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Theorem islnr 27283
Description: Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
islnr  |-  ( A  e. LNoeR 
<->  ( A  e.  Ring  /\  (ringLMod `  A )  e. LNoeM ) )

Proof of Theorem islnr
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . 3  |-  ( a  =  A  ->  (ringLMod `  a )  =  (ringLMod `  A ) )
21eleq1d 2501 . 2  |-  ( a  =  A  ->  (
(ringLMod `  a )  e. LNoeM  <->  (ringLMod `  A )  e. LNoeM )
)
3 df-lnr 27282 . 2  |- LNoeR  =  {
a  e.  Ring  |  (ringLMod `  a )  e. LNoeM }
42, 3elrab2 3086 1  |-  ( A  e. LNoeR 
<->  ( A  e.  Ring  /\  (ringLMod `  A )  e. LNoeM ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5446   Ringcrg 15652  ringLModcrglmod 16233  LNoeMclnm 27141  LNoeRclnr 27281
This theorem is referenced by:  lnrrng  27284  lnrlnm  27285  islnr2  27286
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-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-lnr 27282
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