Theorem lmhmfgsplit 35650

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 1007	. . 3 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> V e. LFinGen )
2		lmhmlmod2 18190	. . . . 5 |- ( F e. ( S LMHom T ) -> T e. LMod )
3	2	3ad2ant1 1026	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> T e. LMod )
4		lmhmrnlss 18208	. . . . 5 |- ( F e. ( S LMHom T ) -> ran F e. ( LSubSp ` T ) )
5	4	3ad2ant1 1026	. . . 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 2429	. . . . 5 |- ( LSubSp ` T ) = ( LSubSp ` T )
8		eqid 2429	. . . . 5 |- ( LSpan ` T ) = ( LSpan ` T )
9	6, 7, 8	islssfg 35634	. . . 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 665	. . 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 213	. 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 1008	. . . . 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 2429	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
14		eqid 2429	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
15	13, 14	lmhmf 18192	. . . . 5 |- ( F e. ( S LMHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
16		ffn 5746	. . . . 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 3994	. . . . 5 |- ( a e. ~P ran F -> a C_ ran F )
19	18	ad2antrl 732	. . . 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 772	. . . 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 7886	. . . 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 1264	. . 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 2429	. . . . . . 7 |- ( LSubSp ` S ) = ( LSubSp ` S )
24		eqid 2429	. . . . . . 7 |- ( LSSum ` S ) = ( LSSum ` S )
25		lmhmfgsplit.z	. . . . . . 7 |- .0. = ( 0g ` T )
26		lmhmfgsplit.k	. . . . . . 7 |- K = ( `' F " { .0. } )
27		simpll1 1044	. . . . . . 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 18191	. . . . . . . . . 10 |- ( F e. ( S LMHom T ) -> S e. LMod )
29	28	3ad2ant1 1026	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> S e. LMod )
30	29	ad2antrr 730	. . . . . . . 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 3688	. . . . . . . . . . 11 |- ( ~P ( Base ` S ) i^i Fin ) C_ ~P ( Base ` S )
32	31	sseli 3466	. . . . . . . . . 10 |- ( b e. ( ~P ( Base ` S ) i^i Fin ) -> b e. ~P ( Base ` S ) )
33		elpwi 3994	. . . . . . . . . 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 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 ) ) -> b C_ ( Base ` S ) )
36		eqid 2429	. . . . . . . . 9 |- ( LSpan ` S ) = ( LSpan ` S )
37	13, 23, 36	lspcl 18134	. . . . . . . 8 |- ( ( S e. LMod /\ b C_ ( Base ` S ) ) -> ( ( LSpan ` S ) ` b ) e. ( LSubSp ` S ) )
38	30, 35, 37	syl2anc 665	. . . . . . 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 18207	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ b C_ ( Base ` S ) ) -> ( F " ( ( LSpan ` S ) ` b ) ) = ( ( LSpan ` T ) ` ( F " b ) ) )
40	27, 35, 39	syl2anc 665	. . . . . . . 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 5881	. . . . . . . . 9 |- ( ( F " b ) = a -> ( ( LSpan ` T ) ` ( F " b ) ) = ( ( LSpan ` T ) ` a ) )
42	41	ad2antll 733	. . . . . . . 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 1075	. . . . . . . . 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 1205	. . . . . . . 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 2474	. . . . . . 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 35647	. . . . . 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 6321	. . . . 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 15146	. . . . . . 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 730	. . . . 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 2471	. . . 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 2429	. . . . 5 |- ( S ↾s ( ( LSpan ` S ) ` b ) ) = ( S ↾s ( ( LSpan ` S ) ` b ) )
54		eqid 2429	. . . . 5 |- ( S ↾s ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) ) = ( S ↾s ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) )
55	26, 25, 23	lmhmkerlss 18209	. . . . . . 7 |- ( F e. ( S LMHom T ) -> K e. ( LSubSp ` S ) )
56	55	3ad2ant1 1026	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> K e. ( LSubSp ` S ) )
57	56	ad2antrr 730	. . . . 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 1045	. . . . 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 3689	. . . . . . . 8 |- ( ~P ( Base ` S ) i^i Fin ) C_ Fin
60	59	sseli 3466	. . . . . . 7 |- ( b e. ( ~P ( Base ` S ) i^i Fin ) -> b e. Fin )
61	60	ad2antrl 732	. . . . . 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 35636	. . . . . 6 |- ( ( S e. LMod /\ b C_ ( Base ` S ) /\ b e. Fin ) -> ( S ↾s ( ( LSpan ` S ) ` b ) ) e. LFinGen )
63	30, 35, 61, 62	syl3anc 1264	. . . . 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 35638	. . . 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 2517	. . 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 2928	. 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 2928	1 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> S e. LFinGen )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1870   E.wrex 2783   i^i cin 3441   C_ wss 3442   ~Pcpw 3985   {csn 4002   `'ccnv 4853   ran crn 4855   "cima 4857   Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   Fincfn 7577   Basecbs 15084   ↾_s cress 15085   0gc0g 15297   LSSumclsm 17221   LModclmod 18026   LSubSpclss 18090   LSpanclspn 18129   LMHom clmhm 18177  LFinGenclfig 35631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-sca 15168  df-vsca 15169  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-ghm 16832  df-cntz 16922  df-lsm 17223  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-lmod 18028  df-lss 18091  df-lsp 18130  df-lmhm 18180  df-lfig 35632

This theorem is referenced by:  lmhmlnmsplit  35651