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Theorem rspectopn 32842
Description: The topology component of the spectrum of a ring. (Contributed by Thierry Arnoux, 4-Jun-2024.)
Hypotheses
Ref Expression
rspecbas.1 𝑆 = (Specβ€˜π‘…)
rspectopn.1 𝐼 = (LIdealβ€˜π‘…)
rspectopn.2 𝑃 = (PrmIdealβ€˜π‘…)
rspectopn.3 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
Assertion
Ref Expression
rspectopn (𝑅 ∈ Ring β†’ 𝐽 = (TopOpenβ€˜π‘†))
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 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rspecval 32839 . . . . 5 (𝑅 ∈ Ring β†’ (Specβ€˜π‘…) = ((IDLsrgβ€˜π‘…) β†Ύs (PrmIdealβ€˜π‘…)))
2 rspecbas.1 . . . . 5 𝑆 = (Specβ€˜π‘…)
3 rspectopn.2 . . . . . 6 𝑃 = (PrmIdealβ€˜π‘…)
43oveq2i 7419 . . . . 5 ((IDLsrgβ€˜π‘…) β†Ύs 𝑃) = ((IDLsrgβ€˜π‘…) β†Ύs (PrmIdealβ€˜π‘…))
51, 2, 43eqtr4g 2797 . . . 4 (𝑅 ∈ Ring β†’ 𝑆 = ((IDLsrgβ€˜π‘…) β†Ύs 𝑃))
65fveq2d 6895 . . 3 (𝑅 ∈ Ring β†’ (TopOpenβ€˜π‘†) = (TopOpenβ€˜((IDLsrgβ€˜π‘…) β†Ύs 𝑃)))
7 eqid 2732 . . . 4 ((IDLsrgβ€˜π‘…) β†Ύs 𝑃) = ((IDLsrgβ€˜π‘…) β†Ύs 𝑃)
8 eqid 2732 . . . 4 (TopOpenβ€˜(IDLsrgβ€˜π‘…)) = (TopOpenβ€˜(IDLsrgβ€˜π‘…))
97, 8resstopn 22689 . . 3 ((TopOpenβ€˜(IDLsrgβ€˜π‘…)) β†Ύt 𝑃) = (TopOpenβ€˜((IDLsrgβ€˜π‘…) β†Ύs 𝑃))
106, 9eqtr4di 2790 . 2 (𝑅 ∈ Ring β†’ (TopOpenβ€˜π‘†) = ((TopOpenβ€˜(IDLsrgβ€˜π‘…)) β†Ύt 𝑃))
11 eqid 2732 . . . . 5 (IDLsrgβ€˜π‘…) = (IDLsrgβ€˜π‘…)
12 rspectopn.1 . . . . 5 𝐼 = (LIdealβ€˜π‘…)
13 eqid 2732 . . . . 5 ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})
1411, 12, 13idlsrgtset 32617 . . . 4 (𝑅 ∈ Ring β†’ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (TopSetβ€˜(IDLsrgβ€˜π‘…)))
1512fvexi 6905 . . . . . . . . . . 11 𝐼 ∈ V
1615rabex 5332 . . . . . . . . . 10 {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ V
1716a1i 11 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) β†’ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ V)
18 simp2 1137 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼 ∧ Β¬ 𝑖 βŠ† 𝑗) β†’ 𝑗 ∈ 𝐼)
1911, 12idlsrgbas 32613 . . . . . . . . . . . . 13 (𝑅 ∈ Ring β†’ 𝐼 = (Baseβ€˜(IDLsrgβ€˜π‘…)))
2019adantr 481 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) β†’ 𝐼 = (Baseβ€˜(IDLsrgβ€˜π‘…)))
21203ad2ant1 1133 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼 ∧ Β¬ 𝑖 βŠ† 𝑗) β†’ 𝐼 = (Baseβ€˜(IDLsrgβ€˜π‘…)))
2218, 21eleqtrd 2835 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 𝐼 ∧ Β¬ 𝑖 βŠ† 𝑗) β†’ 𝑗 ∈ (Baseβ€˜(IDLsrgβ€˜π‘…)))
2322rabssdv 4072 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) β†’ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} βŠ† (Baseβ€˜(IDLsrgβ€˜π‘…)))
2417, 23elpwd 4608 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐼) β†’ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ 𝒫 (Baseβ€˜(IDLsrgβ€˜π‘…)))
2524ralrimiva 3146 . . . . . . 7 (𝑅 ∈ Ring β†’ βˆ€π‘– ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ 𝒫 (Baseβ€˜(IDLsrgβ€˜π‘…)))
