Theorem frlmup3 20139

Description: The range of such an evaluation map is the finite linear combinations of the target vectors and also the span of the target vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.)

Hypotheses

Ref	Expression
frlmup.f	|- F = ( R freeLMod I )
frlmup.b	|- B = ( Base ` F )
frlmup.c	|- C = ( Base ` T )
frlmup.v	|- .x. = ( .s ` T )
frlmup.e	|- E = ( x e. B |-> ( T gsumg ( x oF .x. A ) ) )
frlmup.t	|- ( ph -> T e. LMod )
frlmup.i	|- ( ph -> I e. X )
frlmup.r	|- ( ph -> R = (Scalar ` T ) )
frlmup.a	|- ( ph -> A : I --> C )
frlmup.k	|- K = ( LSpan ` T )

Assertion

Ref	Expression
frlmup3	|- ( ph -> ran E = ( K ` ran A ) )

Distinct variable groups:   x, R   x, I   x, F   x, B   x, C   x, .x.   x, A   x, X   x, K   ph, x   x, T

Allowed substitution hint:   E( x)

Proof of Theorem frlmup3

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frlmup.f	. . . 4 |- F = ( R freeLMod I )
2		frlmup.b	. . . 4 |- B = ( Base ` F )
3		frlmup.c	. . . 4 |- C = ( Base ` T )
4		frlmup.v	. . . 4 |- .x. = ( .s ` T )
5		frlmup.e	. . . 4 |- E = ( x e. B |-> ( T gsumg ( x oF .x. A ) ) )
6		frlmup.t	. . . 4 |- ( ph -> T e. LMod )
7		frlmup.i	. . . 4 |- ( ph -> I e. X )
8		frlmup.r	. . . 4 |- ( ph -> R = (Scalar ` T ) )
9		frlmup.a	. . . 4 |- ( ph -> A : I --> C )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	frlmup1 20137	. . 3 |- ( ph -> E e. ( F LMHom T ) )
11		eqid 2622	. . . . . . . 8 |- (Scalar ` T ) = (Scalar ` T )
12	11	lmodring 18871	. . . . . . 7 |- ( T e. LMod -> (Scalar ` T ) e. Ring )
13	6, 12	syl 17	. . . . . 6 |- ( ph -> (Scalar ` T ) e. Ring )
14	8, 13	eqeltrd 2701	. . . . 5 |- ( ph -> R e. Ring )
15		eqid 2622	. . . . . 6 |- ( R unitVec I ) = ( R unitVec I )
16	15, 1, 2	uvcff 20130	. . . . 5 |- ( ( R e. Ring /\ I e. X ) -> ( R unitVec I ) : I --> B )
17	14, 7, 16	syl2anc 693	. . . 4 |- ( ph -> ( R unitVec I ) : I --> B )
18		frn 6053	. . . 4 |- ( ( R unitVec I ) : I --> B -> ran ( R unitVec I ) C_ B )
19	17, 18	syl 17	. . 3 |- ( ph -> ran ( R unitVec I ) C_ B )
20		eqid 2622	. . . 4 |- ( LSpan ` F ) = ( LSpan ` F )
21		frlmup.k	. . . 4 |- K = ( LSpan ` T )
22	2, 20, 21	lmhmlsp 19049	. . 3 |- ( ( E e. ( F LMHom T ) /\ ran ( R unitVec I ) C_ B ) -> ( E " ( ( LSpan ` F ) ` ran ( R unitVec I ) ) ) = ( K ` ( E " ran ( R unitVec I ) ) ) )
23	10, 19, 22	syl2anc 693	. 2 |- ( ph -> ( E " ( ( LSpan ` F ) ` ran ( R unitVec I ) ) ) = ( K ` ( E " ran ( R unitVec I ) ) ) )
24	2, 3	lmhmf 19034	. . . . . 6 |- ( E e. ( F LMHom T ) -> E : B --> C )
25	10, 24	syl 17	. . . . 5 |- ( ph -> E : B --> C )
26		ffn 6045	. . . . 5 |- ( E : B --> C -> E Fn B )
27	25, 26	syl 17	. . . 4 |- ( ph -> E Fn B )
28		fnima 6010	. . . 4 |- ( E Fn B -> ( E " B ) = ran E )
29	27, 28	syl 17	. . 3 |- ( ph -> ( E " B ) = ran E )
30		eqid 2622	. . . . . . . 8 |- (LBasis ` F ) = (LBasis ` F )
31	1, 15, 30	frlmlbs 20136	. . . . . . 7 |- ( ( R e. Ring /\ I e. X ) -> ran ( R unitVec I ) e. (LBasis ` F ) )
32	14, 7, 31	syl2anc 693	. . . . . 6 |- ( ph -> ran ( R unitVec I ) e. (LBasis ` F ) )
33	2, 30, 20	lbssp 19079	. . . . . 6 |- ( ran ( R unitVec I ) e. (LBasis ` F ) -> ( ( LSpan ` F ) ` ran ( R unitVec I ) ) = B )
34	32, 33	syl 17	. . . . 5 |- ( ph -> ( ( LSpan ` F ) ` ran ( R unitVec I ) ) = B )
