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Theorem lnmlssfg 27178
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 27175 . . 3  |-  ( M  e. LNoeM 
<->  ( M  e.  LMod  /\ 
A. a  e.  S  ( Ms  a )  e. LFinGen ) )
32simprbi 450 . 2  |-  ( M  e. LNoeM  ->  A. a  e.  S  ( Ms  a )  e. LFinGen )
4 oveq2 5866 . . . . 5  |-  ( a  =  U  ->  ( Ms  a )  =  ( Ms  U ) )
5 lnmlssfg.r . . . . 5  |-  R  =  ( Ms  U )
64, 5syl6eqr 2333 . . . 4  |-  ( a  =  U  ->  ( Ms  a )  =  R )
76eleq1d 2349 . . 3  |-  ( a  =  U  ->  (
( Ms  a )  e. LFinGen  <->  R  e. LFinGen ) )
87rspcv 2880 . 2  |-  ( U  e.  S  ->  ( A. a  e.  S  ( Ms  a )  e. LFinGen  ->  R  e. LFinGen ) )
93, 8mpan9 455 1  |-  ( ( M  e. LNoeM  /\  U  e.  S )  ->  R  e. LFinGen )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ 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:  lnmlsslnm  27179  lnmfg  27180  lnmepi  27183  lmhmlnmsplit  27185  lnrfgtr  27324
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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