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Theorem islnm 27152
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 5728 . . . 4  |-  ( w  =  M  ->  ( LSubSp `
 w )  =  ( LSubSp `  M )
)
2 islnm.s . . . 4  |-  S  =  ( LSubSp `  M )
31, 2syl6eqr 2486 . . 3  |-  ( w  =  M  ->  ( LSubSp `
 w )  =  S )
4 oveq1 6088 . . . 4  |-  ( w  =  M  ->  (
ws  i )  =  ( Ms  i ) )
54eleq1d 2502 . . 3  |-  ( w  =  M  ->  (
( ws  i )  e. LFinGen  <->  ( Ms  i )  e. LFinGen )
)
63, 5raleqbidv 2916 . 2  |-  ( w  =  M  ->  ( A. i  e.  ( LSubSp `
 w ) ( ws  i )  e. LFinGen  <->  A. i  e.  S  ( Ms  i
)  e. LFinGen ) )
7 df-lnm 27151 . 2  |- LNoeM  =  {
w  e.  LMod  |  A. i  e.  ( LSubSp `  w ) ( ws  i )  e. LFinGen }
86, 7elrab2 3094 1  |-  ( M  e. LNoeM 
<->  ( M  e.  LMod  /\ 
A. i  e.  S  ( Ms  i )  e. LFinGen ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    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:  islnm2  27153  lnmlmod  27154  lnmlssfg  27155  lnmlsslnm  27156  lnmepi  27160  lmhmlnmsplit  27162
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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