Theorem lmhmlsp 17566

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 2467	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
3	1, 2	lmhmf 17551	. . . . 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 5739	. . . 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 17550	. . . . 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 17549	. . . . . . 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 5354	. . . . . . 7 |- ( F " U ) C_ ran F
12		frn 5743	. . . . . . . 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 3520	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " U ) C_ ( Base ` T ) )
15		eqid 2467	. . . . . . 7 |- ( LSubSp ` T ) = ( LSubSp ` T )
16		lmhmlsp.l	. . . . . . 7 |- L = ( LSpan ` T )
17	2, 15, 16	lspcl 17493	. . . . . 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 2467	. . . . . 6 |- ( LSubSp ` S ) = ( LSubSp ` S )
20	19, 15	lmhmpreima 17565	. . . . 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 3696	. . . . . . 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 5741	. . . . . . . . . 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 3546	. . . . . . . 8 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ dom F )
27		df-ss 3495	. . . . . . . 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 2521	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U = ( dom F i^i U ) )
30		dminss 5426	. . . . . 6 |- ( dom F i^i U ) C_ ( `' F " ( F " U ) )
31	29, 30	syl6eqss 3559	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ ( `' F " ( F " U ) ) )
32	2, 16	lspssid 17502	. . . . . . 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 5378	. . . . . 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 3519	. . . 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 17505	. . . 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 1228	. . 3 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( K ` U ) C_ ( `' F " ( L ` ( F " U ) ) ) )
40		funimass2 5668	. . 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 17493	. . . . 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 17564	. . . 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 17502	. . . . 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 5378	. . . 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 17505	. . 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 1228	. 2 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( L ` ( F " U ) ) C_ ( F " ( K ` U ) ) )
52	41, 51	eqssd 3526	1 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " ( K ` U ) ) = ( L ` ( F " U ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   i^i cin 3480   C_ wss 3481   `'ccnv 5004   dom cdm 5005   ran crn 5006   "cima 5008   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295   Basecbs 14507   LModclmod 17383   LSubSpclss 17449   LSpanclspn 17488   LMHom clmhm 17536

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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-ghm 16137  df-mgp 17014  df-ur 17026  df-ring 17072  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lmhm 17539

This theorem is referenced by:  frlmup3  18703  lindfmm  18731  lmimlbs  18740  lmhmfgima  30958  lmhmfgsplit  30960