Theorem lmhmlsp 17107

Description: Homomorphisms preserve spans. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Hypotheses

Ref	Expression
lmhmlsp.v	|- V = ( Base ` S )
lmhmlsp.k	|- K = ( LSpan ` S )
lmhmlsp.l	|- L = ( LSpan ` T )

Assertion

Ref	Expression
lmhmlsp	|- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " ( K ` U ) ) = ( L ` ( F " U ) ) )

Proof of Theorem lmhmlsp

Step	Hyp	Ref	Expression
1		lmhmlsp.v	. . . . . 6 |- V = ( Base ` S )
2		eqid 2438	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
3	1, 2	lmhmf 17092	. . . . 5 |- ( F e. ( S LMHom T ) -> F : V --> ( Base ` T ) )
4	3	adantr 465	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> F : V --> ( Base ` T ) )
5		ffun 5556	. . . 4 |- ( F : V --> ( Base ` T ) -> Fun F )
6	4, 5	syl 16	. . 3 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> Fun F )
7		lmhmlmod1 17091	. . . . 5 |- ( F e. ( S LMHom T ) -> S e. LMod )
8	7	adantr 465	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> S e. LMod )
9		lmhmlmod2 17090	. . . . . . 7 |- ( F e. ( S LMHom T ) -> T e. LMod )
10	9	adantr 465	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> T e. LMod )
11		imassrn 5175	. . . . . . 7 |- ( F " U ) C_ ran F
12		frn 5560	. . . . . . . 8 |- ( F : V --> ( Base ` T ) -> ran F C_ ( Base ` T ) )
13	4, 12	syl 16	. . . . . . 7 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ran F C_ ( Base ` T ) )
14	11, 13	syl5ss 3362	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " U ) C_ ( Base ` T ) )
15		eqid 2438	. . . . . . 7 |- ( LSubSp ` T ) = ( LSubSp ` T )
16		lmhmlsp.l	. . . . . . 7 |- L = ( LSpan ` T )
17	2, 15, 16	lspcl 17034	. . . . . 6 |- ( ( T e. LMod /\ ( F " U ) C_ ( Base ` T ) ) -> ( L ` ( F " U ) ) e. ( LSubSp ` T ) )
18	10, 14, 17	syl2anc 661	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( L ` ( F " U ) ) e. ( LSubSp ` T ) )
19		eqid 2438	. . . . . 6 |- ( LSubSp ` S ) = ( LSubSp ` S )
20	19, 15	lmhmpreima 17106	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ ( L ` ( F " U ) ) e. ( LSubSp ` T ) ) -> ( `' F " ( L ` ( F " U ) ) ) e. ( LSubSp ` S ) )
21	18, 20	syldan 470	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( `' F " ( L ` ( F " U ) ) ) e. ( LSubSp ` S ) )
22		incom 3538	. . . . . . 7 |- ( dom F i^i U ) = ( U i^i dom F )
23		simpr 461	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ V )
24		fdm 5558	. . . . . . . . . 10 |- ( F : V --> ( Base ` T ) -> dom F = V )
25	4, 24	syl 16	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> dom F = V )
26	23, 25	sseqtr4d 3388	. . . . . . . 8 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ dom F )
27		df-ss 3337	. . . . . . . 8 |- ( U C_ dom F <-> ( U i^i dom F ) = U )
28	26, 27	sylib 196	. . . . . . 7 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( U i^i dom F ) = U )
29	22, 28	syl5req 2483	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U = ( dom F i^i U ) )
30		dminss 5246	. . . . . 6 |- ( dom F i^i U ) C_ ( `' F " ( F " U ) )
31	29, 30	syl6eqss 3401	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ ( `' F " ( F " U ) ) )
32	2, 16	lspssid 17043	. . . . . . 7 |- ( ( T e. LMod /\ ( F " U ) C_ ( Base ` T ) ) -> ( F " U ) C_ ( L ` ( F " U ) ) )
