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Theorem lnmfg 40019
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
StepHypRef Expression
1 eqid 2801 . . 3 (Base‘𝑀) = (Base‘𝑀)
21ressid 16555 . 2 (𝑀 ∈ LNoeM → (𝑀s (Base‘𝑀)) = 𝑀)
3 lnmlmod 40016 . . . 4 (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)
4 eqid 2801 . . . . 5 (LSubSp‘𝑀) = (LSubSp‘𝑀)
51, 4lss1 19707 . . . 4 (𝑀 ∈ LMod → (Base‘𝑀) ∈ (LSubSp‘𝑀))
63, 5syl 17 . . 3 (𝑀 ∈ LNoeM → (Base‘𝑀) ∈ (LSubSp‘𝑀))
7 eqid 2801 . . . 4 (𝑀s (Base‘𝑀)) = (𝑀s (Base‘𝑀))
84, 7lnmlssfg 40017 . . 3 ((𝑀 ∈ LNoeM ∧ (Base‘𝑀) ∈ (LSubSp‘𝑀)) → (𝑀s (Base‘𝑀)) ∈ LFinGen)
96, 8mpdan 686 . 2 (𝑀 ∈ LNoeM → (𝑀s (Base‘𝑀)) ∈ LFinGen)
102, 9eqeltrrd 2894 1 (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112  cfv 6328  (class class class)co 7139  Basecbs 16479  s cress 16480  LModclmod 19631  LSubSpclss 19700  LFinGenclfig 40004  LNoeMclnm 40012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-ress 16487  df-0g 16711  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-grp 18102  df-lmod 19633  df-lss 19701  df-lnm 40013
This theorem is referenced by: (None)
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