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Theorem islnm 39070
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 6499 . . . 4 (𝑤 = 𝑀 → (LSubSp‘𝑤) = (LSubSp‘𝑀))
2 islnm.s . . . 4 𝑆 = (LSubSp‘𝑀)
31, 2syl6eqr 2833 . . 3 (𝑤 = 𝑀 → (LSubSp‘𝑤) = 𝑆)
4 oveq1 6983 . . . 4 (𝑤 = 𝑀 → (𝑤s 𝑖) = (𝑀s 𝑖))
54eleq1d 2851 . . 3 (𝑤 = 𝑀 → ((𝑤s 𝑖) ∈ LFinGen ↔ (𝑀s 𝑖) ∈ LFinGen))
63, 5raleqbidv 3342 . 2 (𝑤 = 𝑀 → (∀𝑖 ∈ (LSubSp‘𝑤)(𝑤s 𝑖) ∈ LFinGen ↔ ∀𝑖𝑆 (𝑀s 𝑖) ∈ LFinGen))
7 df-lnm 39069 . 2 LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤s 𝑖) ∈ LFinGen}
86, 7elrab2 3600 1 (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖𝑆 (𝑀s 𝑖) ∈ LFinGen))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387   = wceq 1507  wcel 2050  wral 3089  cfv 6188  (class class class)co 6976  s cress 16340  LModclmod 19356  LSubSpclss 19425  LFinGenclfig 39060  LNoeMclnm 39068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-iota 6152  df-fv 6196  df-ov 6979  df-lnm 39069
This theorem is referenced by:  islnm2  39071  lnmlmod  39072  lnmlssfg  39073  lnmlsslnm  39074  lnmepi  39078  lmhmlnmsplit  39080
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