Theorem islnm 37647

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 6191	. . . 4 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = (LSubSp‘𝑀))
2		islnm.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑀)
3	1, 2	syl6eqr 2674	. . 3 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = 𝑆)
4		oveq1 6657	. . . 4 ⊢ (𝑤 = 𝑀 → (𝑤 ↾s 𝑖) = (𝑀 ↾s 𝑖))
5	4	eleq1d 2686	. . 3 ⊢ (𝑤 = 𝑀 → ((𝑤 ↾s 𝑖) ∈ LFinGen ↔ (𝑀 ↾s 𝑖) ∈ LFinGen))
6	3, 5	raleqbidv 3152	. 2 ⊢ (𝑤 = 𝑀 → (∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen ↔ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen))
7		df-lnm 37646	. 2 ⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen}
8	6, 7	elrab2 3366	1 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ‘cfv 5888  (class class class)co 6650   ↾_s cress 15858  LModclmod 18863  LSubSpclss 18932  LFinGenclfig 37637  LNoeMclnm 37645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-lnm 37646

This theorem is referenced by:  islnm2  37648  lnmlmod  37649  lnmlssfg  37650  lnmlsslnm  37651  lnmepi  37655  lmhmlnmsplit  37657