Theorem lnmlmod 43196

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 43194	. 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 2113  ∀wral 3048  ‘cfv 6486  (class class class)co 7352   ↾_s cress 17143  LModclmod 20795  LSubSpclss 20866  LFinGenclfig 43184  LNoeMclnm 43192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-ov 7355  df-lnm 43193

This theorem is referenced by:  lnmlsslnm  43198  lnmfg  43199  pwslnmlem1  43209  pwslnm  43211