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Theorem islnm 41433
Description: Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
Hypothesis
Ref Expression
islnm.s 𝑆 = (LSubSp‘𝑀)
Assertion
Ref Expression
islnm (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖𝑆 (𝑀s 𝑖) ∈ LFinGen))
Distinct variable groups:   𝑖,𝑀   𝑆,𝑖

Proof of Theorem islnm
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6847 . . . 4 (𝑤 = 𝑀 → (LSubSp‘𝑤) = (LSubSp‘𝑀))
2 islnm.s . . . 4 𝑆 = (LSubSp‘𝑀)
31, 2eqtr4di 2795 . . 3 (𝑤 = 𝑀 → (LSubSp‘𝑤) = 𝑆)
4 oveq1 7369 . . . 4 (𝑤 = 𝑀 → (𝑤s 𝑖) = (𝑀s 𝑖))
54eleq1d 2823 . . 3 (𝑤 = 𝑀 → ((𝑤s 𝑖) ∈ LFinGen ↔ (𝑀s 𝑖) ∈ LFinGen))
63, 5raleqbidv 3322 . 2 (𝑤 = 𝑀 → (∀𝑖 ∈ (LSubSp‘𝑤)(𝑤s 𝑖) ∈ LFinGen ↔ ∀𝑖𝑆 (𝑀s 𝑖) ∈ LFinGen))
7 df-lnm 41432 . 2 LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤s 𝑖) ∈ LFinGen}
86, 7elrab2 3653 1 (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖𝑆 (𝑀s 𝑖) ∈ LFinGen))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  cfv 6501  (class class class)co 7362  s cress 17119  LModclmod 20338  LSubSpclss 20408  LFinGenclfig 41423  LNoeMclnm 41431
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 2708
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 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-ov 7365  df-lnm 41432
This theorem is referenced by:  islnm2  41434  lnmlmod  41435  lnmlssfg  41436  lnmlsslnm  41437  lnmepi  41441  lmhmlnmsplit  41443
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