Theorem lmhmlnmsplit 30665

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 17479	. . 3 |- ( F e. ( S LMHom T ) -> S e. LMod )
2	1	3ad2ant1 1017	. 2 |- ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) -> S e. LMod )
3		eqid 2467	. . . . . 6 |- ( LSubSp ` S ) = ( LSubSp ` S )
4		eqid 2467	. . . . . 6 |- ( S ↾s a ) = ( S ↾s a )
5	3, 4	reslmhm 17498	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ a e. ( LSubSp ` S ) ) -> ( F |` a ) e. ( ( S ↾s a ) LMHom T ) )
6	5	3ad2antl1 1158	. . . 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 5496	. . . . . . . 8 |- ( `' ( F |` a ) " { .0. } ) = ( ( `' F " { .0. } ) i^i a )
8		lmhmfgsplit.k	. . . . . . . . . 10 |- K = ( `' F " { .0. } )
9	8	eqcomi 2480	. . . . . . . . 9 |- ( `' F " { .0. } ) = K
10	9	ineq1i 3696	. . . . . . . 8 |- ( ( `' F " { .0. } ) i^i a ) = ( K i^i a )
11		incom 3691	. . . . . . . 8 |- ( K i^i a ) = ( a i^i K )
12	7, 10, 11	3eqtri 2500	. . . . . . 7 |- ( `' ( F |` a ) " { .0. } ) = ( a i^i K )
13	12	oveq2i 6295	. . . . . 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 6294	. . . . . . . 8 |- ( U ↾s ( a i^i K ) ) = ( ( S ↾s K ) ↾s ( a i^i K ) )
16		simpl1 999	. . . . . . . . . 10 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> F e. ( S LMHom T ) )
17		cnvexg 6730	. . . . . . . . . . . 12 |- ( F e. ( S LMHom T ) -> `' F e. _V )
18		imaexg 6721	. . . . . . . . . . . 12 |- ( `' F e. _V -> ( `' F " { .0. } ) e. _V )
19	17, 18	syl 16	. . . . . . . . . . 11 |- ( F e. ( S LMHom T ) -> ( `' F " { .0. } ) e. _V )
20	8, 19	syl5eqel 2559	. . . . . . . . . 10 |- ( F e. ( S LMHom T ) -> K e. _V )
21	16, 20	syl 16	. . . . . . . . 9 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> K e. _V )
22		inss2 3719	. . . . . . . . 9 |- ( a i^i K ) C_ K
23		ressabs 14553	. . . . . . . . 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 662	. . . . . . . 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 2520	. . . . . . 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 3116	. . . . . . . 8 |- a e. _V
27		inss1 3718	. . . . . . . 8 |- ( a i^i K ) C_ a
28		ressabs 14553	. . . . . . . 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 672	. . . . . . 7 |- ( ( S ↾s a ) ↾s ( a i^i K ) ) = ( S ↾s ( a i^i K ) )
30	25, 29	syl6reqr 2527	. . . . . 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 2520	. . . . 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 1000	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> U e. LNoeM )
33	2	adantr 465	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> S e. LMod )
34		simpr 461	. . . . . . . 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 17497	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> K e. ( LSubSp ` S ) )
37	16, 36	syl 16	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> K e. ( LSubSp ` S ) )
38	3	lssincl 17411	. . . . . . . 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 1228	. . . . . . 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 2467	. . . . . . . . 9 |- ( LSubSp ` U ) = ( LSubSp ` U )
42	14, 3, 41	lsslss 17407	. . . . . . . 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 661	. . . . . . 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 920	. . . . . 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 2467	. . . . . . 7 |- ( U ↾s ( a i^i K ) ) = ( U ↾s ( a i^i K ) )
46	41, 45	lnmlssfg 30658	. . . . . 6 |- ( ( U e. LNoeM /\ ( a i^i K ) e. ( LSubSp ` U ) ) -> ( U ↾s ( a i^i K ) ) e. LFinGen )
47	32, 44, 46	syl2anc 661	. . . . 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 2555	. . . 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 6294	. . . . . . . 8 |- ( V ↾s ran ( F |` a ) ) = ( ( T ↾s ran F ) ↾s ran ( F |` a ) )
51		rnexg 6716	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> ran F e. _V )
52		resexg 5316	. . . . . . . . . 10 |- ( F e. ( S LMHom T ) -> ( F |` a ) e. _V )
53		rnexg 6716	. . . . . . . . . 10 |- ( ( F |` a ) e. _V -> ran ( F |` a ) e. _V )
54	52, 53	syl 16	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> ran ( F |` a ) e. _V )
