Theorem lmhmlnmsplit 29466

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 17136	. . 3 |- ( F e. ( S LMHom T ) -> S e. LMod )
2	1	3ad2ant1 1009	. 2 |- ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) -> S e. LMod )
3		eqid 2443	. . . . . 6 |- ( LSubSp ` S ) = ( LSubSp ` S )
4		eqid 2443	. . . . . 6 |- ( S ↾s a ) = ( S ↾s a )
5	3, 4	reslmhm 17155	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ a e. ( LSubSp ` S ) ) -> ( F |` a ) e. ( ( S ↾s a ) LMHom T ) )
6	5	3ad2antl1 1150	. . . 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 5348	. . . . . . . 8 |- ( `' ( F |` a ) " { .0. } ) = ( ( `' F " { .0. } ) i^i a )
8		lmhmfgsplit.k	. . . . . . . . . 10 |- K = ( `' F " { .0. } )
9	8	eqcomi 2447	. . . . . . . . 9 |- ( `' F " { .0. } ) = K
10	9	ineq1i 3569	. . . . . . . 8 |- ( ( `' F " { .0. } ) i^i a ) = ( K i^i a )
11		incom 3564	. . . . . . . 8 |- ( K i^i a ) = ( a i^i K )
12	7, 10, 11	3eqtri 2467	. . . . . . 7 |- ( `' ( F |` a ) " { .0. } ) = ( a i^i K )
13	12	oveq2i 6123	. . . . . 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 6122	. . . . . . . 8 |- ( U ↾s ( a i^i K ) ) = ( ( S ↾s K ) ↾s ( a i^i K ) )
16		simpl1 991	. . . . . . . . . 10 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> F e. ( S LMHom T ) )
17		cnvexg 6545	. . . . . . . . . . . 12 |- ( F e. ( S LMHom T ) -> `' F e. _V )
18		imaexg 6536	. . . . . . . . . . . 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 2527	. . . . . . . . . 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 3592	. . . . . . . . 9 |- ( a i^i K ) C_ K
23		ressabs 14257	. . . . . . . . 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 2487	. . . . . . 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 2996	. . . . . . . 8 |- a e. _V
27		inss1 3591	. . . . . . . 8 |- ( a i^i K ) C_ a
28		ressabs 14257	. . . . . . . 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 2494	. . . . . 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 2487	. . . . 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 992	. . . . . 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 17154	. . . . . . . . 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 17068	. . . . . . . 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 1218	. . . . . . 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 2443	. . . . . . . . 9 |- ( LSubSp ` U ) = ( LSubSp ` U )
42	14, 3, 41	lsslss 17064	. . . . . . . 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 913	. . . . . 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 2443	. . . . . . 7 |- ( U ↾s ( a i^i K ) ) = ( U ↾s ( a i^i K ) )
46	41, 45	lnmlssfg 29459	. . . . . 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 2517	. . . 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 6122	. . . . . . . 8 |- ( V ↾s ran ( F |` a ) ) = ( ( T ↾s ran F ) ↾s ran ( F |` a ) )
51		rnexg 6531	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> ran F e. _V )
52		resexg 5170	. . . . . . . . . 10 |- ( F e. ( S LMHom T ) -> ( F |` a ) e. _V )
53		rnexg 6531	. . . . . . . . . 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 14256	. . . . . . . . 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 2487	. . . . . . 7 |- ( F e. ( S LMHom T ) -> ( V ↾s ran ( F |` a ) ) = ( T ↾s ( ran F i^i ran ( F |` a ) ) ) )
58		incom 3564	. . . . . . . . 9 |- ( ran F i^i ran ( F |` a ) ) = ( ran ( F |` a ) i^i ran F )
59		resss 5155	. . . . . . . . . . 11 |- ( F |` a ) C_ F
60		rnss 5089	. . . . . . . . . . 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 3363	. . . . . . . . . 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 2464	. . . . . . . 8 |- ran ( F |` a ) = ( ran F i^i ran ( F |` a ) )
65	64	oveq2i 6123	. . . . . . 7 |- ( T ↾s ran ( F |` a ) ) = ( T ↾s ( ran F i^i ran ( F |` a ) ) )
66	57, 65	syl6reqr 2494	. . . . . 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 993	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> V e. LNoeM )
69		lmhmrnlss 17153	. . . . . . . 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 17135	. . . . . . . . 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 17153	. . . . . . . . 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 2443	. . . . . . . . 9 |- ( LSubSp ` T ) = ( LSubSp ` T )
77		eqid 2443	. . . . . . . . 9 |- ( LSubSp ` V ) = ( LSubSp ` V )
78	49, 76, 77	lsslss 17064	. . . . . . . 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 913	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ran ( F |` a ) e. ( LSubSp ` V ) )
81		eqid 2443	. . . . . . 7 |- ( V ↾s ran ( F |` a ) ) = ( V ↾s ran ( F |` a ) )
82	77, 81	lnmlssfg 29459	. . . . . 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 2517	. . . 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 2443	. . . . 5 |- ( `' ( F |` a ) " { .0. } ) = ( `' ( F |` a ) " { .0. } )
86		eqid 2443	. . . . 5 |- ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) ) = ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) )
87		eqid 2443	. . . . 5 |- ( T ↾s ran ( F |` a ) ) = ( T ↾s ran ( F |` a ) )
88	35, 85, 86, 87	lmhmfgsplit 29465	. . . 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 1218	. . 3 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( S ↾s a ) e. LFinGen )
90	89	ralrimiva 2820	. 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 29456	. 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 965   = wceq 1369   e. wcel 1756   A.wral 2736   _Vcvv 2993   i^i cin 3348   C_ wss 3349   {csn 3898   `'ccnv 4860   ran crn 4862   |` cres 4863   "cima 4864   ` cfv 5439  (class class class)co 6112   ↾_s cress 14196   0gc0g 14399   LModclmod 16970   LSubSpclss 17035   LMHom clmhm 17122  LFinGenclfig 29446  LNoeMclnm 29454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-sca 14275  df-vsca 14276  df-0g 14401  df-mnd 15436  df-submnd 15486  df-grp 15566  df-minusg 15567  df-sbg 15568  df-subg 15699  df-ghm 15766  df-cntz 15856  df-lsm 16156  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-lmod 16972  df-lss 17036  df-lsp 17075  df-lmhm 17125  df-lfig 29447  df-lnm 29455

This theorem is referenced by:  pwslnmlem2  29472