Theorem lnrring 43108

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 43107	. 2 ⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
2	1	simplbi 497	1 ⊢ (𝐴 ∈ LNoeR → 𝐴 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2109  ‘cfv 6514  Ringcrg 20149  ringLModcrglmod 21086  LNoeMclnm 43071  LNoeRclnr 43105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-lnr 43106

This theorem is referenced by:  lnr2i  43112  hbtlem6  43125  hbt  43126