Theorem lnmlssfg 37650

Description: A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Hypotheses

Ref	Expression
lnmlssfg.s	|- S = ( LSubSp ` M )
lnmlssfg.r	|- R = ( M ↾s U )

Assertion

Ref	Expression
lnmlssfg	|- ( ( M e. LNoeM /\ U e. S ) -> R e. LFinGen )

Proof of Theorem lnmlssfg

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnmlssfg.s	. . . 4 |- S = ( LSubSp ` M )
2	1	islnm 37647	. . 3 |- ( M e. LNoeM <-> ( M e. LMod /\ A. a e. S ( M ↾s a ) e. LFinGen ) )
3	2	simprbi 480	. 2 |- ( M e. LNoeM -> A. a e. S ( M ↾s a ) e. LFinGen )
4		oveq2 6658	. . . . 5 |- ( a = U -> ( M ↾s a ) = ( M ↾s U ) )
5		lnmlssfg.r	. . . . 5 |- R = ( M ↾s U )
6	4, 5	syl6eqr 2674	. . . 4 |- ( a = U -> ( M ↾s a ) = R )
7	6	eleq1d 2686	. . 3 |- ( a = U -> ( ( M ↾s a ) e. LFinGen <-> R e. LFinGen ) )
8	7	rspcv 3305	. 2 |- ( U e. S -> ( A. a e. S ( M ↾s a ) e. LFinGen -> R e. LFinGen ) )
9	3, 8	mpan9 486	1 |- ( ( M e. LNoeM /\ U e. S ) -> R e. LFinGen )

Syntax hints:   -> wi 4   /\ 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:  lnmlsslnm  37651  lnmfg  37652  lnmepi  37655  lmhmlnmsplit  37657  lnrfgtr  37690