Theorem lmhmlnmsplit 35989

Description: If the kernel and range of a homomorphism of left modules are Noetherian, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 12-Jun-2015.)

Hypotheses

Ref	Expression
lmhmfgsplit.z	|- .0. = ( 0g ` T )
lmhmfgsplit.k	|- K = ( `' F " { .0. } )
lmhmfgsplit.u	|- U = ( S ↾s K )
lmhmfgsplit.v	|- V = ( T ↾s ran F )

Assertion

Ref	Expression
lmhmlnmsplit	|- ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) -> S e. LNoeM )

Proof of Theorem lmhmlnmsplit

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmhmlmod1 18304	. . 3 |- ( F e. ( S LMHom T ) -> S e. LMod )
2	1	3ad2ant1 1035	. 2 |- ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) -> S e. LMod )
3		eqid 2461	. . . . . 6 |- ( LSubSp ` S ) = ( LSubSp ` S )
4		eqid 2461	. . . . . 6 |- ( S ↾s a ) = ( S ↾s a )
5	3, 4	reslmhm 18323	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ a e. ( LSubSp ` S ) ) -> ( F |` a ) e. ( ( S ↾s a ) LMHom T ) )
6	5	3ad2antl1 1176	. . . 4 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( F |` a ) e. ( ( S ↾s a ) LMHom T ) )
7		cnvresima 5342	. . . . . . . 8 |- ( `' ( F |` a ) " { .0. } ) = ( ( `' F " { .0. } ) i^i a )
8		lmhmfgsplit.k	. . . . . . . . . 10 |- K = ( `' F " { .0. } )
9	8	eqcomi 2470	. . . . . . . . 9 |- ( `' F " { .0. } ) = K
10	9	ineq1i 3641	. . . . . . . 8 |- ( ( `' F " { .0. } ) i^i a ) = ( K i^i a )
11		incom 3636	. . . . . . . 8 |- ( K i^i a ) = ( a i^i K )
12	7, 10, 11	3eqtri 2487	. . . . . . 7 |- ( `' ( F |` a ) " { .0. } ) = ( a i^i K )
13	12	oveq2i 6325	. . . . . 6 |- ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) ) = ( ( S ↾s a ) ↾s ( a i^i K ) )
14		lmhmfgsplit.u	. . . . . . . . 9 |- U = ( S ↾s K )
15	14	oveq1i 6324	. . . . . . . 8 |- ( U ↾s ( a i^i K ) ) = ( ( S ↾s K ) ↾s ( a i^i K ) )
16		simpl1 1017	. . . . . . . . . 10 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> F e. ( S LMHom T ) )
17		cnvexg 6765	. . . . . . . . . . . 12 |- ( F e. ( S LMHom T ) -> `' F e. _V )
18		imaexg 6756	. . . . . . . . . . . 12 |- ( `' F e. _V -> ( `' F " { .0. } ) e. _V )
19	17, 18	syl 17	. . . . . . . . . . 11 |- ( F e. ( S LMHom T ) -> ( `' F " { .0. } ) e. _V )
20	8, 19	syl5eqel 2543	. . . . . . . . . 10 |- ( F e. ( S LMHom T ) -> K e. _V )
21	16, 20	syl 17	. . . . . . . . 9 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> K e. _V )
22		inss2 3664	. . . . . . . . 9 |- ( a i^i K ) C_ K
23		ressabs 15236	. . . . . . . . 9 |- ( ( K e. _V /\ ( a i^i K ) C_ K ) -> ( ( S ↾s K ) ↾s ( a i^i K ) ) = ( S ↾s ( a i^i K ) ) )
24	21, 22, 23	sylancl 673	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( ( S ↾s K ) ↾s ( a i^i K ) ) = ( S ↾s ( a i^i K ) ) )
25	15, 24	syl5eq 2507	. . . . . . 7 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( U ↾s ( a i^i K ) ) = ( S ↾s ( a i^i K ) ) )
26		vex 3059	. . . . . . . 8 |- a e. _V
27		inss1 3663	. . . . . . . 8 |- ( a i^i K ) C_ a
28		ressabs 15236	. . . . . . . 8 |- ( ( a e. _V /\ ( a i^i K ) C_ a ) -> ( ( S ↾s a ) ↾s ( a i^i K ) ) = ( S ↾s ( a i^i K ) ) )
29	26, 27, 28	mp2an 683	. . . . . . 7 |- ( ( S ↾s a ) ↾s ( a i^i K ) ) = ( S ↾s ( a i^i K ) )
30	25, 29	syl6reqr 2514	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( ( S ↾s a ) ↾s ( a i^i K ) ) = ( U ↾s ( a i^i K ) ) )
31	13, 30	syl5eq 2507	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) ) = ( U ↾s ( a i^i K ) ) )
32		simpl2 1018	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> U e. LNoeM )
33	2	adantr 471	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> S e. LMod )
34		simpr 467	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> a e. ( LSubSp ` S ) )
