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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lnmfg | Structured version Visualization version GIF version |
Description: A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
Ref | Expression |
---|---|
lnmfg | ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | 1 | ressid 17251 | . 2 ⊢ (𝑀 ∈ LNoeM → (𝑀 ↾s (Base‘𝑀)) = 𝑀) |
3 | lnmlmod 42775 | . . . 4 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod) | |
4 | eqid 2726 | . . . . 5 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀) | |
5 | 1, 4 | lss1 20909 | . . . 4 ⊢ (𝑀 ∈ LMod → (Base‘𝑀) ∈ (LSubSp‘𝑀)) |
6 | 3, 5 | syl 17 | . . 3 ⊢ (𝑀 ∈ LNoeM → (Base‘𝑀) ∈ (LSubSp‘𝑀)) |
7 | eqid 2726 | . . . 4 ⊢ (𝑀 ↾s (Base‘𝑀)) = (𝑀 ↾s (Base‘𝑀)) | |
8 | 4, 7 | lnmlssfg 42776 | . . 3 ⊢ ((𝑀 ∈ LNoeM ∧ (Base‘𝑀) ∈ (LSubSp‘𝑀)) → (𝑀 ↾s (Base‘𝑀)) ∈ LFinGen) |
9 | 6, 8 | mpdan 685 | . 2 ⊢ (𝑀 ∈ LNoeM → (𝑀 ↾s (Base‘𝑀)) ∈ LFinGen) |
10 | 2, 9 | eqeltrrd 2827 | 1 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ‘cfv 6544 (class class class)co 7414 Basecbs 17206 ↾s cress 17235 LModclmod 20830 LSubSpclss 20902 LFinGenclfig 42763 LNoeMclnm 42771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-iota 6496 df-fun 6546 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-ress 17236 df-0g 17449 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-grp 18924 df-lmod 20832 df-lss 20903 df-lnm 42772 |
This theorem is referenced by: (None) |
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