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Theorem ldualset 29388
Description: Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows us to reuse our existing collection of left vector space theorems. Note the operation reversal in the scalar product to allow us to use the original scalar ring instead of the oppr ring, for convenience. (Contributed by NM, 18-Oct-2014.)
Hypotheses
Ref Expression
ldualset.v  |-  V  =  ( Base `  W
)
ldualset.a  |-  .+  =  ( +g  `  R )
ldualset.p  |-  .+b  =  (  o F  .+  |`  ( F  X.  F ) )
ldualset.f  |-  F  =  (LFnl `  W )
ldualset.d  |-  D  =  (LDual `  W )
ldualset.r  |-  R  =  (Scalar `  W )
ldualset.k  |-  K  =  ( Base `  R
)
ldualset.t  |-  .x.  =  ( .r `  R )
ldualset.o  |-  O  =  (oppr
`  R )
ldualset.s  |-  .xb  =  ( k  e.  K ,  f  e.  F  |->  ( f  o F 
.x.  ( V  X.  { k } ) ) )
ldualset.w  |-  ( ph  ->  W  e.  X )
Assertion
Ref Expression
ldualset  |-  ( ph  ->  D  =  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } ) )
Distinct variable group:    f, k, W
Allowed substitution hints:    ph( f, k)    D( f, k)    .+ ( f, k)    .+b ( f, k)    R( f, k)    .xb ( f, k)    .x. ( f,
k)    F( f, k)    K( f, k)    O( f, k)    V( f, k)    X( f, k)

Proof of Theorem ldualset
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ldualset.w . 2  |-  ( ph  ->  W  e.  X )
2 elex 2798 . 2  |-  ( W  e.  X  ->  W  e.  _V )
3 ldualset.d . . 3  |-  D  =  (LDual `  W )
4 fveq2 5527 . . . . . . . 8  |-  ( w  =  W  ->  (LFnl `  w )  =  (LFnl `  W ) )
5 ldualset.f . . . . . . . 8  |-  F  =  (LFnl `  W )
64, 5syl6eqr 2335 . . . . . . 7  |-  ( w  =  W  ->  (LFnl `  w )  =  F )
76opeq2d 3805 . . . . . 6  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  (LFnl `  w ) >.  =  <. ( Base `  ndx ) ,  F >. )
8 fveq2 5527 . . . . . . . . . . . . 13  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
9 ldualset.r . . . . . . . . . . . . 13  |-  R  =  (Scalar `  W )
108, 9syl6eqr 2335 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (Scalar `  w )  =  R )
1110fveq2d 5531 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( +g  `  (Scalar `  w
) )  =  ( +g  `  R ) )
12 ldualset.a . . . . . . . . . . 11  |-  .+  =  ( +g  `  R )
1311, 12syl6eqr 2335 . . . . . . . . . 10  |-  ( w  =  W  ->  ( +g  `  (Scalar `  w
) )  =  .+  )
14 ofeq 6082 . . . . . . . . . 10  |-  ( ( +g  `  (Scalar `  w ) )  = 
.+  ->  o F ( +g  `  (Scalar `  w ) )  =  o F  .+  )
1513, 14syl 15 . . . . . . . . 9  |-  ( w  =  W  ->  o F ( +g  `  (Scalar `  w ) )  =  o F  .+  )
166, 6xpeq12d 4716 . . . . . . . . 9  |-  ( w  =  W  ->  (
(LFnl `  w )  X.  (LFnl `  w )
)  =  ( F  X.  F ) )
1715, 16reseq12d 4958 . . . . . . . 8  |-  ( w  =  W  ->  (  o F ( +g  `  (Scalar `  w ) )  |`  ( (LFnl `  w )  X.  (LFnl `  w )
) )  =  (  o F  .+  |`  ( F  X.  F ) ) )
18 ldualset.p . . . . . . . 8  |-  .+b  =  (  o F  .+  |`  ( F  X.  F ) )
1917, 18syl6eqr 2335 . . . . . . 7  |-  ( w  =  W  ->  (  o F ( +g  `  (Scalar `  w ) )  |`  ( (LFnl `  w )  X.  (LFnl `  w )
) )  =  .+b  )
