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Theorem ldualfvs 29631
Description: Scalar product operation for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
Hypotheses
Ref Expression
ldualfvs.f  |-  F  =  (LFnl `  W )
ldualfvs.v  |-  V  =  ( Base `  W
)
ldualfvs.r  |-  R  =  (Scalar `  W )
ldualfvs.k  |-  K  =  ( Base `  R
)
ldualfvs.t  |-  .X.  =  ( .r `  R )
ldualfvs.d  |-  D  =  (LDual `  W )
ldualfvs.s  |-  .xb  =  ( .s `  D )
ldualfvs.w  |-  ( ph  ->  W  e.  Y )
ldualfvs.m  |-  .x.  =  ( k  e.  K ,  f  e.  F  |->  ( f  o F 
.X.  ( V  X.  { k } ) ) )
Assertion
Ref Expression
ldualfvs  |-  ( ph  -> 
.xb  =  .x.  )
Distinct variable groups:    f, k, F    f, K, k    .X. , f,
k    f, V, k    f, W, k
Allowed substitution hints:    ph( f, k)    D( f, k)    R( f, k)    .xb ( f, k)    .x. ( f,
k)    Y( f, k)

Proof of Theorem ldualfvs
StepHypRef Expression
1 ldualfvs.v . . . 4  |-  V  =  ( Base `  W
)
2 eqid 2412 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
3 eqid 2412 . . . 4  |-  (  o F ( +g  `  R
)  |`  ( F  X.  F ) )  =  (  o F ( +g  `  R )  |`  ( F  X.  F
) )
4 ldualfvs.f . . . 4  |-  F  =  (LFnl `  W )
5 ldualfvs.d . . . 4  |-  D  =  (LDual `  W )
6 ldualfvs.r . . . 4  |-  R  =  (Scalar `  W )
7 ldualfvs.k . . . 4  |-  K  =  ( Base `  R
)
8 ldualfvs.t . . . 4  |-  .X.  =  ( .r `  R )
9 eqid 2412 . . . 4  |-  (oppr `  R
)  =  (oppr `  R
)
10 eqid 2412 . . . 4  |-  ( k  e.  K ,  f  e.  F  |->  ( f  o F  .X.  ( V  X.  { k } ) ) )  =  ( k  e.  K ,  f  e.  F  |->  ( f  o F 
.X.  ( V  X.  { k } ) ) )
11 ldualfvs.w . . . 4  |-  ( ph  ->  W  e.  Y )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11ldualset 29620 . . 3  |-  ( ph  ->  D  =  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  (  o F ( +g  `  R
)  |`  ( F  X.  F ) ) >. ,  <. (Scalar `  ndx ) ,  (oppr
`  R ) >. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  K ,  f  e.  F  |->  ( f  o F 
.X.  ( V  X.  { k } ) ) ) >. } ) )
1312fveq2d 5699 . 2  |-  ( ph  ->  ( .s `  D
)  =  ( .s
`  ( { <. (
Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  R )  |`  ( F  X.  F
) ) >. ,  <. (Scalar `  ndx ) ,  (oppr `  R ) >. }  u.  {
<. ( .s `  ndx ) ,  ( k  e.  K ,  f  e.  F  |->  ( f  o F  .X.  ( V  X.  { k } ) ) ) >. } ) ) )
14 ldualfvs.s . 2  |-  .xb  =  ( .s `  D )
15 ldualfvs.m . . 3  |-  .x.  =  ( k  e.  K ,  f  e.  F  |->  ( f  o F 
.X.  ( V  X.  { k } ) ) )
16 fvex 5709 . . . . . 6  |-  ( Base `  R )  e.  _V
177, 16eqeltri 2482 . . . . 5  |-  K  e. 
_V
18 fvex 5709 . . . . . 6  |-  (LFnl `  W )  e.  _V
194, 18eqeltri 2482 . . . . 5  |-  F  e. 
_V
2017, 19mpt2ex 6392 . . . 4  |-  ( k  e.  K ,  f  e.  F  |->  ( f  o F  .X.  ( V  X.  { k } ) ) )  e. 
_V
21 eqid 2412 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  (  o F ( +g  `  R
)  |`  ( F  X.  F ) ) >. ,  <. (Scalar `  ndx ) ,  (oppr
`  R ) >. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  K ,  f  e.  F  |->  ( f  o F 
.X.  ( V  X.  { k } ) ) ) >. } )  =  ( { <. (
Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  R )  |`  ( F  X.  F
) ) >. ,  <. (Scalar `  ndx ) ,  (oppr `  R ) >. }  u.  {
<. ( .s `  ndx ) ,  ( k  e.  K ,  f  e.  F  |->  ( f  o F  .X.  ( V  X.  { k } ) ) ) >. } )
2221lmodvsca 13560 . . . 4  |-  ( ( k  e.  K , 
f  e.  F  |->  ( f  o F  .X.  ( V  X.  { k } ) ) )  e.  _V  ->  (
k  e.  K , 
f  e.  F  |->  ( f  o F  .X.  ( V  X.  { k } ) ) )  =  ( .s `  ( { <. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  (  o F ( +g  `  R
)  |`  ( F  X.  F ) ) >. ,  <. (Scalar `  ndx ) ,  (oppr
`  R ) >. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  K ,  f  e.  F  |->  ( f  o F 
.X.  ( V  X.  { k } ) ) ) >. } ) ) )
2320, 22ax-mp 8 . . 3  |-  ( k  e.  K ,  f  e.  F  |->  ( f  o F  .X.  ( V  X.  { k } ) ) )  =  ( .s `  ( { <. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  (  o F ( +g  `  R
)  |`  ( F  X.  F ) ) >. ,  <. (Scalar `  ndx ) ,  (oppr
`  R ) >. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  K ,  f  e.  F  |->  ( f  o F 
.X.  ( V  X.  { k } ) ) ) >. } ) )
2415, 23eqtri 2432 . 2  |-  .x.  =  ( .s `  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  (  o F ( +g  `  R
)  |`  ( F  X.  F ) ) >. ,  <. (Scalar `  ndx ) ,  (oppr
`  R ) >. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  K ,  f  e.  F  |->  ( f  o F 
.X.  ( V  X.  { k } ) ) ) >. } ) )
2513, 14, 243eqtr4g 2469 1  |-  ( ph  -> 
.xb  =  .x.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2924    u. cun 3286   {csn 3782   {ctp 3784   <.cop 3785    X. cxp 4843    |` cres 4847   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050    o Fcof 6270   ndxcnx 13429   Basecbs 13432   +g cplusg 13492   .rcmulr 13493  Scalarcsca 13495   .scvsca 13496  opprcoppr 15690  LFnlclfn 29552  LDualcld 29618
This theorem is referenced by:  ldualvs  29632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-plusg 13505  df-sca 13508  df-vsca 13509  df-ldual 29619
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