26 eqid 2732 . . . . . . . 8 (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})
2726rnmptss 7121 . . . . . . 7 (βˆ€π‘– ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ 𝒫 (Baseβ€˜(IDLsrgβ€˜π‘…)) β†’ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) βŠ† 𝒫 (Baseβ€˜(IDLsrgβ€˜π‘…)))
2825, 27syl 17 . . . . . 6 (𝑅 ∈ Ring β†’ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) βŠ† 𝒫 (Baseβ€˜(IDLsrgβ€˜π‘…)))
2914, 28eqsstrrd 4021 . . . . 5 (𝑅 ∈ Ring β†’ (TopSetβ€˜(IDLsrgβ€˜π‘…)) βŠ† 𝒫 (Baseβ€˜(IDLsrgβ€˜π‘…)))
30 eqid 2732 . . . . . 6 (Baseβ€˜(IDLsrgβ€˜π‘…)) = (Baseβ€˜(IDLsrgβ€˜π‘…))
31 eqid 2732 . . . . . 6 (TopSetβ€˜(IDLsrgβ€˜π‘…)) = (TopSetβ€˜(IDLsrgβ€˜π‘…))
3230, 31topnid 17380 . . . . 5 ((TopSetβ€˜(IDLsrgβ€˜π‘…)) βŠ† 𝒫 (Baseβ€˜(IDLsrgβ€˜π‘…)) β†’ (TopSetβ€˜(IDLsrgβ€˜π‘…)) = (TopOpenβ€˜(IDLsrgβ€˜π‘…)))
3329, 32syl 17 . . . 4 (𝑅 ∈ Ring β†’ (TopSetβ€˜(IDLsrgβ€˜π‘…)) = (TopOpenβ€˜(IDLsrgβ€˜π‘…)))
3414, 33eqtrd 2772 . . 3 (𝑅 ∈ Ring β†’ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (TopOpenβ€˜(IDLsrgβ€˜π‘…)))
3534oveq1d 7423 . 2 (𝑅 ∈ Ring β†’ (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) β†Ύt 𝑃) = ((TopOpenβ€˜(IDLsrgβ€˜π‘…)) β†Ύt 𝑃))
3615mptex 7224 . . . . . . 7 (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) ∈ V
3736rnex 7902 . . . . . 6 ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) ∈ V
383fvexi 6905 . . . . . 6 𝑃 ∈ V
39 elrest 17372 . . . . . 6 ((ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) ∈ V ∧ 𝑃 ∈ V) β†’ (π‘₯ ∈ (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) β†Ύt 𝑃) ↔ βˆƒπ‘¦ ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})π‘₯ = (𝑦 ∩ 𝑃)))
4037, 38, 39mp2an 690 . . . . 5 (π‘₯ ∈ (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) β†Ύt 𝑃) ↔ βˆƒπ‘¦ ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})π‘₯ = (𝑦 ∩ 𝑃))
4116rgenw 3065 . . . . . . 7 βˆ€π‘– ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ V
42 ineq1 4205 . . . . . . . . 9 (𝑦 = {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} β†’ (𝑦 ∩ 𝑃) = ({𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∩ 𝑃))
4342eqeq2d 2743 . . . . . . . 8 (𝑦 = {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} β†’ (π‘₯ = (𝑦 ∩ 𝑃) ↔ π‘₯ = ({𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∩ 𝑃)))
4426, 43rexrnmptw 7096 . . . . . . 7 (βˆ€π‘– ∈ 𝐼 {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ V β†’ (βˆƒπ‘¦ ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})π‘₯ = (𝑦 ∩ 𝑃) ↔ βˆƒπ‘– ∈ 𝐼 π‘₯ = ({𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∩ 𝑃)))
4541, 44ax-mp 5 . . . . . 6 (βˆƒπ‘¦ ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})π‘₯ = (𝑦 ∩ 𝑃) ↔ βˆƒπ‘– ∈ 𝐼 π‘₯ = ({𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∩ 𝑃))
46 inrab2 4307 . . . . . . . . 9 ({𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∩ 𝑃) = {𝑗 ∈ (𝐼 ∩ 𝑃) ∣ Β¬ 𝑖 βŠ† 𝑗}
47 prmidlssidl 32558 . . . . . . . . . . . 12 (𝑅 ∈ Ring β†’ (PrmIdealβ€˜π‘…) βŠ† (LIdealβ€˜π‘…))
