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 e. LNoeM <-> ( M e. LMod /\ A. i e. S ( M ↾s i ) e. 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 ) e. LFinGen <-> ( M ↾s i ) e. LFinGen ) )
6	3, 5	raleqbidv 3152	. 2 |- ( w = M -> ( A. i e. ( LSubSp ` w ) ( w ↾s i ) e. LFinGen <-> A. i e. S ( M ↾s i ) e. LFinGen ) )
7		df-lnm 37646	. 2 |- LNoeM = { w e. LMod | A. i e. ( LSubSp ` w ) ( w ↾s i ) e. LFinGen }
8	6, 7	elrab2 3366	1 |- ( M e. LNoeM <-> ( M e. LMod /\ A. i e. S ( M ↾s i ) e. LFinGen ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.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