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Theorem lnmlmod 27177
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 2283 . . 3  |-  ( LSubSp `  M )  =  (
LSubSp `  M )
21islnm 27175 . 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 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   ↾s cress 13149   LModclmod 15627   LSubSpclss 15689  LFinGenclfig 27165  LNoeMclnm 27173
This theorem is referenced by:  lnmlsslnm  27179  lnmfg  27180  pwslnmlem1  27194  pwslnm  27196
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-lnm 27174
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