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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > drng0mxidl | Structured version Visualization version GIF version |
Description: In a division ring, the zero ideal is a maximal ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.) |
Ref | Expression |
---|---|
drngmxidl.1 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
drng0mxidl | ⊢ (𝑅 ∈ DivRing → { 0 } ∈ (MaxIdeal‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngring 20590 | . 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
2 | eqid 2731 | . . . 4 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
3 | drngmxidl.1 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | lidl0 21085 | . . 3 ⊢ (𝑅 ∈ Ring → { 0 } ∈ (LIdeal‘𝑅)) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝑅 ∈ DivRing → { 0 } ∈ (LIdeal‘𝑅)) |
6 | eqid 2731 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | eqid 2731 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
8 | 6, 7 | ringidcl 20161 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
9 | 1, 8 | syl 17 | . . . 4 ⊢ (𝑅 ∈ DivRing → (1r‘𝑅) ∈ (Base‘𝑅)) |
10 | drngnzr 20603 | . . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing) | |
11 | 7, 3 | nzrnz 20413 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ 0 ) |
12 | nelsn 4668 | . . . . 5 ⊢ ((1r‘𝑅) ≠ 0 → ¬ (1r‘𝑅) ∈ { 0 }) | |
13 | 10, 11, 12 | 3syl 18 | . . . 4 ⊢ (𝑅 ∈ DivRing → ¬ (1r‘𝑅) ∈ { 0 }) |
14 | nelne1 3038 | . . . 4 ⊢ (((1r‘𝑅) ∈ (Base‘𝑅) ∧ ¬ (1r‘𝑅) ∈ { 0 }) → (Base‘𝑅) ≠ { 0 }) | |
15 | 9, 13, 14 | syl2anc 583 | . . 3 ⊢ (𝑅 ∈ DivRing → (Base‘𝑅) ≠ { 0 }) |
16 | 15 | necomd 2995 | . 2 ⊢ (𝑅 ∈ DivRing → { 0 } ≠ (Base‘𝑅)) |
17 | 6, 3, 2 | drngnidl 21097 | . . . . . . 7 ⊢ (𝑅 ∈ DivRing → (LIdeal‘𝑅) = {{ 0 }, (Base‘𝑅)}) |
18 | 17 | eleq2d 2818 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → (𝑗 ∈ (LIdeal‘𝑅) ↔ 𝑗 ∈ {{ 0 }, (Base‘𝑅)})) |
19 | 18 | biimpa 476 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑗 ∈ (LIdeal‘𝑅)) → 𝑗 ∈ {{ 0 }, (Base‘𝑅)}) |
20 | elpri 4650 | . . . . 5 ⊢ (𝑗 ∈ {{ 0 }, (Base‘𝑅)} → (𝑗 = { 0 } ∨ 𝑗 = (Base‘𝑅))) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (𝑗 = { 0 } ∨ 𝑗 = (Base‘𝑅))) |
22 | 21 | a1d 25 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝑗 ∈ (LIdeal‘𝑅)) → ({ 0 } ⊆ 𝑗 → (𝑗 = { 0 } ∨ 𝑗 = (Base‘𝑅)))) |
23 | 22 | ralrimiva 3145 | . 2 ⊢ (𝑅 ∈ DivRing → ∀𝑗 ∈ (LIdeal‘𝑅)({ 0 } ⊆ 𝑗 → (𝑗 = { 0 } ∨ 𝑗 = (Base‘𝑅)))) |
24 | 6 | ismxidl 33020 | . . 3 ⊢ (𝑅 ∈ Ring → ({ 0 } ∈ (MaxIdeal‘𝑅) ↔ ({ 0 } ∈ (LIdeal‘𝑅) ∧ { 0 } ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)({ 0 } ⊆ 𝑗 → (𝑗 = { 0 } ∨ 𝑗 = (Base‘𝑅)))))) |
25 | 24 | biimpar 477 | . 2 ⊢ ((𝑅 ∈ Ring ∧ ({ 0 } ∈ (LIdeal‘𝑅) ∧ { 0 } ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)({ 0 } ⊆ 𝑗 → (𝑗 = { 0 } ∨ 𝑗 = (Base‘𝑅))))) → { 0 } ∈ (MaxIdeal‘𝑅)) |
26 | 1, 5, 16, 23, 25 | syl13anc 1371 | 1 ⊢ (𝑅 ∈ DivRing → { 0 } ∈ (MaxIdeal‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 844 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ⊆ wss 3948 {csn 4628 {cpr 4630 ‘cfv 6543 Basecbs 17151 0gc0g 17392 1rcur 20082 Ringcrg 20134 NzRingcnzr 20410 DivRingcdr 20583 LIdealclidl 21061 MaxIdealcmxidl 33017 |
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 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
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-rmo 3375 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-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-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-nzr 20411 df-subrg 20467 df-drng 20585 df-lmod 20704 df-lss 20775 df-sra 21017 df-rgmod 21018 df-lidl 21063 df-mxidl 33018 |
This theorem is referenced by: drngmxidl 33035 |
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