Theorem zartopn 34132

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 4031	. . . . . . . 8 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃
2		zarcls.1	. . . . . . . . . 10 ⊢ 𝑃 = (PrmIdeal‘𝑅)
3	2	fvexi 6875	. . . . . . . . 9 ⊢ 𝑃 ∈ V
4	3	elpw2 5287	. . . . . . . 8 ⊢ ({𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 ↔ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃)
5	1, 4	mpbir 233	. . . . . . 7 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃
6	5	rgenw 3079	. . . . . 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 7098	. . . . . 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 20281	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	2	rabeqi 3426	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}
13	12	mpteq2i 5193	. . . . . . . 8 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
14	7, 13	eqtri 2784	. . . . . . 7 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
15		eqid 2761	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
16	14, 2, 15	zarcls0 34125	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑉‘{(0g‘𝑅)}) = 𝑃)
17	7	funmpt2 6554	. . . . . . 7 ⊢ Fun 𝑉
18		eqid 2761	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
19	18, 15	lidl0 21287	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ (LIdeal‘𝑅))
20	3	rabex 5292	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ V
21	20, 7	dmmpti 6659	. . . . . . . 8 ⊢ dom 𝑉 = (LIdeal‘𝑅)
22	19, 21	eleqtrrdi 2872	. . . . . . 7 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ dom 𝑉)
23		fvelrn 7051	. . . . . . 7 ⊢ ((Fun 𝑉 ∧ {(0g‘𝑅)} ∈ dom 𝑉) → (𝑉‘{(0g‘𝑅)}) ∈ ran 𝑉)
24	17, 22, 23	sylancr 596	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑉‘{(0g‘𝑅)}) ∈ ran 𝑉)
25	16, 24	eqeltrrd 2862	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ ran 𝑉)
26	11, 25	syl 17	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ ran 𝑉)
27	14	zarclsint 34129	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑧 ⊆ ran 𝑉 ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ ran 𝑉)
28	10, 26, 27	ismred 17620	. . 3 ⊢ (𝑅 ∈ CRing → ran 𝑉 ∈ (Moore‘𝑃))
29		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
30	21, 29	lidl1 21290	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ dom 𝑉)
31	11, 30	syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ dom 𝑉)
32	31, 21	eleqtrdi 2871	. . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ (LIdeal‘𝑅))
33	14, 29	zarcls1 34126	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (LIdeal‘𝑅)) → ((𝑉‘(Base‘𝑅)) = ∅ ↔ (Base‘𝑅) = (Base‘𝑅)))
34	29, 33	mpbiri 260	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (LIdeal‘𝑅)) → (𝑉‘(Base‘𝑅)) = ∅)
35	32, 34	mpdan 697	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑉‘(Base‘𝑅)) = ∅)
36	17	a1i 11	. . . . 5 ⊢ (𝑅 ∈ CRing → Fun 𝑉)
37		fvelrn 7051	. . . . 5 ⊢ ((Fun 𝑉 ∧ (Base‘𝑅) ∈ dom 𝑉) → (𝑉‘(Base‘𝑅)) ∈ ran 𝑉)
38	36, 31, 37	syl2anc 593	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑉‘(Base‘𝑅)) ∈ ran 𝑉)
39	35, 38	eqeltrrd 2862	. . 3 ⊢ (𝑅 ∈ CRing → ∅ ∈ ran 𝑉)
40	14	zarclsun 34127	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ ran 𝑉 ∧ 𝑦 ∈ ran 𝑉) → (𝑥 ∪ 𝑦) ∈ ran 𝑉)
41		eqid 2761	. . 3 ⊢ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}
42	28, 39, 40, 41	mretopd 23139	. 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 34131	. . . . 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 2846	. . 3 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ↔ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} ∈ (TopOn‘𝑃)))
48	46	fveq2d 6865	. . . 4 ⊢ (𝑅 ∈ CRing → (Clsd‘𝐽) = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}))
49	48	eqeq2d 2772	. . 3 ⊢ (𝑅 ∈ CRing → (ran 𝑉 = (Clsd‘𝐽) ↔ ran 𝑉 = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})))
50	47, 49	anbi12d 641	. 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 259	1 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘𝐽)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4552  {csn 4579   ↦ cmpt 5178  dom cdm 5643  ran crn 5644  Fun wfun 6509  ‘cfv 6515  Basecbs 17235  TopOpenctopn 17440  0_gc0g 17458  Ringcrg 20269  CRingccrg 20270  LIdealclidl 21263  TopOnctopon 22957  Clsdccld 23063  PrmIdealcprmidl 33581  Speccrspec 34119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-ac2 10413  ax-cnex 11122  ax-resscn 11123  ax-1cn 11124  ax-icn 11125  ax-addcl 11126  ax-addrcl 11127  ax-mulcl 11128  ax-mulrcl 11129  ax-mulcom 11130  ax-addass 11131  ax-mulass 11132  ax-distr 11133  ax-i2m1 11134  ax-1ne0 11135  ax-1rid 11136  ax-rnegex 11137  ax-rrecex 11138  ax-cnre 11139  ax-pre-lttri 11140  ax-pre-lttrn 11141  ax-pre-ltadd 11142  ax-pre-mulgt0 11143

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-isom 6524  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-rpss 7700  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-oadd 8434  df-er 8671  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-dju 9852  df-card 9890  df-ac 10065  df-pnf 11211  df-mnf 11212  df-xr 11213  df-ltxr 11214  df-le 11215  df-sub 11409  df-neg 11410  df-nn 12204  df-2 12273  df-3 12274  df-4 12275  df-5 12276  df-6 12277  df-7 12278  df-8 12279  df-9 12280  df-n0 12475  df-z 12562  df-dec 12682  df-uz 12833  df-fz 13506  df-struct 17173  df-sets 17190  df-slot 17208  df-ndx 17220  df-base 17236  df-ress 17257  df-plusg 17289  df-mulr 17290  df-sca 17292  df-vsca 17293  df-ip 17294  df-tset 17295  df-ple 17296  df-rest 17441  df-topn 17442  df-0g 17460  df-mre 17604  df-mgm 18664  df-sgrp 18743  df-mnd 18759  df-submnd 18808  df-grp 18968  df-minusg 18969  df-sbg 18970  df-subg 19155  df-cntz 19347  df-lsm 19666  df-cmn 19812  df-abl 19813  df-mgp 20177  df-rng 20189  df-ur 20218  df-ring 20271  df-cring 20272  df-subrg 20606  df-lmod 20916  df-lss 20986  df-lsp 21026  df-sra 21227  df-rgmod 21228  df-lidl 21265  df-rsp 21266  df-lpidl 21379  df-top 22941  df-topon 22958  df-cld 23066  df-prmidl 33582  df-mxidl 33608  df-idlsrg 33657  df-rspec 34120

This theorem is referenced by:  zartop  34133  zartopon  34134  zart0  34136  zarmxt1  34137  zarcmplem  34138  rhmpreimacn  34142