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Theorem islnr 27315
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 5525 . . 3  |-  ( a  =  A  ->  (ringLMod `  a )  =  (ringLMod `  A ) )
21eleq1d 2349 . 2  |-  ( a  =  A  ->  (
(ringLMod `  a )  e. LNoeM  <->  (ringLMod `  A )  e. LNoeM )
)
3 df-lnr 27314 . 2  |- LNoeR  =  {
a  e.  Ring  |  (ringLMod `  a )  e. LNoeM }
42, 3elrab2 2925 1  |-  ( A  e. LNoeR 
<->  ( A  e.  Ring  /\  (ringLMod `  A )  e. LNoeM ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255   Ringcrg 15337  ringLModcrglmod 15922  LNoeMclnm 27173  LNoeRclnr 27313
This theorem is referenced by:  lnrrng  27316  lnrlnm  27317  islnr2  27318
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-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-lnr 27314
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