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Theorem lnmlssfg 29430
Description: A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
lnmlssfg.s  |-  S  =  ( LSubSp `  M )
lnmlssfg.r  |-  R  =  ( Ms  U )
Assertion
Ref Expression
lnmlssfg  |-  ( ( M  e. LNoeM  /\  U  e.  S )  ->  R  e. LFinGen )

Proof of Theorem lnmlssfg
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 lnmlssfg.s . . . 4  |-  S  =  ( LSubSp `  M )
21islnm 29427 . . 3  |-  ( M  e. LNoeM 
<->  ( M  e.  LMod  /\ 
A. a  e.  S  ( Ms  a )  e. LFinGen ) )
32simprbi 464 . 2  |-  ( M  e. LNoeM  ->  A. a  e.  S  ( Ms  a )  e. LFinGen )
4 oveq2 6097 . . . . 5  |-  ( a  =  U  ->  ( Ms  a )  =  ( Ms  U ) )
5 lnmlssfg.r . . . . 5  |-  R  =  ( Ms  U )
64, 5syl6eqr 2491 . . . 4  |-  ( a  =  U  ->  ( Ms  a )  =  R )
76eleq1d 2507 . . 3  |-  ( a  =  U  ->  (
( Ms  a )  e. LFinGen  <->  R  e. LFinGen ) )
87rspcv 3067 . 2  |-  ( U  e.  S  ->  ( A. a  e.  S  ( Ms  a )  e. LFinGen  ->  R  e. LFinGen ) )
93, 8mpan9 469 1  |-  ( ( M  e. LNoeM  /\  U  e.  S )  ->  R  e. LFinGen )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2713   ` cfv 5416  (class class class)co 6089   ↾s cress 14173   LModclmod 16946   LSubSpclss 17011  LFinGenclfig 29417  LNoeMclnm 29425
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 2422
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-iota 5379  df-fv 5424  df-ov 6092  df-lnm 29426
This theorem is referenced by:  lnmlsslnm  29431  lnmfg  29432  lnmepi  29435  lmhmlnmsplit  29437  lnrfgtr  29473
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