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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lnrfg | Structured version Visualization version GIF version |
Description: Finitely-generated modules over a Noetherian ring, being homomorphic images of free modules, are Noetherian. (Contributed by Stefan O'Rear, 7-Feb-2015.) |
Ref | Expression |
---|---|
lnrfg.s | ⊢ 𝑆 = (Scalar‘𝑀) |
Ref | Expression |
---|---|
lnrfg | ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) → 𝑀 ∈ LNoeM) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2772 | . . . 4 ⊢ (𝑆 freeLMod 𝑎) = (𝑆 freeLMod 𝑎) | |
2 | eqid 2772 | . . . 4 ⊢ (Base‘(𝑆 freeLMod 𝑎)) = (Base‘(𝑆 freeLMod 𝑎)) | |
3 | eqid 2772 | . . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
4 | eqid 2772 | . . . 4 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
5 | eqid 2772 | . . . 4 ⊢ (𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘𝑓 ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) = (𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘𝑓 ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) | |
6 | fglmod 39014 | . . . . 5 ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod) | |
7 | 6 | ad3antrrr 717 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑀 ∈ LMod) |
8 | vex 3412 | . . . . 5 ⊢ 𝑎 ∈ V | |
9 | 8 | a1i 11 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑎 ∈ V) |
10 | lnrfg.s | . . . . 5 ⊢ 𝑆 = (Scalar‘𝑀) | |
11 | 10 | a1i 11 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑆 = (Scalar‘𝑀)) |
12 | f1oi 6475 | . . . . . . 7 ⊢ ( I ↾ 𝑎):𝑎–1-1-onto→𝑎 | |
13 | f1of 6438 | . . . . . . 7 ⊢ (( I ↾ 𝑎):𝑎–1-1-onto→𝑎 → ( I ↾ 𝑎):𝑎⟶𝑎) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ ( I ↾ 𝑎):𝑎⟶𝑎 |
15 | elpwi 4426 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 (Base‘𝑀) → 𝑎 ⊆ (Base‘𝑀)) | |
16 | fss 6351 | . . . . . 6 ⊢ ((( I ↾ 𝑎):𝑎⟶𝑎 ∧ 𝑎 ⊆ (Base‘𝑀)) → ( I ↾ 𝑎):𝑎⟶(Base‘𝑀)) | |
17 | 14, 15, 16 | sylancr 578 | . . . . 5 ⊢ (𝑎 ∈ 𝒫 (Base‘𝑀) → ( I ↾ 𝑎):𝑎⟶(Base‘𝑀)) |
18 | 17 | ad2antlr 714 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → ( I ↾ 𝑎):𝑎⟶(Base‘𝑀)) |
19 | 1, 2, 3, 4, 5, 7, 9, 11, 18 | frlmup1 20634 | . . 3 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → (𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘𝑓 ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) ∈ ((𝑆 freeLMod 𝑎) LMHom 𝑀)) |
20 | simpllr 763 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑆 ∈ LNoeR) | |
21 | simprl 758 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑎 ∈ Fin) | |
22 | 1 | lnrfrlm 39059 | . . . 4 ⊢ ((𝑆 ∈ LNoeR ∧ 𝑎 ∈ Fin) → (𝑆 freeLMod 𝑎) ∈ LNoeM) |
23 | 20, 21, 22 | syl2anc 576 | . . 3 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → (𝑆 freeLMod 𝑎) ∈ LNoeM) |
24 | eqid 2772 | . . . . 5 ⊢ (LSpan‘𝑀) = (LSpan‘𝑀) | |
25 | 1, 2, 3, 4, 5, 7, 9, 11, 18, 24 | frlmup3 20636 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → ran (𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘𝑓 ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) = ((LSpan‘𝑀)‘ran ( I ↾ 𝑎))) |
26 | rnresi 5777 | . . . . . 6 ⊢ ran ( I ↾ 𝑎) = 𝑎 | |
27 | 26 | fveq2i 6496 | . . . . 5 ⊢ ((LSpan‘𝑀)‘ran ( I ↾ 𝑎)) = ((LSpan‘𝑀)‘𝑎) |
28 | simprr 760 | . . . . 5 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀)) | |
29 | 27, 28 | syl5eq 2820 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → ((LSpan‘𝑀)‘ran ( I ↾ 𝑎)) = (Base‘𝑀)) |
30 | 25, 29 | eqtrd 2808 | . . 3 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → ran (𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘𝑓 ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) = (Base‘𝑀)) |
31 | 3 | lnmepi 39026 | . . 3 ⊢ (((𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘𝑓 ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) ∈ ((𝑆 freeLMod 𝑎) LMHom 𝑀) ∧ (𝑆 freeLMod 𝑎) ∈ LNoeM ∧ ran (𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘𝑓 ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) = (Base‘𝑀)) → 𝑀 ∈ LNoeM) |
32 | 19, 23, 30, 31 | syl3anc 1351 | . 2 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑀 ∈ LNoeM) |
33 | 3, 24 | islmodfg 39010 | . . . . 5 ⊢ (𝑀 ∈ LMod → (𝑀 ∈ LFinGen ↔ ∃𝑎 ∈ 𝒫 (Base‘𝑀)(𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀)))) |
34 | 6, 33 | syl 17 | . . . 4 ⊢ (𝑀 ∈ LFinGen → (𝑀 ∈ LFinGen ↔ ∃𝑎 ∈ 𝒫 (Base‘𝑀)(𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀)))) |
35 | 34 | ibi 259 | . . 3 ⊢ (𝑀 ∈ LFinGen → ∃𝑎 ∈ 𝒫 (Base‘𝑀)(𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) |
36 | 35 | adantr 473 | . 2 ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) → ∃𝑎 ∈ 𝒫 (Base‘𝑀)(𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) |
37 | 32, 36 | r19.29a 3228 | 1 ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) → 𝑀 ∈ LNoeM) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∃wrex 3083 Vcvv 3409 ⊆ wss 3825 𝒫 cpw 4416 ↦ cmpt 5002 I cid 5304 ran crn 5401 ↾ cres 5402 ⟶wf 6178 –1-1-onto→wf1o 6181 ‘cfv 6182 (class class class)co 6970 ∘𝑓 cof 7219 Fincfn 8298 Basecbs 16329 Scalarcsca 16414 ·𝑠 cvsca 16415 Σg cgsu 16560 LModclmod 19346 LSpanclspn 19455 LMHom clmhm 19503 freeLMod cfrlm 20582 LFinGenclfig 39008 LNoeMclnm 39016 LNoeRclnr 39050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-sup 8693 df-oi 8761 df-card 9154 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-fz 12702 df-fzo 12843 df-seq 13178 df-hash 13499 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-hom 16435 df-cco 16436 df-0g 16561 df-gsum 16562 df-prds 16567 df-pws 16569 df-mre 16705 df-mrc 16706 df-acs 16708 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-mhm 17793 df-submnd 17794 df-grp 17884 df-minusg 17885 df-sbg 17886 df-mulg 18002 df-subg 18050 df-ghm 18117 df-cntz 18208 df-lsm 18512 df-cmn 18658 df-abl 18659 df-mgp 18953 df-ur 18965 df-ring 19012 df-subrg 19246 df-lmod 19348 df-lss 19416 df-lsp 19456 df-lmhm 19506 df-lmim 19507 df-lmic 19508 df-lbs 19559 df-sra 19656 df-rgmod 19657 df-nzr 19742 df-dsmm 20568 df-frlm 20583 df-uvc 20619 df-lfig 39009 df-lnm 39017 df-lnr 39051 |
This theorem is referenced by: lnrfgtr 39061 |
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