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Theorem lnmlmod 27045
Description: A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
Assertion
Ref Expression
lnmlmod  |-  ( M  e. LNoeM  ->  M  e.  LMod )

Proof of Theorem lnmlmod
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . 3  |-  ( LSubSp `  M )  =  (
LSubSp `  M )
21islnm 27043 . 2  |-  ( M  e. LNoeM 
<->  ( M  e.  LMod  /\ 
A. a  e.  (
LSubSp `  M ) ( Ms  a )  e. LFinGen )
)
32simplbi 447 1  |-  ( M  e. LNoeM  ->  M  e.  LMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   A.wral 2666   ` cfv 5413  (class class class)co 6040   ↾s cress 13425   LModclmod 15905   LSubSpclss 15963  LFinGenclfig 27033  LNoeMclnm 27041
This theorem is referenced by:  lnmlsslnm  27047  lnmfg  27048  pwslnmlem1  27062  pwslnm  27064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043  df-lnm 27042
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