| 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 20669 | . . . . . 6 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
| 2 | eqid 2736 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 3 | 2 | mxidlidl 33544 | . . . . . . . 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 3939 | . . . . . 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 21198 | . . . . 5 ⊢ (𝑅 ∈ DivRing → (LIdeal‘𝑅) = {{ 0 }, (Base‘𝑅)}) |
| 10 | 6, 9 | sseqtrd 3970 | . . . 4 ⊢ (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) ⊆ {{ 0 }, (Base‘𝑅)}) |
| 11 | 2 | mxidlnr 33545 | . . . . . 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 3663 | . . . 4 ⊢ (𝑅 ∈ DivRing → ¬ (Base‘𝑅) ∈ (MaxIdeal‘𝑅)) |
| 14 | ssdifsn 4744 | . . . 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 20681 | . . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing) | |
| 17 | 7, 2 | drnglidl1ne0 33556 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (Base‘𝑅) ≠ { 0 }) |
| 18 | 17 | necomd 2987 | . . . 4 ⊢ (𝑅 ∈ NzRing → { 0 } ≠ (Base‘𝑅)) |
| 19 | difprsn2 4757 | . . . 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 3970 | . 2 ⊢ (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) ⊆ {{ 0 }}) |
| 22 | 7 | drng0mxidl 33557 | . . 3 ⊢ (𝑅 ∈ DivRing → { 0 } ∈ (MaxIdeal‘𝑅)) |
| 23 | 22 | snssd 4765 | . 2 ⊢ (𝑅 ∈ DivRing → {{ 0 }} ⊆ (MaxIdeal‘𝑅)) |
| 24 | 21, 23 | eqssd 3951 | 1 ⊢ (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) = {{ 0 }}) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∖ cdif 3898 ⊆ wss 3901 {csn 4580 {cpr 4582 ‘cfv 6492 Basecbs 17136 0gc0g 17359 Ringcrg 20168 NzRingcnzr 20445 DivRingcdr 20662 LIdealclidl 21161 MaxIdealcmxidl 33540 |
| 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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-ip 17195 df-0g 17361 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-nzr 20446 df-subrg 20503 df-drng 20664 df-lmod 20813 df-lss 20883 df-sra 21125 df-rgmod 21126 df-lidl 21163 df-mxidl 33541 |
| This theorem is referenced by: (None) |
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