Theorem ellspdOLD 18340

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 18339 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 5657	. . . . . 6 |- ( F : I --> B -> F Fn I )
3		fnima 5627	. . . . . 6 |- ( F Fn I -> ( F " I ) = ran F )
4	1, 2, 3	3syl 20	. . . . 5 |- ( ph -> ( F " I ) = ran F )
5	4	fveq2d 5793	. . . 4 |- ( ph -> ( N ` ( F " I ) ) = ( N ` ran F ) )
6		eqid 2451	. . . . . 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 5183	. . . . 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 2451	. . . . . 6 |- ( S freeLMod I ) = ( S freeLMod I )
9		eqid 2451	. . . . . 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 18337	. . . . 5 |- ( ph -> ran ( f e. ( Base ` ( S freeLMod I ) ) |-> ( M gsumg ( f oF .x. F ) ) ) = ( N ` ran F ) )
18	7, 17	syl5eqr 2506	. . . 4 |- ( ph -> { a | E. f e. ( Base ` ( S freeLMod I ) ) a = ( M gsumg ( f oF .x. F ) ) } = ( N ` ran F ) )
19	5, 18	eqtr4d 2495	. . 3 |- ( ph -> ( N ` ( F " I ) ) = { a | E. f e. ( Base ` ( S freeLMod I ) ) a = ( M gsumg ( f oF .x. F ) ) } )
20	19	eleq2d 2521	. 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 6215	. . . . . 6 |- ( M gsumg ( f oF .x. F ) ) e. _V
22		eleq1 2523	. . . . . 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 2933	. . . 4 |- ( E. f e. ( Base ` ( S freeLMod I ) ) X = ( M gsumg ( f oF .x. F ) ) -> X e. _V )
25		eqeq1 2455	. . . . 5 |- ( a = X -> ( a = ( M gsumg ( f oF .x. F ) ) <-> X = ( M gsumg ( f oF .x. F ) ) ) )
26	25	rexbidv 2844	. . . 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 3210	. . 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 5799	. . . . . . . 8 |- (Scalar ` M ) e. _V
29	14, 28	eqeltri 2535	. . . . . . 7 |- S e. _V
30		ellspd.k	. . . . . . . 8 |- K = ( Base ` S )
31		ellspd.z	. . . . . . . 8 |- .0. = ( 0g ` S )
32		eqid 2451	. . . . . . . 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 18290	. . . . . . 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 2459	. . . . 5 |- ( ph -> ( Base ` ( S freeLMod I ) ) = { a e. ( K ^m I ) | ( `' a " ( _V \ { .0. } ) ) e. Fin } )
36	35	rexeqdv 3020	. . . 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 5111	. . . . . . 7 |- ( a = f -> `' a = `' f )
38	37	imaeq1d 5266	. . . . . 6 |- ( a = f -> ( `' a " ( _V \ { .0. } ) ) = ( `' f " ( _V \ { .0. } ) ) )
39	38	eleq1d 2520	. . . . 5 |- ( a = f -> ( ( `' a " ( _V \ { .0. } ) ) e. Fin <-> ( `' f " ( _V \ { .0. } ) ) e. Fin ) )
40	39	rexrab 3220	. . . 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 1370   e. wcel 1758   {cab 2436   E.wrex 2796   {crab 2799   _Vcvv 3068   \ cdif 3423   {csn 3975   |-> cmpt 4448   `'ccnv 4937   ran crn 4939   "cima 4941   Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190   oFcof 6418   ^m cmap 7314   Fincfn 7410   Basecbs 14276  Scalarcsca 14343   .scvsca 14344   0gc0g 14480   gsum_g cgsu 14481   LModclmod 17054   LSpanclspn 17158   freeLMod cfrlm 18280

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 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460

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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-fz 11539  df-fzo 11650  df-seq 11908  df-hash 12205  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-hom 14364  df-cco 14365  df-0g 14482  df-gsum 14483  df-prds 14488  df-pws 14490  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-mhm 15566  df-submnd 15567  df-grp 15647  df-minusg 15648  df-sbg 15649  df-mulg 15650  df-subg 15780  df-ghm 15847  df-cntz 15937  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-rng 16753  df-subrg 16969  df-lmod 17056  df-lss 17120  df-lsp 17159  df-lmhm 17209  df-lbs 17262  df-sra 17359  df-rgmod 17360  df-nzr 17446  df-dsmm 18266  df-frlm 18281  df-uvc 18317

This theorem is referenced by: (None)