Theorem lnrlnm 43088

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 43086	. 2 ⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
2	1	simprbi 496	1 ⊢ (𝐴 ∈ LNoeR → (ringLMod‘𝐴) ∈ LNoeM)

Syntax hints:   → wi 4   ∈ wcel 2107  ‘cfv 6541  Ringcrg 20198  ringLModcrglmod 21139  LNoeMclnm 43050  LNoeRclnr 43084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-iota 6494  df-fv 6549  df-lnr 43085

This theorem is referenced by:  lnrfrlm  43093