2019opeq2d 3805 . . . . . 6  |-  ( w  =  W  ->  <. ( +g  `  ndx ) ,  (  o F ( +g  `  (Scalar `  w ) )  |`  ( (LFnl `  w )  X.  (LFnl `  w )
) ) >.  =  <. ( +g  `  ndx ) ,  .+b  >. )
2110fveq2d 5531 . . . . . . . 8  |-  ( w  =  W  ->  (oppr `  (Scalar `  w ) )  =  (oppr
`  R ) )
22 ldualset.o . . . . . . . 8  |-  O  =  (oppr
`  R )
2321, 22syl6eqr 2335 . . . . . . 7  |-  ( w  =  W  ->  (oppr `  (Scalar `  w ) )  =  O )
2423opeq2d 3805 . . . . . 6  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  (oppr `  (Scalar `  w ) )
>.  =  <. (Scalar `  ndx ) ,  O >. )
257, 20, 24tpeq123d 3723 . . . . 5  |-  ( w  =  W  ->  { <. (
Base `  ndx ) ,  (LFnl `  w ) >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  (Scalar `  w ) )  |`  ( (LFnl `  w )  X.  (LFnl `  w )
) ) >. ,  <. (Scalar `  ndx ) ,  (oppr `  (Scalar `  w ) )
>. }  =  { <. (
Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. (Scalar ` 
ndx ) ,  O >. } )
2610fveq2d 5531 . . . . . . . . . 10  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  R )
)
27 ldualset.k . . . . . . . . . 10  |-  K  =  ( Base `  R
)
2826, 27syl6eqr 2335 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  K )
2910fveq2d 5531 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .r `  (Scalar `  w
) )  =  ( .r `  R ) )
30 ldualset.t . . . . . . . . . . . 12  |-  .x.  =  ( .r `  R )
3129, 30syl6eqr 2335 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .r `  (Scalar `  w
) )  =  .x.  )
32 ofeq 6082 . . . . . . . . . . 11  |-  ( ( .r `  (Scalar `  w ) )  = 
.x.  ->  o F ( .r `  (Scalar `  w ) )  =  o F  .x.  )
3331, 32syl 15 . . . . . . . . . 10  |-  ( w  =  W  ->  o F ( .r `  (Scalar `  w ) )  =  o F  .x.  )
34 eqidd 2286 . . . . . . . . . 10  |-  ( w  =  W  ->  f  =  f )
35 fveq2 5527 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
36 ldualset.v . . . . . . . . . . . 12  |-  V  =  ( Base `  W
)
3735, 36syl6eqr 2335 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( Base `  w )  =  V )
3837xpeq1d 4714 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( Base `  w )  X.  { k } )  =  ( V  X.  { k } ) )
3933, 34, 38oveq123d 5881 . . . . . . . . 9  |-  ( w  =  W  ->  (
f  o F ( .r `  (Scalar `  w ) ) ( ( Base `  w
)  X.  { k } ) )  =  ( f  o F 
.x.  ( V  X.  { k } ) ) )
4028, 6, 39mpt2eq123dv 5912 . . . . . . . 8  |-  ( w  =  W  ->  (
k  e.  ( Base `  (Scalar `  w )
) ,  f  e.  (LFnl `  w )  |->  ( f  o F ( .r `  (Scalar `  w ) ) ( ( Base `  w
)  X.  { k } ) ) )  =  ( k  e.  K ,  f  e.  F  |->  ( f  o F  .x.  ( V  X.  { k } ) ) ) )
41 ldualset.s . . . . . . . 8  |-  .xb  =  ( k  e.  K ,  f  e.  F  |->  ( f  o F 
.x.  ( V  X.  { k } ) ) )
4240, 41syl6eqr 2335 . . . . . . 7  |-  ( w  =  W  ->  (
k  e.  ( Base `  (Scalar `  w )
) ,  f  e.  (LFnl `  w )  |->  ( f  o F ( .r `  (Scalar `  w ) ) ( ( Base `  w
)  X.  { k } ) ) )  =  .xb  )
4342opeq2d 3805 . . . . . 6  |-  ( w  =  W  ->  <. ( .s `  ndx ) ,  ( k  e.  (
Base `  (Scalar `  w
) ) ,  f  e.  (LFnl `  w
)  |->  ( f  o F ( .r `  (Scalar `  w ) ) ( ( Base `  w
)  X.  { k } ) ) )
>.  =  <. ( .s
`  ndx ) ,  .xb  >.
