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Theorem islnm 30957
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 5872 . . . 4  |-  ( w  =  M  ->  ( LSubSp `
 w )  =  ( LSubSp `  M )
)
2 islnm.s . . . 4  |-  S  =  ( LSubSp `  M )
31, 2syl6eqr 2526 . . 3  |-  ( w  =  M  ->  ( LSubSp `
 w )  =  S )
4 oveq1 6302 . . . 4  |-  ( w  =  M  ->  (
ws  i )  =  ( Ms  i ) )
54eleq1d 2536 . . 3  |-  ( w  =  M  ->  (
( ws  i )  e. LFinGen  <->  ( Ms  i )  e. LFinGen )
)
63, 5raleqbidv 3077 . 2  |-  ( w  =  M  ->  ( A. i  e.  ( LSubSp `
 w ) ( ws  i )  e. LFinGen  <->  A. i  e.  S  ( Ms  i
)  e. LFinGen ) )
7 df-lnm 30956 . 2  |- LNoeM  =  {
w  e.  LMod  |  A. i  e.  ( LSubSp `  w ) ( ws  i )  e. LFinGen }
86, 7elrab2 3268 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 1379    e. wcel 1767   A.wral 2817   ` cfv 5594  (class class class)co 6295   ↾s cress 14507   LModclmod 17381   LSubSpclss 17447  LFinGenclfig 30947  LNoeMclnm 30955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298  df-lnm 30956
This theorem is referenced by:  islnm2  30958  lnmlmod  30959  lnmlssfg  30960  lnmlsslnm  30961  lnmepi  30965  lmhmlnmsplit  30967
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