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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 2736 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | 1 | ressid 16743 | . 2 ⊢ (𝑀 ∈ LNoeM → (𝑀 ↾s (Base‘𝑀)) = 𝑀) |
3 | lnmlmod 40548 | . . . 4 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod) | |
4 | eqid 2736 | . . . . 5 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀) | |
5 | 1, 4 | lss1 19929 | . . . 4 ⊢ (𝑀 ∈ LMod → (Base‘𝑀) ∈ (LSubSp‘𝑀)) |
6 | 3, 5 | syl 17 | . . 3 ⊢ (𝑀 ∈ LNoeM → (Base‘𝑀) ∈ (LSubSp‘𝑀)) |
7 | eqid 2736 | . . . 4 ⊢ (𝑀 ↾s (Base‘𝑀)) = (𝑀 ↾s (Base‘𝑀)) | |
8 | 4, 7 | lnmlssfg 40549 | . . 3 ⊢ ((𝑀 ∈ LNoeM ∧ (Base‘𝑀) ∈ (LSubSp‘𝑀)) → (𝑀 ↾s (Base‘𝑀)) ∈ LFinGen) |
9 | 6, 8 | mpdan 687 | . 2 ⊢ (𝑀 ∈ LNoeM → (𝑀 ↾s (Base‘𝑀)) ∈ LFinGen) |
10 | 2, 9 | eqeltrrd 2832 | 1 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 ↾s cress 16667 LModclmod 19853 LSubSpclss 19922 LFinGenclfig 40536 LNoeMclnm 40544 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-ress 16674 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-lmod 19855 df-lss 19923 df-lnm 40545 |
This theorem is referenced by: (None) |
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