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Theorem ldualset 28445
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
StepHypRef Expression
1 ldualset.w . 2  |-  ( ph  ->  W  e.  X )
2 elex 2748 . 2  |-  ( W  e.  X  ->  W  e.  _V )
3 ldualset.d . . 3  |-  D  =  (LDual `  W )
4 fveq2 5423 . . . . . . . 8  |-  ( w  =  W  ->  (LFnl `  w )  =  (LFnl `  W ) )
5 ldualset.f . . . . . . . 8  |-  F  =  (LFnl `  W )
64, 5syl6eqr 2306 . . . . . . 7  |-  ( w  =  W  ->  (LFnl `  w )  =  F )
76opeq2d 3744 . . . . . 6  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  (LFnl `  w ) >.  =  <. ( Base `  ndx ) ,  F >. )
8 fveq2 5423 . . . . . . . . . . . . 13  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
9 ldualset.r . . . . . . . . . . . . 13  |-  R  =  (Scalar `  W )
108, 9syl6eqr 2306 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (Scalar `  w )  =  R )
1110fveq2d 5427 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( +g  `  (Scalar `  w
) )  =  ( +g  `  R ) )
12 ldualset.a . . . . . . . . . . 11  |-  .+  =  ( +g  `  R )
1311, 12syl6eqr 2306 . . . . . . . . . 10  |-  ( w  =  W  ->  ( +g  `  (Scalar `  w
) )  =  .+  )
14 ofeq 5979 . . . . . . . . . 10  |-  ( ( +g  `  (Scalar `  w ) )  = 
.+  ->  o F ( +g  `  (Scalar `  w ) )  =  o F  .+  )
1513, 14syl 17 . . . . . . . . 9  |-  ( w  =  W  ->  o F ( +g  `  (Scalar `  w ) )  =  o F  .+  )
166, 6xpeq12d 4667 . . . . . . . . 9  |-  ( w  =  W  ->  (
(LFnl `  w )  X.  (LFnl `  w )
)  =  ( F  X.  F ) )
1715, 16reseq12d 4909 . . . . . . . 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 2306 . . . . . . 7  |-  ( w  =  W  ->  (  o F ( +g  `  (Scalar `  w ) )  |`  ( (LFnl `  w )  X.  (LFnl `  w )
) )  =  .+b  )
2019opeq2d 3744 . . . . . 6  |-  ( w  =  W  ->  <. ( +g  `  ndx ) ,  (  o F ( +g  `  (Scalar `  w ) )  |`  ( (LFnl `  w )  X.  (LFnl `  w )
) ) >.  =  <. ( +g  `  ndx ) ,  .+b  >. )
2110fveq2d 5427 . . . . . . . 8  |-  ( w  =  W  ->  (oppr `  (Scalar `  w ) )  =  (oppr
`  R ) )
22 ldualset.o . . . . . . . 8  |-  O  =  (oppr
`  R )
2321, 22syl6eqr 2306 . . . . . . 7  |-  ( w  =  W  ->  (oppr `  (Scalar `  w ) )  =  O )
2423opeq2d 3744 . . . . . 6  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  (oppr `  (Scalar `  w ) )
>.  =  <. (Scalar `  ndx ) ,  O >. )
257, 20, 24tpeq123d 3662 . . . . 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 5427 . . . . . . . . . 10  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  R )
)
27 ldualset.k . . . . . . . . . 10  |-  K  =  ( Base `  R
)
2826, 27syl6eqr 2306 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  K )
2910fveq2d 5427 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .r `  (Scalar `  w
) )  =  ( .r `  R ) )
30 ldualset.t . . . . . . . . . . . 12  |-  .x.  =  ( .r `  R )
3129, 30syl6eqr 2306 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .r `  (Scalar `  w
) )  =  .x.  )
32 ofeq 5979 . . . . . . . . . . 11  |-  ( ( .r `  (Scalar `  w ) )  = 
.x.  ->  o F ( .r `  (Scalar `  w ) )  =  o F  .x.  )
3331, 32syl 17 . . . . . . . . . 10  |-  ( w  =  W  ->  o F ( .r `  (Scalar `  w ) )  =  o F  .x.  )
34 eqidd 2257 . . . . . . . . . 10  |-  ( w  =  W  ->  f  =  f )
35 fveq2 5423 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
36 ldualset.v . . . . . . . . . . . 12  |-  V  =  ( Base `  W
)
3735, 36syl6eqr 2306 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( Base `  w )  =  V )
3837xpeq1d 4665 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( Base `  w )  X.  { k } )  =  ( V  X.  { k } ) )
3933, 34, 38oveq123d 5778 . . . . . . . . 9  |-  ( w  =  W  ->  (
f  o F ( .r `  (Scalar `  w ) ) ( ( Base `  w
)  X.  { k } ) )  =  ( f  o F 
.x.  ( V  X.  { k } ) ) )
4028, 6, 39mpt2eq123dv 5809 . . . . . . . 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 2306 . . . . . . 7  |-  ( w  =  W  ->  (
k  e.  ( Base `  (Scalar `  w )
) ,  f  e.  (LFnl `  w )  |->  ( f  o F ( .r `  (Scalar `  w ) ) ( ( Base `  w
)  X.  { k } ) ) )  =  .xb  )
4342opeq2d 3744 . . . . . 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 3594 . . . . 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 3272 . . . 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 28444 . . . 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 4456 . . . . 5  |-  { <. (
Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. (Scalar ` 
ndx ) ,  O >. }  e.  _V
48 snex 4154 . . . . 5  |-  { <. ( .s `  ndx ) ,  .xb  >. }  e.  _V
4947, 48unex 4455 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } )  e. 
_V
5045, 46, 49fvmpt 5501 . . 3  |-  ( W  e.  _V  ->  (LDual `  W )  =  ( { <. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } ) )
513, 50syl5eq 2300 . 2  |-  ( W  e.  _V  ->  D  =  ( { <. (
Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. (Scalar ` 
ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } ) )
521, 2, 513syl 20 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 6    = wceq 1619    e. wcel 1621   _Vcvv 2740    u. cun 3092   {csn 3581   {ctp 3583   <.cop 3584    X. cxp 4624    |` cres 4628   ` cfv 4638  (class class class)co 5757    e. cmpt2 5759    o Fcof 5975   ndxcnx 13072   Basecbs 13075   +g cplusg 13135   .rcmulr 13136  Scalarcsca 13138   .scvsca 13139  opprcoppr 15331  LFnlclfn 28377  LDualcld 28443
This theorem is referenced by:  ldualvbase  28446  ldualfvadd  28448  ldualsca  28452  ldualfvs  28456
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-ldual 28444
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