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Theorem lnmlmod 27154
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 2436 . . 3  |-  ( LSubSp `  M )  =  (
LSubSp `  M )
21islnm 27152 . 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 1725   A.wral 2705   ` cfv 5454  (class class class)co 6081   ↾s cress 13470   LModclmod 15950   LSubSpclss 16008  LFinGenclfig 27142  LNoeMclnm 27150
This theorem is referenced by:  lnmlsslnm  27156  lnmfg  27157  pwslnmlem1  27171  pwslnm  27173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-lnm 27151
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