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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pwslnm | Structured version Visualization version GIF version |
Description: Finite powers of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
pwslnm.y | ⊢ 𝑌 = (𝑊 ↑s 𝐼) |
Ref | Expression |
---|---|
pwslnm | ⊢ ((𝑊 ∈ LNoeM ∧ 𝐼 ∈ Fin) → 𝑌 ∈ LNoeM) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwslnm.y | . 2 ⊢ 𝑌 = (𝑊 ↑s 𝐼) | |
2 | oveq2 7370 | . . . . . 6 ⊢ (𝑎 = ∅ → (𝑊 ↑s 𝑎) = (𝑊 ↑s ∅)) | |
3 | 2 | eleq1d 2823 | . . . . 5 ⊢ (𝑎 = ∅ → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s ∅) ∈ LNoeM)) |
4 | 3 | imbi2d 341 | . . . 4 ⊢ (𝑎 = ∅ → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s ∅) ∈ LNoeM))) |
5 | oveq2 7370 | . . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑊 ↑s 𝑎) = (𝑊 ↑s 𝑏)) | |
6 | 5 | eleq1d 2823 | . . . . 5 ⊢ (𝑎 = 𝑏 → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s 𝑏) ∈ LNoeM)) |
7 | 6 | imbi2d 341 | . . . 4 ⊢ (𝑎 = 𝑏 → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s 𝑏) ∈ LNoeM))) |
8 | oveq2 7370 | . . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑊 ↑s 𝑎) = (𝑊 ↑s (𝑏 ∪ {𝑐}))) | |
9 | 8 | eleq1d 2823 | . . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM)) |
10 | 9 | imbi2d 341 | . . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM))) |
11 | oveq2 7370 | . . . . . 6 ⊢ (𝑎 = 𝐼 → (𝑊 ↑s 𝑎) = (𝑊 ↑s 𝐼)) | |
12 | 11 | eleq1d 2823 | . . . . 5 ⊢ (𝑎 = 𝐼 → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s 𝐼) ∈ LNoeM)) |
13 | 12 | imbi2d 341 | . . . 4 ⊢ (𝑎 = 𝐼 → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s 𝐼) ∈ LNoeM))) |
14 | lnmlmod 41435 | . . . . 5 ⊢ (𝑊 ∈ LNoeM → 𝑊 ∈ LMod) | |
15 | eqid 2737 | . . . . . 6 ⊢ (𝑊 ↑s ∅) = (𝑊 ↑s ∅) | |
16 | 15 | pwslnmlem0 41447 | . . . . 5 ⊢ (𝑊 ∈ LMod → (𝑊 ↑s ∅) ∈ LNoeM) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ (𝑊 ∈ LNoeM → (𝑊 ↑s ∅) ∈ LNoeM) |
18 | vex 3452 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
19 | vsnex 5391 | . . . . . . 7 ⊢ {𝑐} ∈ V | |
20 | eqid 2737 | . . . . . . 7 ⊢ (𝑊 ↑s 𝑏) = (𝑊 ↑s 𝑏) | |
21 | eqid 2737 | . . . . . . 7 ⊢ (𝑊 ↑s {𝑐}) = (𝑊 ↑s {𝑐}) | |
22 | eqid 2737 | . . . . . . 7 ⊢ (𝑊 ↑s (𝑏 ∪ {𝑐})) = (𝑊 ↑s (𝑏 ∪ {𝑐})) | |
23 | 14 | ad2antrl 727 | . . . . . . 7 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → 𝑊 ∈ LMod) |
24 | disjsn 4677 | . . . . . . . . 9 ⊢ ((𝑏 ∩ {𝑐}) = ∅ ↔ ¬ 𝑐 ∈ 𝑏) | |
25 | 24 | biimpri 227 | . . . . . . . 8 ⊢ (¬ 𝑐 ∈ 𝑏 → (𝑏 ∩ {𝑐}) = ∅) |
26 | 25 | ad2antlr 726 | . . . . . . 7 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → (𝑏 ∩ {𝑐}) = ∅) |
27 | simprr 772 | . . . . . . 7 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → (𝑊 ↑s 𝑏) ∈ LNoeM) | |
28 | 21 | pwslnmlem1 41448 | . . . . . . . 8 ⊢ (𝑊 ∈ LNoeM → (𝑊 ↑s {𝑐}) ∈ LNoeM) |
29 | 28 | ad2antrl 727 | . . . . . . 7 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → (𝑊 ↑s {𝑐}) ∈ LNoeM) |
30 | 18, 19, 20, 21, 22, 23, 26, 27, 29 | pwslnmlem2 41449 | . . . . . 6 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM) |
31 | 30 | exp32 422 | . . . . 5 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → (𝑊 ∈ LNoeM → ((𝑊 ↑s 𝑏) ∈ LNoeM → (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM))) |
32 | 31 | a2d 29 | . . . 4 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑏) ∈ LNoeM) → (𝑊 ∈ LNoeM → (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM))) |
33 | 4, 7, 10, 13, 17, 32 | findcard2s 9116 | . . 3 ⊢ (𝐼 ∈ Fin → (𝑊 ∈ LNoeM → (𝑊 ↑s 𝐼) ∈ LNoeM)) |
34 | 33 | impcom 409 | . 2 ⊢ ((𝑊 ∈ LNoeM ∧ 𝐼 ∈ Fin) → (𝑊 ↑s 𝐼) ∈ LNoeM) |
35 | 1, 34 | eqeltrid 2842 | 1 ⊢ ((𝑊 ∈ LNoeM ∧ 𝐼 ∈ Fin) → 𝑌 ∈ LNoeM) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∪ cun 3913 ∩ cin 3914 ∅c0 4287 {csn 4591 (class class class)co 7362 Fincfn 8890 ↑s cpws 17335 LModclmod 20338 LNoeMclnm 41431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-fz 13432 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-hom 17164 df-cco 17165 df-0g 17330 df-prds 17336 df-pws 17338 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-submnd 18609 df-grp 18758 df-minusg 18759 df-sbg 18760 df-subg 18932 df-ghm 19013 df-cntz 19104 df-lsm 19425 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-lmod 20340 df-lss 20409 df-lsp 20449 df-lmhm 20499 df-lmim 20500 df-lmic 20501 df-lfig 41424 df-lnm 41432 |
This theorem is referenced by: lnrfrlm 41474 |
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