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Theorem islnr 35404
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 5848 . . 3  |-  ( a  =  A  ->  (ringLMod `  a )  =  (ringLMod `  A ) )
21eleq1d 2471 . 2  |-  ( a  =  A  ->  (
(ringLMod `  a )  e. LNoeM  <->  (ringLMod `  A )  e. LNoeM )
)
3 df-lnr 35403 . 2  |- LNoeR  =  {
a  e.  Ring  |  (ringLMod `  a )  e. LNoeM }
42, 3elrab2 3208 1  |-  ( A  e. LNoeR 
<->  ( A  e.  Ring  /\  (ringLMod `  A )  e. LNoeM ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   ` cfv 5568   Ringcrg 17516  ringLModcrglmod 18133  LNoeMclnm 35363  LNoeRclnr 35402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-lnr 35403
This theorem is referenced by:  lnrring  35405  lnrlnm  35406  islnr2  35407
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