Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ldualset Structured version   Unicode version

Theorem ldualset 29924
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 2965 . 2  |-  ( W  e.  X  ->  W  e.  _V )
3 ldualset.d . . 3  |-  D  =  (LDual `  W )
4 fveq2 5729 . . . . . . . 8  |-  ( w  =  W  ->  (LFnl `  w )  =  (LFnl `  W ) )
5 ldualset.f . . . . . . . 8  |-  F  =  (LFnl `  W )
64, 5syl6eqr 2487 . . . . . . 7  |-  ( w  =  W  ->  (LFnl `  w )  =  F )
76opeq2d 3992 . . . . . 6  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  (LFnl `  w ) >.  =  <. ( Base `  ndx ) ,  F >. )
8 fveq2 5729 . . . . . . . . . . . . 13  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
9 ldualset.r . . . . . . . . . . . . 13  |-  R  =  (Scalar `  W )
108, 9syl6eqr 2487 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (Scalar `  w )  =  R )
1110fveq2d 5733 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( +g  `  (Scalar `  w
) )  =  ( +g  `  R ) )
12 ldualset.a . . . . . . . . . . 11  |-  .+  =  ( +g  `  R )
1311, 12syl6eqr 2487 . . . . . . . . . 10  |-  ( w  =  W  ->  ( +g  `  (Scalar `  w
) )  =  .+  )
14 ofeq 6308 . . . . . . . . . 10  |-  ( ( +g  `  (Scalar `  w ) )  = 
.+  ->  o F ( +g  `  (Scalar `  w ) )  =  o F  .+  )
1513, 14syl 16 . . . . . . . . 9  |-  ( w  =  W  ->  o F ( +g  `  (Scalar `  w ) )  =  o F  .+  )
166, 6xpeq12d 4904 . . . . . . . . 9  |-  ( w  =  W  ->  (
(LFnl `  w )  X.  (LFnl `  w )
)  =  ( F  X.  F ) )
1715, 16reseq12d 5148 . . . . . . . 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 2487 . . . . . . 7  |-  ( w  =  W  ->  (  o F ( +g  `  (Scalar `  w ) )  |`  ( (LFnl `  w )  X.  (LFnl `  w )
) )  =  .+b  )
2019opeq2d 3992 . . . . . 6  |-  ( w  =  W  ->  <. ( +g  `  ndx ) ,  (  o F ( +g  `  (Scalar `  w ) )  |`  ( (LFnl `  w )  X.  (LFnl `  w )
) ) >.  =  <. ( +g  `  ndx ) ,  .+b  >. )
2110fveq2d 5733 . . . . . . . 8  |-  ( w  =  W  ->  (oppr `  (Scalar `  w ) )  =  (oppr
`  R ) )
22 ldualset.o . . . . . . . 8  |-  O  =  (oppr
`  R )
2321, 22syl6eqr 2487 . . . . . . 7  |-  ( w  =  W  ->  (oppr `  (Scalar `  w ) )  =  O )
2423opeq2d 3992 . . . . . 6  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  (oppr `  (Scalar `  w ) )
>.  =  <. (Scalar `  ndx ) ,  O >. )
257, 20, 24tpeq123d 3899 . . . . 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 5733 . . . . . . . . . 10  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  R )
)
27 ldualset.k . . . . . . . . . 10  |-  K  =  ( Base `  R
)
2826, 27syl6eqr 2487 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  K )
2910fveq2d 5733 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .r `  (Scalar `  w
) )  =  ( .r `  R ) )
30 ldualset.t . . . . . . . . . . . 12  |-  .x.  =  ( .r `  R )
3129, 30syl6eqr 2487 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .r `  (Scalar `  w
) )  =  .x.  )
32 ofeq 6308 . . . . . . . . . . 11  |-  ( ( .r `  (Scalar `  w ) )  = 
.x.  ->  o F ( .r `  (Scalar `  w ) )  =  o F  .x.  )
3331, 32syl 16 . . . . . . . . . 10  |-  ( w  =  W  ->  o F ( .r `  (Scalar `  w ) )  =  o F  .x.  )
34 eqidd 2438 . . . . . . . . . 10  |-  ( w  =  W  ->  f  =  f )
35 fveq2 5729 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
36 ldualset.v . . . . . . . . . . . 12  |-  V  =  ( Base `  W
)
3735, 36syl6eqr 2487 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( Base `  w )  =  V )
3837xpeq1d 4902 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( Base `  w )  X.  { k } )  =  ( V  X.  { k } ) )
3933, 34, 38oveq123d 6103 . . . . . . . . 9  |-  ( w  =  W  ->  (
f  o F ( .r `  (Scalar `  w ) ) ( ( Base `  w
)  X.  { k } ) )  =  ( f  o F 
.x.  ( V  X.  { k } ) ) )
4028, 6, 39mpt2eq123dv 6137 . . . . . . . 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 2487 . . . . . . 7  |-  ( w  =  W  ->  (
k  e.  ( Base `  (Scalar `  w )
) ,  f  e.  (LFnl `  w )  |->  ( f  o F ( .r `  (Scalar `  w ) ) ( ( Base `  w
)  X.  { k } ) ) )  =  .xb  )
4342opeq2d 3992 . . . . . 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 3828 . . . . 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 3503 . . . 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 29923 . . . 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 4709 . . . . 5  |-  { <. (
Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. (Scalar ` 
ndx ) ,  O >. }  e.  _V
48 snex 4406 . . . . 5  |-  { <. ( .s `  ndx ) ,  .xb  >. }  e.  _V
4947, 48unex 4708 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } )  e. 
_V
5045, 46, 49fvmpt 5807 . . 3  |-  ( W  e.  _V  ->  (LDual `  W )  =  ( { <. ( Base `  ndx ) ,  F >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } ) )
513, 50syl5eq 2481 . 2  |-  ( W  e.  _V  ->  D  =  ( { <. (
Base `  ndx ) ,  F >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. (Scalar ` 
ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } ) )
521, 2, 513syl 19 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 1653    e. wcel 1726   _Vcvv 2957    u. cun 3319   {csn 3815   {ctp 3817   <.cop 3818    X. cxp 4877    |` cres 4881   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084    o Fcof 6304   ndxcnx 13467   Basecbs 13470   +g cplusg 13530   .rcmulr 13531  Scalarcsca 13533   .scvsca 13534  opprcoppr 15728  LFnlclfn 29856  LDualcld 29922
This theorem is referenced by:  ldualvbase  29925  ldualfvadd  29927  ldualsca  29931  ldualfvs  29935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-res 4891  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-ldual 29923
  Copyright terms: Public domain W3C validator