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Theorem islnm 29456
Description: Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
Hypothesis
Ref Expression
islnm.s  |-  S  =  ( LSubSp `  M )
Assertion
Ref Expression
islnm  |-  ( M  e. LNoeM 
<->  ( M  e.  LMod  /\ 
A. i  e.  S  ( Ms  i )  e. LFinGen ) )
Distinct variable groups:    i, M    S, i

Proof of Theorem islnm
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 fveq2 5712 . . . 4  |-  ( w  =  M  ->  ( LSubSp `
 w )  =  ( LSubSp `  M )
)
2 islnm.s . . . 4  |-  S  =  ( LSubSp `  M )
31, 2syl6eqr 2493 . . 3  |-  ( w  =  M  ->  ( LSubSp `
 w )  =  S )
4 oveq1 6119 . . . 4  |-  ( w  =  M  ->  (
ws  i )  =  ( Ms  i ) )
54eleq1d 2509 . . 3  |-  ( w  =  M  ->  (
( ws  i )  e. LFinGen  <->  ( Ms  i )  e. LFinGen )
)
63, 5raleqbidv 2952 . 2  |-  ( w  =  M  ->  ( A. i  e.  ( LSubSp `
 w ) ( ws  i )  e. LFinGen  <->  A. i  e.  S  ( Ms  i
)  e. LFinGen ) )
7 df-lnm 29455 . 2  |- LNoeM  =  {
w  e.  LMod  |  A. i  e.  ( LSubSp `  w ) ( ws  i )  e. LFinGen }
86, 7elrab2 3140 1  |-  ( M  e. LNoeM 
<->  ( M  e.  LMod  /\ 
A. i  e.  S  ( Ms  i )  e. LFinGen ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   ` cfv 5439  (class class class)co 6112   ↾s cress 14196   LModclmod 16970   LSubSpclss 17035  LFinGenclfig 29446  LNoeMclnm 29454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-iota 5402  df-fv 5447  df-ov 6115  df-lnm 29455
This theorem is referenced by:  islnm2  29457  lnmlmod  29458  lnmlssfg  29459  lnmlsslnm  29460  lnmepi  29464  lmhmlnmsplit  29466
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