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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 7164 | . . . . . 6 ⊢ (𝑎 = ∅ → (𝑊 ↑s 𝑎) = (𝑊 ↑s ∅)) | |
3 | 2 | eleq1d 2897 | . . . . 5 ⊢ (𝑎 = ∅ → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s ∅) ∈ LNoeM)) |
4 | 3 | imbi2d 343 | . . . 4 ⊢ (𝑎 = ∅ → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s ∅) ∈ LNoeM))) |
5 | oveq2 7164 | . . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑊 ↑s 𝑎) = (𝑊 ↑s 𝑏)) | |
6 | 5 | eleq1d 2897 | . . . . 5 ⊢ (𝑎 = 𝑏 → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s 𝑏) ∈ LNoeM)) |
7 | 6 | imbi2d 343 | . . . 4 ⊢ (𝑎 = 𝑏 → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s 𝑏) ∈ LNoeM))) |
8 | oveq2 7164 | . . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑊 ↑s 𝑎) = (𝑊 ↑s (𝑏 ∪ {𝑐}))) | |
9 | 8 | eleq1d 2897 | . . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM)) |
10 | 9 | imbi2d 343 | . . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM))) |
11 | oveq2 7164 | . . . . . 6 ⊢ (𝑎 = 𝐼 → (𝑊 ↑s 𝑎) = (𝑊 ↑s 𝐼)) | |
12 | 11 | eleq1d 2897 | . . . . 5 ⊢ (𝑎 = 𝐼 → ((𝑊 ↑s 𝑎) ∈ LNoeM ↔ (𝑊 ↑s 𝐼) ∈ LNoeM)) |
13 | 12 | imbi2d 343 | . . . 4 ⊢ (𝑎 = 𝐼 → ((𝑊 ∈ LNoeM → (𝑊 ↑s 𝑎) ∈ LNoeM) ↔ (𝑊 ∈ LNoeM → (𝑊 ↑s 𝐼) ∈ LNoeM))) |
14 | lnmlmod 39699 | . . . . 5 ⊢ (𝑊 ∈ LNoeM → 𝑊 ∈ LMod) | |
15 | eqid 2821 | . . . . . 6 ⊢ (𝑊 ↑s ∅) = (𝑊 ↑s ∅) | |
16 | 15 | pwslnmlem0 39711 | . . . . 5 ⊢ (𝑊 ∈ LMod → (𝑊 ↑s ∅) ∈ LNoeM) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ (𝑊 ∈ LNoeM → (𝑊 ↑s ∅) ∈ LNoeM) |
18 | vex 3497 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
19 | snex 5332 | . . . . . . 7 ⊢ {𝑐} ∈ V | |
20 | eqid 2821 | . . . . . . 7 ⊢ (𝑊 ↑s 𝑏) = (𝑊 ↑s 𝑏) | |
21 | eqid 2821 | . . . . . . 7 ⊢ (𝑊 ↑s {𝑐}) = (𝑊 ↑s {𝑐}) | |
22 | eqid 2821 | . . . . . . 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 4647 | . . . . . . . . 9 ⊢ ((𝑏 ∩ {𝑐}) = ∅ ↔ ¬ 𝑐 ∈ 𝑏) | |
25 | 24 | biimpri 230 | . . . . . . . 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 39712 | . . . . . . . 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 39713 | . . . . . 6 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑊 ∈ LNoeM ∧ (𝑊 ↑s 𝑏) ∈ LNoeM)) → (𝑊 ↑s (𝑏 ∪ {𝑐})) ∈ LNoeM) |
31 | 30 | exp32 423 | . . . . 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 8759 | . . 3 ⊢ (𝐼 ∈ Fin → (𝑊 ∈ LNoeM → (𝑊 ↑s 𝐼) ∈ LNoeM)) |
34 | 33 | impcom 410 | . 2 ⊢ ((𝑊 ∈ LNoeM ∧ 𝐼 ∈ Fin) → (𝑊 ↑s 𝐼) ∈ LNoeM) |
35 | 1, 34 | eqeltrid 2917 | 1 ⊢ ((𝑊 ∈ LNoeM ∧ 𝐼 ∈ Fin) → 𝑌 ∈ LNoeM) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 ∩ cin 3935 ∅c0 4291 {csn 4567 (class class class)co 7156 Fincfn 8509 ↑s cpws 16720 LModclmod 19634 LNoeMclnm 39695 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-0g 16715 df-prds 16721 df-pws 16723 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-ghm 18356 df-cntz 18447 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lmhm 19794 df-lmim 19795 df-lmic 19796 df-lfig 39688 df-lnm 39696 |
This theorem is referenced by: lnrfrlm 39738 |
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