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Theorem ellspdOLD 18604
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.) Obsolete version of ellspd 18603 as of 24-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
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
ellspdOLD  |-  ( ph  ->  ( X  e.  ( N `  ( F
" I ) )  <->  E. f  e.  ( K  ^m  I ) ( ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  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 ellspdOLD
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 ellspd.f . . . . . 6  |-  ( ph  ->  F : I --> B )
2 ffn 5729 . . . . . 6  |-  ( F : I --> B  ->  F  Fn  I )
3 fnima 5697 . . . . . 6  |-  ( F  Fn  I  ->  ( F " I )  =  ran  F )
41, 2, 33syl 20 . . . . 5  |-  ( ph  ->  ( F " I
)  =  ran  F
)
54fveq2d 5868 . . . 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 5246 . . . . 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 18601 . . . . 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 6307 . . . . . 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 2952 . . . 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 2973 . . . 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 3257 . . 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 5874 . . . . . . . 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 " ( _V  \  {  .0.  }
) )  e.  Fin }  =  { a  e.  ( K  ^m  I
)  |  ( `' a " ( _V 
\  {  .0.  }
) )  e.  Fin }
338, 30, 31, 32frlmbasOLD 18554 . . . . . . 7  |-  ( ( S  e.  _V  /\  I  e.  _V )  ->  { a  e.  ( K  ^m  I )  |  ( `' a
" ( _V  \  {  .0.  } ) )  e.  Fin }  =  ( Base `  ( S freeLMod  I ) ) )
3429, 13, 33sylancr 663 . . . . . 6  |-  ( ph  ->  { a  e.  ( K  ^m  I )  |  ( `' a
" ( _V  \  {  .0.  } ) )  e.  Fin }  =  ( Base `  ( S freeLMod  I ) ) )
3534eqcomd 2475 . . . . 5  |-  ( ph  ->  ( Base `  ( S freeLMod  I ) )  =  { a  e.  ( K  ^m  I )  |  ( `' a
" ( _V  \  {  .0.  } ) )  e.  Fin } )
3635rexeqdv 3065 . . . 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
" ( _V  \  {  .0.  } ) )  e.  Fin } X  =  ( M  gsumg  ( f  oF  .x.  F
) ) ) )
37 cnveq 5174 . . . . . . 7  |-  ( a  =  f  ->  `' a  =  `' f
)
3837imaeq1d 5334 . . . . . 6  |-  ( a  =  f  ->  ( `' a " ( _V  \  {  .0.  }
) )  =  ( `' f " ( _V  \  {  .0.  }
) ) )
3938eleq1d 2536 . . . . 5  |-  ( a  =  f  ->  (
( `' a "
( _V  \  {  .0.  } ) )  e. 
Fin 
<->  ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin ) )
4039rexrab 3267 . . . 4  |-  ( E. f  e.  { a  e.  ( K  ^m  I )  |  ( `' a " ( _V  \  {  .0.  }
) )  e.  Fin } X  =  ( M 
gsumg  ( f  oF  .x.  F ) )  <->  E. f  e.  ( K  ^m  I ) ( ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  X  =  ( M  gsumg  ( f  oF  .x.  F ) ) ) )
4136, 40syl6bb 261 . . 3  |-  ( ph  ->  ( E. f  e.  ( Base `  ( S freeLMod  I ) ) X  =  ( M  gsumg  ( f  oF  .x.  F
) )  <->  E. f  e.  ( K  ^m  I
) ( ( `' f " ( _V 
\  {  .0.  }
) )  e.  Fin  /\  X  =  ( M 
gsumg  ( f  oF  .x.  F ) ) ) ) )
4227, 41syl5bb 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 " ( _V 
\  {  .0.  }
) )  e.  Fin  /\  X  =  ( M 
gsumg  ( f  oF  .x.  F ) ) ) ) )
4320, 42bitrd 253 1  |-  ( ph  ->  ( X  e.  ( N `  ( F
" I ) )  <->  E. f  e.  ( K  ^m  I ) ( ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  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 2815   {crab 2818   _Vcvv 3113    \ cdif 3473   {csn 4027    |-> cmpt 4505   `'ccnv 4998   ran crn 5000   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282    oFcof 6520    ^m cmap 7417   Fincfn 7513   Basecbs 14486  Scalarcsca 14554   .scvsca 14555   0gc0g 14691    gsumg cgsu 14692   LModclmod 17295   LSpanclspn 17400   freeLMod cfrlm 18544
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-hom 14575  df-cco 14576  df-0g 14693  df-gsum 14694  df-prds 14699  df-pws 14701  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-mhm 15777  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mulg 15861  df-subg 15993  df-ghm 16060  df-cntz 16150  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-subrg 17210  df-lmod 17297  df-lss 17362  df-lsp 17401  df-lmhm 17451  df-lbs 17504  df-sra 17601  df-rgmod 17602  df-nzr 17688  df-dsmm 18530  df-frlm 18545  df-uvc 18581
This theorem is referenced by: (None)
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