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Theorem ellspd 18365
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 5670 . . . . . 6  |-  ( F : I --> B  ->  F  Fn  I )
3 fnima 5640 . . . . . 6  |-  ( F  Fn  I  ->  ( F " I )  =  ran  F )
41, 2, 33syl 20 . . . . 5  |-  ( ph  ->  ( F " I
)  =  ran  F
)
54fveq2d 5806 . . . 4  |-  ( ph  ->  ( N `  ( F " I ) )  =  ( N `  ran  F ) )
6 eqid 2454 . . . . . 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 5196 . . . . 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 2454 . . . . . 6  |-  ( S freeLMod  I )  =  ( S freeLMod  I )
9 eqid 2454 . . . . . 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 18363 . . . . 5  |-  ( ph  ->  ran  ( f  e.  ( Base `  ( S freeLMod  I ) )  |->  ( M  gsumg  ( f  oF  .x.  F ) ) )  =  ( N `
 ran  F )
)
187, 17syl5eqr 2509 . . . 4  |-  ( ph  ->  { a  |  E. f  e.  ( Base `  ( S freeLMod  I )
) a  =  ( M  gsumg  ( f  oF  .x.  F ) ) }  =  ( N `
 ran  F )
)
195, 18eqtr4d 2498 . . 3  |-  ( ph  ->  ( N `  ( F " I ) )  =  { a  |  E. f  e.  (
Base `  ( S freeLMod  I ) ) a  =  ( M  gsumg  ( f  oF  .x.  F ) ) } )
2019eleq2d 2524 . 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 6228 . . . . . 6  |-  ( M 
gsumg  ( f  oF  .x.  F ) )  e.  _V
22 eleq1 2526 . . . . . 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 2943 . . . 4  |-  ( E. f  e.  ( Base `  ( S freeLMod  I )
) X  =  ( M  gsumg  ( f  oF  .x.  F ) )  ->  X  e.  _V )
25 eqeq1 2458 . . . . 5  |-  ( a  =  X  ->  (
a  =  ( M 
gsumg  ( f  oF  .x.  F ) )  <-> 
X  =  ( M 
gsumg  ( f  oF  .x.  F ) ) ) )
2625rexbidv 2868 . . . 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 3220 . . 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 5812 . . . . . . . 8  |-  (Scalar `  M )  e.  _V
2914, 28eqeltri 2538 . . . . . . 7  |-  S  e. 
_V
30 ellspd.k . . . . . . . 8  |-  K  =  ( Base `  S
)
31 ellspd.z . . . . . . . 8  |-  .0.  =  ( 0g `  S )
32 eqid 2454 . . . . . . . 8  |-  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  }  =  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  }
338, 30, 31, 32frlmbas 18315 . . . . . . 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 2462 . . . . 5  |-  ( ph  ->  ( Base `  ( S freeLMod  I ) )  =  { a  e.  ( K  ^m  I )  |  a finSupp  .0.  }
)
3635rexeqdv 3030 . . . 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 4406 . . . . 5  |-  ( a  =  f  ->  (
a finSupp  .0.  <->  f finSupp  .0.  ) )
3837rexrab 3230 . . . 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 1370    e. wcel 1758   {cab 2439   E.wrex 2800   {crab 2803   _Vcvv 3078   class class class wbr 4403    |-> cmpt 4461   ran crn 4952   "cima 4954    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203    oFcof 6431    ^m cmap 7327   finSupp cfsupp 7734   Basecbs 14296  Scalarcsca 14364   .scvsca 14365   0gc0g 14501    gsumg cgsu 14502   LModclmod 17081   LSpanclspn 17185   freeLMod cfrlm 18306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-sup 7806  df-oi 7839  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-fz 11559  df-fzo 11670  df-seq 11928  df-hash 12225  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-mulr 14375  df-sca 14377  df-vsca 14378  df-ip 14379  df-tset 14380  df-ple 14381  df-ds 14383  df-hom 14385  df-cco 14386  df-0g 14503  df-gsum 14504  df-prds 14509  df-pws 14511  df-mre 14647  df-mrc 14648  df-acs 14650  df-mnd 15538  df-mhm 15587  df-submnd 15588  df-grp 15668  df-minusg 15669  df-sbg 15670  df-mulg 15671  df-subg 15801  df-ghm 15868  df-cntz 15958  df-cmn 16404  df-abl 16405  df-mgp 16724  df-ur 16736  df-rng 16780  df-subrg 16996  df-lmod 17083  df-lss 17147  df-lsp 17186  df-lmhm 17236  df-lbs 17289  df-sra 17386  df-rgmod 17387  df-nzr 17473  df-dsmm 18292  df-frlm 18307  df-uvc 18343
This theorem is referenced by:  elfilspd  18367  islindf4  18402
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