)
4443sneqd 3655 . . . . 5  |-  ( w  =  W  ->  { <. ( .s `  ndx ) ,  ( k  e.  ( Base `  (Scalar `  w ) ) ,  f  e.  (LFnl `  w )  |->  ( f  o F ( .r
`  (Scalar `  w )
) ( ( Base `  w )  X.  {
k } ) ) ) >. }  =  { <. ( .s `  ndx ) ,  .xb  >. } )
4525, 44uneq12d 3332 . . . 4  |-  ( w  =  W  ->  ( { <. ( Base `  ndx ) ,  (LFnl `  w
) >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  (Scalar `  w ) )  |`  ( (LFnl `  w )  X.  (LFnl `  w )
) ) >. ,  <. (Scalar `  ndx ) ,  (oppr `  (Scalar `  w ) )
>. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  ( Base `  (Scalar `  w ) ) ,  f  e.  (LFnl `  w )  |->  ( f  o F ( .r
`  (Scalar `  w )
) ( ( Base `  w )  X.  {
k } ) ) ) >. } )  =  ( { <. ( Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. (Scalar ` 
ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } ) )
46 df-ldual 29387 . . . 4  |- LDual  =  ( w  e.  _V  |->  ( { <. ( Base `  ndx ) ,  (LFnl `  w
) >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  (Scalar `  w ) )  |`  ( (LFnl `  w )  X.  (LFnl `  w )
) ) >. ,  <. (Scalar `  ndx ) ,  (oppr `  (Scalar `  w ) )
>. }  u.  { <. ( .s `  ndx ) ,  ( k  e.  ( Base `  (Scalar `  w ) ) ,  f  e.  (LFnl `  w )  |->  ( f  o F ( .r
`  (Scalar `  w )
) ( ( Base `  w )  X.  {
k } ) ) ) >. } ) )
47 tpex 4521 . . . . 5  |-  { <. (
Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. (Scalar ` 
ndx ) ,  O >. }  e.  _V
48 snex 4218 . . . . 5  |-  { <. ( .s `  ndx ) ,  .xb  >. }  e.  _V
4947, 48unex 4520 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } )  e. 
_V
5045, 46, 49fvmpt 5604 . . 3  |-  ( W  e.  _V  ->  (LDual `  W )  =  ( { <. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } ) )
513, 50syl5eq 2329 . 2  |-  ( W  e.  _V  ->  D  =  ( { <. (
Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. (Scalar ` 
ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } ) )
521, 2, 513syl 18 1  |-  ( ph  ->  D  =  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    e. wcel 1686   _Vcvv 2790    u. cun 3152   {csn 3642   {ctp 3644   <.cop 3645    X. cxp 4689    |` cres 4693   ` cfv 5257  (class class class)co 5860    e. cmpt2 5862    o Fcof 6078   ndxcnx 13147   Basecbs 13150   +g cplusg 13210   .rcmulr 13211  Scalarcsca 13213   .scvsca 13214  opprcoppr 15406  LFnlclfn 29320  LDualcld 29386
This theorem is referenced by:  ldualvbase  29389  ldualfvadd  29391  ldualsca  29395  ldualfvs  29399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-res 4703  df-iota 5221  df-fun 5259  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-ldual 29387
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