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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 2724 | . . . . . . . 8 β’ (LSubSpβπ) = (LSubSpβπ) | |
4 | 2, 3 | lssss 20779 | . . . . . . 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 4600 | . . . . . 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 766 | . . . . . 6 β’ (((π β LMod β§ π΅ β Fin) β§ π β (LSubSpβπ)) β π΅ β Fin) | |
9 | ssfi 9170 | . . . . . 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 4187 | . . . 4 β’ (((π β LMod β§ π΅ β Fin) β§ π β (LSubSpβπ)) β π β (π« π΅ β© Fin)) |
12 | eqid 2724 | . . . . . . 7 β’ (LSpanβπ) = (LSpanβπ) | |
13 | 3, 12 | lspid 20825 | . . . . . 6 β’ ((π β LMod β§ π β (LSubSpβπ)) β ((LSpanβπ)βπ) = π) |
14 | 13 | adantlr 712 | . . . . 5 β’ (((π β LMod β§ π΅ β Fin) β§ π β (LSubSpβπ)) β ((LSpanβπ)βπ) = π) |
15 | 14 | eqcomd 2730 | . . . 4 β’ (((π β LMod β§ π΅ β Fin) β§ π β (LSubSpβπ)) β π = ((LSpanβπ)βπ)) |
16 | fveq2 6882 | . . . . 5 β’ (π = π β ((LSpanβπ)βπ) = ((LSpanβπ)βπ)) | |
17 | 16 | rspceeqv 3626 | . . . 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 3138 | . 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 42370 | . 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 1533 β wcel 2098 βwral 3053 βwrex 3062 β© cin 3940 β wss 3941 π« cpw 4595 βcfv 6534 Fincfn 8936 Basecbs 17149 LModclmod 20702 LSubSpclss 20774 LSpanclspn 20814 LNoeMclnm 42367 |
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 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-sca 17218 df-vsca 17219 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-mgp 20036 df-ur 20083 df-ring 20136 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lfig 42360 df-lnm 42368 |
This theorem is referenced by: pwslnmlem0 42383 |
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