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Theorem lnmlmod 40020
Description: A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
Assertion
Ref Expression
lnmlmod (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)

Proof of Theorem lnmlmod
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eqid 2801 . . 3 (LSubSp‘𝑀) = (LSubSp‘𝑀)
21islnm 40018 . 2 (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑀)(𝑀s 𝑎) ∈ LFinGen))
32simplbi 501 1 (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112  wral 3109  cfv 6328  (class class class)co 7139  s cress 16480  LModclmod 19631  LSubSpclss 19700  LFinGenclfig 40008  LNoeMclnm 40016
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-ral 3114  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-ov 7142  df-lnm 40017
This theorem is referenced by:  lnmlsslnm  40022  lnmfg  40023  pwslnmlem1  40033  pwslnm  40035
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