35	34	eqcomd 2628	. . . 4 |- ( ph -> B = ( ( LSpan ` F ) ` ran ( R unitVec I ) ) )
36	35	imaeq2d 5466	. . 3 |- ( ph -> ( E " B ) = ( E " ( ( LSpan ` F ) ` ran ( R unitVec I ) ) ) )
37	29, 36	eqtr3d 2658	. 2 |- ( ph -> ran E = ( E " ( ( LSpan ` F ) ` ran ( R unitVec I ) ) ) )
38		imaco 5640	. . . 4 |- ( ( E o. ( R unitVec I ) ) " I ) = ( E " ( ( R unitVec I ) " I ) )
39		ffn 6045	. . . . . . . 8 |- ( A : I --> C -> A Fn I )
40	9, 39	syl 17	. . . . . . 7 |- ( ph -> A Fn I )
41		ffn 6045	. . . . . . . . 9 |- ( ( R unitVec I ) : I --> B -> ( R unitVec I ) Fn I )
42	17, 41	syl 17	. . . . . . . 8 |- ( ph -> ( R unitVec I ) Fn I )
43		fnco 5999	. . . . . . . 8 |- ( ( E Fn B /\ ( R unitVec I ) Fn I /\ ran ( R unitVec I ) C_ B ) -> ( E o. ( R unitVec I ) ) Fn I )
44	27, 42, 19, 43	syl3anc 1326	. . . . . . 7 |- ( ph -> ( E o. ( R unitVec I ) ) Fn I )
45		fvco2 6273	. . . . . . . . 9 |- ( ( ( R unitVec I ) Fn I /\ u e. I ) -> ( ( E o. ( R unitVec I ) ) ` u ) = ( E ` ( ( R unitVec I ) ` u ) ) )
46	42, 45	sylan 488	. . . . . . . 8 |- ( ( ph /\ u e. I ) -> ( ( E o. ( R unitVec I ) ) ` u ) = ( E ` ( ( R unitVec I ) ` u ) ) )
47	6	adantr 481	. . . . . . . . 9 |- ( ( ph /\ u e. I ) -> T e. LMod )
48	7	adantr 481	. . . . . . . . 9 |- ( ( ph /\ u e. I ) -> I e. X )
49	8	adantr 481	. . . . . . . . 9 |- ( ( ph /\ u e. I ) -> R = (Scalar ` T ) )
50	9	adantr 481	. . . . . . . . 9 |- ( ( ph /\ u e. I ) -> A : I --> C )
51		simpr 477	. . . . . . . . 9 |- ( ( ph /\ u e. I ) -> u e. I )
52	1, 2, 3, 4, 5, 47, 48, 49, 50, 51, 15	frlmup2 20138	. . . . . . . 8 |- ( ( ph /\ u e. I ) -> ( E ` ( ( R unitVec I ) ` u ) ) = ( A ` u ) )
53	46, 52	eqtr2d 2657	. . . . . . 7 |- ( ( ph /\ u e. I ) -> ( A ` u ) = ( ( E o. ( R unitVec I ) ) ` u ) )
54	40, 44, 53	eqfnfvd 6314	. . . . . 6 |- ( ph -> A = ( E o. ( R unitVec I ) ) )
55	54	imaeq1d 5465	. . . . 5 |- ( ph -> ( A " I ) = ( ( E o. ( R unitVec I ) ) " I ) )
56		fnima 6010	. . . . . 6 |- ( A Fn I -> ( A " I ) = ran A )
57	40, 56	syl 17	. . . . 5 |- ( ph -> ( A " I ) = ran A )
58	55, 57	eqtr3d 2658	. . . 4 |- ( ph -> ( ( E o. ( R unitVec I ) ) " I ) = ran A )
59		fnima 6010	. . . . . 6 |- ( ( R unitVec I ) Fn I -> ( ( R unitVec I ) " I ) = ran ( R unitVec I ) )
60	42, 59	syl 17	. . . . 5 |- ( ph -> ( ( R unitVec I ) " I ) = ran ( R unitVec I ) )
61	60	imaeq2d 5466	. . . 4 |- ( ph -> ( E " ( ( R unitVec I ) " I ) ) = ( E " ran ( R unitVec I ) ) )
62	38, 58, 61	3eqtr3a 2680	. . 3 |- ( ph -> ran A = ( E " ran ( R unitVec I ) ) )
63	62	fveq2d 6195	. 2 |- ( ph -> ( K ` ran A ) = ( K ` ( E " ran ( R unitVec I ) ) ) )
64	23, 37, 63	3eqtr4d 2666	1 |- ( ph -> ran E = ( K ` ran A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   |-> cmpt 4729   ran crn 5115   "cima 5117   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   gsum_g cgsu 16101   Ringcrg 18547   LModclmod 18863   LSpanclspn 18971   LMHom clmhm 19019  LBasisclbs 19074   freeLMod cfrlm 20090   unitVec cuvc 20121

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-inf2 8538  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-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lmhm 19022  df-lbs 19075  df-sra 19172  df-rgmod 19173  df-nzr 19258  df-dsmm 20076  df-frlm 20091  df-uvc 20122

This theorem is referenced by:  ellspd  20141  indlcim  20179  lnrfg  37689