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Theorem lnrlnm 40044
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
StepHypRef Expression
1 islnr 40042 . 2 (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
21simprbi 500 1 (𝐴 ∈ LNoeR → (ringLMod‘𝐴) ∈ LNoeM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112  cfv 6328  Ringcrg 19293  ringLModcrglmod 19937  LNoeMclnm 40006  LNoeRclnr 40040
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-lnr 40041
This theorem is referenced by:  lnrfrlm  40049
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