Theorem lnrlnm 43391

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 43389	. 2 ⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
2	1	simprbi 496	1 ⊢ (𝐴 ∈ LNoeR → (ringLMod‘𝐴) ∈ LNoeM)

Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6493  Ringcrg 20172  ringLModcrglmod 21128  LNoeMclnm 43353  LNoeRclnr 43387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6449  df-fv 6501  df-lnr 43388

This theorem is referenced by:  lnrfrlm  43396