Theorem zartopn 31727

Description: The Zariski topology is a topology, and its closed sets are images by 𝑉 of the ideals of 𝑅. (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
zartopn	⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘𝐽)))

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

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

Proof of Theorem zartopn

Dummy variables 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4009	. . . . . . . 8 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃
2		zarcls.1	. . . . . . . . . 10 ⊢ 𝑃 = (PrmIdeal‘𝑅)
3	2	fvexi 6770	. . . . . . . . 9 ⊢ 𝑃 ∈ V
4	3	elpw2 5264	. . . . . . . 8 ⊢ ({𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 ↔ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃)
5	1, 4	mpbir 230	. . . . . . 7 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃
6	5	rgenw 3075	. . . . . 6 ⊢ ∀𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃
7		zarcls.2	. . . . . . 7 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})
8	7	rnmptss 6978	. . . . . 6 ⊢ (∀𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 → ran 𝑉 ⊆ 𝒫 𝑃)
9	6, 8	ax-mp 5	. . . . 5 ⊢ ran 𝑉 ⊆ 𝒫 𝑃
10	9	a1i 11	. . . 4 ⊢ (𝑅 ∈ CRing → ran 𝑉 ⊆ 𝒫 𝑃)
11		crngring 19710	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	2	rabeqi 3406	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}
13	12	mpteq2i 5175	. . . . . . . 8 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
14	7, 13	eqtri 2766	. . . . . . 7 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
15		eqid 2738	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
16	14, 2, 15	zarcls0 31720	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑉‘{(0g‘𝑅)}) = 𝑃)
17	7	funmpt2 6457	. . . . . . 7 ⊢ Fun 𝑉
18		eqid 2738	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
19	18, 15	lidl0 20403	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ (LIdeal‘𝑅))
20	3	rabex 5251	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ V
21	20, 7	dmmpti 6561	. . . . . . . 8 ⊢ dom 𝑉 = (LIdeal‘𝑅)
22	19, 21	eleqtrrdi 2850	. . . . . . 7 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ dom 𝑉)
23		fvelrn 6936	. . . . . . 7 ⊢ ((Fun 𝑉 ∧ {(0g‘𝑅)} ∈ dom 𝑉) → (𝑉‘{(0g‘𝑅)}) ∈ ran 𝑉)
24	17, 22, 23	sylancr 586	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑉‘{(0g‘𝑅)}) ∈ ran 𝑉)
25	16, 24	eqeltrrd 2840	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ ran 𝑉)
26	11, 25	syl 17	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ ran 𝑉)
27	14	zarclsint 31724	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑧 ⊆ ran 𝑉 ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ ran 𝑉)
28	10, 26, 27	ismred 17228	. . 3 ⊢ (𝑅 ∈ CRing → ran 𝑉 ∈ (Moore‘𝑃))
29		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
30	21, 29	lidl1 20404	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ dom 𝑉)
31	11, 30	syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ dom 𝑉)
32	31, 21	eleqtrdi 2849	. . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ (LIdeal‘𝑅))
33	14, 29	zarcls1 31721	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (LIdeal‘𝑅)) → ((𝑉‘(Base‘𝑅)) = ∅ ↔ (Base‘𝑅) = (Base‘𝑅)))
34	29, 33	mpbiri 257	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (LIdeal‘𝑅)) → (𝑉‘(Base‘𝑅)) = ∅)
35	32, 34	mpdan 683	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑉‘(Base‘𝑅)) = ∅)
36	17	a1i 11	. . . . 5 ⊢ (𝑅 ∈ CRing → Fun 𝑉)
37		fvelrn 6936	. . . . 5 ⊢ ((Fun 𝑉 ∧ (Base‘𝑅) ∈ dom 𝑉) → (𝑉‘(Base‘𝑅)) ∈ ran 𝑉)
38	36, 31, 37	syl2anc 583	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑉‘(Base‘𝑅)) ∈ ran 𝑉)
39	35, 38	eqeltrrd 2840	. . 3 ⊢ (𝑅 ∈ CRing → ∅ ∈ ran 𝑉)
40	14	zarclsun 31722	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ ran 𝑉 ∧ 𝑦 ∈ ran 𝑉) → (𝑥 ∪ 𝑦) ∈ ran 𝑉)
41		eqid 2738	. . 3 ⊢ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}
42	28, 39, 40, 41	mretopd 22151	. 2 ⊢ (𝑅 ∈ CRing → ({𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})))
43		zartop.1	. . . . . 6 ⊢ 𝑆 = (Spec‘𝑅)
44		zartop.2	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑆)
45	43, 44, 2, 7	zarcls 31726	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})
46	11, 45	syl 17	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})
47	46	eleq1d 2823	. . 3 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ↔ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} ∈ (TopOn‘𝑃)))
48	46	fveq2d 6760	. . . 4 ⊢ (𝑅 ∈ CRing → (Clsd‘𝐽) = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}))
49	48	eqeq2d 2749	. . 3 ⊢ (𝑅 ∈ CRing → (ran 𝑉 = (Clsd‘𝐽) ↔ ran 𝑉 = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})))
50	47, 49	anbi12d 630	. 2 ⊢ (𝑅 ∈ CRing → ((𝐽 ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘𝐽)) ↔ ({𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}))))
51	42, 50	mpbird 256	1 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘𝐽)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558   ↦ cmpt 5153  dom cdm 5580  ran crn 5581  Fun wfun 6412  ‘cfv 6418  Basecbs 16840  TopOpenctopn 17049  0_gc0g 17067  Ringcrg 19698  CRingccrg 19699  LIdealclidl 20347  TopOnctopon 21967  Clsdccld 22075  PrmIdealcprmidl 31512  Speccrspec 31714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-ac2 10150  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-rpss 7554  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-ac 9803  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-rest 17050  df-topn 17051  df-0g 17069  df-mre 17212  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-cntz 18838  df-lsm 19156  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-cring 19701  df-subrg 19937  df-lmod 20040  df-lss 20109  df-lsp 20149  df-sra 20349  df-rgmod 20350  df-lidl 20351  df-rsp 20352  df-lpidl 20427  df-top 21951  df-topon 21968  df-cld 22078  df-prmidl 31513  df-mxidl 31534  df-idlsrg 31548  df-rspec 31715

This theorem is referenced by:  zartop  31728  zartopon  31729  zart0  31731  zarmxt1  31732  zarcmplem  31733  rhmpreimacn  31737