35		lmhmfgsplit.z	. . . . . . . . . 10 |- .0. = ( 0g ` T )
36	8, 35, 3	lmhmkerlss 18322	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> K e. ( LSubSp ` S ) )
37	16, 36	syl 17	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> K e. ( LSubSp ` S ) )
38	3	lssincl 18236	. . . . . . . 8 |- ( ( S e. LMod /\ a e. ( LSubSp ` S ) /\ K e. ( LSubSp ` S ) ) -> ( a i^i K ) e. ( LSubSp ` S ) )
39	33, 34, 37, 38	syl3anc 1276	. . . . . . 7 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( a i^i K ) e. ( LSubSp ` S ) )
40	22	a1i 11	. . . . . . 7 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( a i^i K ) C_ K )
41		eqid 2461	. . . . . . . . 9 |- ( LSubSp ` U ) = ( LSubSp ` U )
42	14, 3, 41	lsslss 18232	. . . . . . . 8 |- ( ( S e. LMod /\ K e. ( LSubSp ` S ) ) -> ( ( a i^i K ) e. ( LSubSp ` U ) <-> ( ( a i^i K ) e. ( LSubSp ` S ) /\ ( a i^i K ) C_ K ) ) )
43	33, 37, 42	syl2anc 671	. . . . . . 7 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( ( a i^i K ) e. ( LSubSp ` U ) <-> ( ( a i^i K ) e. ( LSubSp ` S ) /\ ( a i^i K ) C_ K ) ) )
44	39, 40, 43	mpbir2and 938	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( a i^i K ) e. ( LSubSp ` U ) )
45		eqid 2461	. . . . . . 7 |- ( U ↾s ( a i^i K ) ) = ( U ↾s ( a i^i K ) )
46	41, 45	lnmlssfg 35982	. . . . . 6 |- ( ( U e. LNoeM /\ ( a i^i K ) e. ( LSubSp ` U ) ) -> ( U ↾s ( a i^i K ) ) e. LFinGen )
47	32, 44, 46	syl2anc 671	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( U ↾s ( a i^i K ) ) e. LFinGen )
48	31, 47	eqeltrd 2539	. . . 4 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) ) e. LFinGen )
49		lmhmfgsplit.v	. . . . . . . . 9 |- V = ( T ↾s ran F )
50	49	oveq1i 6324	. . . . . . . 8 |- ( V ↾s ran ( F |` a ) ) = ( ( T ↾s ran F ) ↾s ran ( F |` a ) )
51		rnexg 6751	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> ran F e. _V )
52		resexg 5165	. . . . . . . . . 10 |- ( F e. ( S LMHom T ) -> ( F |` a ) e. _V )
53		rnexg 6751	. . . . . . . . . 10 |- ( ( F |` a ) e. _V -> ran ( F |` a ) e. _V )
54	52, 53	syl 17	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> ran ( F |` a ) e. _V )
55		ressress 15235	. . . . . . . . 9 |- ( ( ran F e. _V /\ ran ( F |` a ) e. _V ) -> ( ( T ↾s ran F ) ↾s ran ( F |` a ) ) = ( T ↾s ( ran F i^i ran ( F |` a ) ) ) )
56	51, 54, 55	syl2anc 671	. . . . . . . 8 |- ( F e. ( S LMHom T ) -> ( ( T ↾s ran F ) ↾s ran ( F |` a ) ) = ( T ↾s ( ran F i^i ran ( F |` a ) ) ) )
57	50, 56	syl5eq 2507	. . . . . . 7 |- ( F e. ( S LMHom T ) -> ( V ↾s ran ( F |` a ) ) = ( T ↾s ( ran F i^i ran ( F |` a ) ) ) )
58		incom 3636	. . . . . . . . 9 |- ( ran F i^i ran ( F |` a ) ) = ( ran ( F |` a ) i^i ran F )
59		resss 5146	. . . . . . . . . . 11 |- ( F |` a ) C_ F
60		rnss 5081	. . . . . . . . . . 11 |- ( ( F |` a ) C_ F -> ran ( F |` a ) C_ ran F )
61	59, 60	ax-mp 5	. . . . . . . . . 10 |- ran ( F |` a ) C_ ran F
62		df-ss 3429	. . . . . . . . . 10 |- ( ran ( F |` a ) C_ ran F <-> ( ran ( F |` a ) i^i ran F ) = ran ( F |` a ) )
63	61, 62	mpbi 213	. . . . . . . . 9 |- ( ran ( F |` a ) i^i ran F ) = ran ( F |` a )
64	58, 63	eqtr2i 2484	. . . . . . . 8 |- ran ( F |` a ) = ( ran F i^i ran ( F |` a ) )
65	64	oveq2i 6325	. . . . . . 7 |- ( T ↾s ran ( F |` a ) ) = ( T ↾s ( ran F i^i ran ( F |` a ) ) )
66	57, 65	syl6reqr 2514	. . . . . 6 |- ( F e. ( S LMHom T ) -> ( T ↾s ran ( F |` a ) ) = ( V ↾s ran ( F |` a ) ) )
67	16, 66	syl 17	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( T ↾s ran ( F |` a ) ) = ( V ↾s ran ( F |` a ) ) )
68		simpl3 1019	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> V e. LNoeM )
