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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspecbas | Structured version Visualization version GIF version |
Description: The prime ideals form the base of the spectrum of a ring. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
Ref | Expression |
---|---|
rspecbas.1 | ⊢ 𝑆 = (Spec‘𝑅) |
Ref | Expression |
---|---|
rspecbas | ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = (Base‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmidlssidl 33440 | . . . 4 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅)) | |
2 | eqid 2740 | . . . . 5 ⊢ (IDLsrg‘𝑅) = (IDLsrg‘𝑅) | |
3 | eqid 2740 | . . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
4 | 2, 3 | idlsrgbas 33499 | . . . 4 ⊢ (𝑅 ∈ Ring → (LIdeal‘𝑅) = (Base‘(IDLsrg‘𝑅))) |
5 | 1, 4 | sseqtrd 4049 | . . 3 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (Base‘(IDLsrg‘𝑅))) |
6 | eqid 2740 | . . . 4 ⊢ ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)) | |
7 | eqid 2740 | . . . 4 ⊢ (Base‘(IDLsrg‘𝑅)) = (Base‘(IDLsrg‘𝑅)) | |
8 | 6, 7 | ressbas2 17298 | . . 3 ⊢ ((PrmIdeal‘𝑅) ⊆ (Base‘(IDLsrg‘𝑅)) → (PrmIdeal‘𝑅) = (Base‘((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))) |
9 | 5, 8 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = (Base‘((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))) |
10 | rspecbas.1 | . . . 4 ⊢ 𝑆 = (Spec‘𝑅) | |
11 | rspecval 33812 | . . . 4 ⊢ (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅))) | |
12 | 10, 11 | eqtrid 2792 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑆 = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅))) |
13 | 12 | fveq2d 6926 | . 2 ⊢ (𝑅 ∈ Ring → (Base‘𝑆) = (Base‘((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))) |
14 | 9, 13 | eqtr4d 2783 | 1 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = (Base‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 ‘cfv 6575 (class class class)co 7450 Basecbs 17260 ↾s cress 17289 Ringcrg 20262 LIdealclidl 21241 PrmIdealcprmidl 33430 IDLsrgcidlsrg 33495 Speccrspec 33810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-z 12642 df-dec 12761 df-uz 12906 df-fz 13570 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-tset 17332 df-ple 17333 df-prmidl 33431 df-idlsrg 33496 df-rspec 33811 |
This theorem is referenced by: rspectps 33831 |
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