Theorem zartopn 34019

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 4020	. . . . . . . 8 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃
2		zarcls.1	. . . . . . . . . 10 ⊢ 𝑃 = (PrmIdeal‘𝑅)
3	2	fvexi 6854	. . . . . . . . 9 ⊢ 𝑃 ∈ V
4	3	elpw2 5275	. . . . . . . 8 ⊢ ({𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 ↔ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃)
5	1, 4	mpbir 231	. . . . . . 7 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃
6	5	rgenw 3055	. . . . . 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 7075	. . . . . 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 20226	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	2	rabeqi 3402	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}
13	12	mpteq2i 5181	. . . . . . . 8 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
14	7, 13	eqtri 2759	. . . . . . 7 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
15		eqid 2736	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
16	14, 2, 15	zarcls0 34012	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑉‘{(0g‘𝑅)}) = 𝑃)
17	7	funmpt2 6537	. . . . . . 7 ⊢ Fun 𝑉
18		eqid 2736	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
19	18, 15	lidl0 21228	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ (LIdeal‘𝑅))
20	3	rabex 5280	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ V
21	20, 7	dmmpti 6642	. . . . . . . 8 ⊢ dom 𝑉 = (LIdeal‘𝑅)
22	19, 21	eleqtrrdi 2847	. . . . . . 7 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ dom 𝑉)
23		fvelrn 7028	. . . . . . 7 ⊢ ((Fun 𝑉 ∧ {(0g‘𝑅)} ∈ dom 𝑉) → (𝑉‘{(0g‘𝑅)}) ∈ ran 𝑉)
24	17, 22, 23	sylancr 588	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑉‘{(0g‘𝑅)}) ∈ ran 𝑉)
25	16, 24	eqeltrrd 2837	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ ran 𝑉)
26	11, 25	syl 17	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ ran 𝑉)
27	14	zarclsint 34016	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑧 ⊆ ran 𝑉 ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ ran 𝑉)
28	10, 26, 27	ismred 17564	. . 3 ⊢ (𝑅 ∈ CRing → ran 𝑉 ∈ (Moore‘𝑃))
29		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
30	21, 29	lidl1 21231	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ dom 𝑉)
31	11, 30	syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ dom 𝑉)
32	31, 21	eleqtrdi 2846	. . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ (LIdeal‘𝑅))
33	14, 29	zarcls1 34013	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (LIdeal‘𝑅)) → ((𝑉‘(Base‘𝑅)) = ∅ ↔ (Base‘𝑅) = (Base‘𝑅)))
34	29, 33	mpbiri 258	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (LIdeal‘𝑅)) → (𝑉‘(Base‘𝑅)) = ∅)
35	32, 34	mpdan 688	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑉‘(Base‘𝑅)) = ∅)
36	17	a1i 11	. . . . 5 ⊢ (𝑅 ∈ CRing → Fun 𝑉)
37		fvelrn 7028	. . . . 5 ⊢ ((Fun 𝑉 ∧ (Base‘𝑅) ∈ dom 𝑉) → (𝑉‘(Base‘𝑅)) ∈ ran 𝑉)
38	36, 31, 37	syl2anc 585	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑉‘(Base‘𝑅)) ∈ ran 𝑉)
39	35, 38	eqeltrrd 2837	. . 3 ⊢ (𝑅 ∈ CRing → ∅ ∈ ran 𝑉)
40	14	zarclsun 34014	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ ran 𝑉 ∧ 𝑦 ∈ ran 𝑉) → (𝑥 ∪ 𝑦) ∈ ran 𝑉)
41		eqid 2736	. . 3 ⊢ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}
42	28, 39, 40, 41	mretopd 23057	. 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 34018	. . . . 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 2821	. . 3 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ↔ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} ∈ (TopOn‘𝑃)))
48	46	fveq2d 6844	. . . 4 ⊢ (𝑅 ∈ CRing → (Clsd‘𝐽) = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}))
49	48	eqeq2d 2747	. . 3 ⊢ (𝑅 ∈ CRing → (ran 𝑉 = (Clsd‘𝐽) ↔ ran 𝑉 = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})))
50	47, 49	anbi12d 633	. 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 257	1 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘𝐽)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389   ∖ cdif 3886   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  {csn 4567   ↦ cmpt 5166  dom cdm 5631  ran crn 5632  Fun wfun 6492  ‘cfv 6498  Basecbs 17179  TopOpenctopn 17384  0_gc0g 17402  Ringcrg 20214  CRingccrg 20215  LIdealclidl 21204  TopOnctopon 22875  Clsdccld 22981  PrmIdealcprmidl 33495  Speccrspec 34006

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-rpss 7677  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-ac 10038  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  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-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-rest 17385  df-topn 17386  df-0g 17404  df-mre 17548  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-subg 19099  df-cntz 19292  df-lsm 19611  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-cring 20217  df-subrg 20547  df-lmod 20857  df-lss 20927  df-lsp 20967  df-sra 21168  df-rgmod 21169  df-lidl 21206  df-rsp 21207  df-lpidl 21320  df-top 22859  df-topon 22876  df-cld 22984  df-prmidl 33496  df-mxidl 33520  df-idlsrg 33561  df-rspec 34007

This theorem is referenced by:  zartop  34020  zartopon  34021  zart0  34023  zarmxt1  34024  zarcmplem  34025  rhmpreimacn  34029