Theorem lmhmfgsplit 30664

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 998	. . 3 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> V e. LFinGen )
2		lmhmlmod2 17478	. . . . 5 |- ( F e. ( S LMHom T ) -> T e. LMod )
3	2	3ad2ant1 1017	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> T e. LMod )
4		lmhmrnlss 17496	. . . . 5 |- ( F e. ( S LMHom T ) -> ran F e. ( LSubSp ` T ) )
5	4	3ad2ant1 1017	. . . 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 2467	. . . . 5 |- ( LSubSp ` T ) = ( LSubSp ` T )
8		eqid 2467	. . . . 5 |- ( LSpan ` T ) = ( LSpan ` T )
9	6, 7, 8	islssfg 30648	. . . 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 661	. . 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 210	. 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 999	. . . . 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 2467	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
14		eqid 2467	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
15	13, 14	lmhmf 17480	. . . . 5 |- ( F e. ( S LMHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
16		ffn 5731	. . . . 5 |- ( F : ( Base ` S ) --> ( Base ` T ) -> F Fn ( Base ` S ) )
17	12, 15, 16	3syl 20	. . . 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 4019	. . . . 5 |- ( a e. ~P ran F -> a C_ ran F )
19	18	ad2antrl 727	. . . 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 763	. . . 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 7826	. . . 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 1228	. . 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 2467	. . . . . . 7 |- ( LSubSp ` S ) = ( LSubSp ` S )
24		eqid 2467	. . . . . . 7 |- ( LSSum ` S ) = ( LSSum ` S )
25		lmhmfgsplit.z	. . . . . . 7 |- .0. = ( 0g ` T )
26		lmhmfgsplit.k	. . . . . . 7 |- K = ( `' F " { .0. } )
27		simpll1 1035	. . . . . . 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 17479	. . . . . . . . . 10 |- ( F e. ( S LMHom T ) -> S e. LMod )
29	28	3ad2ant1 1017	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> S e. LMod )
30	29	ad2antrr 725	. . . . . . . 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 3718	. . . . . . . . . . 11 |- ( ~P ( Base ` S ) i^i Fin ) C_ ~P ( Base ` S )
32	31	sseli 3500	. . . . . . . . . 10 |- ( b e. ( ~P ( Base ` S ) i^i Fin ) -> b e. ~P ( Base ` S ) )
33		elpwi 4019	. . . . . . . . . 10 |- ( b e. ~P ( Base ` S ) -> b C_ ( Base ` S ) )
34	32, 33	syl 16	. . . . . . . . 9 |- ( b e. ( ~P ( Base ` S ) i^i Fin ) -> b C_ ( Base ` S ) )
35	34	ad2antrl 727	. . . . . . . 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 2467	. . . . . . . . 9 |- ( LSpan ` S ) = ( LSpan ` S )
37	13, 23, 36	lspcl 17422	. . . . . . . 8 |- ( ( S e. LMod /\ b C_ ( Base ` S ) ) -> ( ( LSpan ` S ) ` b ) e. ( LSubSp ` S ) )
38	30, 35, 37	syl2anc 661	. . . . . . 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 17495	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ b C_ ( Base ` S ) ) -> ( F " ( ( LSpan ` S ) ` b ) ) = ( ( LSpan ` T ) ` ( F " b ) ) )
40	27, 35, 39	syl2anc 661	. . . . . . . 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 5866	. . . . . . . . 9 |- ( ( F " b ) = a -> ( ( LSpan ` T ) ` ( F " b ) ) = ( ( LSpan ` T ) ` a ) )
42	41	ad2antll 728	. . . . . . . 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 1066	. . . . . . . . 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 1196	. . . . . . . 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 2512	. . . . . . 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 30661	. . . . . 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 6300	. . . . 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 14550	. . . . . . 7 |- ( S e. LMod -> ( S ↾s ( Base ` S ) ) = S )
49	29, 48	syl 16	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> ( S ↾s ( Base ` S ) ) = S )
50	49	ad2antrr 725	. . . . 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 2509	. . . 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 2467	. . . . 5 |- ( S ↾s ( ( LSpan ` S ) ` b ) ) = ( S ↾s ( ( LSpan ` S ) ` b ) )
54		eqid 2467	. . . . 5 |- ( S ↾s ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) ) = ( S ↾s ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) )
55	26, 25, 23	lmhmkerlss 17497	. . . . . . 7 |- ( F e. ( S LMHom T ) -> K e. ( LSubSp ` S ) )
56	55	3ad2ant1 1017	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> K e. ( LSubSp ` S ) )
57	56	ad2antrr 725	. . . . 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 1036	. . . . 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 3719	. . . . . . . 8 |- ( ~P ( Base ` S ) i^i Fin ) C_ Fin
60	59	sseli 3500	. . . . . . 7 |- ( b e. ( ~P ( Base ` S ) i^i Fin ) -> b e. Fin )
61	60	ad2antrl 727	. . . . . 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 30650	. . . . . 6 |- ( ( S e. LMod /\ b C_ ( Base ` S ) /\ b e. Fin ) -> ( S ↾s ( ( LSpan ` S ) ` b ) ) e. LFinGen )
63	30, 35, 61, 62	syl3anc 1228	. . . . 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 30652	. . . 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 2555	. . 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 2959	. 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 2959	1 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> S e. LFinGen )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   E.wrex 2815   i^i cin 3475   C_ wss 3476   ~Pcpw 4010   {csn 4027   `'ccnv 4998   ran crn 5000   "cima 5002   Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   Fincfn 7516   Basecbs 14490   ↾_s cress 14491   0gc0g 14695   LSSumclsm 16460   LModclmod 17312   LSubSpclss 17378   LSpanclspn 17417   LMHom clmhm 17465  LFinGenclfig 30645

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

This theorem is referenced by:  lmhmlnmsplit  30665