69		lmhmrnlss 18321	. . . . . . . 8 |- ( ( F |` a ) e. ( ( S ↾s a ) LMHom T ) -> ran ( F |` a ) e. ( LSubSp ` T ) )
70	6, 69	syl 17	. . . . . . 7 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ran ( F |` a ) e. ( LSubSp ` T ) )
71	61	a1i 11	. . . . . . 7 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ran ( F |` a ) C_ ran F )
72		lmhmlmod2 18303	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> T e. LMod )
73	16, 72	syl 17	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> T e. LMod )
74		lmhmrnlss 18321	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> ran F e. ( LSubSp ` T ) )
75	16, 74	syl 17	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ran F e. ( LSubSp ` T ) )
76		eqid 2461	. . . . . . . . 9 |- ( LSubSp ` T ) = ( LSubSp ` T )
77		eqid 2461	. . . . . . . . 9 |- ( LSubSp ` V ) = ( LSubSp ` V )
78	49, 76, 77	lsslss 18232	. . . . . . . 8 |- ( ( T e. LMod /\ ran F e. ( LSubSp ` T ) ) -> ( ran ( F |` a ) e. ( LSubSp ` V ) <-> ( ran ( F |` a ) e. ( LSubSp ` T ) /\ ran ( F |` a ) C_ ran F ) ) )
79	73, 75, 78	syl2anc 671	. . . . . . 7 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( ran ( F |` a ) e. ( LSubSp ` V ) <-> ( ran ( F |` a ) e. ( LSubSp ` T ) /\ ran ( F |` a ) C_ ran F ) ) )
80	70, 71, 79	mpbir2and 938	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ran ( F |` a ) e. ( LSubSp ` V ) )
81		eqid 2461	. . . . . . 7 |- ( V ↾s ran ( F |` a ) ) = ( V ↾s ran ( F |` a ) )
82	77, 81	lnmlssfg 35982	. . . . . 6 |- ( ( V e. LNoeM /\ ran ( F |` a ) e. ( LSubSp ` V ) ) -> ( V ↾s ran ( F |` a ) ) e. LFinGen )
83	68, 80, 82	syl2anc 671	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( V ↾s ran ( F |` a ) ) e. LFinGen )
84	67, 83	eqeltrd 2539	. . . 4 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( T ↾s ran ( F |` a ) ) e. LFinGen )
85		eqid 2461	. . . . 5 |- ( `' ( F |` a ) " { .0. } ) = ( `' ( F |` a ) " { .0. } )
86		eqid 2461	. . . . 5 |- ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) ) = ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) )
87		eqid 2461	. . . . 5 |- ( T ↾s ran ( F |` a ) ) = ( T ↾s ran ( F |` a ) )
88	35, 85, 86, 87	lmhmfgsplit 35988	. . . 4 |- ( ( ( F |` a ) e. ( ( S ↾s a ) LMHom T ) /\ ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) ) e. LFinGen /\ ( T ↾s ran ( F |` a ) ) e. LFinGen ) -> ( S ↾s a ) e. LFinGen )
89	6, 48, 84, 88	syl3anc 1276	. . 3 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( S ↾s a ) e. LFinGen )
90	89	ralrimiva 2813	. 2 |- ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) -> A. a e. ( LSubSp ` S ) ( S ↾s a ) e. LFinGen )
91	3	islnm 35979	. 2 |- ( S e. LNoeM <-> ( S e. LMod /\ A. a e. ( LSubSp ` S ) ( S ↾s a ) e. LFinGen ) )
92	2, 90, 91	sylanbrc 675	1 |- ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) -> S e. LNoeM )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1454   e. wcel 1897   A.wral 2748   _Vcvv 3056   i^i cin 3414   C_ wss 3415   {csn 3979   `'ccnv 4851   ran crn 4853   |` cres 4854   "cima 4855   ` cfv 5600  (class class class)co 6314   ↾_s cress 15170   0gc0g 15386   LModclmod 18139   LSubSpclss 18203   LMHom clmhm 18290  LFinGenclfig 35969  LNoeMclnm 35977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-sca 15254  df-vsca 15255  df-0g 15388  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-submnd 16631  df-grp 16721  df-minusg 16722  df-sbg 16723  df-subg 16862  df-ghm 16929  df-cntz 17019  df-lsm 17336  df-cmn 17480  df-abl 17481  df-mgp 17772  df-ur 17784  df-ring 17830  df-lmod 18141  df-lss 18204  df-lsp 18243  df-lmhm 18293  df-lfig 35970  df-lnm 35978

This theorem is referenced by:  pwslnmlem2  35995