Theorem lnmlmod 43068

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 2735	. . 3 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀)
2	1	islnm 43066	. 2 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑀)(𝑀 ↾s 𝑎) ∈ LFinGen))
3	2	simplbi 497	1 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)

Syntax hints:   → wi 4   ∈ wcel 2106  ∀wral 3059  ‘cfv 6563  (class class class)co 7431   ↾_s cress 17274  LModclmod 20875  LSubSpclss 20947  LFinGenclfig 43056  LNoeMclnm 43064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-lnm 43065

This theorem is referenced by:  lnmlsslnm  43070  lnmfg  43071  pwslnmlem1  43081  pwslnm  43083