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Theorem ellspd 26924
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.)
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 "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  X  =  ( M  gsumg  ( f  o F 
.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.
StepHypRef Expression
1 ellspd.f . . . . . 6  |-  ( ph  ->  F : I --> B )
2 ffn 5532 . . . . . 6  |-  ( F : I --> B  ->  F  Fn  I )
3 fnima 5504 . . . . . 6  |-  ( F  Fn  I  ->  ( F " I )  =  ran  F )
41, 2, 33syl 19 . . . . 5  |-  ( ph  ->  ( F " I
)  =  ran  F
)
54fveq2d 5673 . . . 4  |-  ( ph  ->  ( N `  ( F " I ) )  =  ( N `  ran  F ) )
6 eqid 2388 . . . . . 6  |-  ( f  e.  ( Base `  ( S freeLMod  I ) )  |->  ( M  gsumg  ( f  o F 
.x.  F ) ) )  =  ( f  e.  ( Base `  ( S freeLMod  I ) )  |->  ( M  gsumg  ( f  o F 
.x.  F ) ) )
76rnmpt 5057 . . . . 5  |-  ran  (
f  e.  ( Base `  ( S freeLMod  I )
)  |->  ( M  gsumg  ( f  o F  .x.  F
) ) )  =  { a  |  E. f  e.  ( Base `  ( S freeLMod  I )
) a  =  ( M  gsumg  ( f  o F 
.x.  F ) ) }
8 eqid 2388 . . . . . 6  |-  ( S freeLMod  I )  =  ( S freeLMod  I )
9 eqid 2388 . . . . . 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 )
1514a1i 11 . . . . . 6  |-  ( ph  ->  S  =  (Scalar `  M ) )
16 ellspd.n . . . . . 6  |-  N  =  ( LSpan `  M )
178, 9, 10, 11, 6, 12, 13, 15, 1, 16frlmup3 26922 . . . . 5  |-  ( ph  ->  ran  ( f  e.  ( Base `  ( S freeLMod  I ) )  |->  ( M  gsumg  ( f  o F 
.x.  F ) ) )  =  ( N `
 ran  F )
)
187, 17syl5eqr 2434 . . . 4  |-  ( ph  ->  { a  |  E. f  e.  ( Base `  ( S freeLMod  I )
) a  =  ( M  gsumg  ( f  o F 
.x.  F ) ) }  =  ( N `
 ran  F )
)
195, 18eqtr4d 2423 . . 3  |-  ( ph  ->  ( N `  ( F " I ) )  =  { a  |  E. f  e.  (
Base `  ( S freeLMod  I ) ) a  =  ( M  gsumg  ( f  o F 
.x.  F ) ) } )
2019eleq2d 2455 . 2  |-  ( ph  ->  ( X  e.  ( N `  ( F
" I ) )  <-> 
X  e.  { a  |  E. f  e.  ( Base `  ( S freeLMod  I ) ) a  =  ( M  gsumg  ( f  o F  .x.  F
) ) } ) )
21 ovex 6046 . . . . . 6  |-  ( M 
gsumg  ( f  o F 
.x.  F ) )  e.  _V
22 eleq1 2448 . . . . . 6  |-  ( X  =  ( M  gsumg  ( f  o F  .x.  F
) )  ->  ( X  e.  _V  <->  ( M  gsumg  ( f  o F  .x.  F ) )  e. 
_V ) )
2321, 22mpbiri 225 . . . . 5  |-  ( X  =  ( M  gsumg  ( f  o F  .x.  F
) )  ->  X  e.  _V )
2423rexlimivw 2770 . . . 4  |-  ( E. f  e.  ( Base `  ( S freeLMod  I )
) X  =  ( M  gsumg  ( f  o F 
.x.  F ) )  ->  X  e.  _V )
25 eqeq1 2394 . . . . 5  |-  ( a  =  X  ->  (
a  =  ( M 
gsumg  ( f  o F 
.x.  F ) )  <-> 
X  =  ( M 
gsumg  ( f  o F 
.x.  F ) ) ) )
2625rexbidv 2671 . . . 4  |-  ( a  =  X  ->  ( E. f  e.  ( Base `  ( S freeLMod  I ) ) a  =  ( M  gsumg  ( f  o F 
.x.  F ) )  <->  E. f  e.  ( Base `  ( S freeLMod  I ) ) X  =  ( M  gsumg  ( f  o F 
.x.  F ) ) ) )
2724, 26elab3 3033 . . 3  |-  ( X  e.  { a  |  E. f  e.  (
Base `  ( S freeLMod  I ) ) a  =  ( M  gsumg  ( f  o F 
.x.  F ) ) }  <->  E. f  e.  (
Base `  ( S freeLMod  I ) ) X  =  ( M  gsumg  ( f  o F 
.x.  F ) ) )
28 fvex 5683 . . . . . . . 8  |-  (Scalar `  M )  e.  _V
2914, 28eqeltri 2458 . . . . . . 7  |-  S  e. 
