Theorem lmhmlsp 18272

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 2451	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
3	1, 2	lmhmf 18257	. . . . 5 |- ( F e. ( S LMHom T ) -> F : V --> ( Base ` T ) )
4	3	adantr 467	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> F : V --> ( Base ` T ) )
5		ffun 5731	. . . 4 |- ( F : V --> ( Base ` T ) -> Fun F )
6	4, 5	syl 17	. . 3 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> Fun F )
7		lmhmlmod1 18256	. . . . 5 |- ( F e. ( S LMHom T ) -> S e. LMod )
8	7	adantr 467	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> S e. LMod )
9		lmhmlmod2 18255	. . . . . . 7 |- ( F e. ( S LMHom T ) -> T e. LMod )
10	9	adantr 467	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> T e. LMod )
11		imassrn 5179	. . . . . . 7 |- ( F " U ) C_ ran F
12		frn 5735	. . . . . . . 8 |- ( F : V --> ( Base ` T ) -> ran F C_ ( Base ` T ) )
13	4, 12	syl 17	. . . . . . 7 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ran F C_ ( Base ` T ) )
14	11, 13	syl5ss 3443	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " U ) C_ ( Base ` T ) )
15		eqid 2451	. . . . . . 7 |- ( LSubSp ` T ) = ( LSubSp ` T )
16		lmhmlsp.l	. . . . . . 7 |- L = ( LSpan ` T )
17	2, 15, 16	lspcl 18199	. . . . . 6 |- ( ( T e. LMod /\ ( F " U ) C_ ( Base ` T ) ) -> ( L ` ( F " U ) ) e. ( LSubSp ` T ) )
18	10, 14, 17	syl2anc 667	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( L ` ( F " U ) ) e. ( LSubSp ` T ) )
19		eqid 2451	. . . . . 6 |- ( LSubSp ` S ) = ( LSubSp ` S )
20	19, 15	lmhmpreima 18271	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ ( L ` ( F " U ) ) e. ( LSubSp ` T ) ) -> ( `' F " ( L ` ( F " U ) ) ) e. ( LSubSp ` S ) )
21	18, 20	syldan 473	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( `' F " ( L ` ( F " U ) ) ) e. ( LSubSp ` S ) )
22		incom 3625	. . . . . . 7 |- ( dom F i^i U ) = ( U i^i dom F )
23		simpr 463	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ V )
24		fdm 5733	. . . . . . . . . 10 |- ( F : V --> ( Base ` T ) -> dom F = V )
25	4, 24	syl 17	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> dom F = V )
26	23, 25	sseqtr4d 3469	. . . . . . . 8 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ dom F )
27		df-ss 3418	. . . . . . . 8 |- ( U C_ dom F <-> ( U i^i dom F ) = U )
28	26, 27	sylib 200	. . . . . . 7 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( U i^i dom F ) = U )
29	22, 28	syl5req 2498	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U = ( dom F i^i U ) )
30		dminss 5250	. . . . . 6 |- ( dom F i^i U ) C_ ( `' F " ( F " U ) )
31	29, 30	syl6eqss 3482	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ ( `' F " ( F " U ) ) )
32	2, 16	lspssid 18208	. . . . . . 7 |- ( ( T e. LMod /\ ( F " U ) C_ ( Base ` T ) ) -> ( F " U ) C_ ( L ` ( F " U ) ) )
33	10, 14, 32	syl2anc 667	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " U ) C_ ( L ` ( F " U ) ) )
34		imass2 5204	. . . . . 6 |- ( ( F " U ) C_ ( L ` ( F " U ) ) -> ( `' F " ( F " U ) ) C_ ( `' F " ( L ` ( F " U ) ) ) )
35	33, 34	syl 17	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( `' F " ( F " U ) ) C_ ( `' F " ( L ` ( F " U ) ) ) )
36	31, 35	sstrd 3442	. . . 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 18211	. . . 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 1268	. . 3 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( K ` U ) C_ ( `' F " ( L ` ( F " U ) ) ) )
40		funimass2 5657	. . 3 |- ( ( Fun F /\ ( K ` U ) C_ ( `' F " ( L ` ( F " U ) ) ) ) -> ( F " ( K ` U ) ) C_ ( L ` ( F " U ) ) )
41	6, 39, 40	syl2anc 667	. 2 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " ( K ` U ) ) C_ ( L ` ( F " U ) ) )
42	1, 19, 37	lspcl 18199	. . . . 5 |- ( ( S e. LMod /\ U C_ V ) -> ( K ` U ) e. ( LSubSp ` S ) )
43	8, 23, 42	syl2anc 667	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( K ` U ) e. ( LSubSp ` S ) )
44	19, 15	lmhmima 18270	. . . 4 |- ( ( F e. ( S LMHom T ) /\ ( K ` U ) e. ( LSubSp ` S ) ) -> ( F " ( K ` U ) ) e. ( LSubSp ` T ) )
45	43, 44	syldan 473	. . 3 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " ( K ` U ) ) e. ( LSubSp ` T ) )
46	1, 37	lspssid 18208	. . . . 5 |- ( ( S e. LMod /\ U C_ V ) -> U C_ ( K ` U ) )
47	8, 23, 46	syl2anc 667	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ ( K ` U ) )
48		imass2 5204	. . . 4 |- ( U C_ ( K ` U ) -> ( F " U ) C_ ( F " ( K ` U ) ) )
49	47, 48	syl 17	. . 3 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " U ) C_ ( F " ( K ` U ) ) )
50	15, 16	lspssp 18211	. . 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 1268	. 2 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( L ` ( F " U ) ) C_ ( F " ( K ` U ) ) )
52	41, 51	eqssd 3449	1 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " ( K ` U ) ) = ( L ` ( F " U ) ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   i^i cin 3403   C_ wss 3404   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837   Fun wfun 5576   -->wf 5578   ` cfv 5582  (class class class)co 6290   Basecbs 15121   LModclmod 18091   LSubSpclss 18155   LSpanclspn 18194   LMHom clmhm 18242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-minusg 16674  df-sbg 16675  df-subg 16814  df-ghm 16881  df-mgp 17724  df-ur 17736  df-ring 17782  df-lmod 18093  df-lss 18156  df-lsp 18195  df-lmhm 18245

This theorem is referenced by:  frlmup3  19358  lindfmm  19385  lmimlbs  19394  lmhmfgima  35942  lmhmfgsplit  35944