Theorem lmhmlsp 17256

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 2454	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
3	1, 2	lmhmf 17241	. . . . 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 5672	. . . 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 17240	. . . . 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 17239	. . . . . . 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 5291	. . . . . . 7 |- ( F " U ) C_ ran F
12		frn 5676	. . . . . . . 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 3478	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " U ) C_ ( Base ` T ) )
15		eqid 2454	. . . . . . 7 |- ( LSubSp ` T ) = ( LSubSp ` T )
16		lmhmlsp.l	. . . . . . 7 |- L = ( LSpan ` T )
17	2, 15, 16	lspcl 17183	. . . . . 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 2454	. . . . . 6 |- ( LSubSp ` S ) = ( LSubSp ` S )
20	19, 15	lmhmpreima 17255	. . . . 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 3654	. . . . . . 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 5674	. . . . . . . . . 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 3504	. . . . . . . 8 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ dom F )
27		df-ss 3453	. . . . . . . 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 2508	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U = ( dom F i^i U ) )
30		dminss 5362	. . . . . 6 |- ( dom F i^i U ) C_ ( `' F " ( F " U ) )
31	29, 30	syl6eqss 3517	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ ( `' F " ( F " U ) ) )
32	2, 16	lspssid 17192	. . . . . . 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 5315	. . . . . 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 3477	. . . 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 17195	. . . 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 1219	. . 3 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( K ` U ) C_ ( `' F " ( L ` ( F " U ) ) ) )
40		funimass2 5603	. . 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 17183	. . . . 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 17254	. . . 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 17192	. . . . 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 5315	. . . 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 17195	. . 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 1219	. 2 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( L ` ( F " U ) ) C_ ( F " ( K ` U ) ) )
52	41, 51	eqssd 3484	1 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " ( K ` U ) ) = ( L ` ( F " U ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   i^i cin 3438   C_ wss 3439   `'ccnv 4950   dom cdm 4951   ran crn 4952   "cima 4954   Fun wfun 5523   -->wf 5525   ` cfv 5529  (class class class)co 6203   Basecbs 14295   LModclmod 17074   LSubSpclss 17139   LSpanclspn 17178   LMHom clmhm 17226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-0g 14502  df-mnd 15537  df-grp 15667  df-minusg 15668  df-sbg 15669  df-subg 15800  df-ghm 15867  df-mgp 16717  df-ur 16729  df-rng 16773  df-lmod 17076  df-lss 17140  df-lsp 17179  df-lmhm 17229

This theorem is referenced by:  frlmup3  18356  lindfmm  18384  lmimlbs  18393  lmhmfgima  29605  lmhmfgsplit  29607