Theorem lnrlnm 43220

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

Assertion

Ref	Expression
lnrlnm	⊢ (𝐴 ∈ LNoeR → (ringLMod‘𝐴) ∈ LNoeM)

Proof of Theorem lnrlnm

Step	Hyp	Ref	Expression
1		islnr 43218	. 2 ⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
2	1	simprbi 496	1 ⊢ (𝐴 ∈ LNoeR → (ringLMod‘𝐴) ∈ LNoeM)

Syntax hints:   → wi 4   ∈ wcel 2113  ‘cfv 6489  Ringcrg 20161  ringLModcrglmod 21116  LNoeMclnm 43182  LNoeRclnr 43216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6445  df-fv 6497  df-lnr 43217

This theorem is referenced by:  lnrfrlm  43225