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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lidlunitel | Structured version Visualization version GIF version |
Description: If an ideal πΌ contains a unit π½, then it is the whole ring. (Contributed by Thierry Arnoux, 19-Mar-2025.) |
Ref | Expression |
---|---|
lidlunitel.1 | β’ π΅ = (Baseβπ ) |
lidlunitel.2 | β’ π = (Unitβπ ) |
lidlunitel.3 | β’ (π β π½ β π) |
lidlunitel.4 | β’ (π β π½ β πΌ) |
lidlunitel.5 | β’ (π β π β Ring) |
lidlunitel.6 | β’ (π β πΌ β (LIdealβπ )) |
Ref | Expression |
---|---|
lidlunitel | β’ (π β πΌ = π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lidlunitel.5 | . 2 β’ (π β π β Ring) | |
2 | lidlunitel.6 | . 2 β’ (π β πΌ β (LIdealβπ )) | |
3 | lidlunitel.3 | . . . 4 β’ (π β π½ β π) | |
4 | lidlunitel.2 | . . . . 5 β’ π = (Unitβπ ) | |
5 | eqid 2726 | . . . . 5 β’ (invrβπ ) = (invrβπ ) | |
6 | eqid 2726 | . . . . 5 β’ (.rβπ ) = (.rβπ ) | |
7 | eqid 2726 | . . . . 5 β’ (1rβπ ) = (1rβπ ) | |
8 | 4, 5, 6, 7 | unitlinv 20295 | . . . 4 β’ ((π β Ring β§ π½ β π) β (((invrβπ )βπ½)(.rβπ )π½) = (1rβπ )) |
9 | 1, 3, 8 | syl2anc 583 | . . 3 β’ (π β (((invrβπ )βπ½)(.rβπ )π½) = (1rβπ )) |
10 | lidlunitel.1 | . . . . . 6 β’ π΅ = (Baseβπ ) | |
11 | 10, 4 | unitss 20278 | . . . . 5 β’ π β π΅ |
12 | 4, 5 | unitinvcl 20292 | . . . . . 6 β’ ((π β Ring β§ π½ β π) β ((invrβπ )βπ½) β π) |
13 | 1, 3, 12 | syl2anc 583 | . . . . 5 β’ (π β ((invrβπ )βπ½) β π) |
14 | 11, 13 | sselid 3975 | . . . 4 β’ (π β ((invrβπ )βπ½) β π΅) |
15 | lidlunitel.4 | . . . 4 β’ (π β π½ β πΌ) | |
16 | eqid 2726 | . . . . 5 β’ (LIdealβπ ) = (LIdealβπ ) | |
17 | 16, 10, 6 | lidlmcl 21084 | . . . 4 β’ (((π β Ring β§ πΌ β (LIdealβπ )) β§ (((invrβπ )βπ½) β π΅ β§ π½ β πΌ)) β (((invrβπ )βπ½)(.rβπ )π½) β πΌ) |
18 | 1, 2, 14, 15, 17 | syl22anc 836 | . . 3 β’ (π β (((invrβπ )βπ½)(.rβπ )π½) β πΌ) |
19 | 9, 18 | eqeltrrd 2828 | . 2 β’ (π β (1rβπ ) β πΌ) |
20 | 16, 10, 7 | lidl1el 21085 | . . 3 β’ ((π β Ring β§ πΌ β (LIdealβπ )) β ((1rβπ ) β πΌ β πΌ = π΅)) |
21 | 20 | biimpa 476 | . 2 β’ (((π β Ring β§ πΌ β (LIdealβπ )) β§ (1rβπ ) β πΌ) β πΌ = π΅) |
22 | 1, 2, 19, 21 | syl21anc 835 | 1 β’ (π β πΌ = π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βcfv 6537 (class class class)co 7405 Basecbs 17153 .rcmulr 17207 1rcur 20086 Ringcrg 20138 Unitcui 20257 invrcinvr 20289 LIdealclidl 21065 |
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-subrg 20471 df-lmod 20708 df-lss 20779 df-sra 21021 df-rgmod 21022 df-lidl 21067 |
This theorem is referenced by: unitpidl1 33048 ig1pnunit 33176 |
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