![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > zar0ring | Structured version Visualization version GIF version |
Description: The Zariski Topology of the trivial ring. (Contributed by Thierry Arnoux, 1-Jul-2024.) |
Ref | Expression |
---|---|
zartop.1 | ⊢ 𝑆 = (Spec‘𝑅) |
zartop.2 | ⊢ 𝐽 = (TopOpen‘𝑆) |
zar0ring.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
zar0ring | ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐽 = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zartop.2 | . . 3 ⊢ 𝐽 = (TopOpen‘𝑆) | |
2 | zartop.1 | . . . . 5 ⊢ 𝑆 = (Spec‘𝑅) | |
3 | eqid 2735 | . . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
4 | eqid 2735 | . . . . 5 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅) | |
5 | eqid 2735 | . . . . 5 ⊢ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) = ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) | |
6 | 2, 3, 4, 5 | rspectopn 33828 | . . . 4 ⊢ (𝑅 ∈ Ring → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopOpen‘𝑆)) |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) = (TopOpen‘𝑆)) |
8 | 1, 7 | eqtr4id 2794 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐽 = ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗})) |
9 | fvex 6920 | . . . . . 6 ⊢ (PrmIdeal‘𝑅) ∈ V | |
10 | 9 | rabex 5345 | . . . . 5 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗} ∈ V |
11 | eqid 2735 | . . . . 5 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) | |
12 | 10, 11 | fnmpti 6712 | . . . 4 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) Fn (LIdeal‘𝑅) |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) Fn (LIdeal‘𝑅)) |
14 | zar0ring.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
15 | eqid 2735 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
16 | 14, 15 | 0ringidl 33429 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (LIdeal‘𝑅) = {{(0g‘𝑅)}}) |
17 | snex 5442 | . . . . . 6 ⊢ {(0g‘𝑅)} ∈ V | |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → {(0g‘𝑅)} ∈ V) |
19 | 18 | snn0d 4780 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → {{(0g‘𝑅)}} ≠ ∅) |
20 | 16, 19 | eqnetrd 3006 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (LIdeal‘𝑅) ≠ ∅) |
21 | 14 | 0ringprmidl 33457 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (PrmIdeal‘𝑅) = ∅) |
22 | 21 | rabeqdv 3449 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗} = {𝑗 ∈ ∅ ∣ ¬ 𝑖 ⊆ 𝑗}) |
23 | rab0 4392 | . . . . . 6 ⊢ {𝑗 ∈ ∅ ∣ ¬ 𝑖 ⊆ 𝑗} = ∅ | |
24 | 22, 23 | eqtrdi 2791 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗} = ∅) |
25 | 24 | mpteq2dv 5250 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ ∅)) |
26 | fconstmpt 5751 | . . . 4 ⊢ ((LIdeal‘𝑅) × {∅}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ ∅) | |
27 | 25, 26 | eqtr4di 2793 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) = ((LIdeal‘𝑅) × {∅})) |
28 | fconst5 7226 | . . . 4 ⊢ (((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) Fn (LIdeal‘𝑅) ∧ (LIdeal‘𝑅) ≠ ∅) → ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) = ((LIdeal‘𝑅) × {∅}) ↔ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) = {∅})) | |
29 | 28 | biimpa 476 | . . 3 ⊢ ((((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) Fn (LIdeal‘𝑅) ∧ (LIdeal‘𝑅) ≠ ∅) ∧ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) = ((LIdeal‘𝑅) × {∅})) → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) = {∅}) |
30 | 13, 20, 27, 29 | syl21anc 838 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗}) = {∅}) |
31 | 8, 30 | eqtrd 2775 | 1 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐽 = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 {crab 3433 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 {csn 4631 ↦ cmpt 5231 × cxp 5687 ran crn 5690 Fn wfn 6558 ‘cfv 6563 1c1 11154 ♯chash 14366 Basecbs 17245 TopOpenctopn 17468 0gc0g 17486 Ringcrg 20251 LIdealclidl 21234 PrmIdealcprmidl 33443 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-rmo 3378 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-int 4952 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-card 9977 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-hash 14367 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-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-rest 17469 df-topn 17470 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-subrg 20587 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-prmidl 33444 df-idlsrg 33509 df-rspec 33824 |
This theorem is referenced by: zarcmplem 33842 |
Copyright terms: Public domain | W3C validator |