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Theorem ellspd 27222
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 5583 . . . . . 6  |-  ( F : I --> B  ->  F  Fn  I )
3 fnima 5555 . . . . . 6  |-  ( F  Fn  I  ->  ( F " I )  =  ran  F )
41, 2, 33syl 19 . . . . 5  |-  ( ph  ->  ( F " I
)  =  ran  F
)
54fveq2d 5724 . . . 4  |-  ( ph  ->  ( N `  ( F " I ) )  =  ( N `  ran  F ) )
6 eqid 2435 . . . . . 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 5108 . . . . 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 2435 . . . . . 6  |-  ( S freeLMod  I )  =  ( S freeLMod  I )
9 eqid 2435 . . . . . 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 27220 . . . . 5  |-  ( ph  ->  ran  ( f  e.  ( Base `  ( S freeLMod  I ) )  |->  ( M  gsumg  ( f  o F 
.x.  F ) ) )  =  ( N `
 ran  F )
)
187, 17syl5eqr 2481 . . . 4  |-  ( ph  ->  { a  |  E. f  e.  ( Base `  ( S freeLMod  I )
) a  =  ( M  gsumg  ( f  o F 
.x.  F ) ) }  =  ( N `
 ran  F )
)
195, 18eqtr4d 2470 . . 3  |-  ( ph  ->  ( N `  ( F " I ) )  =  { a  |  E. f  e.  (
Base `  ( S freeLMod  I ) ) a  =  ( M  gsumg  ( f  o F 
.x.  F ) ) } )
2019eleq2d 2502 . 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 6098 . . . . . 6  |-  ( M 
gsumg  ( f  o F 
.x.  F ) )  e.  _V
22 eleq1 2495 . . . . . 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 2818 . . . 4  |-  ( E. f  e.  ( Base `  ( S freeLMod  I )
) X  =  ( M  gsumg  ( f  o F 
.x.  F ) )  ->  X  e.  _V )
25 eqeq1 2441 . . . . 5  |-  ( a  =  X  ->  (
a  =  ( M 
gsumg  ( f  o F 
.x.  F ) )  <-> 
X  =  ( M 
gsumg  ( f  o F 
.x.  F ) ) ) )
2625rexbidv 2718 . . . 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 3081 . . 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 5734 . . . . . . . 8  |-  (Scalar `  M )  e.  _V
2914, 28eqeltri 2505 . . . . . . 7  |-  S  e. 
_V
30 ellspd.k . . . . . . . 8  |-  K  =  ( Base `  S
)
31 ellspd.z . . . . . . . 8  |-  .0.  =  ( 0g `  S )
32 eqid 2435 . . . . . . . 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 27191 . . . . . . 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 2440 . . . . 5  |-  ( ph  ->  ( Base `  ( S freeLMod  I ) )  =  { a  e.  ( K  ^m  I )  |  ( `' a
" ( _V  \  {  .0.  } ) )  e.  Fin } )
3635rexeqdv 2903 . . . 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 5038 . . . . . . 7  |-  ( a  =  f  ->  `' a  =  `' f
)
3837imaeq1d 5194 . . . . . 6  |-  ( a  =  f  ->  ( `' a " ( _V  \  {  .0.  }
) )  =  ( `' f " ( _V  \  {  .0.  }
) ) )
3938eleq1d 2501 . . . . 5  |-  ( a  =  f  ->  (
( `' a "
( _V  \  {  .0.  } ) )  e. 
Fin 
<->  ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin ) )
4039rexrab 3090 . . . 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 1652    e. wcel 1725   {cab 2421   E.wrex 2698   {crab 2701   _Vcvv 2948    \ cdif 3309   {csn 3806    e. cmpt 4258   `'ccnv 4869   ran crn 4871   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295    ^m cmap 7010   Fincfn 7101   Basecbs 13461  Scalarcsca 13524   .scvsca 13525   0gc0g 13715    gsumg cgsu 13716   LModclmod 15942   LSpanclspn 16039   freeLMod cfrlm 27180
This theorem is referenced by:  elfilspd  27223  islindf4  27276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-hom 13545  df-cco 13546  df-prds 13663  df-pws 13665  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-ghm 14996  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-subrg 15858  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lmhm 16090  df-lbs 16139  df-sra 16236  df-rgmod 16237  df-nzr 16321  df-dsmm 27166  df-frlm 27182  df-uvc 27183
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