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Theorem ldualfvs 29948
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 2296 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
3 eqid 2296 . . . 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 2296 . . . 4  |-  (oppr `  R
)  =  (oppr `  R
)
10 eqid 2296 . . . 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 29937 . . 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 5545 . 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 5555 . . . . . 6  |-  ( Base `  R )  e.  _V
177, 16eqeltri 2366 . . . . 5  |-  K  e. 
_V
18 fvex 5555 . . . . . 6  |-  (LFnl `  W )  e.  _V
194, 18eqeltri 2366 . . . . 5  |-  F  e. 
_V
2017, 19mpt2ex 6214 . . . 4  |-  ( k  e.  K ,  f  e.  F  |->  ( f  o F  .X.  ( V  X.  { k } ) ) )  e. 
_V
21 eqid 2296 . . . . 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 13292 . . . 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 2316 . 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 2353 1  |-  ( ph  -> 
.xb  =  .x.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163   {csn 3653   {ctp 3655   <.cop 3656    X. cxp 4703    |` cres 4707   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    o Fcof 6092   ndxcnx 13161   Basecbs 13164   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228  opprcoppr 15420  LFnlclfn 29869  LDualcld 29935
This theorem is referenced by:  ldualvs  29949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-sca 13240  df-vsca 13241  df-ldual 29936
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