_V
30 ellspd.k . . . . . . . 8  |-  K  =  ( Base `  S
)
31 ellspd.z . . . . . . . 8  |-  .0.  =  ( 0g `  S )
32 eqid 2388 . . . . . . . 8  |-  { a  e.  ( K  ^m  I )  |  ( `' a " ( _V  \  {  .0.  }
) )  e.  Fin }  =  { a  e.  ( K  ^m  I
)  |  ( `' a " ( _V 
\  {  .0.  }
) )  e.  Fin }
338, 30, 31, 32frlmbas 26893 . . . . . . 7  |-  ( ( S  e.  _V  /\  I  e.  _V )  ->  { a  e.  ( K  ^m  I )  |  ( `' a
" ( _V  \  {  .0.  } ) )  e.  Fin }  =  ( Base `  ( S freeLMod  I ) ) )
3429, 13, 33sylancr 645 . . . . . 6  |-  ( ph  ->  { a  e.  ( K  ^m  I )  |  ( `' a
" ( _V  \  {  .0.  } ) )  e.  Fin }  =  ( Base `  ( S freeLMod  I ) ) )
3534eqcomd 2393 . . . . 5  |-  ( ph  ->  ( Base `  ( S freeLMod  I ) )  =  { a  e.  ( K  ^m  I )  |  ( `' a
" ( _V  \  {  .0.  } ) )  e.  Fin } )
3635rexeqdv 2855 . . . 4  |-  ( ph  ->  ( E. f  e.  ( Base `  ( S freeLMod  I ) ) X  =  ( M  gsumg  ( f  o F  .x.  F
) )  <->  E. f  e.  { a  e.  ( K  ^m  I )  |  ( `' a
" ( _V  \  {  .0.  } ) )  e.  Fin } X  =  ( M  gsumg  ( f  o F  .x.  F
) ) ) )
37 cnveq 4987 . . . . . . 7  |-  ( a  =  f  ->  `' a  =  `' f
)
3837imaeq1d 5143 . . . . . 6  |-  ( a  =  f  ->  ( `' a " ( _V  \  {  .0.  }
) )  =  ( `' f " ( _V  \  {  .0.  }
) ) )
3938eleq1d 2454 . . . . 5  |-  ( a  =  f  ->  (
( `' a "
( _V  \  {  .0.  } ) )  e. 
Fin 
<->  ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin ) )
4039rexrab 3042 . . . 4  |-  ( E. f  e.  { a  e.  ( K  ^m  I )  |  ( `' a " ( _V  \  {  .0.  }
) )  e.  Fin } X  =  ( M 
gsumg  ( f  o F 
.x.  F ) )  <->  E. f  e.  ( K  ^m  I ) ( ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  X  =  ( M  gsumg  ( f  o F 
.x.  F ) ) ) )
4136, 40syl6bb 253 . . 3  |-  ( ph  ->  ( E. f  e.  ( Base `  ( S freeLMod  I ) ) X  =  ( M  gsumg  ( f  o F  .x.  F
) )  <->  E. f  e.  ( K  ^m  I
) ( ( `' f " ( _V 
\  {  .0.  }
) )  e.  Fin  /\  X  =  ( M 
gsumg  ( f  o F 
.x.  F ) ) ) ) )
4227, 41syl5bb 249 . 2  |-  ( ph  ->  ( X  e.  {
a  |  E. f  e.  ( Base `  ( S freeLMod  I ) ) a  =  ( M  gsumg  ( f  o F  .x.  F
) ) }  <->  E. f  e.  ( K  ^m  I
) ( ( `' f " ( _V 
\  {  .0.  }
) )  e.  Fin  /\  X  =  ( M 
gsumg  ( f  o F 
.x.  F ) ) ) ) )
4320, 42bitrd 245 1  |-  ( ph  ->  ( X  e.  ( N `  ( F
" I ) )  <->  E. f  e.  ( K  ^m  I ) ( ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  X  =  ( M  gsumg  ( f  o F 
.x.  F ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2374   E.wrex 2651   {crab 2654   _Vcvv 2900    \ cdif 3261   {csn 3758    e. cmpt 4208   `'ccnv 4818   ran crn 4820   "cima 4822    Fn wfn 5390   -->wf 5391   ` cfv 5395  (class class class)co 6021    o Fcof 6243    ^m cmap 6955   Fincfn 7046   Basecbs 13397  Scalarcsca 13460   .scvsca 13461   0gc0g 13651    gsumg cgsu 13652   LModclmod 15878   LSpanclspn 15975   freeLMod cfrlm 26882
This theorem is referenced by:  elfilspd  26925  islindf4  26978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-fz 10977  df-fzo 11067  df-seq 11252  df-hash 11547  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-hom 13481  df-cco 13482  df-prds 13599  df-pws 13601  df-0g 13655  df-gsum 13656  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-mhm 14666  df-submnd 14667  df-grp 14740  df-minusg 14741  df-sbg 14742  df-mulg 14743  df-subg 14869  df-ghm 14932  df-cntz 15044  df-cmn 15342  df-abl 15343  df-mgp 15577  df-rng 15591  df-ur 15593  df-subrg 15794  df-lmod 15880  df-lss 15937  df-lsp 15976  df-lmhm 16026  df-lbs 16075  df-sra 16172  df-rgmod 16173  df-nzr 16257  df-dsmm 26868  df-frlm 26884  df-uvc 26885
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