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Theorem ellspd 27254
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 5389 . . . . . 6  |-  ( F : I --> B  ->  F  Fn  I )
3 fnima 5362 . . . . . 6  |-  ( F  Fn  I  ->  ( F " I )  =  ran  F )
41, 2, 33syl 18 . . . . 5  |-  ( ph  ->  ( F " I
)  =  ran  F
)
54fveq2d 5529 . . . 4  |-  ( ph  ->  ( N `  ( F " I ) )  =  ( N `  ran  F ) )
6 eqid 2283 . . . . . 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 4925 . . . . 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 2283 . . . . . 6  |-  ( S freeLMod  I )  =  ( S freeLMod  I )
9 eqid 2283 . . . . . 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 10 . . . . . 6  |-  ( ph  ->  S  =  (Scalar `  M ) )
16 ellspd.n . . . . . 6  |-  N  =  ( LSpan `  M )
178, 9, 10, 11, 6, 12, 13, 15, 1, 16frlmup3 27252 . . . . 5  |-  ( ph  ->  ran  ( f  e.  ( Base `  ( S freeLMod  I ) )  |->  ( M  gsumg  ( f  o F 
.x.  F ) ) )  =  ( N `
 ran  F )
)
187, 17syl5eqr 2329 . . . 4  |-  ( ph  ->  { a  |  E. f  e.  ( Base `  ( S freeLMod  I )
) a  =  ( M  gsumg  ( f  o F 
.x.  F ) ) }  =  ( N `
 ran  F )
)
195, 18eqtr4d 2318 . . 3  |-  ( ph  ->  ( N `  ( F " I ) )  =  { a  |  E. f  e.  (
Base `  ( S freeLMod  I ) ) a  =  ( M  gsumg  ( f  o F 
.x.  F ) ) } )
2019eleq2d 2350 . 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 5883 . . . . . 6  |-  ( M 
gsumg  ( f  o F 
.x.  F ) )  e.  _V
22 eleq1 2343 . . . . . 6  |-  ( X  =  ( M  gsumg  ( f  o F  .x.  F
) )  ->  ( X  e.  _V  <->  ( M  gsumg  ( f  o F  .x.  F ) )  e. 
_V ) )
2321, 22mpbiri 224 . . . . 5  |-  ( X  =  ( M  gsumg  ( f  o F  .x.  F
) )  ->  X  e.  _V )
2423rexlimivw 2663 . . . 4  |-  ( E. f  e.  ( Base `  ( S freeLMod  I )
) X  =  ( M  gsumg  ( f  o F 
.x.  F ) )  ->  X  e.  _V )
25 eqeq1 2289 . . . . 5  |-  ( a  =  X  ->  (
a  =  ( M 
gsumg  ( f  o F 
.x.  F ) )  <-> 
X  =  ( M 
gsumg  ( f  o F 
.x.  F ) ) ) )
2625rexbidv 2564 . . . 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 2921 . . 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 5539 . . . . . . . 8  |-  (Scalar `  M )  e.  _V
2914, 28eqeltri 2353 . . . . . . 7  |-  S  e. 
_V
30 ellspd.k . . . . . . . 8  |-  K  =  ( Base `  S
)
31 ellspd.z . . . . . . . 8  |-  .0.  =  ( 0g `  S )
32 eqid 2283 . . . . . . . 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 27223 . . . . . . 7  |-  ( ( S  e.  _V  /\  I  e.  _V )  ->  { a  e.  ( K  ^m  I )  |  ( `' a
" ( _V  \  {  .0.  } ) )  e.  Fin }  =  ( Base `  ( S freeLMod  I ) ) )
3429, 13, 33sylancr 644 . . . . . 6  |-  ( ph  ->  { a  e.  ( K  ^m  I )  |  ( `' a
" ( _V  \  {  .0.  } ) )  e.  Fin }  =  ( Base `  ( S freeLMod  I ) ) )
3534eqcomd 2288 . . . . 5  |-  ( ph  ->  ( Base `  ( S freeLMod  I ) )  =  { a  e.  ( K  ^m  I )  |  ( `' a
" ( _V  \  {  .0.  } ) )  e.  Fin } )
3635rexeqdv 2743 . . . 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 4855 . . . . . . 7  |-  ( a  =  f  ->  `' a  =  `' f
)
3837imaeq1d 5011 . . . . . 6  |-  ( a  =  f  ->  ( `' a " ( _V  \  {  .0.  }
) )  =  ( `' f " ( _V  \  {  .0.  }
) ) )
3938eleq1d 2349 . . . . 5  |-  ( a  =  f  ->  (
( `' a "
( _V  \  {  .0.  } ) )  e. 
Fin 
<->  ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin ) )
4039rexrab 2929 . . . 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 252 . . 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 248 . 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 244 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149   {csn 3640    e. cmpt 4077   `'ccnv 4688   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076    ^m cmap 6772   Fincfn 6863   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400    gsumg cgsu 13401   LModclmod 15627   LSpanclspn 15728   freeLMod cfrlm 27212
This theorem is referenced by:  elfilspd  27255  islindf4  27308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lmhm 15779  df-lbs 15828  df-sra 15925  df-rgmod 15926  df-nzr 16010  df-dsmm 27198  df-frlm 27214  df-uvc 27215
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