Theorem lmhmlnmsplit 37657

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 19033	. . 3 |- ( F e. ( S LMHom T ) -> S e. LMod )
2	1	3ad2ant1 1082	. 2 |- ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) -> S e. LMod )
3		eqid 2622	. . . . . 6 |- ( LSubSp ` S ) = ( LSubSp ` S )
4		eqid 2622	. . . . . 6 |- ( S ↾s a ) = ( S ↾s a )
5	3, 4	reslmhm 19052	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ a e. ( LSubSp ` S ) ) -> ( F |` a ) e. ( ( S ↾s a ) LMHom T ) )
6	5	3ad2antl1 1223	. . . 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 5623	. . . . . . . 8 |- ( `' ( F |` a ) " { .0. } ) = ( ( `' F " { .0. } ) i^i a )
8		lmhmfgsplit.k	. . . . . . . . . 10 |- K = ( `' F " { .0. } )
9	8	eqcomi 2631	. . . . . . . . 9 |- ( `' F " { .0. } ) = K
10	9	ineq1i 3810	. . . . . . . 8 |- ( ( `' F " { .0. } ) i^i a ) = ( K i^i a )
11		incom 3805	. . . . . . . 8 |- ( K i^i a ) = ( a i^i K )
12	7, 10, 11	3eqtri 2648	. . . . . . 7 |- ( `' ( F |` a ) " { .0. } ) = ( a i^i K )
13	12	oveq2i 6661	. . . . . 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 6660	. . . . . . . 8 |- ( U ↾s ( a i^i K ) ) = ( ( S ↾s K ) ↾s ( a i^i K ) )
16		simpl1 1064	. . . . . . . . . 10 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> F e. ( S LMHom T ) )
17		cnvexg 7112	. . . . . . . . . . . 12 |- ( F e. ( S LMHom T ) -> `' F e. _V )
18		imaexg 7103	. . . . . . . . . . . 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 2705	. . . . . . . . . 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 3834	. . . . . . . . 9 |- ( a i^i K ) C_ K
23		ressabs 15939	. . . . . . . . 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 694	. . . . . . . 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 2668	. . . . . . 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 3203	. . . . . . . 8 |- a e. _V
27		inss1 3833	. . . . . . . 8 |- ( a i^i K ) C_ a
28		ressabs 15939	. . . . . . . 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 708	. . . . . . 7 |- ( ( S ↾s a ) ↾s ( a i^i K ) ) = ( S ↾s ( a i^i K ) )
30	25, 29	syl6reqr 2675	. . . . . 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 2668	. . . . 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 1065	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> U e. LNoeM )
33	2	adantr 481	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> S e. LMod )
34		simpr 477	. . . . . . . 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 19051	. . . . . . . . 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 18965	. . . . . . . 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 1326	. . . . . . 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 2622	. . . . . . . . 9 |- ( LSubSp ` U ) = ( LSubSp ` U )
42	14, 3, 41	lsslss 18961	. . . . . . . 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 693	. . . . . . 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 957	. . . . . 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 2622	. . . . . . 7 |- ( U ↾s ( a i^i K ) ) = ( U ↾s ( a i^i K ) )
46	41, 45	lnmlssfg 37650	. . . . . 6 |- ( ( U e. LNoeM /\ ( a i^i K ) e. ( LSubSp ` U ) ) -> ( U ↾s ( a i^i K ) ) e. LFinGen )
47	32, 44, 46	syl2anc 693	. . . . 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 2701	. . . 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 6660	. . . . . . . 8 |- ( V ↾s ran ( F |` a ) ) = ( ( T ↾s ran F ) ↾s ran ( F |` a ) )
51		rnexg 7098	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> ran F e. _V )
52		resexg 5442	. . . . . . . . . 10 |- ( F e. ( S LMHom T ) -> ( F |` a ) e. _V )
53		rnexg 7098	. . . . . . . . . 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 15938	. . . . . . . . 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 693	. . . . . . . 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 2668	. . . . . . 7 |- ( F e. ( S LMHom T ) -> ( V ↾s ran ( F |` a ) ) = ( T ↾s ( ran F i^i ran ( F |` a ) ) ) )
58		incom 3805	. . . . . . . . 9 |- ( ran F i^i ran ( F |` a ) ) = ( ran ( F |` a ) i^i ran F )
59		resss 5422	. . . . . . . . . . 11 |- ( F |` a ) C_ F
60		rnss 5354	. . . . . . . . . . 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 3588	. . . . . . . . . 10 |- ( ran ( F |` a ) C_ ran F <-> ( ran ( F |` a ) i^i ran F ) = ran ( F |` a ) )
63	61, 62	mpbi 220	. . . . . . . . 9 |- ( ran ( F |` a ) i^i ran F ) = ran ( F |` a )
64	58, 63	eqtr2i 2645	. . . . . . . 8 |- ran ( F |` a ) = ( ran F i^i ran ( F |` a ) )
65	64	oveq2i 6661	. . . . . . 7 |- ( T ↾s ran ( F |` a ) ) = ( T ↾s ( ran F i^i ran ( F |` a ) ) )
66	57, 65	syl6reqr 2675	. . . . . 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 1066	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> V e. LNoeM )
69		lmhmrnlss 19050	. . . . . . . 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 19032	. . . . . . . . 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 19050	. . . . . . . . 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 2622	. . . . . . . . 9 |- ( LSubSp ` T ) = ( LSubSp ` T )
77		eqid 2622	. . . . . . . . 9 |- ( LSubSp ` V ) = ( LSubSp ` V )
78	49, 76, 77	lsslss 18961	. . . . . . . 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 693	. . . . . . 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 957	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ran ( F |` a ) e. ( LSubSp ` V ) )
81		eqid 2622	. . . . . . 7 |- ( V ↾s ran ( F |` a ) ) = ( V ↾s ran ( F |` a ) )
82	77, 81	lnmlssfg 37650	. . . . . 6 |- ( ( V e. LNoeM /\ ran ( F |` a ) e. ( LSubSp ` V ) ) -> ( V ↾s ran ( F |` a ) ) e. LFinGen )
83	68, 80, 82	syl2anc 693	. . . . 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 2701	. . . 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 2622	. . . . 5 |- ( `' ( F |` a ) " { .0. } ) = ( `' ( F |` a ) " { .0. } )
86		eqid 2622	. . . . 5 |- ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) ) = ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) )
87		eqid 2622	. . . . 5 |- ( T ↾s ran ( F |` a ) ) = ( T ↾s ran ( F |` a ) )
88	35, 85, 86, 87	lmhmfgsplit 37656	. . . 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 1326	. . 3 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( S ↾s a ) e. LFinGen )
90	89	ralrimiva 2966	. 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 37647	. 2 |- ( S e. LNoeM <-> ( S e. LMod /\ A. a e. ( LSubSp ` S ) ( S ↾s a ) e. LFinGen ) )
92	2, 90, 91	sylanbrc 698	1 |- ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) -> S e. LNoeM )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   ` cfv 5888  (class class class)co 6650   ↾_s cress 15858   0gc0g 16100   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-oadd 7564  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-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  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:  pwslnmlem2  37663