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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mxidlnzr | Structured version Visualization version GIF version |
Description: A ring with a maximal ideal is a nonzero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
Ref | Expression |
---|---|
mxidlval.1 | β’ π΅ = (Baseβπ ) |
Ref | Expression |
---|---|
mxidlnzr | β’ ((π β Ring β§ π β (MaxIdealβπ )) β π β NzRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mxidlval.1 | . . . . . 6 β’ π΅ = (Baseβπ ) | |
2 | 1 | mxidlidl 33209 | . . . . 5 β’ ((π β Ring β§ π β (MaxIdealβπ )) β π β (LIdealβπ )) |
3 | eqid 2728 | . . . . . 6 β’ (LIdealβπ ) = (LIdealβπ ) | |
4 | eqid 2728 | . . . . . 6 β’ (0gβπ ) = (0gβπ ) | |
5 | 3, 4 | lidl0cl 21130 | . . . . 5 β’ ((π β Ring β§ π β (LIdealβπ )) β (0gβπ ) β π) |
6 | 2, 5 | syldan 589 | . . . 4 β’ ((π β Ring β§ π β (MaxIdealβπ )) β (0gβπ ) β π) |
7 | eqid 2728 | . . . . 5 β’ (1rβπ ) = (1rβπ ) | |
8 | 1, 7 | mxidln1 33212 | . . . 4 β’ ((π β Ring β§ π β (MaxIdealβπ )) β Β¬ (1rβπ ) β π) |
9 | nelne2 3037 | . . . 4 β’ (((0gβπ ) β π β§ Β¬ (1rβπ ) β π) β (0gβπ ) β (1rβπ )) | |
10 | 6, 8, 9 | syl2anc 582 | . . 3 β’ ((π β Ring β§ π β (MaxIdealβπ )) β (0gβπ ) β (1rβπ )) |
11 | 10 | necomd 2993 | . 2 β’ ((π β Ring β§ π β (MaxIdealβπ )) β (1rβπ ) β (0gβπ )) |
12 | 7, 4 | isnzr 20467 | . . 3 β’ (π β NzRing β (π β Ring β§ (1rβπ ) β (0gβπ ))) |
13 | 12 | biimpri 227 | . 2 β’ ((π β Ring β§ (1rβπ ) β (0gβπ )) β π β NzRing) |
14 | 11, 13 | syldan 589 | 1 β’ ((π β Ring β§ π β (MaxIdealβπ )) β π β NzRing) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2937 βcfv 6553 Basecbs 17189 0gc0g 17430 1rcur 20135 Ringcrg 20187 NzRingcnzr 20465 LIdealclidl 21116 MaxIdealcmxidl 33205 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-sca 17258 df-vsca 17259 df-ip 17260 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19092 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-nzr 20466 df-subrg 20522 df-lmod 20759 df-lss 20830 df-sra 21072 df-rgmod 21073 df-lidl 21118 df-mxidl 33206 |
This theorem is referenced by: mxidlnzrb 33225 mxidlprmALT 33243 |
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