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Theorem lnmlmod 42534
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 2728 . . 3 (LSubSp‘𝑀) = (LSubSp‘𝑀)
21islnm 42532 . 2 (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑀)(𝑀s 𝑎) ∈ LFinGen))
32simplbi 496 1 (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wral 3058  cfv 6553  (class class class)co 7426  s cress 17216  LModclmod 20750  LSubSpclss 20822  LFinGenclfig 42522  LNoeMclnm 42530
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 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-lnm 42531
This theorem is referenced by:  lnmlsslnm  42536  lnmfg  42537  pwslnmlem1  42547  pwslnm  42549
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