lnmfg - Mathbox for Stefan O'Rear

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Theorem lnmfg 39075

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 2779	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2	1	ressid 16415	. 2 ⊢ (𝑀 ∈ LNoeM → (𝑀 ↾_s (Base‘𝑀)) = 𝑀)
3		lnmlmod 39072	. . . 4 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)
4		eqid 2779	. . . . 5 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀)
5	1, 4	lss1 19432	. . . 4 ⊢ (𝑀 ∈ LMod → (Base‘𝑀) ∈ (LSubSp‘𝑀))
6	3, 5	syl 17	. . 3 ⊢ (𝑀 ∈ LNoeM → (Base‘𝑀) ∈ (LSubSp‘𝑀))
7		eqid 2779	. . . 4 ⊢ (𝑀 ↾_s (Base‘𝑀)) = (𝑀 ↾_s (Base‘𝑀))
8	4, 7	lnmlssfg 39073	. . 3 ⊢ ((𝑀 ∈ LNoeM ∧ (Base‘𝑀) ∈ (LSubSp‘𝑀)) → (𝑀 ↾_s (Base‘𝑀)) ∈ LFinGen)
9	6, 8	mpdan 674	. 2 ⊢ (𝑀 ∈ LNoeM → (𝑀 ↾_s (Base‘𝑀)) ∈ LFinGen)
10	2, 9	eqeltrrd 2868	1 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2050 ‘cfv 6188 (class class class)co 6976 Basecbs 16339 ↾_s cress 16340 LModclmod 19356 LSubSpclss 19425 LFinGenclfig 39060 LNoeMclnm 39068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-iota 6152 df-fun 6190 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-ress 16347 df-0g 16571 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-grp 17894 df-lmod 19358 df-lss 19426 df-lnm 39069

This theorem is referenced by: (None)

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