Theorem lnrring 43859

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 43858	. 2 ⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
2	1	simplbi 501	1 ⊢ (𝐴 ∈ LNoeR → 𝐴 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2143  ‘cfv 6536  Ringcrg 20310  ringLModcrglmod 21293  LNoeMclnm 43822  LNoeRclnr 43856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-lnr 43857

This theorem is referenced by:  lnr2i  43863  hbtlem6  43876  hbt  43877