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Theorem islnm 40014
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 6649 . . . 4 (𝑤 = 𝑀 → (LSubSp‘𝑤) = (LSubSp‘𝑀))
2 islnm.s . . . 4 𝑆 = (LSubSp‘𝑀)
31, 2eqtr4di 2854 . . 3 (𝑤 = 𝑀 → (LSubSp‘𝑤) = 𝑆)
4 oveq1 7146 . . . 4 (𝑤 = 𝑀 → (𝑤s 𝑖) = (𝑀s 𝑖))
54eleq1d 2877 . . 3 (𝑤 = 𝑀 → ((𝑤s 𝑖) ∈ LFinGen ↔ (𝑀s 𝑖) ∈ LFinGen))
63, 5raleqbidv 3357 . 2 (𝑤 = 𝑀 → (∀𝑖 ∈ (LSubSp‘𝑤)(𝑤s 𝑖) ∈ LFinGen ↔ ∀𝑖𝑆 (𝑀s 𝑖) ∈ LFinGen))
7 df-lnm 40013 . 2 LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤s 𝑖) ∈ LFinGen}
86, 7elrab2 3634 1 (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖𝑆 (𝑀s 𝑖) ∈ LFinGen))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  cfv 6328  (class class class)co 7139  s cress 16480  LModclmod 19631  LSubSpclss 19700  LFinGenclfig 40004  LNoeMclnm 40012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-lnm 40013
This theorem is referenced by:  islnm2  40015  lnmlmod  40016  lnmlssfg  40017  lnmlsslnm  40018  lnmepi  40022  lmhmlnmsplit  40024
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