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Theorem rspectopn 33828
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 33825 . . . . 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 2800 . . . 4 (𝑅 ∈ Ring → 𝑆 = ((IDLsrg‘𝑅) ↾s 𝑃))
65fveq2d 6911 . . 3 (𝑅 ∈ Ring → (TopOpen‘𝑆) = (TopOpen‘((IDLsrg‘𝑅) ↾s 𝑃)))
7 eqid 2735 . . . 4 ((IDLsrg‘𝑅) ↾s 𝑃) = ((IDLsrg‘𝑅) ↾s 𝑃)
8 eqid 2735 . . . 4 (TopOpen‘(IDLsrg‘𝑅)) = (TopOpen‘(IDLsrg‘𝑅))
97, 8resstopn 23210 . . 3 ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃) = (TopOpen‘((IDLsrg‘𝑅) ↾s 𝑃))
106, 9eqtr4di 2793 . 2 (𝑅 ∈ Ring → (TopOpen‘𝑆) = ((TopOpen‘(IDLsrg‘𝑅)) ↾t 𝑃))
11 eqid 2735 . . . . 5 (IDLsrg‘𝑅) = (IDLsrg‘𝑅)
12 rspectopn.1 . . . . 5 𝐼 = (LIdeal‘𝑅)
13 eqid 2735 . . . . 5 ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) = ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})
1411, 12, 13idlsrgtset 33516 . . . 4 (𝑅 ∈ Ring → ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) = (TopSet‘(IDLsrg‘𝑅)))
1512fvexi 6921 . . . . . . . . . . 11 𝐼 ∈ V
1615rabex 5345 . . . . . . . . . 10 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ V
1716a1i 11 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ V)
18 simp2 1136 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑖𝐼) ∧ 𝑗𝐼 ∧ ¬ 𝑖𝑗) → 𝑗𝐼)
1911, 12idlsrgbas 33512 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝐼 = (Base‘(IDLsrg‘𝑅)))
2019adantr 480 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
21203ad2ant1 1132 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑖𝐼) ∧ 𝑗𝐼 ∧ ¬ 𝑖𝑗) → 𝐼 = (Base‘(IDLsrg‘𝑅)))
2218, 21eleqtrd 2841 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑖𝐼) ∧ 𝑗𝐼 ∧ ¬ 𝑖𝑗) → 𝑗 ∈ (Base‘(IDLsrg‘𝑅)))
2322rabssdv 4085 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → {𝑗𝐼 ∣ ¬ 𝑖𝑗} ⊆ (Base‘(IDLsrg‘𝑅)))
2417, 23elpwd 4611 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑖𝐼) → {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
2524ralrimiva 3144 . . . . . . 7 (𝑅 ∈ Ring → ∀𝑖𝐼 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ 𝒫 (Base‘(IDLsrg‘𝑅)))
26 eqid 2735 . . . . . . . 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 4035 . . . . 5 (𝑅 ∈ Ring → (TopSet‘(IDLsrg‘𝑅)) ⊆ 𝒫 (Base‘(IDLsrg‘𝑅)))
30 eqid 2735 . . . . . 6 (Base‘(IDLsrg‘𝑅)) = (Base‘(IDLsrg‘𝑅))
31 eqid 2735 . . . . . 6 (TopSet‘(IDLsrg‘𝑅)) = (TopSet‘(IDLsrg‘𝑅))
3230, 31topnid 17482 . . . . 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 2775 . . 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 7933 . . . . . 6 ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ∈ V
383fvexi 6921 . . . . . 6 𝑃 ∈ V
39 elrest 17474 . . . . . 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 3063 . . . . . . 7 𝑖𝐼 {𝑗𝐼 ∣ ¬ 𝑖𝑗} ∈ V
42 ineq1 4221 . . . . . . . . 9 (𝑦 = {𝑗𝐼 ∣ ¬ 𝑖𝑗} → (𝑦𝑃) = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃))
4342eqeq2d 2746 . . . . . . . 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 4323 . . . . . . . . 9 ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) = {𝑗 ∈ (𝐼𝑃) ∣ ¬ 𝑖𝑗}
47 prmidlssidl 33453 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
4847, 3, 123sstr4g 4041 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑃𝐼)
49 sseqin2 4231 . . . . . . . . . . 11 (𝑃𝐼 ↔ (𝐼𝑃) = 𝑃)
5048, 49sylib 218 . . . . . . . . . 10 (𝑅 ∈ Ring → (𝐼𝑃) = 𝑃)
5150rabeqdv 3449 . . . . . . . . 9 (𝑅 ∈ Ring → {𝑗 ∈ (𝐼𝑃) ∣ ¬ 𝑖𝑗} = {𝑗𝑃 ∣ ¬ 𝑖𝑗})
5246, 51eqtrid 2787 . . . . . . . 8 (𝑅 ∈ Ring → ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) = {𝑗𝑃 ∣ ¬ 𝑖𝑗})
5352eqeq2d 2746 . . . . . . 7 (𝑅 ∈ Ring → (𝑥 = ({𝑗𝐼 ∣ ¬ 𝑖𝑗} ∩ 𝑃) ↔ 𝑥 = {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
5453rexbidv 3177 . . . . . 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 2831 . . . . 5 (𝑥𝐽𝑥 ∈ ran (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗}))
59 eqid 2735 . . . . . 6 (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗}) = (𝑖𝐼 ↦ {𝑗𝑃 ∣ ¬ 𝑖𝑗})
6038rabex 5345 . . . . . 6 {𝑗𝑃 ∣ ¬ 𝑖𝑗} ∈ V
6159, 60elrnmpti 5976 . . . . 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 2733 . 2 (𝑅 ∈ Ring → (ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗}) ↾t 𝑃) = 𝐽)
6510, 35, 643eqtr2rd 2782 1 (𝑅 ∈ Ring → 𝐽 = (TopOpen‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cin 3962  wss 3963  𝒫 cpw 4605  cmpt 5231  ran crn 5690  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  TopSetcts 17304  t crest 17467  TopOpenctopn 17468  Ringcrg 20251  LIdealclidl 21234  PrmIdealcprmidl 33443  IDLsrgcidlsrg 33508  Speccrspec 33823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-tset 17317  df-ple 17318  df-rest 17469  df-topn 17470  df-prmidl 33444  df-idlsrg 33509  df-rspec 33824
This theorem is referenced by:  zarcls  33835  zar0ring  33839
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