Theorem lmhmfgsplit 35944

Description: If the kernel and range of a homomorphism of left modules are finitely generated, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 6-May-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
lmhmfgsplit	|- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> S e. LFinGen )

Proof of Theorem lmhmfgsplit

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1010	. . 3 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> V e. LFinGen )
2		lmhmlmod2 18255	. . . . 5 |- ( F e. ( S LMHom T ) -> T e. LMod )
3	2	3ad2ant1 1029	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> T e. LMod )
4		lmhmrnlss 18273	. . . . 5 |- ( F e. ( S LMHom T ) -> ran F e. ( LSubSp ` T ) )
5	4	3ad2ant1 1029	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> ran F e. ( LSubSp ` T ) )
6		lmhmfgsplit.v	. . . . 5 |- V = ( T ↾s ran F )
7		eqid 2451	. . . . 5 |- ( LSubSp ` T ) = ( LSubSp ` T )
8		eqid 2451	. . . . 5 |- ( LSpan ` T ) = ( LSpan ` T )
9	6, 7, 8	islssfg 35928	. . . 4 |- ( ( T e. LMod /\ ran F e. ( LSubSp ` T ) ) -> ( V e. LFinGen <-> E. a e. ~P ran F ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) )
10	3, 5, 9	syl2anc 667	. . 3 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> ( V e. LFinGen <-> E. a e. ~P ran F ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) )
11	1, 10	mpbid 214	. 2 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> E. a e. ~P ran F ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) )
12		simpl1 1011	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) -> F e. ( S LMHom T ) )
13		eqid 2451	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
14		eqid 2451	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
15	13, 14	lmhmf 18257	. . . . 5 |- ( F e. ( S LMHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
16		ffn 5728	. . . . 5 |- ( F : ( Base ` S ) --> ( Base ` T ) -> F Fn ( Base ` S ) )
17	12, 15, 16	3syl 18	. . . 4 |- ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) -> F Fn ( Base ` S ) )
18		elpwi 3960	. . . . 5 |- ( a e. ~P ran F -> a C_ ran F )
19	18	ad2antrl 734	. . . 4 |- ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) -> a C_ ran F )
20		simprrl 774	. . . 4 |- ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) -> a e. Fin )
21		fipreima 7880	. . . 4 |- ( ( F Fn ( Base ` S ) /\ a C_ ran F /\ a e. Fin ) -> E. b e. ( ~P ( Base ` S ) i^i Fin ) ( F " b ) = a )
22	17, 19, 20, 21	syl3anc 1268	. . 3 |- ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) -> E. b e. ( ~P ( Base ` S ) i^i Fin ) ( F " b ) = a )
23		eqid 2451	. . . . . . 7 |- ( LSubSp ` S ) = ( LSubSp ` S )
24		eqid 2451	. . . . . . 7 |- ( LSSum ` S ) = ( LSSum ` S )
25		lmhmfgsplit.z	. . . . . . 7 |- .0. = ( 0g ` T )
26		lmhmfgsplit.k	. . . . . . 7 |- K = ( `' F " { .0. } )
27		simpll1 1047	. . . . . . 7 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> F e. ( S LMHom T ) )
28		lmhmlmod1 18256	. . . . . . . . . 10 |- ( F e. ( S LMHom T ) -> S e. LMod )
29	28	3ad2ant1 1029	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> S e. LMod )
30	29	ad2antrr 732	. . . . . . . 8 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> S e. LMod )
31		inss1 3652	. . . . . . . . . . 11 |- ( ~P ( Base ` S ) i^i Fin ) C_ ~P ( Base ` S )
32	31	sseli 3428	. . . . . . . . . 10 |- ( b e. ( ~P ( Base ` S ) i^i Fin ) -> b e. ~P ( Base ` S ) )
33		elpwi 3960	. . . . . . . . . 10 |- ( b e. ~P ( Base ` S ) -> b C_ ( Base ` S ) )
34	32, 33	syl 17	. . . . . . . . 9 |- ( b e. ( ~P ( Base ` S ) i^i Fin ) -> b C_ ( Base ` S ) )
35	34	ad2antrl 734	. . . . . . . 8 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> b C_ ( Base ` S ) )
36		eqid 2451	. . . . . . . . 9 |- ( LSpan ` S ) = ( LSpan ` S )
37	13, 23, 36	lspcl 18199	. . . . . . . 8 |- ( ( S e. LMod /\ b C_ ( Base ` S ) ) -> ( ( LSpan ` S ) ` b ) e. ( LSubSp ` S ) )
38	30, 35, 37	syl2anc 667	. . . . . . 7 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> ( ( LSpan ` S ) ` b ) e. ( LSubSp ` S ) )
39	13, 36, 8	lmhmlsp 18272	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ b C_ ( Base ` S ) ) -> ( F " ( ( LSpan ` S ) ` b ) ) = ( ( LSpan ` T ) ` ( F " b ) ) )
