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Theorem lnmlssfg 27146
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 27143 . . 3  |-  ( M  e. LNoeM 
<->  ( M  e.  LMod  /\ 
A. a  e.  S  ( Ms  a )  e. LFinGen ) )
32simprbi 451 . 2  |-  ( M  e. LNoeM  ->  A. a  e.  S  ( Ms  a )  e. LFinGen )
4 oveq2 6081 . . . . 5  |-  ( a  =  U  ->  ( Ms  a )  =  ( Ms  U ) )
5 lnmlssfg.r . . . . 5  |-  R  =  ( Ms  U )
64, 5syl6eqr 2485 . . . 4  |-  ( a  =  U  ->  ( Ms  a )  =  R )
76eleq1d 2501 . . 3  |-  ( a  =  U  ->  (
( Ms  a )  e. LFinGen  <->  R  e. LFinGen ) )
87rspcv 3040 . 2  |-  ( U  e.  S  ->  ( A. a  e.  S  ( Ms  a )  e. LFinGen  ->  R  e. LFinGen ) )
93, 8mpan9 456 1  |-  ( ( M  e. LNoeM  /\  U  e.  S )  ->  R  e. LFinGen )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   ` cfv 5446  (class class class)co 6073   ↾s cress 13462   LModclmod 15942   LSubSpclss 16000  LFinGenclfig 27133  LNoeMclnm 27141
This theorem is referenced by:  lnmlsslnm  27147  lnmfg  27148  lnmepi  27151  lmhmlnmsplit  27153  lnrfgtr  27292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-lnm 27142
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