2760	. . . . . . . . 9 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})
34	33	rnmptss 7124	. . . . . . . 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 3904	. . . . . 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 3454	. . . . . . 7 ⊢ 𝑠 ∈ V
39	33	elrnmpt 5939	. . . . . . 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 3432	. . . 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 3927	. 2 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})
46	7, 45	eqtrd 2795	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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∖ cdif 4045  𝒫 cpw 4556   ↦ cmpt 5185  ran crn 5651  ‘cfv 6539  TopOpenctopn 17598  Ringcrg 20465  LIdealclidl 21490  PrmIdealcprmidl 21622  Speccrspec 34541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-er 8721  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-nn 12340  df-2 12409  df-3 12410  df-4 12411  df-5 12412  df-6 12413  df-7 12414  df-8 12415  df-9 12416  df-n0 12611  df-z 12698  df-dec 12819  df-uz 12970  df-fz 13644  df-struct 17331  df-sets 17348  df-slot 17366  df-ndx 17378  df-base 17394  df-ress 17415  df-plusg 17447  df-mulr 17448  df-tset 17453  df-ple 17454  df-rest 17599  df-topn 17600  df-prmidl 21623  df-idlsrg 34080  df-rspec 34542

This theorem is used by:  zartopn  34554