Theorem lnmlmod 43921

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 2760	. . 3 ⊢ (LSubSp‘𝑀) = (LSubSp‘𝑀)
2	1	islnm 43919	. 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 3076  ‘cfv 6533  (class class class)co 7414   ↾_s cress 17323  LModclmod 21045  LSubSpclss 21116  LFinGenclfig 43909  LNoeMclnm 43917

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 2732

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  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 6489  df-fv 6541  df-ov 7417  df-lnm 43918

This theorem is used by:  lnmlsslnm  43923  lnmfg  43924  pwslnmlem1  43934  pwslnm  43936