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Theorem lnmlssfg 27157
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 27154 . . 3  |-  ( M  e. LNoeM 
<->  ( M  e.  LMod  /\ 
A. a  e.  S  ( Ms  a )  e. LFinGen ) )
32simprbi 452 . 2  |-  ( M  e. LNoeM  ->  A. a  e.  S  ( Ms  a )  e. LFinGen )
4 oveq2 6091 . . . . 5  |-  ( a  =  U  ->  ( Ms  a )  =  ( Ms  U ) )
5 lnmlssfg.r . . . . 5  |-  R  =  ( Ms  U )
64, 5syl6eqr 2488 . . . 4  |-  ( a  =  U  ->  ( Ms  a )  =  R )
76eleq1d 2504 . . 3  |-  ( a  =  U  ->  (
( Ms  a )  e. LFinGen  <->  R  e. LFinGen ) )
87rspcv 3050 . 2  |-  ( U  e.  S  ->  ( A. a  e.  S  ( Ms  a )  e. LFinGen  ->  R  e. LFinGen ) )
93, 8mpan9 457 1  |-  ( ( M  e. LNoeM  /\  U  e.  S )  ->  R  e. LFinGen )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   ` cfv 5456  (class class class)co 6083   ↾s cress 13472   LModclmod 15952   LSubSpclss 16010  LFinGenclfig 27144  LNoeMclnm 27152
This theorem is referenced by:  lnmlsslnm  27158  lnmfg  27159  lnmepi  27162  lmhmlnmsplit  27164  lnrfgtr  27303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-lnm 27153
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