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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspectset | Structured version Visualization version GIF version |
Description: Topology component of the spectrum of a ring. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
Ref | Expression |
---|---|
rspecbas.1 | ⊢ 𝑆 = (Spec‘𝑅) |
rspectset.1 | ⊢ 𝐼 = (LIdeal‘𝑅) |
rspectset.2 | ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) |
Ref | Expression |
---|---|
rspectset | ⊢ (𝑅 ∈ Ring → 𝐽 = (TopSet‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6904 | . . 3 ⊢ (PrmIdeal‘𝑅) ∈ V | |
2 | eqid 2731 | . . . 4 ⊢ ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)) | |
3 | eqid 2731 | . . . 4 ⊢ (TopSet‘(IDLsrg‘𝑅)) = (TopSet‘(IDLsrg‘𝑅)) | |
4 | 2, 3 | resstset 17315 | . . 3 ⊢ ((PrmIdeal‘𝑅) ∈ V → (TopSet‘(IDLsrg‘𝑅)) = (TopSet‘((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))) |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ (TopSet‘(IDLsrg‘𝑅)) = (TopSet‘((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅))) |
6 | eqid 2731 | . . 3 ⊢ (IDLsrg‘𝑅) = (IDLsrg‘𝑅) | |
7 | rspectset.1 | . . 3 ⊢ 𝐼 = (LIdeal‘𝑅) | |
8 | rspectset.2 | . . 3 ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) | |
9 | 6, 7, 8 | idlsrgtset 32897 | . 2 ⊢ (𝑅 ∈ Ring → 𝐽 = (TopSet‘(IDLsrg‘𝑅))) |
10 | rspecbas.1 | . . . 4 ⊢ 𝑆 = (Spec‘𝑅) | |
11 | rspecval 33143 | . . . 4 ⊢ (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅))) | |
12 | 10, 11 | eqtrid 2783 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑆 = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅))) |
13 | 12 | fveq2d 6895 | . 2 ⊢ (𝑅 ∈ Ring → (TopSet‘𝑆) = (TopSet‘((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))) |
14 | 5, 9, 13 | 3eqtr4a 2797 | 1 ⊢ (𝑅 ∈ Ring → 𝐽 = (TopSet‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2105 {crab 3431 Vcvv 3473 ⊆ wss 3948 ↦ cmpt 5231 ran crn 5677 ‘cfv 6543 (class class class)co 7412 ↾s cress 17178 TopSetcts 17208 Ringcrg 20128 LIdealclidl 20929 PrmIdealcprmidl 32828 IDLsrgcidlsrg 32889 Speccrspec 33141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-tset 17221 df-ple 17222 df-idlsrg 32890 df-rspec 33142 |
This theorem is referenced by: (None) |
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