Theorem lmhmfgsplit 29465

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 990	. . 3 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> V e. LFinGen )
2		lmhmlmod2 17135	. . . . 5 |- ( F e. ( S LMHom T ) -> T e. LMod )
3	2	3ad2ant1 1009	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> T e. LMod )
4		lmhmrnlss 17153	. . . . 5 |- ( F e. ( S LMHom T ) -> ran F e. ( LSubSp ` T ) )
5	4	3ad2ant1 1009	. . . 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 2443	. . . . 5 |- ( LSubSp ` T ) = ( LSubSp ` T )
8		eqid 2443	. . . . 5 |- ( LSpan ` T ) = ( LSpan ` T )
9	6, 7, 8	islssfg 29449	. . . 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 991	. . . . 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 2443	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
14		eqid 2443	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
15	13, 14	lmhmf 17137	. . . . 5 |- ( F e. ( S LMHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
16		ffn 5580	. . . . 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 3890	. . . . 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 7638	. . . 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 1218	. . 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 2443	. . . . . . 7 |- ( LSubSp ` S ) = ( LSubSp ` S )
24		eqid 2443	. . . . . . 7 |- ( LSSum ` S ) = ( LSSum ` S )
25		lmhmfgsplit.z	. . . . . . 7 |- .0. = ( 0g ` T )
26		lmhmfgsplit.k	. . . . . . 7 |- K = ( `' F " { .0. } )
27		simpll1 1027	. . . . . . 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 17136	. . . . . . . . . 10 |- ( F e. ( S LMHom T ) -> S e. LMod )
29	28	3ad2ant1 1009	. . . . . . . . 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 3591	. . . . . . . . . . 11 |- ( ~P ( Base ` S ) i^i Fin ) C_ ~P ( Base ` S )
32	31	sseli 3373	. . . . . . . . . 10 |- ( b e. ( ~P ( Base ` S ) i^i Fin ) -> b e. ~P ( Base ` S ) )
33		elpwi 3890	. . . . . . . . . 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 2443	. . . . . . . . 9 |- ( LSpan ` S ) = ( LSpan ` S )
37	13, 23, 36	lspcl 17079	. . . . . . . 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 17152	. . . . . . . . 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 5712	. . . . . . . . 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 1058	. . . . . . . . 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 1187	. . . . . . . 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 2479	. . . . . . 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 29462	. . . . . 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 6128	. . . . 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 14254	. . . . . . 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 2476	. . . 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 2443	. . . . 5 |- ( S ↾s ( ( LSpan ` S ) ` b ) ) = ( S ↾s ( ( LSpan ` S ) ` b ) )
54		eqid 2443	. . . . 5 |- ( S ↾s ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) ) = ( S ↾s ( K ( LSSum ` S ) ( ( LSpan ` S ) ` b ) ) )
55	26, 25, 23	lmhmkerlss 17154	. . . . . . 7 |- ( F e. ( S LMHom T ) -> K e. ( LSubSp ` S ) )
56	55	3ad2ant1 1009	. . . . . 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 1028	. . . . 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 3592	. . . . . . . 8 |- ( ~P ( Base ` S ) i^i Fin ) C_ Fin
60	59	sseli 3373	. . . . . . 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 29451	. . . . . 6 |- ( ( S e. LMod /\ b C_ ( Base ` S ) /\ b e. Fin ) -> ( S ↾s ( ( LSpan ` S ) ` b ) ) e. LFinGen )
63	30, 35, 61, 62	syl3anc 1218	. . . . 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 29453	. . . 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 2866	. 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 2866	1 |- ( ( F e. ( S LMHom T ) /\ U e. LFinGen /\ V e. LFinGen ) -> S e. LFinGen )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   E.wrex 2737   i^i cin 3348   C_ wss 3349   ~Pcpw 3881   {csn 3898   `'ccnv 4860   ran crn 4862   "cima 4864   Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112   Fincfn 7331   Basecbs 14195   ↾_s cress 14196   0gc0g 14399   LSSumclsm 16154   LModclmod 16970   LSubSpclss 17035   LSpanclspn 17074   LMHom clmhm 17122  LFinGenclfig 29446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-sca 14275  df-vsca 14276  df-0g 14401  df-mnd 15436  df-submnd 15486  df-grp 15566  df-minusg 15567  df-sbg 15568  df-subg 15699  df-ghm 15766  df-cntz 15856  df-lsm 16156  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-lmod 16972  df-lss 17036  df-lsp 17075  df-lmhm 17125  df-lfig 29447

This theorem is referenced by:  lmhmlnmsplit  29466