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Theorem islnm 38173
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 6332 . . . 4 (𝑤 = 𝑀 → (LSubSp‘𝑤) = (LSubSp‘𝑀))
2 islnm.s . . . 4 𝑆 = (LSubSp‘𝑀)
31, 2syl6eqr 2823 . . 3 (𝑤 = 𝑀 → (LSubSp‘𝑤) = 𝑆)
4 oveq1 6800 . . . 4 (𝑤 = 𝑀 → (𝑤s 𝑖) = (𝑀s 𝑖))
54eleq1d 2835 . . 3 (𝑤 = 𝑀 → ((𝑤s 𝑖) ∈ LFinGen ↔ (𝑀s 𝑖) ∈ LFinGen))
63, 5raleqbidv 3301 . 2 (𝑤 = 𝑀 → (∀𝑖 ∈ (LSubSp‘𝑤)(𝑤s 𝑖) ∈ LFinGen ↔ ∀𝑖𝑆 (𝑀s 𝑖) ∈ LFinGen))
7 df-lnm 38172 . 2 LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤s 𝑖) ∈ LFinGen}
86, 7elrab2 3518 1 (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖𝑆 (𝑀s 𝑖) ∈ LFinGen))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  cfv 6031  (class class class)co 6793  s cress 16065  LModclmod 19073  LSubSpclss 19142  LFinGenclfig 38163  LNoeMclnm 38171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-lnm 38172
This theorem is referenced by:  islnm2  38174  lnmlmod  38175  lnmlssfg  38176  lnmlsslnm  38177  lnmepi  38181  lmhmlnmsplit  38183
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