Theorem islnm 40546

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 6695	. . . 4 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = (LSubSp‘𝑀))
2		islnm.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑀)
3	1, 2	eqtr4di 2789	. . 3 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = 𝑆)
4		oveq1 7198	. . . 4 ⊢ (𝑤 = 𝑀 → (𝑤 ↾s 𝑖) = (𝑀 ↾s 𝑖))
5	4	eleq1d 2815	. . 3 ⊢ (𝑤 = 𝑀 → ((𝑤 ↾s 𝑖) ∈ LFinGen ↔ (𝑀 ↾s 𝑖) ∈ LFinGen))
6	3, 5	raleqbidv 3303	. 2 ⊢ (𝑤 = 𝑀 → (∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen ↔ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen))
7		df-lnm 40545	. 2 ⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen}
8	6, 7	elrab2 3594	1 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ‘cfv 6358  (class class class)co 7191   ↾_s cress 16667  LModclmod 19853  LSubSpclss 19922  LFinGenclfig 40536  LNoeMclnm 40544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-iota 6316  df-fv 6366  df-ov 7194  df-lnm 40545

This theorem is referenced by:  islnm2  40547  lnmlmod  40548  lnmlssfg  40549  lnmlsslnm  40550  lnmepi  40554  lmhmlnmsplit  40556