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Theorem rspectopn 31235
 Description: The topology component of the spectrum of a ring. (Contributed by Thierry Arnoux, 4-Jun-2024.)
Hypotheses
Ref Expression
rspecbas.1 𝑆 = (Spec‘𝑅)
rspectopn.1 𝐼 = (LIdeal‘𝑅)
rspectopn.2 𝑃 = (PrmIdeal‘𝑅)
rspectopn.3 𝐽 = ran (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗})
Assertion
Ref Expression
rspectopn (𝑅 ∈ Ring → 𝐽 = (TopOpen‘𝑆))
Distinct variable groups:   𝑖,𝐼,𝑗   𝑃,𝑖,𝑗   𝑅,𝑖,𝑗
Allowed substitution hints:   𝑆(𝑖,𝑗)   𝐽(𝑖,𝑗)

Proof of Theorem rspectopn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rspecval 31232 . . . . 5 (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))
2 rspecbas.1 . . . . 5 𝑆 = (Spec‘𝑅)
3 rspectopn.2 . . . . . 6 𝑃 = (PrmIdeal‘𝑅)
43oveq2i 7147 . . . . 5 ((IDLsrg‘𝑅) ↾s 𝑃) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅))
51, 2, 43eqtr4g 2858 . . . 4 (𝑅 ∈ Ring → 𝑆 = ((IDLsrg‘𝑅) ↾s 𝑃))
65fveq2d 6650 . . 3 (𝑅 ∈ Ring → (TopOpen‘𝑆) = (TopOpen‘((IDLsrg‘𝑅) ↾s 𝑃)))
7 eqid 2798 . . . 4 ((IDLsrg‘𝑅) ↾s 𝑃) = ((IDLsrg‘𝑅) ↾s 𝑃)
8 eqid 2798 . . . 4 (TopOpen‘(IDLsrg‘𝑅)) = (TopOpen‘(IDLsrg‘𝑅))
97, 8resstopn 21801 . . 3 ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃) = (TopOpen‘((IDLsrg‘𝑅) ↾s 𝑃))
106, 9eqtr4di 2851 . 2 (𝑅 ∈ Ring → (TopOpen‘𝑆) = ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃))
11 eqid 2798 . . . . 5 (IDLsrg‘𝑅) = (IDLsrg‘𝑅)
12 rspectopn.1 . . . . 5 𝐼 = (LIdeal‘𝑅)
13 eqid 2798 . . . . 5 ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) = ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})
1411, 12, 13idlsrgtset 31071 . . . 4 (𝑅 ∈ Ring → ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) = (TopSet‘(IDLsrg‘𝑅)))
1512fvexi 6660 . . . . . . . . . . 11 𝐼 ∈ V
1615rabex 5200 . . . . . . . . . 10 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ V
1716a1i 11 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ V)
18 simp2 1134 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑖𝐼) ∧ 𝑗𝐼 ∧ ¬ 𝑖𝑗) → 𝑗𝐼)
1911, 12idlsrgbas 31067 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝐼 = (Base‘(IDLsrg‘𝑅)))
2019adantr 484 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
21203ad2ant1 1130 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑖𝐼) ∧ 𝑗𝐼 ∧ ¬ 𝑖𝑗) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
2218, 21eleqtrd 2892 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑖𝐼) ∧ 𝑗𝐼 ∧ ¬ 𝑖𝑗) → 𝑗 ∈ (Base‘(IDLsrg‘𝑅)))
2322rabssdv 4002 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → {𝑗𝐼 ∣ ¬ 𝑖𝑗} ⊆ (Base‘(IDLsrg‘𝑅)))
2417, 23elpwd 4505 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
2524ralrimiva 3149 . . . . . . 7 (𝑅 ∈ Ring → ∀𝑖𝐼 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
26 eqid 2798 . . . . . . . 8 (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) = (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})
2726rnmptss 6864 . . . . . . 7 (∀𝑖𝐼 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)) → ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)))
2825, 27syl 17 . . . . . 6 (𝑅 ∈ Ring → ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)))
2914, 28eqsstrrd 3954 . . . . 5 (𝑅 ∈ Ring → (TopSet‘(IDLsrg‘𝑅)) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)))
30 eqid 2798 . . . . . 6 (Base‘(IDLsrg‘𝑅)) = (Base‘(IDLsrg‘𝑅))
31 eqid 2798 . . . . . 6 (TopSet‘(IDLsrg‘𝑅)) = (TopSet‘(IDLsrg‘𝑅))
3230, 31topnid 16704 . . . . 5 ((TopSet‘(IDLsrg‘𝑅)) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)) → (TopSet‘(IDLsrg‘𝑅)) = (TopOpen‘(IDLsrg‘𝑅)))
3329, 32syl 17 . . . 4 (𝑅 ∈ Ring → (TopSet‘(IDLsrg‘𝑅)) = (TopOpen‘(IDLsrg‘𝑅)))
3414, 33eqtrd 2833 . . 3 (𝑅 ∈ Ring → ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) = (TopOpen‘(IDLsrg‘𝑅)))
3534oveq1d 7151 . 2 (𝑅 ∈ Ring → (ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ↾t 𝑃) = ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃))
3615mptex 6964 . . . . . . 7 (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ∈ V
3736rnex 7602 . . . . . 6 ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ∈ V
383fvexi 6660 . . . . . 6 𝑃 ∈ V
