Theorem lnmepi 37655

Description: Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Hypothesis

Ref	Expression
lnmepi.b	|- B = ( Base ` T )

Assertion

Ref	Expression
lnmepi	|- ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) -> T e. LNoeM )

Proof of Theorem lnmepi

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmhmlmod2 19032	. . 3 |- ( F e. ( S LMHom T ) -> T e. LMod )
2	1	3ad2ant1 1082	. 2 |- ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) -> T e. LMod )
3		eqid 2622	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
4		lnmepi.b	. . . . . . . . 9 |- B = ( Base ` T )
5	3, 4	lmhmf 19034	. . . . . . . 8 |- ( F e. ( S LMHom T ) -> F : ( Base ` S ) --> B )
6	5	3ad2ant1 1082	. . . . . . 7 |- ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) -> F : ( Base ` S ) --> B )
7		simp3 1063	. . . . . . 7 |- ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) -> ran F = B )
8		dffo2 6119	. . . . . . 7 |- ( F : ( Base ` S ) -onto-> B <-> ( F : ( Base ` S ) --> B /\ ran F = B ) )
9	6, 7, 8	sylanbrc 698	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) -> F : ( Base ` S ) -onto-> B )
10		eqid 2622	. . . . . . 7 |- ( LSubSp ` T ) = ( LSubSp ` T )
11	4, 10	lssss 18937	. . . . . 6 |- ( a e. ( LSubSp ` T ) -> a C_ B )
12		foimacnv 6154	. . . . . 6 |- ( ( F : ( Base ` S ) -onto-> B /\ a C_ B ) -> ( F " ( `' F " a ) ) = a )
13	9, 11, 12	syl2an 494	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) /\ a e. ( LSubSp ` T ) ) -> ( F " ( `' F " a ) ) = a )
14	13	oveq2d 6666	. . . 4 |- ( ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) /\ a e. ( LSubSp ` T ) ) -> ( T ↾s ( F " ( `' F " a ) ) ) = ( T ↾s a ) )
15		eqid 2622	. . . . 5 |- ( T ↾s ( F " ( `' F " a ) ) ) = ( T ↾s ( F " ( `' F " a ) ) )
16		eqid 2622	. . . . 5 |- ( S ↾s ( `' F " a ) ) = ( S ↾s ( `' F " a ) )
17		eqid 2622	. . . . 5 |- ( LSubSp ` S ) = ( LSubSp ` S )
18		simpl2 1065	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) /\ a e. ( LSubSp ` T ) ) -> S e. LNoeM )
19	17, 10	lmhmpreima 19048	. . . . . . 7 |- ( ( F e. ( S LMHom T ) /\ a e. ( LSubSp ` T ) ) -> ( `' F " a ) e. ( LSubSp ` S ) )
20	19	3ad2antl1 1223	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) /\ a e. ( LSubSp ` T ) ) -> ( `' F " a ) e. ( LSubSp ` S ) )
21	17, 16	lnmlssfg 37650	. . . . . 6 |- ( ( S e. LNoeM /\ ( `' F " a ) e. ( LSubSp ` S ) ) -> ( S ↾s ( `' F " a ) ) e. LFinGen )
22	18, 20, 21	syl2anc 693	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) /\ a e. ( LSubSp ` T ) ) -> ( S ↾s ( `' F " a ) ) e. LFinGen )
23		simpl1 1064	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) /\ a e. ( LSubSp ` T ) ) -> F e. ( S LMHom T ) )
24	15, 16, 17, 22, 20, 23	lmhmfgima 37654	. . . 4 |- ( ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) /\ a e. ( LSubSp ` T ) ) -> ( T ↾s ( F " ( `' F " a ) ) ) e. LFinGen )
25	14, 24	eqeltrrd 2702	. . 3 |- ( ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) /\ a e. ( LSubSp ` T ) ) -> ( T ↾s a ) e. LFinGen )
26	25	ralrimiva 2966	. 2 |- ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) -> A. a e. ( LSubSp ` T ) ( T ↾s a ) e. LFinGen )
27	10	islnm 37647	. 2 |- ( T e. LNoeM <-> ( T e. LMod /\ A. a e. ( LSubSp ` T ) ( T ↾s a ) e. LFinGen ) )
28	2, 26, 27	sylanbrc 698	1 |- ( ( F e. ( S LMHom T ) /\ S e. LNoeM /\ ran F = B ) -> T e. LNoeM )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   `'ccnv 5113   ran crn 5115   "cima 5117   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   LModclmod 18863   LSubSpclss 18932   LMHom clmhm 19019  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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-sca 15957  df-vsca 15958  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lmhm 19022  df-lfig 37638  df-lnm 37646

This theorem is referenced by:  lnmlmic  37658  pwslnmlem1  37662  lnrfg  37689