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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opprmxidlabs | Structured version Visualization version GIF version |
Description: The maximal ideal of the opposite ring's opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
Ref | Expression |
---|---|
oppreqg.o | β’ π = (opprβπ ) |
oppr2idl.2 | β’ (π β π β Ring) |
opprmxidl.3 | β’ (π β π β (MaxIdealβπ )) |
Ref | Expression |
---|---|
opprmxidlabs | β’ (π β π β (MaxIdealβ(opprβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppr2idl.2 | . . 3 β’ (π β π β Ring) | |
2 | oppreqg.o | . . . 4 β’ π = (opprβπ ) | |
3 | 2 | opprring 20239 | . . 3 β’ (π β Ring β π β Ring) |
4 | eqid 2724 | . . . 4 β’ (opprβπ) = (opprβπ) | |
5 | 4 | opprring 20239 | . . 3 β’ (π β Ring β (opprβπ) β Ring) |
6 | 1, 3, 5 | 3syl 18 | . 2 β’ (π β (opprβπ) β Ring) |
7 | opprmxidl.3 | . . . 4 β’ (π β π β (MaxIdealβπ )) | |
8 | eqid 2724 | . . . . 5 β’ (Baseβπ ) = (Baseβπ ) | |
9 | 8 | mxidlidl 33048 | . . . 4 β’ ((π β Ring β§ π β (MaxIdealβπ )) β π β (LIdealβπ )) |
10 | 1, 7, 9 | syl2anc 583 | . . 3 β’ (π β π β (LIdealβπ )) |
11 | 2, 1 | opprlidlabs 33068 | . . 3 β’ (π β (LIdealβπ ) = (LIdealβ(opprβπ))) |
12 | 10, 11 | eleqtrd 2827 | . 2 β’ (π β π β (LIdealβ(opprβπ))) |
13 | 8 | mxidlnr 33049 | . . 3 β’ ((π β Ring β§ π β (MaxIdealβπ )) β π β (Baseβπ )) |
14 | 1, 7, 13 | syl2anc 583 | . 2 β’ (π β π β (Baseβπ )) |
15 | 1 | ad2antrr 723 | . . . . 5 β’ (((π β§ π β (LIdealβ(opprβπ))) β§ π β π) β π β Ring) |
16 | 7 | ad2antrr 723 | . . . . 5 β’ (((π β§ π β (LIdealβ(opprβπ))) β§ π β π) β π β (MaxIdealβπ )) |
17 | simplr 766 | . . . . . 6 β’ (((π β§ π β (LIdealβ(opprβπ))) β§ π β π) β π β (LIdealβ(opprβπ))) | |
18 | 11 | ad2antrr 723 | . . . . . 6 β’ (((π β§ π β (LIdealβ(opprβπ))) β§ π β π) β (LIdealβπ ) = (LIdealβ(opprβπ))) |
19 | 17, 18 | eleqtrrd 2828 | . . . . 5 β’ (((π β§ π β (LIdealβ(opprβπ))) β§ π β π) β π β (LIdealβπ )) |
20 | simpr 484 | . . . . 5 β’ (((π β§ π β (LIdealβ(opprβπ))) β§ π β π) β π β π) | |
21 | 8 | mxidlmax 33050 | . . . . 5 β’ (((π β Ring β§ π β (MaxIdealβπ )) β§ (π β (LIdealβπ ) β§ π β π)) β (π = π β¨ π = (Baseβπ ))) |
22 | 15, 16, 19, 20, 21 | syl22anc 836 | . . . 4 β’ (((π β§ π β (LIdealβ(opprβπ))) β§ π β π) β (π = π β¨ π = (Baseβπ ))) |
23 | 22 | ex 412 | . . 3 β’ ((π β§ π β (LIdealβ(opprβπ))) β (π β π β (π = π β¨ π = (Baseβπ )))) |
24 | 23 | ralrimiva 3138 | . 2 β’ (π β βπ β (LIdealβ(opprβπ))(π β π β (π = π β¨ π = (Baseβπ )))) |
25 | 2, 8 | opprbas 20233 | . . . . 5 β’ (Baseβπ ) = (Baseβπ) |
26 | 4, 25 | opprbas 20233 | . . . 4 β’ (Baseβπ ) = (Baseβ(opprβπ)) |
27 | 26 | ismxidl 33047 | . . 3 β’ ((opprβπ) β Ring β (π β (MaxIdealβ(opprβπ)) β (π β (LIdealβ(opprβπ)) β§ π β (Baseβπ ) β§ βπ β (LIdealβ(opprβπ))(π β π β (π = π β¨ π = (Baseβπ )))))) |
28 | 27 | biimpar 477 | . 2 β’ (((opprβπ) β Ring β§ (π β (LIdealβ(opprβπ)) β§ π β (Baseβπ ) β§ βπ β (LIdealβ(opprβπ))(π β π β (π = π β¨ π = (Baseβπ ))))) β π β (MaxIdealβ(opprβπ))) |
29 | 6, 12, 14, 24, 28 | syl13anc 1369 | 1 β’ (π β π β (MaxIdealβ(opprβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β¨ wo 844 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2932 βwral 3053 β wss 3940 βcfv 6533 Basecbs 17143 Ringcrg 20128 opprcoppr 20225 LIdealclidl 21055 MaxIdealcmxidl 33044 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-lss 20769 df-sra 21011 df-rgmod 21012 df-lidl 21057 df-mxidl 33045 |
This theorem is referenced by: qsdrngi 33078 |
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