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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islnr3 | Structured version Visualization version GIF version |
Description: Relate left-Noetherian rings to Noetherian-type closure property of the left ideal system. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
Ref | Expression |
---|---|
islnr3.b | β’ π΅ = (Baseβπ ) |
islnr3.u | β’ π = (LIdealβπ ) |
Ref | Expression |
---|---|
islnr3 | β’ (π β LNoeR β (π β Ring β§ π β (NoeACSβπ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islnr3.b | . . 3 β’ π΅ = (Baseβπ ) | |
2 | islnr3.u | . . 3 β’ π = (LIdealβπ ) | |
3 | eqid 2728 | . . 3 β’ (RSpanβπ ) = (RSpanβπ ) | |
4 | 1, 2, 3 | islnr2 42538 | . 2 β’ (π β LNoeR β (π β Ring β§ βπ₯ β π βπ¦ β (π« π΅ β© Fin)π₯ = ((RSpanβπ )βπ¦))) |
5 | eqid 2728 | . . . . . . . . . 10 β’ (mrClsβπ) = (mrClsβπ) | |
6 | 2, 3, 5 | mrcrsp 21136 | . . . . . . . . 9 β’ (π β Ring β (RSpanβπ ) = (mrClsβπ)) |
7 | 6 | fveq1d 6899 | . . . . . . . 8 β’ (π β Ring β ((RSpanβπ )βπ¦) = ((mrClsβπ)βπ¦)) |
8 | 7 | eqeq2d 2739 | . . . . . . 7 β’ (π β Ring β (π₯ = ((RSpanβπ )βπ¦) β π₯ = ((mrClsβπ)βπ¦))) |
9 | 8 | rexbidv 3175 | . . . . . 6 β’ (π β Ring β (βπ¦ β (π« π΅ β© Fin)π₯ = ((RSpanβπ )βπ¦) β βπ¦ β (π« π΅ β© Fin)π₯ = ((mrClsβπ)βπ¦))) |
10 | 9 | ralbidv 3174 | . . . . 5 β’ (π β Ring β (βπ₯ β π βπ¦ β (π« π΅ β© Fin)π₯ = ((RSpanβπ )βπ¦) β βπ₯ β π βπ¦ β (π« π΅ β© Fin)π₯ = ((mrClsβπ)βπ¦))) |
11 | 1, 2 | lidlacs 21130 | . . . . . 6 β’ (π β Ring β π β (ACSβπ΅)) |
12 | 11 | biantrurd 532 | . . . . 5 β’ (π β Ring β (βπ₯ β π βπ¦ β (π« π΅ β© Fin)π₯ = ((mrClsβπ)βπ¦) β (π β (ACSβπ΅) β§ βπ₯ β π βπ¦ β (π« π΅ β© Fin)π₯ = ((mrClsβπ)βπ¦)))) |
13 | 10, 12 | bitrd 279 | . . . 4 β’ (π β Ring β (βπ₯ β π βπ¦ β (π« π΅ β© Fin)π₯ = ((RSpanβπ )βπ¦) β (π β (ACSβπ΅) β§ βπ₯ β π βπ¦ β (π« π΅ β© Fin)π₯ = ((mrClsβπ)βπ¦)))) |
14 | 5 | isnacs 42124 | . . . 4 β’ (π β (NoeACSβπ΅) β (π β (ACSβπ΅) β§ βπ₯ β π βπ¦ β (π« π΅ β© Fin)π₯ = ((mrClsβπ)βπ¦))) |
15 | 13, 14 | bitr4di 289 | . . 3 β’ (π β Ring β (βπ₯ β π βπ¦ β (π« π΅ β© Fin)π₯ = ((RSpanβπ )βπ¦) β π β (NoeACSβπ΅))) |
16 | 15 | pm5.32i 574 | . 2 β’ ((π β Ring β§ βπ₯ β π βπ¦ β (π« π΅ β© Fin)π₯ = ((RSpanβπ )βπ¦)) β (π β Ring β§ π β (NoeACSβπ΅))) |
17 | 4, 16 | bitri 275 | 1 β’ (π β LNoeR β (π β Ring β§ π β (NoeACSβπ΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 βwral 3058 βwrex 3067 β© cin 3946 π« cpw 4603 βcfv 6548 Fincfn 8964 Basecbs 17180 mrClscmrc 17563 ACScacs 17565 Ringcrg 20173 LIdealclidl 21102 RSpancrsp 21103 NoeACScnacs 42122 LNoeRclnr 42533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-0g 17423 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-subg 19078 df-mgp 20075 df-ur 20122 df-ring 20175 df-subrg 20508 df-lmod 20745 df-lss 20816 df-lsp 20856 df-sra 21058 df-rgmod 21059 df-lidl 21104 df-rsp 21105 df-nacs 42123 df-lfig 42492 df-lnm 42500 df-lnr 42534 |
This theorem is referenced by: hbt 42554 |
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