Theorem lnrring 36701

Description: Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Assertion

Ref	Expression
lnrring	⊢ (𝐴 ∈ LNoeR → 𝐴 ∈ Ring)

Proof of Theorem lnrring

Step	Hyp	Ref	Expression
1		islnr 36700	. 2 ⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
2	1	simplbi 475	1 ⊢ (𝐴 ∈ LNoeR → 𝐴 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 1977  ‘cfv 5804  Ringcrg 18370  ringLModcrglmod 18990  LNoeMclnm 36663  LNoeRclnr 36698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-lnr 36699

This theorem is referenced by:  lnr2i  36705  hbtlem6  36718  hbt  36719