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Theorem ellspd 18130
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.) (Revised by AV, 24-Jun-2019.)
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 finSupp  .0.  /\  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 ellspd
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 ellspd.f . . . . . 6  |-  ( ph  ->  F : I --> B )
2 ffn 5556 . . . . . 6  |-  ( F : I --> B  ->  F  Fn  I )
3 fnima 5526 . . . . . 6  |-  ( F  Fn  I  ->  ( F " I )  =  ran  F )
41, 2, 33syl 20 . . . . 5  |-  ( ph  ->  ( F " I
)  =  ran  F
)
54fveq2d 5692 . . . 4  |-  ( ph  ->  ( N `  ( F " I ) )  =  ( N `  ran  F ) )
6 eqid 2441 . . . . . 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 ) ) )
76rnmpt 5081 . . . . 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 2441 . . . . . 6  |-  ( S freeLMod  I )  =  ( S freeLMod  I )
9 eqid 2441 . . . . . 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 18128 . . . . 5  |-  ( ph  ->  ran  ( f  e.  ( Base `  ( S freeLMod  I ) )  |->  ( M  gsumg  ( f  oF  .x.  F ) ) )  =  ( N `
 ran  F )
)
187, 17syl5eqr 2487 . . . 4  |-  ( ph  ->  { a  |  E. f  e.  ( Base `  ( S freeLMod  I )
) a  =  ( M  gsumg  ( f  oF  .x.  F ) ) }  =  ( N `
 ran  F )
)
195, 18eqtr4d 2476 . . 3  |-  ( ph  ->  ( N `  ( F " I ) )  =  { a  |  E. f  e.  (
Base `  ( S freeLMod  I ) ) a  =  ( M  gsumg  ( f  oF  .x.  F ) ) } )
2019eleq2d 2508 . 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 6115 . . . . . 6  |-  ( M 
gsumg  ( f  oF  .x.  F ) )  e.  _V
22 eleq1 2501 . . . . . 6  |-  ( X  =  ( M  gsumg  ( f  oF  .x.  F
) )  ->  ( X  e.  _V  <->  ( M  gsumg  ( f  oF  .x.  F ) )  e. 
_V ) )
2321, 22mpbiri 233 . . . . 5  |-  ( X  =  ( M  gsumg  ( f  oF  .x.  F
) )  ->  X  e.  _V )
2423rexlimivw 2835 . . . 4  |-  ( E. f  e.  ( Base `  ( S freeLMod  I )
) X  =  ( M  gsumg  ( f  oF  .x.  F ) )  ->  X  e.  _V )
25 eqeq1 2447 . . . . 5  |-  ( a  =  X  ->  (
a  =  ( M 
gsumg  ( f  oF  .x.  F ) )  <-> 
X  =  ( M 
gsumg  ( f  oF  .x.  F ) ) ) )
2625rexbidv 2734 . . . 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 ) ) ) )
2724, 26elab3 3110 . . 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 5698 . . . . . . . 8  |-  (Scalar `  M )  e.  _V
2914, 28eqeltri 2511 . . . . . . 7  |-  S  e. 
_V
30 ellspd.k . . . . . . . 8  |-  K  =  ( Base `  S
)
31 ellspd.z . . . . . . . 8  |-  .0.  =  ( 0g `  S )
32 eqid 2441 . . . . . . . 8  |-  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  }  =  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  }
338, 30, 31, 32frlmbas 18080 . . . . . . 7  |-  ( ( S  e.  _V  /\  I  e.  _V )  ->  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  }  =  ( Base `  ( S freeLMod  I ) ) )
3429, 13, 33sylancr 658 . . . . . 6  |-  ( ph  ->  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  }  =  ( Base `  ( S freeLMod  I ) ) )
3534eqcomd 2446 . . . . 5  |-  ( ph  ->  ( Base `  ( S freeLMod  I ) )  =  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  }
)
3635rexeqdv 2922 . . . 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 finSupp  .0.  } X  =  ( M  gsumg  ( f  oF  .x.  F ) ) ) )
37 breq1 4292 . . . . 5  |-  ( a  =  f  ->  (
a finSupp  .0.  <->  f finSupp  .0.  ) )
3837rexrab 3120 . . . 4  |-  ( E. f  e.  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  } X  =  ( M  gsumg  ( f  oF  .x.  F ) )  <->  E. f  e.  ( K  ^m  I ) ( f finSupp  .0.  /\  X  =  ( M  gsumg  ( f  oF  .x.  F ) ) ) )
3936, 38syl6bb 261 . . 3  |-  ( ph  ->  ( E. f  e.  ( Base `  ( S freeLMod  I ) ) X  =  ( M  gsumg  ( f  oF  .x.  F
) )  <->  E. f  e.  ( K  ^m  I
) ( f finSupp  .0.  /\  X  =  ( M 
gsumg  ( f  oF  .x.  F ) ) ) ) )
4027, 39syl5bb 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 finSupp  .0.  /\  X  =  ( M 
gsumg  ( f  oF  .x.  F ) ) ) ) )
4120, 40bitrd 253 1  |-  ( ph  ->  ( X  e.  ( N `  ( F
" I ) )  <->  E. f  e.  ( K  ^m  I ) ( f finSupp  .0.  /\  X  =  ( M  gsumg  ( f  oF  .x.  F ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   {cab 2427   E.wrex 2714   {crab 2717   _Vcvv 2970   class class class wbr 4289    e. cmpt 4347   ran crn 4837   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317    ^m cmap 7210   finSupp cfsupp 7616   Basecbs 14170  Scalarcsca 14237   .scvsca 14238   0gc0g 14374    gsumg cgsu 14375   LModclmod 16928   LSpanclspn 17030   freeLMod cfrlm 18071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-hom 14258  df-cco 14259  df-0g 14376  df-gsum 14377  df-prds 14382  df-pws 14384  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-ghm 15738  df-cntz 15828  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-subrg 16843  df-lmod 16930  df-lss 16992  df-lsp 17031  df-lmhm 17081  df-lbs 17134  df-sra 17231  df-rgmod 17232  df-nzr 17318  df-dsmm 18057  df-frlm 18072  df-uvc 18108
This theorem is referenced by:  elfilspd  18132  islindf4  18167
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