Theorem ellspdOLD 18604

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.) Obsolete version of ellspd 18603 as of 24-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)

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
ellspdOLD	|- ( ph -> ( X e. ( N ` ( F " I ) ) <-> E. f e. ( K ^m I ) ( ( `' f " ( _V \ { .0. } ) ) e. Fin /\ 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 ellspdOLD

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ellspd.f	. . . . . 6 |- ( ph -> F : I --> B )
2		ffn 5729	. . . . . 6 |- ( F : I --> B -> F Fn I )
3		fnima 5697	. . . . . 6 |- ( F Fn I -> ( F " I ) = ran F )
4	1, 2, 3	3syl 20	. . . . 5 |- ( ph -> ( F " I ) = ran F )
5	4	fveq2d 5868	. . . 4 |- ( ph -> ( N ` ( F " I ) ) = ( N ` ran F ) )
6		eqid 2467	. . . . . 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 5246	. . . . 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 2467	. . . . . 6 |- ( S freeLMod I ) = ( S freeLMod I )
9		eqid 2467	. . . . . 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 18601	. . . . 5 |- ( ph -> ran ( f e. ( Base ` ( S freeLMod I ) ) |-> ( M gsumg ( f oF .x. F ) ) ) = ( N ` ran F ) )
18	7, 17	syl5eqr 2522	. . . 4 |- ( ph -> { a | E. f e. ( Base ` ( S freeLMod I ) ) a = ( M gsumg ( f oF .x. F ) ) } = ( N ` ran F ) )
19	5, 18	eqtr4d 2511	. . 3 |- ( ph -> ( N ` ( F " I ) ) = { a | E. f e. ( Base ` ( S freeLMod I ) ) a = ( M gsumg ( f oF .x. F ) ) } )
20	19	eleq2d 2537	. 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 6307	. . . . . 6 |- ( M gsumg ( f oF .x. F ) ) e. _V
22		eleq1 2539	. . . . . 6 |- ( X = ( M gsumg ( f oF .x. F ) ) -> ( X e. _V <-> ( M gsumg ( f oF .x. F ) ) e. _V ) )
23	21, 22	mpbiri 233	. . . . 5 |- ( X = ( M gsumg ( f oF .x. F ) ) -> X e. _V )
24	23	rexlimivw 2952	. . . 4 |- ( E. f e. ( Base ` ( S freeLMod I ) ) X = ( M gsumg ( f oF .x. F ) ) -> X e. _V )
25		eqeq1 2471	. . . . 5 |- ( a = X -> ( a = ( M gsumg ( f oF .x. F ) ) <-> X = ( M gsumg ( f oF .x. F ) ) ) )
26	25	rexbidv 2973	. . . 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 3257	. . 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 5874	. . . . . . . 8 |- (Scalar ` M ) e. _V
29	14, 28	eqeltri 2551	. . . . . . 7 |- S e. _V
30		ellspd.k	. . . . . . . 8 |- K = ( Base ` S )
31		ellspd.z	. . . . . . . 8 |- .0. = ( 0g ` S )
32		eqid 2467	. . . . . . . 8 |- { a e. ( K ^m I ) | ( `' a " ( _V \ { .0. } ) ) e. Fin } = { a e. ( K ^m I ) | ( `' a " ( _V \ { .0. } ) ) e. Fin }
33	8, 30, 31, 32	frlmbasOLD 18554	. . . . . . 7 |- ( ( S e. _V /\ I e. _V ) -> { a e. ( K ^m I ) | ( `' a " ( _V \ { .0. } ) ) e. Fin } = ( Base ` ( S freeLMod I ) ) )
34	29, 13, 33	sylancr 663	. . . . . 6 |- ( ph -> { a e. ( K ^m I ) | ( `' a " ( _V \ { .0. } ) ) e. Fin } = ( Base ` ( S freeLMod I ) ) )
35	34	eqcomd 2475	. . . . 5 |- ( ph -> ( Base ` ( S freeLMod I ) ) = { a e. ( K ^m I ) | ( `' a " ( _V \ { .0. } ) ) e. Fin } )
36	35	rexeqdv 3065	. . . 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 " ( _V \ { .0. } ) ) e. Fin } X = ( M gsumg ( f oF .x. F ) ) ) )
37		cnveq 5174	. . . . . . 7 |- ( a = f -> `' a = `' f )
38	37	imaeq1d 5334	. . . . . 6 |- ( a = f -> ( `' a " ( _V \ { .0. } ) ) = ( `' f " ( _V \ { .0. } ) ) )
39	38	eleq1d 2536	. . . . 5 |- ( a = f -> ( ( `' a " ( _V \ { .0. } ) ) e. Fin <-> ( `' f " ( _V \ { .0. } ) ) e. Fin ) )
40	39	rexrab 3267	. . . 4 |- ( E. f e. { a e. ( K ^m I ) | ( `' a " ( _V \ { .0. } ) ) e. Fin } X = ( M gsumg ( f oF .x. F ) ) <-> E. f e. ( K ^m I ) ( ( `' f " ( _V \ { .0. } ) ) e. Fin /\ X = ( M gsumg ( f oF .x. F ) ) ) )
41	36, 40	syl6bb 261	. . 3 |- ( ph -> ( E. f e. ( Base ` ( S freeLMod I ) ) X = ( M gsumg ( f oF .x. F ) ) <-> E. f e. ( K ^m I ) ( ( `' f " ( _V \ { .0. } ) ) e. Fin /\ X = ( M gsumg ( f oF .x. F ) ) ) ) )
42	27, 41	syl5bb 257	. 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 " ( _V \ { .0. } ) ) e. Fin /\ X = ( M gsumg ( f oF .x. F ) ) ) ) )
43	20, 42	bitrd 253	1 |- ( ph -> ( X e. ( N ` ( F " I ) ) <-> E. f e. ( K ^m I ) ( ( `' f " ( _V \ { .0. } ) ) e. Fin /\ X = ( M gsumg ( f oF .x. F ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   {cab 2452   E.wrex 2815   {crab 2818   _Vcvv 3113   \ cdif 3473   {csn 4027   |-> cmpt 4505   `'ccnv 4998   ran crn 5000   "cima 5002   Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   oFcof 6520   ^m cmap 7417   Fincfn 7513   Basecbs 14486  Scalarcsca 14554   .scvsca 14555   0gc0g 14691   gsum_g cgsu 14692   LModclmod 17295   LSpanclspn 17400   freeLMod cfrlm 18544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-hom 14575  df-cco 14576  df-0g 14693  df-gsum 14694  df-prds 14699  df-pws 14701  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-mhm 15777  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mulg 15861  df-subg 15993  df-ghm 16060  df-cntz 16150  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-subrg 17210  df-lmod 17297  df-lss 17362  df-lsp 17401  df-lmhm 17451  df-lbs 17504  df-sra 17601  df-rgmod 17602  df-nzr 17688  df-dsmm 18530  df-frlm 18545  df-uvc 18581

This theorem is referenced by: (None)