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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 | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
drng0mxidl | β’ (π β DivRing β { 0 } β (MaxIdealβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngring 20594 | . 2 β’ (π β DivRing β π β Ring) | |
2 | eqid 2726 | . . . 4 β’ (LIdealβπ ) = (LIdealβπ ) | |
3 | drngmxidl.1 | . . . 4 β’ 0 = (0gβπ ) | |
4 | 2, 3 | lidl0 21089 | . . 3 β’ (π β Ring β { 0 } β (LIdealβπ )) |
5 | 1, 4 | syl 17 | . 2 β’ (π β DivRing β { 0 } β (LIdealβπ )) |
6 | eqid 2726 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
7 | eqid 2726 | . . . . . 6 β’ (1rβπ ) = (1rβπ ) | |
8 | 6, 7 | ringidcl 20165 | . . . . 5 β’ (π β Ring β (1rβπ ) β (Baseβπ )) |
9 | 1, 8 | syl 17 | . . . 4 β’ (π β DivRing β (1rβπ ) β (Baseβπ )) |
10 | drngnzr 20607 | . . . . 5 β’ (π β DivRing β π β NzRing) | |
11 | 7, 3 | nzrnz 20417 | . . . . 5 β’ (π β NzRing β (1rβπ ) β 0 ) |
12 | nelsn 4663 | . . . . 5 β’ ((1rβπ ) β 0 β Β¬ (1rβπ ) β { 0 }) | |
13 | 10, 11, 12 | 3syl 18 | . . . 4 β’ (π β DivRing β Β¬ (1rβπ ) β { 0 }) |
14 | nelne1 3033 | . . . 4 β’ (((1rβπ ) β (Baseβπ ) β§ Β¬ (1rβπ ) β { 0 }) β (Baseβπ ) β { 0 }) | |
15 | 9, 13, 14 | syl2anc 583 | . . 3 β’ (π β DivRing β (Baseβπ ) β { 0 }) |
16 | 15 | necomd 2990 | . 2 β’ (π β DivRing β { 0 } β (Baseβπ )) |
17 | 6, 3, 2 | drngnidl 21101 | . . . . . . 7 β’ (π β DivRing β (LIdealβπ ) = {{ 0 }, (Baseβπ )}) |
18 | 17 | eleq2d 2813 | . . . . . 6 β’ (π β DivRing β (π β (LIdealβπ ) β π β {{ 0 }, (Baseβπ )})) |
19 | 18 | biimpa 476 | . . . . 5 β’ ((π β DivRing β§ π β (LIdealβπ )) β π β {{ 0 }, (Baseβπ )}) |
20 | elpri 4645 | . . . . 5 β’ (π β {{ 0 }, (Baseβπ )} β (π = { 0 } β¨ π = (Baseβπ ))) | |
21 | 19, 20 | syl 17 | . . . 4 β’ ((π β DivRing β§ π β (LIdealβπ )) β (π = { 0 } β¨ π = (Baseβπ ))) |
22 | 21 | a1d 25 | . . 3 β’ ((π β DivRing β§ π β (LIdealβπ )) β ({ 0 } β π β (π = { 0 } β¨ π = (Baseβπ )))) |
23 | 22 | ralrimiva 3140 | . 2 β’ (π β DivRing β βπ β (LIdealβπ )({ 0 } β π β (π = { 0 } β¨ π = (Baseβπ )))) |
24 | 6 | ismxidl 33084 | . . 3 β’ (π β Ring β ({ 0 } β (MaxIdealβπ ) β ({ 0 } β (LIdealβπ ) β§ { 0 } β (Baseβπ ) β§ βπ β (LIdealβπ )({ 0 } β π β (π = { 0 } β¨ π = (Baseβπ )))))) |
25 | 24 | biimpar 477 | . 2 β’ ((π β Ring β§ ({ 0 } β (LIdealβπ ) β§ { 0 } β (Baseβπ ) β§ βπ β (LIdealβπ )({ 0 } β π β (π = { 0 } β¨ π = (Baseβπ ))))) β { 0 } β (MaxIdealβπ )) |
26 | 1, 5, 16, 23, 25 | syl13anc 1369 | 1 β’ (π β DivRing β { 0 } β (MaxIdealβπ )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β¨ wo 844 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 β wss 3943 {csn 4623 {cpr 4625 βcfv 6537 Basecbs 17153 0gc0g 17394 1rcur 20086 Ringcrg 20138 NzRingcnzr 20414 DivRingcdr 20587 LIdealclidl 21065 MaxIdealcmxidl 33081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-nzr 20415 df-subrg 20471 df-drng 20589 df-lmod 20708 df-lss 20779 df-sra 21021 df-rgmod 21022 df-lidl 21067 df-mxidl 33082 |
This theorem is referenced by: drngmxidl 33099 |
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