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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > drngmxidl | Structured version Visualization version GIF version |
Description: The zero ideal is the only ideal of a division ring. (Contributed by Thierry Arnoux, 16-Mar-2025.) |
Ref | Expression |
---|---|
drngmxidl.1 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
drngmxidl | ⊢ (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) = {{ 0 }}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngring 20728 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
2 | eqid 2736 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 2 | mxidlidl 33478 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (MaxIdeal‘𝑅)) → 𝑖 ∈ (LIdeal‘𝑅)) |
4 | 3 | ex 412 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑖 ∈ (MaxIdeal‘𝑅) → 𝑖 ∈ (LIdeal‘𝑅))) |
5 | 4 | ssrdv 3988 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (MaxIdeal‘𝑅) ⊆ (LIdeal‘𝑅)) |
6 | 1, 5 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) ⊆ (LIdeal‘𝑅)) |
7 | drngmxidl.1 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
8 | eqid 2736 | . . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
9 | 2, 7, 8 | drngnidl 21245 | . . . . 5 ⊢ (𝑅 ∈ DivRing → (LIdeal‘𝑅) = {{ 0 }, (Base‘𝑅)}) |
10 | 6, 9 | sseqtrd 4019 | . . . 4 ⊢ (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) ⊆ {{ 0 }, (Base‘𝑅)}) |
11 | 2 | mxidlnr 33479 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (MaxIdeal‘𝑅)) → 𝑖 ≠ (Base‘𝑅)) |
12 | 1, 11 | sylan 580 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (MaxIdeal‘𝑅)) → 𝑖 ≠ (Base‘𝑅)) |
13 | 12 | nelrdva 3710 | . . . 4 ⊢ (𝑅 ∈ DivRing → ¬ (Base‘𝑅) ∈ (MaxIdeal‘𝑅)) |
14 | ssdifsn 4786 | . . . 4 ⊢ ((MaxIdeal‘𝑅) ⊆ ({{ 0 }, (Base‘𝑅)} ∖ {(Base‘𝑅)}) ↔ ((MaxIdeal‘𝑅) ⊆ {{ 0 }, (Base‘𝑅)} ∧ ¬ (Base‘𝑅) ∈ (MaxIdeal‘𝑅))) | |
15 | 10, 13, 14 | sylanbrc 583 | . . 3 ⊢ (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) ⊆ ({{ 0 }, (Base‘𝑅)} ∖ {(Base‘𝑅)})) |
16 | drngnzr 20740 | . . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing) | |
17 | 7, 2 | drnglidl1ne0 33490 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (Base‘𝑅) ≠ { 0 }) |
18 | 17 | necomd 2995 | . . . 4 ⊢ (𝑅 ∈ NzRing → { 0 } ≠ (Base‘𝑅)) |
19 | difprsn2 4799 | . . . 4 ⊢ ({ 0 } ≠ (Base‘𝑅) → ({{ 0 }, (Base‘𝑅)} ∖ {(Base‘𝑅)}) = {{ 0 }}) | |
20 | 16, 18, 19 | 3syl 18 | . . 3 ⊢ (𝑅 ∈ DivRing → ({{ 0 }, (Base‘𝑅)} ∖ {(Base‘𝑅)}) = {{ 0 }}) |
21 | 15, 20 | sseqtrd 4019 | . 2 ⊢ (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) ⊆ {{ 0 }}) |
22 | 7 | drng0mxidl 33491 | . . 3 ⊢ (𝑅 ∈ DivRing → { 0 } ∈ (MaxIdeal‘𝑅)) |
23 | 22 | snssd 4807 | . 2 ⊢ (𝑅 ∈ DivRing → {{ 0 }} ⊆ (MaxIdeal‘𝑅)) |
24 | 21, 23 | eqssd 4000 | 1 ⊢ (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) = {{ 0 }}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2939 ∖ cdif 3947 ⊆ wss 3950 {csn 4624 {cpr 4626 ‘cfv 6559 Basecbs 17243 0gc0g 17480 Ringcrg 20226 NzRingcnzr 20504 DivRingcdr 20721 LIdealclidl 21208 MaxIdealcmxidl 33474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-tpos 8247 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-sca 17309 df-vsca 17310 df-ip 17311 df-0g 17482 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-grp 18950 df-minusg 18951 df-sbg 18952 df-subg 19137 df-cmn 19796 df-abl 19797 df-mgp 20134 df-rng 20146 df-ur 20175 df-ring 20228 df-oppr 20326 df-dvdsr 20349 df-unit 20350 df-invr 20380 df-nzr 20505 df-subrg 20562 df-drng 20723 df-lmod 20852 df-lss 20922 df-sra 21164 df-rgmod 21165 df-lidl 21210 df-mxidl 33475 |
This theorem is referenced by: (None) |
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