Theorem islnm 41819

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 ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen))

Distinct variable groups:   𝑖,𝑀   𝑆,𝑖

Proof of Theorem islnm

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6892	. . . 4 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = (LSubSp‘𝑀))
2		islnm.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑀)
3	1, 2	eqtr4di 2791	. . 3 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = 𝑆)
4		oveq1 7416	. . . 4 ⊢ (𝑤 = 𝑀 → (𝑤 ↾s 𝑖) = (𝑀 ↾s 𝑖))
5	4	eleq1d 2819	. . 3 ⊢ (𝑤 = 𝑀 → ((𝑤 ↾s 𝑖) ∈ LFinGen ↔ (𝑀 ↾s 𝑖) ∈ LFinGen))
6	3, 5	raleqbidv 3343	. 2 ⊢ (𝑤 = 𝑀 → (∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen ↔ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen))
7		df-lnm 41818	. 2 ⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen}
8	6, 7	elrab2 3687	1 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ‘cfv 6544  (class class class)co 7409   ↾_s cress 17173  LModclmod 20471  LSubSpclss 20542  LFinGenclfig 41809  LNoeMclnm 41817

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 2704

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 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-lnm 41818

This theorem is referenced by:  islnm2  41820  lnmlmod  41821  lnmlssfg  41822  lnmlsslnm  41823  lnmepi  41827  lmhmlnmsplit  41829