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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 7221 | . . . . . 6 ⊢ (𝑎 = ∅ → (𝑊 ↑s 𝑎) = (𝑊 ↑s ∅)) | |
3 | 2 | eleq1d 2822 | . . . . 5 ⊢ (𝑎 = ∅ → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s ∅) ∈ LNoeM)) |
4 | 3 | imbi2d 344 | . . . 4 ⊢ (𝑎 = ∅ → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s ∅) ∈ LNoeM))) |
5 | oveq2 7221 | . . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑊 ↑s 𝑎) = (𝑊 ↑s 𝑏)) | |
6 | 5 | eleq1d 2822 | . . . . 5 ⊢ (𝑎 = 𝑏 → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s 𝑏) ∈ LNoeM)) |
7 | 6 | imbi2d 344 | . . . 4 ⊢ (𝑎 = 𝑏 → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s 𝑏) ∈ LNoeM))) |
8 | oveq2 7221 | . . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑊 ↑s 𝑎) = (𝑊 ↑s (𝑏 ∪ {𝑐}))) | |
9 | 8 | eleq1d 2822 | . . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM)) |
10 | 9 | imbi2d 344 | . . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM))) |
11 | oveq2 7221 | . . . . . 6 ⊢ (𝑎 = 𝐼 → (𝑊 ↑s 𝑎) = (𝑊 ↑s 𝐼)) | |
12 | 11 | eleq1d 2822 | . . . . 5 ⊢ (𝑎 = 𝐼 → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s 𝐼) ∈ LNoeM)) |
13 | 12 | imbi2d 344 | . . . 4 ⊢ (𝑎 = 𝐼 → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s 𝐼) ∈ LNoeM))) |
14 | lnmlmod 40607 | . . . . 5 ⊢ (𝑊 ∈ LNoeM → 𝑊 ∈ LMod) | |
15 | eqid 2737 | . . . . . 6 ⊢ (𝑊 ↑s ∅) = (𝑊 ↑s ∅) | |
16 | 15 | pwslnmlem0 40619 | . . . . 5 ⊢ (𝑊 ∈ LMod → (𝑊 ↑s ∅) ∈ LNoeM) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ (𝑊 ∈ LNoeM → (𝑊 ↑s ∅) ∈ LNoeM) |
18 | vex 3412 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
19 | snex 5324 | . . . . . . 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 728 | . . . . . . 7 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → 𝑊 ∈ LMod) |
24 | disjsn 4627 | . . . . . . . . 9 ⊢ ((𝑏 ∩ {𝑐}) = ∅ ↔ ¬ 𝑐 ∈ 𝑏) | |
25 | 24 | biimpri 231 | . . . . . . . 8 ⊢ (¬ 𝑐 ∈ 𝑏 → (𝑏 ∩ {𝑐}) = ∅) |
26 | 25 | ad2antlr 727 | . . . . . . 7 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → (𝑏 ∩ {𝑐}) = ∅) |
27 | simprr 773 | . . . . . . 7 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → (𝑊 ↑s 𝑏) ∈ LNoeM) | |
28 | 21 | pwslnmlem1 40620 | . . . . . . . 8 ⊢ (𝑊 ∈ LNoeM → (𝑊 ↑s {𝑐}) ∈ LNoeM) |
29 | 28 | ad2antrl 728 | . . . . . . 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 40621 | . . . . . 6 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM) |
31 | 30 | exp32 424 | . . . . 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 8843 | . . 3 ⊢ (𝐼 ∈ Fin → (𝑊 ∈ LNoeM → (𝑊 ↑s 𝐼) ∈ LNoeM)) |
34 | 33 | impcom 411 | . 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 399 = wceq 1543 ∈ wcel 2110 ∪ cun 3864 ∩ cin 3865 ∅c0 4237 {csn 4541 (class class class)co 7213 Fincfn 8626 ↑s cpws 16951 LModclmod 19899 LNoeMclnm 40603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-hom 16826 df-cco 16827 df-0g 16946 df-prds 16952 df-pws 16954 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-ghm 18620 df-cntz 18711 df-lsm 19025 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-lmod 19901 df-lss 19969 df-lsp 20009 df-lmhm 20059 df-lmim 20060 df-lmic 20061 df-lfig 40596 df-lnm 40604 |
This theorem is referenced by: lnrfrlm 40646 |
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