55		ressress 14552	. . . . . . . . 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 661	. . . . . . . 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 2520	. . . . . . 7 |- ( F e. ( S LMHom T ) -> ( V ↾s ran ( F |` a ) ) = ( T ↾s ( ran F i^i ran ( F |` a ) ) ) )
58		incom 3691	. . . . . . . . 9 |- ( ran F i^i ran ( F |` a ) ) = ( ran ( F |` a ) i^i ran F )
59		resss 5297	. . . . . . . . . . 11 |- ( F |` a ) C_ F
60		rnss 5231	. . . . . . . . . . 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 3490	. . . . . . . . . 10 |- ( ran ( F |` a ) C_ ran F <-> ( ran ( F |` a ) i^i ran F ) = ran ( F |` a ) )
63	61, 62	mpbi 208	. . . . . . . . 9 |- ( ran ( F |` a ) i^i ran F ) = ran ( F |` a )
64	58, 63	eqtr2i 2497	. . . . . . . 8 |- ran ( F |` a ) = ( ran F i^i ran ( F |` a ) )
65	64	oveq2i 6295	. . . . . . 7 |- ( T ↾s ran ( F |` a ) ) = ( T ↾s ( ran F i^i ran ( F |` a ) ) )
66	57, 65	syl6reqr 2527	. . . . . 6 |- ( F e. ( S LMHom T ) -> ( T ↾s ran ( F |` a ) ) = ( V ↾s ran ( F |` a ) ) )
67	16, 66	syl 16	. . . . 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 1001	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> V e. LNoeM )
69		lmhmrnlss 17496	. . . . . . . 8 |- ( ( F |` a ) e. ( ( S ↾s a ) LMHom T ) -> ran ( F |` a ) e. ( LSubSp ` T ) )
70	6, 69	syl 16	. . . . . . 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 17478	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> T e. LMod )
73	16, 72	syl 16	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> T e. LMod )
74		lmhmrnlss 17496	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> ran F e. ( LSubSp ` T ) )
75	16, 74	syl 16	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ran F e. ( LSubSp ` T ) )
76		eqid 2467	. . . . . . . . 9 |- ( LSubSp ` T ) = ( LSubSp ` T )
77		eqid 2467	. . . . . . . . 9 |- ( LSubSp ` V ) = ( LSubSp ` V )
78	49, 76, 77	lsslss 17407	. . . . . . . 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 661	. . . . . . 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 920	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ran ( F |` a ) e. ( LSubSp ` V ) )
81		eqid 2467	. . . . . . 7 |- ( V ↾s ran ( F |` a ) ) = ( V ↾s ran ( F |` a ) )
82	77, 81	lnmlssfg 30658	. . . . . 6 |- ( ( V e. LNoeM /\ ran ( F |` a ) e. ( LSubSp ` V ) ) -> ( V ↾s ran ( F |` a ) ) e. LFinGen )
83	68, 80, 82	syl2anc 661	. . . . 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 2555	. . . 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 2467	. . . . 5 |- ( `' ( F |` a ) " { .0. } ) = ( `' ( F |` a ) " { .0. } )
86		eqid 2467	. . . . 5 |- ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) ) = ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) )
87		eqid 2467	. . . . 5 |- ( T ↾s ran ( F |` a ) ) = ( T ↾s ran ( F |` a ) )
88	35, 85, 86, 87	lmhmfgsplit 30664	. . . 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 1228	. . 3 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( S ↾s a ) e. LFinGen )
90	89	ralrimiva 2878	. 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 30655	. 2 |- ( S e. LNoeM <-> ( S e. LMod /\ A. a e. ( LSubSp ` S ) ( S ↾s a ) e. LFinGen ) )
92	2, 90, 91	sylanbrc 664	1 |- ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) -> S e. LNoeM )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   A.wral 2814   _Vcvv 3113   i^i cin 3475   C_ wss 3476   {csn 4027   `'ccnv 4998   ran crn 5000   |` cres 5001   "cima 5002   ` cfv 5588  (class class class)co 6284   ↾_s cress 14491   0gc0g 14695   LModclmod 17312   LSubSpclss 17378   LMHom clmhm 17465  LFinGenclfig 30645  LNoeMclnm 30653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-sca 14571  df-vsca 14572  df-0g 14697  df-mnd 15732  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-subg 16003  df-ghm 16070  df-cntz 16160  df-lsm 16462  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-lmod 17314  df-lss 17379  df-lsp 17418  df-lmhm 17468  df-lfig 30646  df-lnm 30654

This theorem is referenced by:  pwslnmlem2  30671