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Theorem lnmlmod 39072
 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 2779 . . 3 (LSubSp‘𝑀) = (LSubSp‘𝑀)
21islnm 39070 . 2 (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑀)(𝑀s 𝑎) ∈ LFinGen))
32simplbi 490 1 (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2050  ∀wral 3089  ‘cfv 6188  (class class class)co 6976   ↾s cress 16340  LModclmod 19356  LSubSpclss 19425  LFinGenclfig 39060  LNoeMclnm 39068 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-iota 6152  df-fv 6196  df-ov 6979  df-lnm 39069 This theorem is referenced by:  lnmlsslnm  39074  lnmfg  39075  pwslnmlem1  39085  pwslnm  39087
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