Theorem lmhmfgsplit 37656

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 1063	. . 3 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> V e. LFinGen )
2		lmhmlmod2 19032	. . . . 5 |- ( F e. ( S LMHom T ) -> T e. LMod )
3	2	3ad2ant1 1082	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> T e. LMod )
4		lmhmrnlss 19050	. . . . 5 |- ( F e. ( S LMHom T ) -> ran F e. ( LSubSp ` T ) )
5	4	3ad2ant1 1082	. . . 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 2622	. . . . 5 |- ( LSubSp ` T ) = ( LSubSp ` T )
8		eqid 2622	. . . . 5 |- ( LSpan ` T ) = ( LSpan ` T )
9	6, 7, 8	islssfg 37640	. . . 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 693	. . 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 222	. 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 1064	. . . . 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 2622	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
14		eqid 2622	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
15	13, 14	lmhmf 19034	. . . . 5 |- ( F e. ( S LMHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
16		ffn 6045	. . . . 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 4168	. . . . 5 |- ( a e. ~P ran F -> a C_ ran F )
19	18	ad2antrl 764	. . . 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 804	. . . 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 8272	. . . 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 1326	. . 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 2622	. . . . . . 7 |- ( LSubSp ` S ) = ( LSubSp ` S )
24		eqid 2622	. . . . . . 7 |- ( LSSum ` S ) = ( LSSum ` S )
25		lmhmfgsplit.z	. . . . . . 7 |- .0. = ( 0g ` T )
26		lmhmfgsplit.k	. . . . . . 7 |- K = ( `' F " { .0. } )
27		simpll1 1100	. . . . . . 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 19033	. . . . . . . . . 10 |- ( F e. ( S LMHom T ) -> S e. LMod )
29	28	3ad2ant1 1082	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> S e. LMod )
30	29	ad2antrr 762	. . . . . . . 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 3833	. . . . . . . . . . 11 |- ( ~P ( Base ` S ) i^i Fin ) C_ ~P ( Base ` S )
32	31	sseli 3599	. . . . . . . . . 10 |- ( b e. ( ~P ( Base ` S ) i^i Fin ) -> b e. ~P ( Base ` S ) )
33		elpwi 4168	. . . . . . . . . 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 764	. . . . . . . 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 2622	. . . . . . . . 9 |- ( LSpan ` S ) = ( LSpan ` S )
37	13, 23, 36	lspcl 18976	. . . . . . . 8 |- ( ( S e. LMod /\ b C_ ( Base ` S ) ) -> ( ( LSpan ` S ) ` b ) e. ( LSubSp ` S ) )
38	30, 35, 37	syl2anc 693	. . . . . . 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 19049	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ b C_ ( Base ` S ) ) -> ( F " ( ( LSpan ` S ) ` b ) ) = ( ( LSpan ` T ) ` ( F " b ) ) )
40	27, 35, 39	syl2anc 693	. . . . . . . 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 6191	. . . . . . . . 9 |- ( ( F " b ) = a -> ( ( LSpan ` T ) ` ( F " b ) ) = ( ( LSpan ` T ) ` a ) )
42	41	ad2antll 765	. . . . . . . 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 1131	. . . . . . . . 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 1265	. . . . . . . 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 2660	. . . . . . 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 37653	. . . . . 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 6666	. . . . 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 15935	. . . . . . 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 762	. . . . 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 2657	. . . 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 2622	. . . . 5 |- ( S ↾s ( ( LSpan ` S ) ` b ) ) = ( S ↾s ( ( LSpan ` S ) ` b ) )
54		eqid 2622	. . . . 5 |- ( S ↾s ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) ) = ( S ↾s ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) )
55	26, 25, 23	lmhmkerlss 19051	. . . . . . 7 |- ( F e. ( S LMHom T ) -> K e. ( LSubSp ` S ) )
56	55	3ad2ant1 1082	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> K e. ( LSubSp ` S ) )
57	56	ad2antrr 762	. . . . 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 1101	. . . . 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 3834	. . . . . . . 8 |- ( ~P ( Base ` S ) i^i Fin ) C_ Fin
60	59	sseli 3599	. . . . . . 7 |- ( b e. ( ~P ( Base ` S ) i^i Fin ) -> b e. Fin )
61	60	ad2antrl 764	. . . . . 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 37642	. . . . . 6 |- ( ( S e. LMod /\ b C_ ( Base ` S ) /\ b e. Fin ) -> ( S ↾s ( ( LSpan ` S ) ` b ) ) e. LFinGen )
63	30, 35, 61, 62	syl3anc 1326	. . . . 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 37644	. . . 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 2701	. . 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 3035	. 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 3035	1 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> S e. LFinGen )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   `'ccnv 5113   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857   ↾_s cress 15858   0gc0g 16100   LSSumclsm 18049   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971   LMHom clmhm 19019  LFinGenclfig 37637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-sca 15957  df-vsca 15958  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lmhm 19022  df-lfig 37638

This theorem is referenced by:  lmhmlnmsplit  37657