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Theorem lnmlssfg 31188
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 31185 . . 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 6304 . . . . 5  |-  ( a  =  U  ->  ( Ms  a )  =  ( Ms  U ) )
5 lnmlssfg.r . . . . 5  |-  R  =  ( Ms  U )
64, 5syl6eqr 2516 . . . 4  |-  ( a  =  U  ->  ( Ms  a )  =  R )
76eleq1d 2526 . . 3  |-  ( a  =  U  ->  (
( Ms  a )  e. LFinGen  <->  R  e. LFinGen ) )
87rspcv 3206 . 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 1395    e. wcel 1819   A.wral 2807   ` cfv 5594  (class class class)co 6296   ↾s cress 14644   LModclmod 17638   LSubSpclss 17704  LFinGenclfig 31175  LNoeMclnm 31183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-lnm 31184
This theorem is referenced by:  lnmlsslnm  31189  lnmfg  31190  lnmepi  31193  lmhmlnmsplit  31195  lnrfgtr  31231
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