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Theorem ldualset 28565
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 2771 . 2  |-  ( W  e.  X  ->  W  e.  _V )
3 ldualset.d . . 3  |-  D  =  (LDual `  W )
4 fveq2 5458 . . . . . . . 8  |-  ( w  =  W  ->  (LFnl `  w )  =  (LFnl `  W ) )
5 ldualset.f . . . . . . . 8  |-  F  =  (LFnl `  W )
64, 5syl6eqr 2308 . . . . . . 7  |-  ( w  =  W  ->  (LFnl `  w )  =  F )
76opeq2d 3777 . . . . . 6  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  (LFnl `  w ) >.  =  <. ( Base `  ndx ) ,  F >. )
8 fveq2 5458 . . . . . . . . . . . . 13  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
9 ldualset.r . . . . . . . . . . . . 13  |-  R  =  (Scalar `  W )
108, 9syl6eqr 2308 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (Scalar `  w )  =  R )
1110fveq2d 5462 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( +g  `  (Scalar `  w
) )  =  ( +g  `  R ) )
12 ldualset.a . . . . . . . . . . 11  |-  .+  =  ( +g  `  R )
1311, 12syl6eqr 2308 . . . . . . . . . 10  |-  ( w  =  W  ->  ( +g  `  (Scalar `  w
) )  =  .+  )
14 ofeq 6014 . . . . . . . . . 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 4702 . . . . . . . . 9  |-  ( w  =  W  ->  (
(LFnl `  w )  X.  (LFnl `  w )
)  =  ( F  X.  F ) )
1715, 16reseq12d 4944 . . . . . . . 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 2308 . . . . . . 7  |-  ( w  =  W  ->  (  o F ( +g  `  (Scalar `  w ) )  |`  ( (LFnl `  w )  X.  (LFnl `  w )
) )  =  .+b  )
2019opeq2d 3777 . . . . . 6  |-  ( w  =  W  ->  <. ( +g  `  ndx ) ,  (  o F ( +g  `  (Scalar `  w ) )  |`  ( (LFnl `  w )  X.  (LFnl `  w )
) ) >.  =  <. ( +g  `  ndx ) ,  .+b  >. )
2110fveq2d 5462 . . . . . . . 8  |-  ( w  =  W  ->  (oppr `  (Scalar `  w ) )  =  (oppr
`  R ) )
22 ldualset.o . . . . . . . 8  |-  O  =  (oppr
`  R )
2321, 22syl6eqr 2308 . . . . . . 7  |-  ( w  =  W  ->  (oppr `  (Scalar `  w ) )  =  O )
2423opeq2d 3777 . . . . . 6  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  (oppr `  (Scalar `  w ) )
>.  =  <. (Scalar `  ndx ) ,  O >. )
257, 20, 24tpeq123d 3695 . . . . 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 5462 . . . . . . . . . 10  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  R )
)
27 ldualset.k . . . . . . . . . 10  |-  K  =  ( Base `  R
)
2826, 27syl6eqr 2308 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  K )
2910fveq2d 5462 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .r `  (Scalar `  w
) )  =  ( .r `  R ) )
30 ldualset.t . . . . . . . . . . . 12  |-  .x.  =  ( .r `  R )
3129, 30syl6eqr 2308 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .r `  (Scalar `  w
) )  =  .x.  )
32 ofeq 6014 . . . . . . . . . . 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 2259 . . . . . . . . . 10  |-  ( w  =  W  ->  f  =  f )
35 fveq2 5458 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
36 ldualset.v . . . . . . . . . . . 12  |-  V  =  ( Base `  W
)
3735, 36syl6eqr 2308 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( Base `  w )  =  V )
3837xpeq1d 4700 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( Base `  w )  X.  { k } )  =  ( V  X.  { k } ) )
3933, 34, 38oveq123d 5813 . . . . . . . . 9  |-  ( w  =  W  ->  (
f  o F ( .r `  (Scalar `  w ) ) ( ( Base `  w
)  X.  { k } ) )  =  ( f  o F 
.x.  ( V  X.  { k } ) ) )
4028, 6, 39mpt2eq123dv 5844 . . . . . . . 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 2308 . . . . . . 7  |-  ( w  =  W  ->  (
k  e.  ( Base `  (Scalar `  w )
) ,  f  e.  (LFnl `  w )  |->  ( f  o F ( .r `  (Scalar `  w ) ) ( ( Base `  w
)  X.  { k } ) ) )  =  .xb  )
4342opeq2d 3777 . . . . . 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 3627 . . . . 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 3305 . . . 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 28564 . . . 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 4491 . . . . 5  |-  { <. (
Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. (Scalar ` 
ndx ) ,  O >. }  e.  _V
48 snex 4188 . . . . 5  |-  { <. ( .s `  ndx ) ,  .xb  >. }  e.  _V
4947, 48unex 4490 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } )  e. 
_V
5045, 46, 49fvmpt 5536 . . 3  |-  ( W  e.  _V  ->  (LDual `  W )  =  ( { <. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } ) )
513, 50syl5eq 2302 . 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 2763    u. cun 3125   {csn 3614   {ctp 3616   <.cop 3617    X. cxp 4659    |` cres 4663   ` cfv 4673  (class class class)co 5792    e. cmpt2 5794    o Fcof 6010   ndxcnx 13108   Basecbs 13111   +g cplusg 13171   .rcmulr 13172  Scalarcsca 13174   .scvsca 13175  opprcoppr 15367  LFnlclfn 28497  LDualcld 28563
This theorem is referenced by:  ldualvbase  28566  ldualfvadd  28568  ldualsca  28572  ldualfvs  28576
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 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-ldual 28564
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