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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 2736 | . . . 4 ⊢ (𝑆 freeLMod 𝑎) = (𝑆 freeLMod 𝑎) | |
2 | eqid 2736 | . . . 4 ⊢ (Base‘(𝑆 freeLMod 𝑎)) = (Base‘(𝑆 freeLMod 𝑎)) | |
3 | eqid 2736 | . . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
4 | eqid 2736 | . . . 4 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
5 | eqid 2736 | . . . 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 41149 | . . . . 5 ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod) | |
7 | 6 | ad3antrrr 727 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑀 ∈ LMod) |
8 | vex 3445 | . . . . 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 6799 | . . . . . . 7 ⊢ ( I ↾ 𝑎):𝑎–1-1-onto→𝑎 | |
13 | f1of 6761 | . . . . . . 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 4553 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 (Base‘𝑀) → 𝑎 ⊆ (Base‘𝑀)) | |
16 | fss 6662 | . . . . . 6 ⊢ ((( I ↾ 𝑎):𝑎⟶𝑎 ∧ 𝑎 ⊆ (Base‘𝑀)) → ( I ↾ 𝑎):𝑎⟶(Base‘𝑀)) | |
17 | 14, 15, 16 | sylancr 587 | . . . . 5 ⊢ (𝑎 ∈ 𝒫 (Base‘𝑀) → ( I ↾ 𝑎):𝑎⟶(Base‘𝑀)) |
18 | 17 | ad2antlr 724 | . . . 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 21103 | . . 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 773 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑆 ∈ LNoeR) | |
21 | simprl 768 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑎 ∈ Fin) | |
22 | 1 | lnrfrlm 41194 | . . . 4 ⊢ ((𝑆 ∈ LNoeR ∧ 𝑎 ∈ Fin) → (𝑆 freeLMod 𝑎) ∈ LNoeM) |
23 | 20, 21, 22 | syl2anc 584 | . . 3 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → (𝑆 freeLMod 𝑎) ∈ LNoeM) |
24 | eqid 2736 | . . . . 5 ⊢ (LSpan‘𝑀) = (LSpan‘𝑀) | |
25 | 1, 2, 3, 4, 5, 7, 9, 11, 18, 24 | frlmup3 21105 | . . . 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 6007 | . . . . . 6 ⊢ ran ( I ↾ 𝑎) = 𝑎 | |
27 | 26 | fveq2i 6822 | . . . . 5 ⊢ ((LSpan‘𝑀)‘ran ( I ↾ 𝑎)) = ((LSpan‘𝑀)‘𝑎) |
28 | simprr 770 | . . . . 5 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀)) | |
29 | 27, 28 | eqtrid 2788 | . . . 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 2776 | . . 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 41161 | . . 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 1370 | . 2 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑀 ∈ LNoeM) |
33 | 3, 24 | islmodfg 41145 | . . . . 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 266 | . . 3 ⊢ (𝑀 ∈ LFinGen → ∃𝑎 ∈ 𝒫 (Base‘𝑀)(𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) |
36 | 35 | adantr 481 | . 2 ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) → ∃𝑎 ∈ 𝒫 (Base‘𝑀)(𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) |
37 | 32, 36 | r19.29a 3155 | 1 ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) → 𝑀 ∈ LNoeM) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∃wrex 3070 Vcvv 3441 ⊆ wss 3897 𝒫 cpw 4546 ↦ cmpt 5172 I cid 5511 ran crn 5615 ↾ cres 5616 ⟶wf 6469 –1-1-onto→wf1o 6472 ‘cfv 6473 (class class class)co 7329 ∘f cof 7585 Fincfn 8796 Basecbs 17001 Scalarcsca 17054 ·𝑠 cvsca 17055 Σg cgsu 17240 LModclmod 20221 LSpanclspn 20331 LMHom clmhm 20379 freeLMod cfrlm 21051 LFinGenclfig 41143 LNoeMclnm 41151 LNoeRclnr 41185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-om 7773 df-1st 7891 df-2nd 7892 df-supp 8040 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-ixp 8749 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-fsupp 9219 df-sup 9291 df-oi 9359 df-card 9788 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-fz 13333 df-fzo 13476 df-seq 13815 df-hash 14138 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-sca 17067 df-vsca 17068 df-ip 17069 df-tset 17070 df-ple 17071 df-ds 17073 df-hom 17075 df-cco 17076 df-0g 17241 df-gsum 17242 df-prds 17247 df-pws 17249 df-mre 17384 df-mrc 17385 df-acs 17387 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-mhm 18519 df-submnd 18520 df-grp 18668 df-minusg 18669 df-sbg 18670 df-mulg 18789 df-subg 18840 df-ghm 18920 df-cntz 19011 df-lsm 19329 df-cmn 19475 df-abl 19476 df-mgp 19808 df-ur 19825 df-ring 19872 df-subrg 20119 df-lmod 20223 df-lss 20292 df-lsp 20332 df-lmhm 20382 df-lmim 20383 df-lmic 20384 df-lbs 20435 df-sra 20532 df-rgmod 20533 df-nzr 20627 df-dsmm 21037 df-frlm 21052 df-uvc 21088 df-lfig 41144 df-lnm 41152 df-lnr 41186 |
This theorem is referenced by: lnrfgtr 41196 |
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