Theorem lmhmlnmsplit 31272

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 17874	. . 3 |- ( F e. ( S LMHom T ) -> S e. LMod )
2	1	3ad2ant1 1015	. 2 |- ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) -> S e. LMod )
3		eqid 2454	. . . . . 6 |- ( LSubSp ` S ) = ( LSubSp ` S )
4		eqid 2454	. . . . . 6 |- ( S ↾s a ) = ( S ↾s a )
5	3, 4	reslmhm 17893	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ a e. ( LSubSp ` S ) ) -> ( F |` a ) e. ( ( S ↾s a ) LMHom T ) )
6	5	3ad2antl1 1156	. . . 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 5479	. . . . . . . 8 |- ( `' ( F |` a ) " { .0. } ) = ( ( `' F " { .0. } ) i^i a )
8		lmhmfgsplit.k	. . . . . . . . . 10 |- K = ( `' F " { .0. } )
9	8	eqcomi 2467	. . . . . . . . 9 |- ( `' F " { .0. } ) = K
10	9	ineq1i 3682	. . . . . . . 8 |- ( ( `' F " { .0. } ) i^i a ) = ( K i^i a )
11		incom 3677	. . . . . . . 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 6281	. . . . . 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 6280	. . . . . . . 8 |- ( U ↾s ( a i^i K ) ) = ( ( S ↾s K ) ↾s ( a i^i K ) )
16		simpl1 997	. . . . . . . . . 10 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> F e. ( S LMHom T ) )
17		cnvexg 6719	. . . . . . . . . . . 12 |- ( F e. ( S LMHom T ) -> `' F e. _V )
18		imaexg 6710	. . . . . . . . . . . 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 2546	. . . . . . . . . 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 3705	. . . . . . . . 9 |- ( a i^i K ) C_ K
23		ressabs 14782	. . . . . . . . 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 660	. . . . . . . 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 3109	. . . . . . . 8 |- a e. _V
27		inss1 3704	. . . . . . . 8 |- ( a i^i K ) C_ a
28		ressabs 14782	. . . . . . . 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 670	. . . . . . 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 998	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> U e. LNoeM )
33	2	adantr 463	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> S e. LMod )
34		simpr 459	. . . . . . . 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 17892	. . . . . . . . 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 17806	. . . . . . . 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 1226	. . . . . . 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 2454	. . . . . . . . 9 |- ( LSubSp ` U ) = ( LSubSp ` U )
42	14, 3, 41	lsslss 17802	. . . . . . . 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 659	. . . . . . 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 2454	. . . . . . 7 |- ( U ↾s ( a i^i K ) ) = ( U ↾s ( a i^i K ) )
46	41, 45	lnmlssfg 31265	. . . . . 6 |- ( ( U e. LNoeM /\ ( a i^i K ) e. ( LSubSp ` U ) ) -> ( U ↾s ( a i^i K ) ) e. LFinGen )
47	32, 44, 46	syl2anc 659	. . . . 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 2542	. . . 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 6280	. . . . . . . 8 |- ( V ↾s ran ( F |` a ) ) = ( ( T ↾s ran F ) ↾s ran ( F |` a ) )
51		rnexg 6705	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> ran F e. _V )
52		resexg 5304	. . . . . . . . . 10 |- ( F e. ( S LMHom T ) -> ( F |` a ) e. _V )
53		rnexg 6705	. . . . . . . . . 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 14781	. . . . . . . . 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 659	. . . . . . . 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 3677	. . . . . . . . 9 |- ( ran F i^i ran ( F |` a ) ) = ( ran ( F |` a ) i^i ran F )
59		resss 5285	. . . . . . . . . . 11 |- ( F |` a ) C_ F
60		rnss 5220	. . . . . . . . . . 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 3475	. . . . . . . . . 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 2484	. . . . . . . 8 |- ran ( F |` a ) = ( ran F i^i ran ( F |` a ) )
65	64	oveq2i 6281	. . . . . . 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 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 999	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> V e. LNoeM )
69		lmhmrnlss 17891	. . . . . . . 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 17873	. . . . . . . . 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 17891	. . . . . . . . 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 2454	. . . . . . . . 9 |- ( LSubSp ` T ) = ( LSubSp ` T )
77		eqid 2454	. . . . . . . . 9 |- ( LSubSp ` V ) = ( LSubSp ` V )
78	49, 76, 77	lsslss 17802	. . . . . . . 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 659	. . . . . . 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 2454	. . . . . . 7 |- ( V ↾s ran ( F |` a ) ) = ( V ↾s ran ( F |` a ) )
82	77, 81	lnmlssfg 31265	. . . . . 6 |- ( ( V e. LNoeM /\ ran ( F |` a ) e. ( LSubSp ` V ) ) -> ( V ↾s ran ( F |` a ) ) e. LFinGen )
83	68, 80, 82	syl2anc 659	. . . . 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 2542	. . . 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 2454	. . . . 5 |- ( `' ( F |` a ) " { .0. } ) = ( `' ( F |` a ) " { .0. } )
86		eqid 2454	. . . . 5 |- ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) ) = ( ( S ↾s a ) ↾s ( `' ( F |` a ) " { .0. } ) )
87		eqid 2454	. . . . 5 |- ( T ↾s ran ( F |` a ) ) = ( T ↾s ran ( F |` a ) )
88	35, 85, 86, 87	lmhmfgsplit 31271	. . . 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 1226	. . 3 |- ( ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) /\ a e. ( LSubSp ` S ) ) -> ( S ↾s a ) e. LFinGen )
90	89	ralrimiva 2868	. 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 31262	. 2 |- ( S e. LNoeM <-> ( S e. LMod /\ A. a e. ( LSubSp ` S ) ( S ↾s a ) e. LFinGen ) )
92	2, 90, 91	sylanbrc 662	1 |- ( ( F e. ( S LMHom T ) /\ U e. LNoeM /\ V e. LNoeM ) -> S e. LNoeM )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   A.wral 2804   _Vcvv 3106   i^i cin 3460   C_ wss 3461   {csn 4016   `'ccnv 4987   ran crn 4989   |` cres 4990   "cima 4991   ` cfv 5570  (class class class)co 6270   ↾_s cress 14717   0gc0g 14929   LModclmod 17707   LSubSpclss 17773   LMHom clmhm 17860  LFinGenclfig 31252  LNoeMclnm 31260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-sca 14800  df-vsca 14801  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-ghm 16464  df-cntz 16554  df-lsm 16855  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-lmod 17709  df-lss 17774  df-lsp 17813  df-lmhm 17863  df-lfig 31253  df-lnm 31261

This theorem is referenced by:  pwslnmlem2  31278