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Theorem islnm 27175
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 5525 . . . 4  |-  ( w  =  M  ->  ( LSubSp `
 w )  =  ( LSubSp `  M )
)
2 islnm.s . . . 4  |-  S  =  ( LSubSp `  M )
31, 2syl6eqr 2333 . . 3  |-  ( w  =  M  ->  ( LSubSp `
 w )  =  S )
4 oveq1 5865 . . . 4  |-  ( w  =  M  ->  (
ws  i )  =  ( Ms  i ) )
54eleq1d 2349 . . 3  |-  ( w  =  M  ->  (
( ws  i )  e. LFinGen  <->  ( Ms  i )  e. LFinGen )
)
63, 5raleqbidv 2748 . 2  |-  ( w  =  M  ->  ( A. i  e.  ( LSubSp `
 w ) ( ws  i )  e. LFinGen  <->  A. i  e.  S  ( Ms  i
)  e. LFinGen ) )
7 df-lnm 27174 . 2  |- LNoeM  =  {
w  e.  LMod  |  A. i  e.  ( LSubSp `  w ) ( ws  i )  e. LFinGen }
86, 7elrab2 2925 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 1623    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:  islnm2  27176  lnmlmod  27177  lnmlssfg  27178  lnmlsslnm  27179  lnmepi  27183  lmhmlnmsplit  27185
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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