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Theorem islnm 27278
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 5541 . . . 4  |-  ( w  =  M  ->  ( LSubSp `
 w )  =  ( LSubSp `  M )
)
2 islnm.s . . . 4  |-  S  =  ( LSubSp `  M )
31, 2syl6eqr 2346 . . 3  |-  ( w  =  M  ->  ( LSubSp `
 w )  =  S )
4 oveq1 5881 . . . 4  |-  ( w  =  M  ->  (
ws  i )  =  ( Ms  i ) )
54eleq1d 2362 . . 3  |-  ( w  =  M  ->  (
( ws  i )  e. LFinGen  <->  ( Ms  i )  e. LFinGen )
)
63, 5raleqbidv 2761 . 2  |-  ( w  =  M  ->  ( A. i  e.  ( LSubSp `
 w ) ( ws  i )  e. LFinGen  <->  A. i  e.  S  ( Ms  i
)  e. LFinGen ) )
7 df-lnm 27277 . 2  |- LNoeM  =  {
w  e.  LMod  |  A. i  e.  ( LSubSp `  w ) ( ws  i )  e. LFinGen }
86, 7elrab2 2938 1  |-  ( M  e. LNoeM 
<->  ( M  e.  LMod  /\ 
A. i  e.  S  ( Ms  i )  e. LFinGen ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   ↾s cress 13165   LModclmod 15643   LSubSpclss 15705  LFinGenclfig 27268  LNoeMclnm 27276
This theorem is referenced by:  islnm2  27279  lnmlmod  27280  lnmlssfg  27281  lnmlsslnm  27282  lnmepi  27286  lmhmlnmsplit  27288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-lnm 27277
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