Theorem islnm 43531

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 6828	. . . 4 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = (LSubSp‘𝑀))
2		islnm.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑀)
3	1, 2	eqtr4di 2792	. . 3 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = 𝑆)
4		oveq1 7364	. . . 4 ⊢ (𝑤 = 𝑀 → (𝑤 ↾s 𝑖) = (𝑀 ↾s 𝑖))
5	4	eleq1d 2824	. . 3 ⊢ (𝑤 = 𝑀 → ((𝑤 ↾s 𝑖) ∈ LFinGen ↔ (𝑀 ↾s 𝑖) ∈ LFinGen))
6	3, 5	raleqbidv 3313	. 2 ⊢ (𝑤 = 𝑀 → (∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen ↔ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen))
7		df-lnm 43530	. 2 ⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen}
8	6, 7	elrab2 3632	1 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ‘cfv 6486  (class class class)co 7357   ↾_s cress 17192  LModclmod 20851  LSubSpclss 20922  LFinGenclfig 43521  LNoeMclnm 43529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-iota 6442  df-fv 6494  df-ov 7360  df-lnm 43530

This theorem is referenced by:  islnm2  43532  lnmlmod  43533  lnmlssfg  43534  lnmlsslnm  43535  lnmepi  43539  lmhmlnmsplit  43541