Theorem lnmlmod 44080

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 2761	. . 3 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀)
2	1	islnm 44078	. 2 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑀)(𝑀 ↾s 𝑎) ∈ LFinGen))
3	2	simplbi 502	1 ⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ∀wral 3077  ‘cfv 6538  (class class class)co 7420   ↾_s cress 17408  LModclmod 21135  LSubSpclss 21206  LFinGenclfig 44068  LNoeMclnm 44076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546  df-ov 7423  df-lnm 44077

This theorem is used by:  lnmlsslnm  44082  lnmfg  44083  pwslnmlem1  44093  pwslnm  44095