Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lnmlsslnm | Structured version Visualization version GIF version |
Description: All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
Ref | Expression |
---|---|
lnmlssfg.s | ⊢ 𝑆 = (LSubSp‘𝑀) |
lnmlssfg.r | ⊢ 𝑅 = (𝑀 ↾s 𝑈) |
Ref | Expression |
---|---|
lnmlsslnm | ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LNoeM) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lnmlmod 39685 | . . 3 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod) | |
2 | lnmlssfg.r | . . . 4 ⊢ 𝑅 = (𝑀 ↾s 𝑈) | |
3 | lnmlssfg.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑀) | |
4 | 2, 3 | lsslmod 19735 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LMod) |
5 | 1, 4 | sylan 582 | . 2 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LMod) |
6 | 2 | oveq1i 7169 | . . . . 5 ⊢ (𝑅 ↾s 𝑎) = ((𝑀 ↾s 𝑈) ↾s 𝑎) |
7 | simplr 767 | . . . . . 6 ⊢ (((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) ∧ 𝑎 ∈ (LSubSp‘𝑅)) → 𝑈 ∈ 𝑆) | |
8 | eqid 2824 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | eqid 2824 | . . . . . . . . 9 ⊢ (LSubSp‘𝑅) = (LSubSp‘𝑅) | |
10 | 8, 9 | lssss 19711 | . . . . . . . 8 ⊢ (𝑎 ∈ (LSubSp‘𝑅) → 𝑎 ⊆ (Base‘𝑅)) |
11 | 10 | adantl 484 | . . . . . . 7 ⊢ (((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) ∧ 𝑎 ∈ (LSubSp‘𝑅)) → 𝑎 ⊆ (Base‘𝑅)) |
12 | eqid 2824 | . . . . . . . . . 10 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
13 | 12, 3 | lssss 19711 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑀)) |
14 | 2, 12 | ressbas2 16558 | . . . . . . . . 9 ⊢ (𝑈 ⊆ (Base‘𝑀) → 𝑈 = (Base‘𝑅)) |
15 | 13, 14 | syl 17 | . . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 = (Base‘𝑅)) |
16 | 15 | ad2antlr 725 | . . . . . . 7 ⊢ (((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) ∧ 𝑎 ∈ (LSubSp‘𝑅)) → 𝑈 = (Base‘𝑅)) |
17 | 11, 16 | sseqtrrd 4011 | . . . . . 6 ⊢ (((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) ∧ 𝑎 ∈ (LSubSp‘𝑅)) → 𝑎 ⊆ 𝑈) |
18 | ressabs 16566 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑎 ⊆ 𝑈) → ((𝑀 ↾s 𝑈) ↾s 𝑎) = (𝑀 ↾s 𝑎)) | |
19 | 7, 17, 18 | syl2anc 586 | . . . . 5 ⊢ (((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) ∧ 𝑎 ∈ (LSubSp‘𝑅)) → ((𝑀 ↾s 𝑈) ↾s 𝑎) = (𝑀 ↾s 𝑎)) |
20 | 6, 19 | syl5eq 2871 | . . . 4 ⊢ (((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) ∧ 𝑎 ∈ (LSubSp‘𝑅)) → (𝑅 ↾s 𝑎) = (𝑀 ↾s 𝑎)) |
21 | simpll 765 | . . . . 5 ⊢ (((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) ∧ 𝑎 ∈ (LSubSp‘𝑅)) → 𝑀 ∈ LNoeM) | |
22 | 2, 3, 9 | lsslss 19736 | . . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑎 ∈ (LSubSp‘𝑅) ↔ (𝑎 ∈ 𝑆 ∧ 𝑎 ⊆ 𝑈))) |
23 | 1, 22 | sylan 582 | . . . . . 6 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → (𝑎 ∈ (LSubSp‘𝑅) ↔ (𝑎 ∈ 𝑆 ∧ 𝑎 ⊆ 𝑈))) |
24 | 23 | simprbda 501 | . . . . 5 ⊢ (((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) ∧ 𝑎 ∈ (LSubSp‘𝑅)) → 𝑎 ∈ 𝑆) |
25 | eqid 2824 | . . . . . 6 ⊢ (𝑀 ↾s 𝑎) = (𝑀 ↾s 𝑎) | |
26 | 3, 25 | lnmlssfg 39686 | . . . . 5 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑎 ∈ 𝑆) → (𝑀 ↾s 𝑎) ∈ LFinGen) |
27 | 21, 24, 26 | syl2anc 586 | . . . 4 ⊢ (((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) ∧ 𝑎 ∈ (LSubSp‘𝑅)) → (𝑀 ↾s 𝑎) ∈ LFinGen) |
28 | 20, 27 | eqeltrd 2916 | . . 3 ⊢ (((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) ∧ 𝑎 ∈ (LSubSp‘𝑅)) → (𝑅 ↾s 𝑎) ∈ LFinGen) |
29 | 28 | ralrimiva 3185 | . 2 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → ∀𝑎 ∈ (LSubSp‘𝑅)(𝑅 ↾s 𝑎) ∈ LFinGen) |
30 | 9 | islnm 39683 | . 2 ⊢ (𝑅 ∈ LNoeM ↔ (𝑅 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑅)(𝑅 ↾s 𝑎) ∈ LFinGen)) |
31 | 5, 29, 30 | sylanbrc 585 | 1 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LNoeM) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ⊆ wss 3939 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 ↾s cress 16487 LModclmod 19637 LSubSpclss 19706 LFinGenclfig 39673 LNoeMclnm 39681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-sca 16584 df-vsca 16585 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-mgp 19243 df-ur 19255 df-ring 19302 df-lmod 19639 df-lss 19707 df-lnm 39682 |
This theorem is referenced by: (None) |
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