Theorem ellspd 20141

Description: The elements of the span of an indexed collection of basic vectors are those vectors which can be written as finite linear combinations of basic vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by AV, 24-Jun-2019.)

Hypotheses

Ref	Expression
ellspd.n	|- N = ( LSpan ` M )
ellspd.v	|- B = ( Base ` M )
ellspd.k	|- K = ( Base ` S )
ellspd.s	|- S = (Scalar ` M )
ellspd.z	|- .0. = ( 0g ` S )
ellspd.t	|- .x. = ( .s ` M )
ellspd.f	|- ( ph -> F : I --> B )
ellspd.m	|- ( ph -> M e. LMod )
ellspd.i	|- ( ph -> I e. _V )

Assertion

Ref	Expression
ellspd	|- ( ph -> ( X e. ( N ` ( F " I ) ) <-> E. f e. ( K ^m I ) ( f finSupp .0. /\ X = ( M gsumg ( f oF .x. F ) ) ) ) )

Distinct variable groups:   f, M   B, f   f, N   f, K   S, f   .0. , f   .x. , f   f, F   f, I   f, X   ph, f

Proof of Theorem ellspd

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ellspd.f	. . . . . 6 |- ( ph -> F : I --> B )
2		ffn 6045	. . . . . 6 |- ( F : I --> B -> F Fn I )
3		fnima 6010	. . . . . 6 |- ( F Fn I -> ( F " I ) = ran F )
4	1, 2, 3	3syl 18	. . . . 5 |- ( ph -> ( F " I ) = ran F )
5	4	fveq2d 6195	. . . 4 |- ( ph -> ( N ` ( F " I ) ) = ( N ` ran F ) )
6		eqid 2622	. . . . . 6 |- ( f e. ( Base ` ( S freeLMod I ) ) |-> ( M gsumg ( f oF .x. F ) ) ) = ( f e. ( Base ` ( S freeLMod I ) ) |-> ( M gsumg ( f oF .x. F ) ) )
7	6	rnmpt 5371	. . . . 5 |- ran ( f e. ( Base ` ( S freeLMod I ) ) |-> ( M gsumg ( f oF .x. F ) ) ) = { a | E. f e. ( Base ` ( S freeLMod I ) ) a = ( M gsumg ( f oF .x. F ) ) }
8		eqid 2622	. . . . . 6 |- ( S freeLMod I ) = ( S freeLMod I )
9		eqid 2622	. . . . . 6 |- ( Base ` ( S freeLMod I ) ) = ( Base ` ( S freeLMod I ) )
10		ellspd.v	. . . . . 6 |- B = ( Base ` M )
11		ellspd.t	. . . . . 6 |- .x. = ( .s ` M )
12		ellspd.m	. . . . . 6 |- ( ph -> M e. LMod )
13		ellspd.i	. . . . . 6 |- ( ph -> I e. _V )
14		ellspd.s	. . . . . . 7 |- S = (Scalar ` M )
15	14	a1i 11	. . . . . 6 |- ( ph -> S = (Scalar ` M ) )
16		ellspd.n	. . . . . 6 |- N = ( LSpan ` M )
17	8, 9, 10, 11, 6, 12, 13, 15, 1, 16	frlmup3 20139	. . . . 5 |- ( ph -> ran ( f e. ( Base ` ( S freeLMod I ) ) |-> ( M gsumg ( f oF .x. F ) ) ) = ( N ` ran F ) )
18	7, 17	syl5eqr 2670	. . . 4 |- ( ph -> { a | E. f e. ( Base ` ( S freeLMod I ) ) a = ( M gsumg ( f oF .x. F ) ) } = ( N ` ran F ) )
19	5, 18	eqtr4d 2659	. . 3 |- ( ph -> ( N ` ( F " I ) ) = { a | E. f e. ( Base ` ( S freeLMod I ) ) a = ( M gsumg ( f oF .x. F ) ) } )
20	19	eleq2d 2687	. 2 |- ( ph -> ( X e. ( N ` ( F " I ) ) <-> X e. { a | E. f e. ( Base ` ( S freeLMod I ) ) a = ( M gsumg ( f oF .x. F ) ) } ) )
21		ovex 6678	. . . . . 6 |- ( M gsumg ( f oF .x. F ) ) e. _V
22		eleq1 2689	. . . . . 6 |- ( X = ( M gsumg ( f oF .x. F ) ) -> ( X e. _V <-> ( M gsumg ( f oF .x. F ) ) e. _V ) )
