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Theorem lnrring 41468
Description: Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
lnrring (𝐴 ∈ LNoeR β†’ 𝐴 ∈ Ring)

Proof of Theorem lnrring
StepHypRef Expression
1 islnr 41467 . 2 (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLModβ€˜π΄) ∈ LNoeM))
21simplbi 499 1 (𝐴 ∈ LNoeR β†’ 𝐴 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107  β€˜cfv 6501  Ringcrg 19971  ringLModcrglmod 20646  LNoeMclnm 41431  LNoeRclnr 41465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-lnr 41466
This theorem is referenced by:  lnr2i  41472  hbtlem6  41485  hbt  41486
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