4847, 3, 123sstr4g 4027 . . . . . . . . . . 11 (𝑅 ∈ Ring β†’ 𝑃 βŠ† 𝐼)
49 sseqin2 4215 . . . . . . . . . . 11 (𝑃 βŠ† 𝐼 ↔ (𝐼 ∩ 𝑃) = 𝑃)
5048, 49sylib 217 . . . . . . . . . 10 (𝑅 ∈ Ring β†’ (𝐼 ∩ 𝑃) = 𝑃)
5150rabeqdv 3447 . . . . . . . . 9 (𝑅 ∈ Ring β†’ {𝑗 ∈ (𝐼 ∩ 𝑃) ∣ Β¬ 𝑖 βŠ† 𝑗} = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
5246, 51eqtrid 2784 . . . . . . . 8 (𝑅 ∈ Ring β†’ ({𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∩ 𝑃) = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
5352eqeq2d 2743 . . . . . . 7 (𝑅 ∈ Ring β†’ (π‘₯ = ({𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∩ 𝑃) ↔ π‘₯ = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}))
5453rexbidv 3178 . . . . . 6 (𝑅 ∈ Ring β†’ (βˆƒπ‘– ∈ 𝐼 π‘₯ = ({𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗} ∩ 𝑃) ↔ βˆƒπ‘– ∈ 𝐼 π‘₯ = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}))
5545, 54bitrid 282 . . . . 5 (𝑅 ∈ Ring β†’ (βˆƒπ‘¦ ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗})π‘₯ = (𝑦 ∩ 𝑃) ↔ βˆƒπ‘– ∈ 𝐼 π‘₯ = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}))
5640, 55bitrid 282 . . . 4 (𝑅 ∈ Ring β†’ (π‘₯ ∈ (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) β†Ύt 𝑃) ↔ βˆƒπ‘– ∈ 𝐼 π‘₯ = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}))
57 rspectopn.3 . . . . . 6 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
5857eleq2i 2825 . . . . 5 (π‘₯ ∈ 𝐽 ↔ π‘₯ ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}))
59 eqid 2732 . . . . . 6 (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
6038rabex 5332 . . . . . 6 {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ V
6159, 60elrnmpti 5959 . . . . 5 (π‘₯ ∈ ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) ↔ βˆƒπ‘– ∈ 𝐼 π‘₯ = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
6258, 61bitri 274 . . . 4 (π‘₯ ∈ 𝐽 ↔ βˆƒπ‘– ∈ 𝐼 π‘₯ = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
6356, 62bitr4di 288 . . 3 (𝑅 ∈ Ring β†’ (π‘₯ ∈ (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) β†Ύt 𝑃) ↔ π‘₯ ∈ 𝐽))
6463eqrdv 2730 . 2 (𝑅 ∈ Ring β†’ (ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ Β¬ 𝑖 βŠ† 𝑗}) β†Ύt 𝑃) = 𝐽)
6510, 35, 643eqtr2rd 2779 1 (𝑅 ∈ Ring β†’ 𝐽 = (TopOpenβ€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602   ↦ cmpt 5231  ran crn 5677  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143   β†Ύs cress 17172  TopSetcts 17202   β†Ύt crest 17365  TopOpenctopn 17366  Ringcrg 20055  LIdealclidl 20782  PrmIdealcprmidl 32548  IDLsrgcidlsrg 32609  Speccrspec 32837
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-dec 12677  df-uz 12822  df-fz 13484  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-tset 17215  df-ple 17216  df-rest 17367  df-topn 17368  df-prmidl 32549  df-idlsrg 32610  df-rspec 32838
This theorem is referenced by:  zarcls  32849  zar0ring  32853
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