33	10, 14, 32	syl2anc 661	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " U ) C_ ( L ` ( F " U ) ) )
34		imass2 5199	. . . . . 6 |- ( ( F " U ) C_ ( L ` ( F " U ) ) -> ( `' F " ( F " U ) ) C_ ( `' F " ( L ` ( F " U ) ) ) )
35	33, 34	syl 16	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( `' F " ( F " U ) ) C_ ( `' F " ( L ` ( F " U ) ) ) )
36	31, 35	sstrd 3361	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ ( `' F " ( L ` ( F " U ) ) ) )
37		lmhmlsp.k	. . . . 5 |- K = ( LSpan ` S )
38	19, 37	lspssp 17046	. . . 4 |- ( ( S e. LMod /\ ( `' F " ( L ` ( F " U ) ) ) e. ( LSubSp ` S ) /\ U C_ ( `' F " ( L ` ( F " U ) ) ) ) -> ( K ` U ) C_ ( `' F " ( L ` ( F " U ) ) ) )
39	8, 21, 36, 38	syl3anc 1218	. . 3 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( K ` U ) C_ ( `' F " ( L ` ( F " U ) ) ) )
40		funimass2 5487	. . 3 |- ( ( Fun F /\ ( K ` U ) C_ ( `' F " ( L ` ( F " U ) ) ) ) -> ( F " ( K ` U ) ) C_ ( L ` ( F " U ) ) )
41	6, 39, 40	syl2anc 661	. 2 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " ( K ` U ) ) C_ ( L ` ( F " U ) ) )
42	1, 19, 37	lspcl 17034	. . . . 5 |- ( ( S e. LMod /\ U C_ V ) -> ( K ` U ) e. ( LSubSp ` S ) )
43	8, 23, 42	syl2anc 661	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( K ` U ) e. ( LSubSp ` S ) )
44	19, 15	lmhmima 17105	. . . 4 |- ( ( F e. ( S LMHom T ) /\ ( K ` U ) e. ( LSubSp ` S ) ) -> ( F " ( K ` U ) ) e. ( LSubSp ` T ) )
45	43, 44	syldan 470	. . 3 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " ( K ` U ) ) e. ( LSubSp ` T ) )
46	1, 37	lspssid 17043	. . . . 5 |- ( ( S e. LMod /\ U C_ V ) -> U C_ ( K ` U ) )
47	8, 23, 46	syl2anc 661	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ ( K ` U ) )
48		imass2 5199	. . . 4 |- ( U C_ ( K ` U ) -> ( F " U ) C_ ( F " ( K ` U ) ) )
49	47, 48	syl 16	. . 3 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " U ) C_ ( F " ( K ` U ) ) )
50	15, 16	lspssp 17046	. . 3 |- ( ( T e. LMod /\ ( F " ( K ` U ) ) e. ( LSubSp ` T ) /\ ( F " U ) C_ ( F " ( K ` U ) ) ) -> ( L ` ( F " U ) ) C_ ( F " ( K ` U ) ) )
51	10, 45, 49, 50	syl3anc 1218	. 2 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( L ` ( F " U ) ) C_ ( F " ( K ` U ) ) )
52	41, 51	eqssd 3368	1 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " ( K ` U ) ) = ( L ` ( F " U ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   i^i cin 3322   C_ wss 3323   `'ccnv 4834   dom cdm 4835   ran crn 4836   "cima 4838   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6086   Basecbs 14166   LModclmod 16926   LSubSpclss 16990   LSpanclspn 17029   LMHom clmhm 17077

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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351

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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-0g 14372  df-mnd 15407  df-grp 15536  df-minusg 15537  df-sbg 15538  df-subg 15669  df-ghm 15736  df-mgp 16580  df-ur 16592  df-rng 16635  df-lmod 16928  df-lss 16991  df-lsp 17030  df-lmhm 17080

This theorem is referenced by:  frlmup3  18203  lindfmm  18231  lmimlbs  18240  lmhmfgima  29390  lmhmfgsplit  29392