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Theorem ellspd 18705
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 5737 . . . . . 6  |-  ( F : I --> B  ->  F  Fn  I )
3 fnima 5705 . . . . . 6  |-  ( F  Fn  I  ->  ( F " I )  =  ran  F )
41, 2, 33syl 20 . . . . 5  |-  ( ph  ->  ( F " I
)  =  ran  F
)
54fveq2d 5876 . . . 4  |-  ( ph  ->  ( N `  ( F " I ) )  =  ( N `  ran  F ) )
6 eqid 2467 . . . . . 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 5254 . . . . 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 2467 . . . . . 6  |-  ( S freeLMod  I )  =  ( S freeLMod  I )
9 eqid 2467 . . . . . 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 18703 . . . . 5  |-  ( ph  ->  ran  ( f  e.  ( Base `  ( S freeLMod  I ) )  |->  ( M  gsumg  ( f  oF  .x.  F ) ) )  =  ( N `
 ran  F )
)
187, 17syl5eqr 2522 . . . 4  |-  ( ph  ->  { a  |  E. f  e.  ( Base `  ( S freeLMod  I )
) a  =  ( M  gsumg  ( f  oF  .x.  F ) ) }  =  ( N `
 ran  F )
)
195, 18eqtr4d 2511 . . 3  |-  ( ph  ->  ( N `  ( F " I ) )  =  { a  |  E. f  e.  (
Base `  ( S freeLMod  I ) ) a  =  ( M  gsumg  ( f  oF  .x.  F ) ) } )
2019eleq2d 2537 . 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 6320 . . . . . 6  |-  ( M 
gsumg  ( f  oF  .x.  F ) )  e.  _V
22 eleq1 2539 . . . . . 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 2956 . . . 4  |-  ( E. f  e.  ( Base `  ( S freeLMod  I )
) X  =  ( M  gsumg  ( f  oF  .x.  F ) )  ->  X  e.  _V )
25 eqeq1 2471 . . . . 5  |-  ( a  =  X  ->  (
a  =  ( M 
gsumg  ( f  oF  .x.  F ) )  <-> 
X  =  ( M 
gsumg  ( f  oF  .x.  F ) ) ) )
2625rexbidv 2978 . . . 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 3262 . . 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 5882 . . . . . . . 8  |-  (Scalar `  M )  e.  _V
2914, 28eqeltri 2551 . . . . . . 7  |-  S  e. 
_V
30 ellspd.k . . . . . . . 8  |-  K  =  ( Base `  S
)
31 ellspd.z . . . . . . . 8  |-  .0.  =  ( 0g `  S )
32 eqid 2467 . . . . . . . 8  |-  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  }  =  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  }
338, 30, 31, 32frlmbas 18655 . . . . . . 7  |-  ( ( S  e.  _V  /\  I  e.  _V )  ->  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  }  =  ( Base `  ( S freeLMod  I ) ) )
3429, 13, 33sylancr 663 . . . . . 6  |-  ( ph  ->  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  }  =  ( Base `  ( S freeLMod  I ) ) )
3534eqcomd 2475 . . . . 5  |-  ( ph  ->  ( Base `  ( S freeLMod  I ) )  =  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  }
)
3635rexeqdv 3070 . . . 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 4456 . . . . 5  |-  ( a  =  f  ->  (
a finSupp  .0.  <->  f finSupp  .0.  ) )
3837rexrab 3272 . . . 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 1379    e. wcel 1767   {cab 2452   E.wrex 2818   {crab 2821   _Vcvv 3118   class class class wbr 4453    |-> cmpt 4511   ran crn 5006   "cima 5008    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    oFcof 6533    ^m cmap 7432   finSupp cfsupp 7841   Basecbs 14507  Scalarcsca 14575   .scvsca 14576   0gc0g 14712    gsumg cgsu 14713   LModclmod 17383   LSpanclspn 17488   freeLMod cfrlm 18646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-0g 14714  df-gsum 14715  df-prds 14720  df-pws 14722  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-ghm 16137  df-cntz 16227  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-subrg 17298  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lmhm 17539  df-lbs 17592  df-sra 17689  df-rgmod 17690  df-nzr 17776  df-dsmm 18632  df-frlm 18647  df-uvc 18683
This theorem is referenced by:  elfilspd  18707  islindf4  18742
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