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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > filnm | Structured version Visualization version GIF version |
Description: Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
filnm.b | β’ π΅ = (Baseβπ) |
Ref | Expression |
---|---|
filnm | β’ ((π β LMod β§ π΅ β Fin) β π β LNoeM) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . 2 β’ ((π β LMod β§ π΅ β Fin) β π β LMod) | |
2 | filnm.b | . . . . . . . 8 β’ π΅ = (Baseβπ) | |
3 | eqid 2728 | . . . . . . . 8 β’ (LSubSpβπ) = (LSubSpβπ) | |
4 | 2, 3 | lssss 20819 | . . . . . . 7 β’ (π β (LSubSpβπ) β π β π΅) |
5 | 4 | adantl 481 | . . . . . 6 β’ (((π β LMod β§ π΅ β Fin) β§ π β (LSubSpβπ)) β π β π΅) |
6 | velpw 4608 | . . . . . 6 β’ (π β π« π΅ β π β π΅) | |
7 | 5, 6 | sylibr 233 | . . . . 5 β’ (((π β LMod β§ π΅ β Fin) β§ π β (LSubSpβπ)) β π β π« π΅) |
8 | simplr 768 | . . . . . 6 β’ (((π β LMod β§ π΅ β Fin) β§ π β (LSubSpβπ)) β π΅ β Fin) | |
9 | ssfi 9197 | . . . . . 6 β’ ((π΅ β Fin β§ π β π΅) β π β Fin) | |
10 | 8, 5, 9 | syl2anc 583 | . . . . 5 β’ (((π β LMod β§ π΅ β Fin) β§ π β (LSubSpβπ)) β π β Fin) |
11 | 7, 10 | elind 4194 | . . . 4 β’ (((π β LMod β§ π΅ β Fin) β§ π β (LSubSpβπ)) β π β (π« π΅ β© Fin)) |
12 | eqid 2728 | . . . . . . 7 β’ (LSpanβπ) = (LSpanβπ) | |
13 | 3, 12 | lspid 20865 | . . . . . 6 β’ ((π β LMod β§ π β (LSubSpβπ)) β ((LSpanβπ)βπ) = π) |
14 | 13 | adantlr 714 | . . . . 5 β’ (((π β LMod β§ π΅ β Fin) β§ π β (LSubSpβπ)) β ((LSpanβπ)βπ) = π) |
15 | 14 | eqcomd 2734 | . . . 4 β’ (((π β LMod β§ π΅ β Fin) β§ π β (LSubSpβπ)) β π = ((LSpanβπ)βπ)) |
16 | fveq2 6897 | . . . . 5 β’ (π = π β ((LSpanβπ)βπ) = ((LSpanβπ)βπ)) | |
17 | 16 | rspceeqv 3631 | . . . 4 β’ ((π β (π« π΅ β© Fin) β§ π = ((LSpanβπ)βπ)) β βπ β (π« π΅ β© Fin)π = ((LSpanβπ)βπ)) |
18 | 11, 15, 17 | syl2anc 583 | . . 3 β’ (((π β LMod β§ π΅ β Fin) β§ π β (LSubSpβπ)) β βπ β (π« π΅ β© Fin)π = ((LSpanβπ)βπ)) |
19 | 18 | ralrimiva 3143 | . 2 β’ ((π β LMod β§ π΅ β Fin) β βπ β (LSubSpβπ)βπ β (π« π΅ β© Fin)π = ((LSpanβπ)βπ)) |
20 | 2, 3, 12 | islnm2 42502 | . 2 β’ (π β LNoeM β (π β LMod β§ βπ β (LSubSpβπ)βπ β (π« π΅ β© Fin)π = ((LSpanβπ)βπ))) |
21 | 1, 19, 20 | sylanbrc 582 | 1 β’ ((π β LMod β§ π΅ β Fin) β π β LNoeM) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βwral 3058 βwrex 3067 β© cin 3946 β wss 3947 π« cpw 4603 βcfv 6548 Fincfn 8963 Basecbs 17179 LModclmod 20742 LSubSpclss 20814 LSpanclspn 20854 LNoeMclnm 42499 |
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 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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-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 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-sca 17248 df-vsca 17249 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-mgp 20074 df-ur 20121 df-ring 20174 df-lmod 20744 df-lss 20815 df-lsp 20855 df-lfig 42492 df-lnm 42500 |
This theorem is referenced by: pwslnmlem0 42515 |
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