Theorem lnmlmod 41821

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.

Step	Hyp	Ref	Expression
1		eqid 2733	. . 3 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀)
2	1	islnm 41819	. 2 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑀)(𝑀 ↾s 𝑎) ∈ LFinGen))
3	2	simplbi 499	1 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)

Syntax hints:   → wi 4   ∈ wcel 2107  ∀wral 3062  ‘cfv 6544  (class class class)co 7409   ↾_s cress 17173  LModclmod 20471  LSubSpclss 20542  LFinGenclfig 41809  LNoeMclnm 41817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-lnm 41818

This theorem is referenced by:  lnmlsslnm  41823  lnmfg  41824  pwslnmlem1  41834  pwslnm  41836