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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 7432 | . . . . . 6 ⊢ (𝑎 = ∅ → (𝑊 ↑s 𝑎) = (𝑊 ↑s ∅)) | |
3 | 2 | eleq1d 2813 | . . . . 5 ⊢ (𝑎 = ∅ → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s ∅) ∈ LNoeM)) |
4 | 3 | imbi2d 339 | . . . 4 ⊢ (𝑎 = ∅ → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s ∅) ∈ LNoeM))) |
5 | oveq2 7432 | . . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑊 ↑s 𝑎) = (𝑊 ↑s 𝑏)) | |
6 | 5 | eleq1d 2813 | . . . . 5 ⊢ (𝑎 = 𝑏 → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s 𝑏) ∈ LNoeM)) |
7 | 6 | imbi2d 339 | . . . 4 ⊢ (𝑎 = 𝑏 → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s 𝑏) ∈ LNoeM))) |
8 | oveq2 7432 | . . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑊 ↑s 𝑎) = (𝑊 ↑s (𝑏 ∪ {𝑐}))) | |
9 | 8 | eleq1d 2813 | . . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM)) |
10 | 9 | imbi2d 339 | . . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM))) |
11 | oveq2 7432 | . . . . . 6 ⊢ (𝑎 = 𝐼 → (𝑊 ↑s 𝑎) = (𝑊 ↑s 𝐼)) | |
12 | 11 | eleq1d 2813 | . . . . 5 ⊢ (𝑎 = 𝐼 → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s 𝐼) ∈ LNoeM)) |
13 | 12 | imbi2d 339 | . . . 4 ⊢ (𝑎 = 𝐼 → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s 𝐼) ∈ LNoeM))) |
14 | lnmlmod 42506 | . . . . 5 ⊢ (𝑊 ∈ LNoeM → 𝑊 ∈ LMod) | |
15 | eqid 2727 | . . . . . 6 ⊢ (𝑊 ↑s ∅) = (𝑊 ↑s ∅) | |
16 | 15 | pwslnmlem0 42518 | . . . . 5 ⊢ (𝑊 ∈ LMod → (𝑊 ↑s ∅) ∈ LNoeM) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ (𝑊 ∈ LNoeM → (𝑊 ↑s ∅) ∈ LNoeM) |
18 | vex 3475 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
19 | vsnex 5433 | . . . . . . 7 ⊢ {𝑐} ∈ V | |
20 | eqid 2727 | . . . . . . 7 ⊢ (𝑊 ↑s 𝑏) = (𝑊 ↑s 𝑏) | |
21 | eqid 2727 | . . . . . . 7 ⊢ (𝑊 ↑s {𝑐}) = (𝑊 ↑s {𝑐}) | |
22 | eqid 2727 | . . . . . . 7 ⊢ (𝑊 ↑s (𝑏 ∪ {𝑐})) = (𝑊 ↑s (𝑏 ∪ {𝑐})) | |
23 | 14 | ad2antrl 726 | . . . . . . 7 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → 𝑊 ∈ LMod) |
24 | disjsn 4718 | . . . . . . . . 9 ⊢ ((𝑏 ∩ {𝑐}) = ∅ ↔ ¬ 𝑐 ∈ 𝑏) | |
25 | 24 | biimpri 227 | . . . . . . . 8 ⊢ (¬ 𝑐 ∈ 𝑏 → (𝑏 ∩ {𝑐}) = ∅) |
26 | 25 | ad2antlr 725 | . . . . . . 7 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → (𝑏 ∩ {𝑐}) = ∅) |
27 | simprr 771 | . . . . . . 7 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → (𝑊 ↑s 𝑏) ∈ LNoeM) | |
28 | 21 | pwslnmlem1 42519 | . . . . . . . 8 ⊢ (𝑊 ∈ LNoeM → (𝑊 ↑s {𝑐}) ∈ LNoeM) |
29 | 28 | ad2antrl 726 | . . . . . . 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 42520 | . . . . . 6 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM) |
31 | 30 | exp32 419 | . . . . 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 9194 | . . 3 ⊢ (𝐼 ∈ Fin → (𝑊 ∈ LNoeM → (𝑊 ↑s 𝐼) ∈ LNoeM)) |
34 | 33 | impcom 406 | . 2 ⊢ ((𝑊 ∈ LNoeM ∧ 𝐼 ∈ Fin) → (𝑊 ↑s 𝐼) ∈ LNoeM) |
35 | 1, 34 | eqeltrid 2832 | 1 ⊢ ((𝑊 ∈ LNoeM ∧ 𝐼 ∈ Fin) → 𝑌 ∈ LNoeM) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∪ cun 3945 ∩ cin 3946 ∅c0 4324 {csn 4630 (class class class)co 7424 Fincfn 8968 ↑s cpws 17433 LModclmod 20748 LNoeMclnm 42502 |
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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-map 8851 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-fz 13523 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-hom 17262 df-cco 17263 df-0g 17428 df-prds 17434 df-pws 17436 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-mhm 18745 df-submnd 18746 df-grp 18898 df-minusg 18899 df-sbg 18900 df-subg 19083 df-ghm 19173 df-cntz 19273 df-lsm 19596 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180 df-lmod 20750 df-lss 20821 df-lsp 20861 df-lmhm 20912 df-lmim 20913 df-lmic 20914 df-lfig 42495 df-lnm 42503 |
This theorem is referenced by: lnrfrlm 42545 |
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