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Theorem ellspdOLD 18340
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 18339 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 5657 . . . . . 6  |-  ( F : I --> B  ->  F  Fn  I )
3 fnima 5627 . . . . . 6  |-  ( F  Fn  I  ->  ( F " I )  =  ran  F )
41, 2, 33syl 20 . . . . 5  |-  ( ph  ->  ( F " I
)  =  ran  F
)
54fveq2d 5793 . . . 4  |-  ( ph  ->  ( N `  ( F " I ) )  =  ( N `  ran  F ) )
6 eqid 2451 . . . . . 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 5183 . . . . 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 2451 . . . . . 6  |-  ( S freeLMod  I )  =  ( S freeLMod  I )
9 eqid 2451 . . . . . 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 18337 . . . . 5  |-  ( ph  ->  ran  ( f  e.  ( Base `  ( S freeLMod  I ) )  |->  ( M  gsumg  ( f  oF  .x.  F ) ) )  =  ( N `
 ran  F )
)
187, 17syl5eqr 2506 . . . 4  |-  ( ph  ->  { a  |  E. f  e.  ( Base `  ( S freeLMod  I )
) a  =  ( M  gsumg  ( f  oF  .x.  F ) ) }  =  ( N `
 ran  F )
)
195, 18eqtr4d 2495 . . 3  |-  ( ph  ->  ( N `  ( F " I ) )  =  { a  |  E. f  e.  (
Base `  ( S freeLMod  I ) ) a  =  ( M  gsumg  ( f  oF  .x.  F ) ) } )
2019eleq2d 2521 . 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 6215 . . . . . 6  |-  ( M 
gsumg  ( f  oF  .x.  F ) )  e.  _V
22 eleq1 2523 . . . . . 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 2933 . . . 4  |-  ( E. f  e.  ( Base `  ( S freeLMod  I )
) X  =  ( M  gsumg  ( f  oF  .x.  F ) )  ->  X  e.  _V )
25 eqeq1 2455 . . . . 5  |-  ( a  =  X  ->  (
a  =  ( M 
gsumg  ( f  oF  .x.  F ) )  <-> 
X  =  ( M 
gsumg  ( f  oF  .x.  F ) ) ) )
2625rexbidv 2844 . . . 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 3210 . . 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 5799 . . . . . . . 8  |-  (Scalar `  M )  e.  _V
2914, 28eqeltri 2535 . . . . . . 7  |-  S  e. 
_V
30 ellspd.k . . . . . . . 8  |-  K  =  ( Base `  S
)
31 ellspd.z . . . . . . . 8  |-  .0.  =  ( 0g `  S )
32 eqid 2451 . . . . . . . 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 18290 . . . . . . 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 2459 . . . . 5  |-  ( ph  ->  ( Base `  ( S freeLMod  I ) )  =  { a  e.  ( K  ^m  I )  |  ( `' a
" ( _V  \  {  .0.  } ) )  e.  Fin } )
3635rexeqdv 3020 . . . 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 5111 . . . . . . 7  |-  ( a  =  f  ->  `' a  =  `' f
)
3837imaeq1d 5266 . . . . . 6  |-  ( a  =  f  ->  ( `' a " ( _V  \  {  .0.  }
) )  =  ( `' f " ( _V  \  {  .0.  }
) ) )
3938eleq1d 2520 . . . . 5  |-  ( a  =  f  ->  (
( `' a "
( _V  \  {  .0.  } ) )  e. 
Fin 
<->  ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin ) )
4039rexrab 3220 . . . 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 1370    e. wcel 1758   {cab 2436   E.wrex 2796   {crab 2799   _Vcvv 3068    \ cdif 3423   {csn 3975    |-> cmpt 4448   `'ccnv 4937   ran crn 4939   "cima 4941    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190    oFcof 6418    ^m cmap 7314   Fincfn 7410   Basecbs 14276  Scalarcsca 14343   .scvsca 14344   0gc0g 14480    gsumg cgsu 14481   LModclmod 17054   LSpanclspn 17158   freeLMod cfrlm 18280
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 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-fz 11539  df-fzo 11650  df-seq 11908  df-hash 12205  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-hom 14364  df-cco 14365  df-0g 14482  df-gsum 14483  df-prds 14488  df-pws 14490  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-mhm 15566  df-submnd 15567  df-grp 15647  df-minusg 15648  df-sbg 15649  df-mulg 15650  df-subg 15780  df-ghm 15847  df-cntz 15937  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-rng 16753  df-subrg 16969  df-lmod 17056  df-lss 17120  df-lsp 17159  df-lmhm 17209  df-lbs 17262  df-sra 17359  df-rgmod 17360  df-nzr 17446  df-dsmm 18266  df-frlm 18281  df-uvc 18317
This theorem is referenced by: (None)
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