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Theorem rspectopn 33866
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 33863 . . . . 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 7442 . . . . 5 ((IDLsrg‘𝑅) ↾s 𝑃) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅))
51, 2, 43eqtr4g 2802 . . . 4 (𝑅 ∈ Ring → 𝑆 = ((IDLsrg‘𝑅) ↾s 𝑃))
65fveq2d 6910 . . 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 23194 . . 3 ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃) = (TopOpen‘((IDLsrg‘𝑅) ↾s 𝑃))
106, 9eqtr4di 2795 . 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 33536 . . . 4 (𝑅 ∈ Ring → ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) = (TopSet‘(IDLsrg‘𝑅)))
1512fvexi 6920 . . . . . . . . . . 11 𝐼 ∈ V
1615rabex 5339 . . . . . . . . . 10 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ V
1716a1i 11 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ V)
18 simp2 1138 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑖𝐼) ∧ 𝑗𝐼 ∧ ¬ 𝑖𝑗) → 𝑗𝐼)
1911, 12idlsrgbas 33532 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝐼 = (Base‘(IDLsrg‘𝑅)))
2019adantr 480 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
21203ad2ant1 1134 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑖𝐼) ∧ 𝑗𝐼 ∧ ¬ 𝑖𝑗) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
2218, 21eleqtrd 2843 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑖𝐼) ∧ 𝑗𝐼 ∧ ¬ 𝑖𝑗) → 𝑗 ∈ (Base‘(IDLsrg‘𝑅)))
2322rabssdv 4075 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → {𝑗𝐼 ∣ ¬ 𝑖𝑗} ⊆ (Base‘(IDLsrg‘𝑅)))
2417, 23elpwd 4606 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
2524ralrimiva 3146 . . . . . . 7 (𝑅 ∈ Ring → ∀𝑖𝐼 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
26 eqid 2737 . . . . . . . 8 (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) = (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})
2726rnmptss 7143 . . . . . . 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 4019 . . . . 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 17480 . . . . 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 7446 . 2 (𝑅 ∈ Ring → (ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ↾t 𝑃) = ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃))
3615mptex 7243 . . . . . . 7 (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ∈ V
3736rnex 7932 . . . . . 6 ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ∈ V
383fvexi 6920 . . . . . 6 𝑃 ∈ V
39 elrest 17472 . . . . . 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 3065 . . . . . . 7 𝑖𝐼 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ V
42 ineq1 4213 . . . . . . . . 9 (𝑦 = {𝑗𝐼 ∣ ¬ 𝑖𝑗} → (𝑦𝑃) = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃))
4342eqeq2d 2748 . . . . . . . 8 (𝑦 = {𝑗𝐼 ∣ ¬ 𝑖𝑗} → (𝑥 = (𝑦𝑃) ↔ 𝑥 = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃)))
4426, 43rexrnmptw 7115 . . . . . . 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 4317 . . . . . . . . 9 ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) = {𝑗 ∈ (𝐼𝑃) ∣ ¬ 𝑖𝑗}
47 prmidlssidl 33473 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
4847, 3, 123sstr4g 4037 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑃𝐼)
49 sseqin2 4223 . . . . . . . . . . 11 (𝑃𝐼 ↔ (𝐼𝑃) = 𝑃)
5048, 49sylib 218 . . . . . . . . . 10 (𝑅 ∈ Ring → (𝐼𝑃) = 𝑃)
5150rabeqdv 3452 . . . . . . . . 9 (𝑅 ∈ Ring → {𝑗 ∈ (𝐼𝑃) ∣ ¬ 𝑖𝑗} = {𝑗𝑃 ∣ ¬ 𝑖𝑗})
5246, 51eqtrid 2789 . . . . . . . 8 (𝑅 ∈ Ring → ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) = {𝑗𝑃 ∣ ¬ 𝑖𝑗})
5352eqeq2d 2748 . . . . . . 7 (𝑅 ∈ Ring → (𝑥 = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) ↔ 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
5453rexbidv 3179 . . . . . 6 (𝑅 ∈ Ring → (∃𝑖𝐼 𝑥 = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) ↔ ∃𝑖𝐼 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
5545, 54bitrid 283 . . . . 5 (𝑅 ∈ Ring → (∃𝑦 ∈ ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})𝑥 = (𝑦𝑃) ↔ ∃𝑖𝐼 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
5640, 55bitrid 283 . . . 4 (𝑅 ∈ Ring → (𝑥 ∈ (ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ↾t 𝑃) ↔ ∃𝑖𝐼 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
57 rspectopn.3 . . . . . 6 𝐽 = ran (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗})
5857eleq2i 2833 . . . . 5 (𝑥𝐽𝑥 ∈ ran (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
59 eqid 2737 . . . . . 6 (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗}) = (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗})
6038rabex 5339 . . . . . 6 {𝑗𝑃 ∣ ¬ 𝑖𝑗} ∈ V
6159, 60elrnmpti 5973 . . . . 5 (𝑥 ∈ ran (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗}) ↔ ∃𝑖𝐼 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗})
6258, 61bitri 275 . . . 4 (𝑥𝐽 ↔ ∃𝑖𝐼 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗})
6356, 62bitr4di 289 . . 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600  cmpt 5225  ran crn 5686  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  TopSetcts 17303  t crest 17465  TopOpenctopn 17466  Ringcrg 20230  LIdealclidl 21216  PrmIdealcprmidl 33463  IDLsrgcidlsrg 33528  Speccrspec 33861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-tset 17316  df-ple 17317  df-rest 17467  df-topn 17468  df-prmidl 33464  df-idlsrg 33529  df-rspec 33862
This theorem is referenced by:  zarcls  33873  zar0ring  33877
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