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Theorem islnr 26985
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 5669 . . 3  |-  ( a  =  A  ->  (ringLMod `  a )  =  (ringLMod `  A ) )
21eleq1d 2454 . 2  |-  ( a  =  A  ->  (
(ringLMod `  a )  e. LNoeM  <->  (ringLMod `  A )  e. LNoeM )
)
3 df-lnr 26984 . 2  |- LNoeR  =  {
a  e.  Ring  |  (ringLMod `  a )  e. LNoeM }
42, 3elrab2 3038 1  |-  ( A  e. LNoeR 
<->  ( A  e.  Ring  /\  (ringLMod `  A )  e. LNoeM ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   ` cfv 5395   Ringcrg 15588  ringLModcrglmod 16169  LNoeMclnm 26843  LNoeRclnr 26983
This theorem is referenced by:  lnrrng  26986  lnrlnm  26987  islnr2  26988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-lnr 26984
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