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Theorem lnmlmod 42381
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 2726 . . 3 (LSubSp‘𝑀) = (LSubSp‘𝑀)
21islnm 42379 . 2 (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑀)(𝑀s 𝑎) ∈ LFinGen))
32simplbi 497 1 (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wral 3055  cfv 6536  (class class class)co 7404  s cress 17179  LModclmod 20703  LSubSpclss 20775  LFinGenclfig 42369  LNoeMclnm 42377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-lnm 42378
This theorem is referenced by:  lnmlsslnm  42383  lnmfg  42384  pwslnmlem1  42394  pwslnm  42396
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