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Theorem rspectopn 31531
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 31528 . . . . 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 7224 . . . . 5 ((IDLsrg‘𝑅) ↾s 𝑃) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅))
51, 2, 43eqtr4g 2803 . . . 4 (𝑅 ∈ Ring → 𝑆 = ((IDLsrg‘𝑅) ↾s 𝑃))
65fveq2d 6721 . . 3 (𝑅 ∈ Ring → (TopOpen‘𝑆) = (TopOpen‘((IDLsrg‘𝑅) ↾s 𝑃)))
7 eqid 2737 . . . 4 ((IDLsrg‘𝑅) ↾s 𝑃) = ((IDLsrg‘𝑅) ↾s 𝑃)
8 eqid 2737 . . . 4 (TopOpen‘(IDLsrg‘𝑅)) = (TopOpen‘(IDLsrg‘𝑅))
97, 8resstopn 22083 . . 3 ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃) = (TopOpen‘((IDLsrg‘𝑅) ↾s 𝑃))
106, 9eqtr4di 2796 . 2 (𝑅 ∈ Ring → (TopOpen‘𝑆) = ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃))
11 eqid 2737 . . . . 5 (IDLsrg‘𝑅) = (IDLsrg‘𝑅)
12 rspectopn.1 . . . . 5 𝐼 = (LIdeal‘𝑅)
13 eqid 2737 . . . . 5 ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) = ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})
1411, 12, 13idlsrgtset 31367 . . . 4 (𝑅 ∈ Ring → ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) = (TopSet‘(IDLsrg‘𝑅)))
1512fvexi 6731 . . . . . . . . . . 11 𝐼 ∈ V
1615rabex 5225 . . . . . . . . . 10 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ V
1716a1i 11 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ V)
18 simp2 1139 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑖𝐼) ∧ 𝑗𝐼 ∧ ¬ 𝑖𝑗) → 𝑗𝐼)
1911, 12idlsrgbas 31363 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝐼 = (Base‘(IDLsrg‘𝑅)))
2019adantr 484 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
21203ad2ant1 1135 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑖𝐼) ∧ 𝑗𝐼 ∧ ¬ 𝑖𝑗) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
2218, 21eleqtrd 2840 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑖𝐼) ∧ 𝑗𝐼 ∧ ¬ 𝑖𝑗) → 𝑗 ∈ (Base‘(IDLsrg‘𝑅)))
2322rabssdv 3988 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → {𝑗𝐼 ∣ ¬ 𝑖𝑗} ⊆ (Base‘(IDLsrg‘𝑅)))
2417, 23elpwd 4521 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
2524ralrimiva 3105 . . . . . . 7 (𝑅 ∈ Ring → ∀𝑖𝐼 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
26 eqid 2737 . . . . . . . 8 (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) = (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})
2726rnmptss 6939 . . . . . . 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 3940 . . . . 5 (𝑅 ∈ Ring → (TopSet‘(IDLsrg‘𝑅)) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)))
30 eqid 2737 . . . . . 6 (Base‘(IDLsrg‘𝑅)) = (Base‘(IDLsrg‘𝑅))
31 eqid 2737 . . . . . 6 (TopSet‘(IDLsrg‘𝑅)) = (TopSet‘(IDLsrg‘𝑅))
3230, 31topnid 16940 . . . . 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 2777 . . 3 (𝑅 ∈ Ring → ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) = (TopOpen‘(IDLsrg‘𝑅)))
3534oveq1d 7228 . 2 (𝑅 ∈ Ring → (ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ↾t 𝑃) = ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃))
3615mptex 7039 . . . . . . 7 (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ∈ V
3736rnex 7690 . . . . . 6 ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ∈ V
383fvexi 6731 . . . . . 6 𝑃 ∈ V
39 elrest 16932 . . . . . 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 692 . . . . 5 (𝑥 ∈ (ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ↾t 𝑃) ↔ ∃𝑦 ∈ ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})𝑥 = (𝑦𝑃))
4116rgenw 3073 . . . . . . 7 𝑖𝐼 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ V
42 ineq1 4120 . . . . . . . . 9 (𝑦 = {𝑗𝐼 ∣ ¬ 𝑖𝑗} → (𝑦𝑃) = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃))
4342eqeq2d 2748 . . . . . . . 8 (𝑦 = {𝑗𝐼 ∣ ¬ 𝑖𝑗} → (𝑥 = (𝑦𝑃) ↔ 𝑥 = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃)))
4426, 43rexrnmptw 6914 . . . . . . 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 4222 . . . . . . . . 9 ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) = {𝑗 ∈ (𝐼𝑃) ∣ ¬ 𝑖𝑗}
47 prmidlssidl 31334 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
4847, 3, 123sstr4g 3946 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑃𝐼)
49 sseqin2 4130 . . . . . . . . . . 11 (𝑃𝐼 ↔ (𝐼𝑃) = 𝑃)
5048, 49sylib 221 . . . . . . . . . 10 (𝑅 ∈ Ring → (𝐼𝑃) = 𝑃)
5150rabeqdv 3395 . . . . . . . . 9 (𝑅 ∈ Ring → {𝑗 ∈ (𝐼𝑃) ∣ ¬ 𝑖𝑗} = {𝑗𝑃 ∣ ¬ 𝑖𝑗})
5246, 51syl5eq 2790 . . . . . . . 8 (𝑅 ∈ Ring → ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) = {𝑗𝑃 ∣ ¬ 𝑖𝑗})
5352eqeq2d 2748 . . . . . . 7 (𝑅 ∈ Ring → (𝑥 = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) ↔ 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
5453rexbidv 3216 . . . . . 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 2829 . . . . 5 (𝑥𝐽𝑥 ∈ ran (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
59 eqid 2737 . . . . . 6 (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗}) = (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗})
6038rabex 5225 . . . . . 6 {𝑗𝑃 ∣ ¬ 𝑖𝑗} ∈ V
6159, 60elrnmpti 5829 . . . . 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 2735 . 2 (𝑅 ∈ Ring → (ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ↾t 𝑃) = 𝐽)
6510, 35, 643eqtr2rd 2784 1 (𝑅 ∈ Ring → 𝐽 = (TopOpen‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wrex 3062  {crab 3065  Vcvv 3408  cin 3865  wss 3866  𝒫 cpw 4513  cmpt 5135  ran crn 5552  cfv 6380  (class class class)co 7213  Basecbs 16760  s cress 16784  TopSetcts 16808  t crest 16925  TopOpenctopn 16926  Ringcrg 19562  LIdealclidl 20207  PrmIdealcprmidl 31324  IDLsrgcidlsrg 31359  Speccrspec 31526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-fz 13096  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-tset 16821  df-ple 16822  df-rest 16927  df-topn 16928  df-prmidl 31325  df-idlsrg 31360  df-rspec 31527
This theorem is referenced by:  zarcls  31538  zar0ring  31542
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