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Theorem lnmlmod 26500
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 2358 . . 3  |-  ( LSubSp `  M )  =  (
LSubSp `  M )
21islnm 26498 . 2  |-  ( M  e. LNoeM 
<->  ( M  e.  LMod  /\ 
A. a  e.  (
LSubSp `  M ) ( Ms  a )  e. LFinGen )
)
32simplbi 446 1  |-  ( M  e. LNoeM  ->  M  e.  LMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1710   A.wral 2619   ` cfv 5337  (class class class)co 5945   ↾s cress 13246   LModclmod 15726   LSubSpclss 15788  LFinGenclfig 26488  LNoeMclnm 26496
This theorem is referenced by:  lnmlsslnm  26502  lnmfg  26503  pwslnmlem1  26517  pwslnm  26519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-iota 5301  df-fv 5345  df-ov 5948  df-lnm 26497
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