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Theorem lnrrng 27293
Description: Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
lnrrng  |-  ( A  e. LNoeR  ->  A  e.  Ring )

Proof of Theorem lnrrng
StepHypRef Expression
1 islnr 27292 . 2  |-  ( A  e. LNoeR 
<->  ( A  e.  Ring  /\  (ringLMod `  A )  e. LNoeM ) )
21simplbi 447 1  |-  ( A  e. LNoeR  ->  A  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   ` cfv 5454   Ringcrg 15660  ringLModcrglmod 16241  LNoeMclnm 27150  LNoeRclnr 27290
This theorem is referenced by:  lnr2i  27297  hbtlem6  27310  hbt  27311
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-lnr 27291
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