Theorem lmhmlsp 19049

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 2622	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
3	1, 2	lmhmf 19034	. . . . 5 |- ( F e. ( S LMHom T ) -> F : V --> ( Base ` T ) )
4	3	adantr 481	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> F : V --> ( Base ` T ) )
5		ffun 6048	. . . 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 19033	. . . . 5 |- ( F e. ( S LMHom T ) -> S e. LMod )
8	7	adantr 481	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> S e. LMod )
9		lmhmlmod2 19032	. . . . . . 7 |- ( F e. ( S LMHom T ) -> T e. LMod )
10	9	adantr 481	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> T e. LMod )
11		imassrn 5477	. . . . . . 7 |- ( F " U ) C_ ran F
12		frn 6053	. . . . . . . 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 3614	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " U ) C_ ( Base ` T ) )
15		eqid 2622	. . . . . . 7 |- ( LSubSp ` T ) = ( LSubSp ` T )
16		lmhmlsp.l	. . . . . . 7 |- L = ( LSpan ` T )
17	2, 15, 16	lspcl 18976	. . . . . 6 |- ( ( T e. LMod /\ ( F " U ) C_ ( Base ` T ) ) -> ( L ` ( F " U ) ) e. ( LSubSp ` T ) )
18	10, 14, 17	syl2anc 693	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( L ` ( F " U ) ) e. ( LSubSp ` T ) )
19		eqid 2622	. . . . . 6 |- ( LSubSp ` S ) = ( LSubSp ` S )
20	19, 15	lmhmpreima 19048	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ ( L ` ( F " U ) ) e. ( LSubSp ` T ) ) -> ( `' F " ( L ` ( F " U ) ) ) e. ( LSubSp ` S ) )
21	18, 20	syldan 487	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( `' F " ( L ` ( F " U ) ) ) e. ( LSubSp ` S ) )
22		incom 3805	. . . . . . 7 |- ( dom F i^i U ) = ( U i^i dom F )
23		simpr 477	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ V )
24		fdm 6051	. . . . . . . . . 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 3642	. . . . . . . 8 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ dom F )
27		df-ss 3588	. . . . . . . 8 |- ( U C_ dom F <-> ( U i^i dom F ) = U )
28	26, 27	sylib 208	. . . . . . 7 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( U i^i dom F ) = U )
29	22, 28	syl5req 2669	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U = ( dom F i^i U ) )
30		dminss 5547	. . . . . 6 |- ( dom F i^i U ) C_ ( `' F " ( F " U ) )
31	29, 30	syl6eqss 3655	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ ( `' F " ( F " U ) ) )
32	2, 16	lspssid 18985	. . . . . . 7 |- ( ( T e. LMod /\ ( F " U ) C_ ( Base ` T ) ) -> ( F " U ) C_ ( L ` ( F " U ) ) )
33	10, 14, 32	syl2anc 693	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " U ) C_ ( L ` ( F " U ) ) )
34		imass2 5501	. . . . . 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 3613	. . . 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 18988	. . . 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 1326	. . 3 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( K ` U ) C_ ( `' F " ( L ` ( F " U ) ) ) )
40		funimass2 5972	. . 3 |- ( ( Fun F /\ ( K ` U ) C_ ( `' F " ( L ` ( F " U ) ) ) ) -> ( F " ( K ` U ) ) C_ ( L ` ( F " U ) ) )
41	6, 39, 40	syl2anc 693	. 2 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " ( K ` U ) ) C_ ( L ` ( F " U ) ) )
42	1, 19, 37	lspcl 18976	. . . . 5 |- ( ( S e. LMod /\ U C_ V ) -> ( K ` U ) e. ( LSubSp ` S ) )
43	8, 23, 42	syl2anc 693	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( K ` U ) e. ( LSubSp ` S ) )
44	19, 15	lmhmima 19047	. . . 4 |- ( ( F e. ( S LMHom T ) /\ ( K ` U ) e. ( LSubSp ` S ) ) -> ( F " ( K ` U ) ) e. ( LSubSp ` T ) )
45	43, 44	syldan 487	. . 3 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " ( K ` U ) ) e. ( LSubSp ` T ) )
46	1, 37	lspssid 18985	. . . . 5 |- ( ( S e. LMod /\ U C_ V ) -> U C_ ( K ` U ) )
47	8, 23, 46	syl2anc 693	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> U C_ ( K ` U ) )
48		imass2 5501	. . . 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 18988	. . 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 1326	. 2 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( L ` ( F " U ) ) C_ ( F " ( K ` U ) ) )
52	41, 51	eqssd 3620	1 |- ( ( F e. ( S LMHom T ) /\ U C_ V ) -> ( F " ( K ` U ) ) = ( L ` ( F " U ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971   LMHom clmhm 19019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lmhm 19022

This theorem is referenced by:  frlmup3  20139  lindfmm  20166  lmimlbs  20175  lmhmfgima  37654  lmhmfgsplit  37656