40	27, 35, 39	syl2anc 667	. . . . . . . 8 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> ( F " ( ( LSpan ` S ) ` b ) ) = ( ( LSpan ` T ) ` ( F " b ) ) )
41		fveq2 5865	. . . . . . . . 9 |- ( ( F " b ) = a -> ( ( LSpan ` T ) ` ( F " b ) ) = ( ( LSpan ` T ) ` a ) )
42	41	ad2antll 735	. . . . . . . 8 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> ( ( LSpan ` T ) ` ( F " b ) ) = ( ( LSpan ` T ) ` a ) )
43		simp2rr 1078	. . . . . . . . 9 |- ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> ( ( LSpan ` T ) ` a ) = ran F )
44	43	3expa 1208	. . . . . . . 8 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> ( ( LSpan ` T ) ` a ) = ran F )
45	40, 42, 44	3eqtrd 2489	. . . . . . 7 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> ( F " ( ( LSpan ` S ) ` b ) ) = ran F )
46	23, 24, 25, 26, 13, 27, 38, 45	kercvrlsm 35941	. . . . . 6 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) = ( Base ` S ) )
47	46	oveq2d 6306	. . . . 5 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> ( S ↾s ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) ) = ( S ↾s ( Base ` S ) ) )
48	13	ressid 15184	. . . . . . 7 |- ( S e. LMod -> ( S ↾s ( Base ` S ) ) = S )
49	29, 48	syl 17	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> ( S ↾s ( Base ` S ) ) = S )
50	49	ad2antrr 732	. . . . 5 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> ( S ↾s ( Base ` S ) ) = S )
51	47, 50	eqtr2d 2486	. . . 4 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> S = ( S ↾s ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) ) )
52		lmhmfgsplit.u	. . . . 5 |- U = ( S ↾s K )
53		eqid 2451	. . . . 5 |- ( S ↾s ( ( LSpan ` S ) ` b ) ) = ( S ↾s ( ( LSpan ` S ) ` b ) )
54		eqid 2451	. . . . 5 |- ( S ↾s ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) ) = ( S ↾s ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) )
55	26, 25, 23	lmhmkerlss 18274	. . . . . . 7 |- ( F e. ( S LMHom T ) -> K e. ( LSubSp ` S ) )
56	55	3ad2ant1 1029	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> K e. ( LSubSp ` S ) )
57	56	ad2antrr 732	. . . . 5 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> K e. ( LSubSp ` S ) )
58		simpll2 1048	. . . . 5 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> U e. LFinGen )
59		inss2 3653	. . . . . . . 8 |- ( ~P ( Base ` S ) i^i Fin ) C_ Fin
60	59	sseli 3428	. . . . . . 7 |- ( b e. ( ~P ( Base ` S ) i^i Fin ) -> b e. Fin )
61	60	ad2antrl 734	. . . . . 6 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> b e. Fin )
62	36, 13, 53	islssfgi 35930	. . . . . 6 |- ( ( S e. LMod /\ b C_ ( Base ` S ) /\ b e. Fin ) -> ( S ↾s ( ( LSpan ` S ) ` b ) ) e. LFinGen )
63	30, 35, 61, 62	syl3anc 1268	. . . . 5 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> ( S ↾s ( ( LSpan ` S ) ` b ) ) e. LFinGen )
64	23, 24, 52, 53, 54, 30, 57, 38, 58, 63	lsmfgcl 35932	. . . 4 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> ( S ↾s ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) ) e. LFinGen )
65	51, 64	eqeltrd 2529	. . 3 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) /\ ( b e. ( ~P ( Base ` S ) i^i Fin ) /\ ( F " b ) = a ) ) -> S e. LFinGen )
66	22, 65	rexlimddv 2883	. 2 |- ( ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) /\ ( a e. ~P ran F /\ ( a e. Fin /\ ( ( LSpan ` T ) ` a ) = ran F ) ) ) -> S e. LFinGen )
67	11, 66	rexlimddv 2883	1 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> S e. LFinGen )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   E.wrex 2738   i^i cin 3403   C_ wss 3404   ~Pcpw 3951   {csn 3968   `'ccnv 4833   ran crn 4835   "cima 4837   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   Fincfn 7569   Basecbs 15121   ↾_s cress 15122   0gc0g 15338   LSSumclsm 17286   LModclmod 18091   LSubSpclss 18155   LSpanclspn 18194   LMHom clmhm 18242  LFinGenclfig 35925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-sca 15206  df-vsca 15207  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-subg 16814  df-ghm 16881  df-cntz 16971  df-lsm 17288  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-lmod 18093  df-lss 18156  df-lsp 18195  df-lmhm 18245  df-lfig 35926

This theorem is referenced by:  lmhmlnmsplit  35945