39 elrest 16696 . . . . . 6 ((ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ∈ V ∧ 𝑃 ∈ V) → (𝑥 ∈ (ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ↾t 𝑃) ↔ ∃𝑦 ∈ ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})𝑥 = (𝑦𝑃)))
4037, 38, 39mp2an 691 . . . . 5 (𝑥 ∈ (ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ↾t 𝑃) ↔ ∃𝑦 ∈ ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})𝑥 = (𝑦𝑃))
4116rgenw 3118 . . . . . . 7 𝑖𝐼 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ V
42 ineq1 4131 . . . . . . . . 9 (𝑦 = {𝑗𝐼 ∣ ¬ 𝑖𝑗} → (𝑦𝑃) = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃))
4342eqeq2d 2809 . . . . . . . 8 (𝑦 = {𝑗𝐼 ∣ ¬ 𝑖𝑗} → (𝑥 = (𝑦𝑃) ↔ 𝑥 = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃)))
4426, 43rexrnmptw 6839 . . . . . . 7 (∀𝑖𝐼 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ V → (∃𝑦 ∈ ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})𝑥 = (𝑦𝑃) ↔ ∃𝑖𝐼 𝑥 = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃)))
4541, 44ax-mp 5 . . . . . 6 (∃𝑦 ∈ ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})𝑥 = (𝑦𝑃) ↔ ∃𝑖𝐼 𝑥 = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃))
46 inrab2 4228 . . . . . . . . 9 ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) = {𝑗 ∈ (𝐼𝑃) ∣ ¬ 𝑖𝑗}
47 prmidlssidl 31038 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
4847, 3, 123sstr4g 3960 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑃𝐼)
49 sseqin2 4142 . . . . . . . . . . 11 (𝑃𝐼 ↔ (𝐼𝑃) = 𝑃)
5048, 49sylib 221 . . . . . . . . . 10 (𝑅 ∈ Ring → (𝐼𝑃) = 𝑃)
5150rabeqdv 3432 . . . . . . . . 9 (𝑅 ∈ Ring → {𝑗 ∈ (𝐼𝑃) ∣ ¬ 𝑖𝑗} = {𝑗𝑃 ∣ ¬ 𝑖𝑗})
5246, 51syl5eq 2845 . . . . . . . 8 (𝑅 ∈ Ring → ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) = {𝑗𝑃 ∣ ¬ 𝑖𝑗})
5352eqeq2d 2809 . . . . . . 7 (𝑅 ∈ Ring → (𝑥 = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) ↔ 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
5453rexbidv 3256 . . . . . 6 (𝑅 ∈ Ring → (∃𝑖𝐼 𝑥 = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) ↔ ∃𝑖𝐼 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
5545, 54syl5bb 286 . . . . 5 (𝑅 ∈ Ring → (∃𝑦 ∈ ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})𝑥 = (𝑦𝑃) ↔ ∃𝑖𝐼 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
5640, 55syl5bb 286 . . . 4 (𝑅 ∈ Ring → (𝑥 ∈ (ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ↾t 𝑃) ↔ ∃𝑖𝐼 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
57 rspectopn.3 . . . . . 6 𝐽 = ran (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗})
5857eleq2i 2881 . . . . 5 (𝑥𝐽𝑥 ∈ ran (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
59 eqid 2798 . . . . . 6 (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗}) = (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗})
6038rabex 5200 . . . . . 6 {𝑗𝑃 ∣ ¬ 𝑖𝑗} ∈ V
6159, 60elrnmpti 5797 . . . . 5 (𝑥 ∈ ran (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗}) ↔ ∃𝑖𝐼 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗})
6258, 61bitri 278 . . . 4 (𝑥𝐽 ↔ ∃𝑖𝐼 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗})
6356, 62bitr4di 292 . . 3 (𝑅 ∈ Ring → (𝑥 ∈ (ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ↾t 𝑃) ↔ 𝑥𝐽))
6463eqrdv 2796 . 2 (𝑅 ∈ Ring → (ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ↾t 𝑃) = 𝐽)
6510, 35, 643eqtr2rd 2840 1 (𝑅 ∈ Ring → 𝐽 = (TopOpen‘𝑆))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497   ↦ cmpt 5111  ran crn 5521  ‘cfv 6325  (class class class)co 7136  Basecbs 16478   ↾s cress 16479  TopSetcts 16566   ↾t crest 16689  TopOpenctopn 16690  Ringcrg 19294  LIdealclidl 19939  PrmIdealcprmidl 31028  IDLsrgcidlsrg 31063  Speccrspec 31230 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-2 11691  df-3 11692  df-4 11693  df-5 11694  df-6 11695  df-7 11696  df-8 11697  df-9 11698  df-n0 11889  df-z 11973  df-dec 12090  df-uz 12235  df-fz 12889  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-tset 16579  df-ple 16580  df-rest 16691  df-topn 16692  df-prmidl 31029  df-idlsrg 31064  df-rspec 31231 This theorem is referenced by:  zarcls  31242  zar0ring  31246
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