lnmfg - Mathbox for Stefan O'Rear

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Theorem lnmfg 39675

Description: A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)

**Assertion**
Ref	Expression
lnmfg	⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen)

**Proof of Theorem lnmfg**
Step	Hyp	Ref	Expression
1		eqid 2821	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2	1	ressid 16553	. 2 ⊢ (𝑀 ∈ LNoeM → (𝑀 ↾_s (Base‘𝑀)) = 𝑀)
3		lnmlmod 39672	. . . 4 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)
4		eqid 2821	. . . . 5 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀)
5	1, 4	lss1 19704	. . . 4 ⊢ (𝑀 ∈ LMod → (Base‘𝑀) ∈ (LSubSp‘𝑀))
6	3, 5	syl 17	. . 3 ⊢ (𝑀 ∈ LNoeM → (Base‘𝑀) ∈ (LSubSp‘𝑀))
7		eqid 2821	. . . 4 ⊢ (𝑀 ↾_s (Base‘𝑀)) = (𝑀 ↾_s (Base‘𝑀))
8	4, 7	lnmlssfg 39673	. . 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 2914	1 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 ↾_s cress 16478 LModclmod 19628 LSubSpclss 19697 LFinGenclfig 39660 LNoeMclnm 39668

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-ress 16485 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-lmod 19630 df-lss 19698 df-lnm 39669

This theorem is referenced by: (None)

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