23	21, 22	mpbiri 248	. . . . 5 |- ( X = ( M gsumg ( f oF .x. F ) ) -> X e. _V )
24	23	rexlimivw 3029	. . . 4 |- ( E. f e. ( Base ` ( S freeLMod I ) ) X = ( M gsumg ( f oF .x. F ) ) -> X e. _V )
25		eqeq1 2626	. . . . 5 |- ( a = X -> ( a = ( M gsumg ( f oF .x. F ) ) <-> X = ( M gsumg ( f oF .x. F ) ) ) )
26	25	rexbidv 3052	. . . 4 |- ( a = X -> ( E. f e. ( Base ` ( S freeLMod I ) ) a = ( M gsumg ( f oF .x. F ) ) <-> E. f e. ( Base ` ( S freeLMod I ) ) X = ( M gsumg ( f oF .x. F ) ) ) )
27	24, 26	elab3 3358	. . 3 |- ( X e. { a | E. f e. ( Base ` ( S freeLMod I ) ) a = ( M gsumg ( f oF .x. F ) ) } <-> E. f e. ( Base ` ( S freeLMod I ) ) X = ( M gsumg ( f oF .x. F ) ) )
28		fvex 6201	. . . . . . . 8 |- (Scalar ` M ) e. _V
29	14, 28	eqeltri 2697	. . . . . . 7 |- S e. _V
30		ellspd.k	. . . . . . . 8 |- K = ( Base ` S )
31		ellspd.z	. . . . . . . 8 |- .0. = ( 0g ` S )
32		eqid 2622	. . . . . . . 8 |- { a e. ( K ^m I ) | a finSupp .0. } = { a e. ( K ^m I ) | a finSupp .0. }
33	8, 30, 31, 32	frlmbas 20099	. . . . . . 7 |- ( ( S e. _V /\ I e. _V ) -> { a e. ( K ^m I ) | a finSupp .0. } = ( Base ` ( S freeLMod I ) ) )
34	29, 13, 33	sylancr 695	. . . . . 6 |- ( ph -> { a e. ( K ^m I ) | a finSupp .0. } = ( Base ` ( S freeLMod I ) ) )
35	34	eqcomd 2628	. . . . 5 |- ( ph -> ( Base ` ( S freeLMod I ) ) = { a e. ( K ^m I ) | a finSupp .0. } )
36	35	rexeqdv 3145	. . . 4 |- ( ph -> ( E. f e. ( Base ` ( S freeLMod I ) ) X = ( M gsumg ( f oF .x. F ) ) <-> E. f e. { a e. ( K ^m I ) | a finSupp .0. } X = ( M gsumg ( f oF .x. F ) ) ) )
37		breq1 4656	. . . . 5 |- ( a = f -> ( a finSupp .0. <-> f finSupp .0. ) )
38	37	rexrab 3370	. . . 4 |- ( E. f e. { a e. ( K ^m I ) | a finSupp .0. } X = ( M gsumg ( f oF .x. F ) ) <-> E. f e. ( K ^m I ) ( f finSupp .0. /\ X = ( M gsumg ( f oF .x. F ) ) ) )
39	36, 38	syl6bb 276	. . 3 |- ( ph -> ( E. f e. ( Base ` ( S freeLMod I ) ) X = ( M gsumg ( f oF .x. F ) ) <-> E. f e. ( K ^m I ) ( f finSupp .0. /\ X = ( M gsumg ( f oF .x. F ) ) ) ) )
40	27, 39	syl5bb 272	. 2 |- ( ph -> ( X e. { a | E. f e. ( Base ` ( S freeLMod I ) ) a = ( M gsumg ( f oF .x. F ) ) } <-> E. f e. ( K ^m I ) ( f finSupp .0. /\ X = ( M gsumg ( f oF .x. F ) ) ) ) )
41	20, 40	bitrd 268	1 |- ( ph -> ( X e. ( N ` ( F " I ) ) <-> E. f e. ( K ^m I ) ( f finSupp .0. /\ X = ( M gsumg ( f oF .x. F ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   {crab 2916   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   ^m cmap 7857   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   gsum_g cgsu 16101   LModclmod 18863   LSpanclspn 18971   freeLMod cfrlm 20090

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:  elfilspd  20142  islindf4  20177