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Theorem hvmapfval 32557
Description: Map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)
Hypotheses
Ref Expression
hvmapval.h  |-  H  =  ( LHyp `  K
)
hvmapval.u  |-  U  =  ( ( DVecH `  K
) `  W )
hvmapval.o  |-  O  =  ( ( ocH `  K
) `  W )
hvmapval.v  |-  V  =  ( Base `  U
)
hvmapval.p  |-  .+  =  ( +g  `  U )
hvmapval.t  |-  .x.  =  ( .s `  U )
hvmapval.z  |-  .0.  =  ( 0g `  U )
hvmapval.s  |-  S  =  (Scalar `  U )
hvmapval.r  |-  R  =  ( Base `  S
)
hvmapval.m  |-  M  =  ( (HVMap `  K
) `  W )
hvmapval.k  |-  ( ph  ->  ( K  e.  A  /\  W  e.  H
) )
Assertion
Ref Expression
hvmapfval  |-  ( ph  ->  M  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  (
iota_ j  e.  R E. t  e.  ( O `  { x } ) v  =  ( t  .+  (
j  .x.  x )
) ) ) ) )
Distinct variable groups:    t, j,
v, x, K    t, W    t, O    R, j    x, V    j, W, v, x    x,  .0.
Allowed substitution hints:    ph( x, v, t, j)    A( x, v, t, j)    .+ ( x, v, t, j)    R( x, v, t)    S( x, v, t, j)    .x. ( x, v, t, j)    U( x, v, t, j)    H( x, v, t, j)    M( x, v, t, j)    O( x, v, j)    V( v, t, j)    .0. ( v,
t, j)

Proof of Theorem hvmapfval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 hvmapval.k . 2  |-  ( ph  ->  ( K  e.  A  /\  W  e.  H
) )
2 hvmapval.m . . . 4  |-  M  =  ( (HVMap `  K
) `  W )
3 hvmapval.h . . . . . 6  |-  H  =  ( LHyp `  K
)
43hvmapffval 32556 . . . . 5  |-  ( K  e.  A  ->  (HVMap `  K )  =  ( w  e.  H  |->  ( x  e.  ( (
Base `  ( ( DVecH `  K ) `  w ) )  \  { ( 0g `  ( ( DVecH `  K
) `  w )
) } )  |->  ( v  e.  ( Base `  ( ( DVecH `  K
) `  w )
)  |->  ( iota_ j  e.  ( Base `  (Scalar `  ( ( DVecH `  K
) `  w )
) ) E. t  e.  ( ( ( ocH `  K ) `  w
) `  { x } ) v  =  ( t ( +g  `  ( ( DVecH `  K
) `  w )
) ( j ( .s `  ( (
DVecH `  K ) `  w ) ) x ) ) ) ) ) ) )
54fveq1d 5730 . . . 4  |-  ( K  e.  A  ->  (
(HVMap `  K ) `  W )  =  ( ( w  e.  H  |->  ( x  e.  ( ( Base `  (
( DVecH `  K ) `  w ) )  \  { ( 0g `  ( ( DVecH `  K
) `  w )
) } )  |->  ( v  e.  ( Base `  ( ( DVecH `  K
) `  w )
)  |->  ( iota_ j  e.  ( Base `  (Scalar `  ( ( DVecH `  K
) `  w )
) ) E. t  e.  ( ( ( ocH `  K ) `  w
) `  { x } ) v  =  ( t ( +g  `  ( ( DVecH `  K
) `  w )
) ( j ( .s `  ( (
DVecH `  K ) `  w ) ) x ) ) ) ) ) ) `  W
) )
62, 5syl5eq 2480 . . 3  |-  ( K  e.  A  ->  M  =  ( ( w  e.  H  |->  ( x  e.  ( ( Base `  ( ( DVecH `  K
) `  w )
)  \  { ( 0g `  ( ( DVecH `  K ) `  w
) ) } ) 
|->  ( v  e.  (
Base `  ( ( DVecH `  K ) `  w ) )  |->  (
iota_ j  e.  ( Base `  (Scalar `  (
( DVecH `  K ) `  w ) ) ) E. t  e.  ( ( ( ocH `  K
) `  w ) `  { x } ) v  =  ( t ( +g  `  (
( DVecH `  K ) `  w ) ) ( j ( .s `  ( ( DVecH `  K
) `  w )
) x ) ) ) ) ) ) `
 W ) )
7 fveq2 5728 . . . . . . . . 9  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
8 hvmapval.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
97, 8syl6eqr 2486 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
109fveq2d 5732 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  (
Base `  U )
)
11 hvmapval.v . . . . . . 7  |-  V  =  ( Base `  U
)
1210, 11syl6eqr 2486 . . . . . 6  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  V )
139fveq2d 5732 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  ( ( DVecH `  K ) `  w
) )  =  ( 0g `  U ) )
14 hvmapval.z . . . . . . . 8  |-  .0.  =  ( 0g `  U )
1513, 14syl6eqr 2486 . . . . . . 7  |-  ( w  =  W  ->  ( 0g `  ( ( DVecH `  K ) `  w
) )  =  .0.  )
1615sneqd 3827 . . . . . 6  |-  ( w  =  W  ->  { ( 0g `  ( (
DVecH `  K ) `  w ) ) }  =  {  .0.  }
)
1712, 16difeq12d 3466 . . . . 5  |-  ( w  =  W  ->  (
( Base `  ( ( DVecH `  K ) `  w ) )  \  { ( 0g `  ( ( DVecH `  K
) `  w )
) } )  =  ( V  \  {  .0.  } ) )
189fveq2d 5732 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  ( ( DVecH `  K
) `  w )
)  =  (Scalar `  U ) )
19 hvmapval.s . . . . . . . . . 10  |-  S  =  (Scalar `  U )
2018, 19syl6eqr 2486 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  ( ( DVecH `  K
) `  w )
)  =  S )
2120fveq2d 5732 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  (Scalar `  (
( DVecH `  K ) `  w ) ) )  =  ( Base `  S
) )
22 hvmapval.r . . . . . . . 8  |-  R  =  ( Base `  S
)
2321, 22syl6eqr 2486 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  (Scalar `  (
( DVecH `  K ) `  w ) ) )  =  R )
24 fveq2 5728 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( ocH `  K
) `  w )  =  ( ( ocH `  K ) `  W
) )
25 hvmapval.o . . . . . . . . . 10  |-  O  =  ( ( ocH `  K
) `  W )
2624, 25syl6eqr 2486 . . . . . . . . 9  |-  ( w  =  W  ->  (
( ocH `  K
) `  w )  =  O )
2726fveq1d 5730 . . . . . . . 8  |-  ( w  =  W  ->  (
( ( ocH `  K
) `  w ) `  { x } )  =  ( O `  { x } ) )
289fveq2d 5732 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( +g  `  ( ( DVecH `  K ) `  w
) )  =  ( +g  `  U ) )
29 hvmapval.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  U )
3028, 29syl6eqr 2486 . . . . . . . . . 10  |-  ( w  =  W  ->  ( +g  `  ( ( DVecH `  K ) `  w
) )  =  .+  )
31 eqidd 2437 . . . . . . . . . 10  |-  ( w  =  W  ->  t  =  t )
329fveq2d 5732 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .s `  ( ( DVecH `  K ) `  w
) )  =  ( .s `  U ) )
33 hvmapval.t . . . . . . . . . . . 12  |-  .x.  =  ( .s `  U )
3432, 33syl6eqr 2486 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .s `  ( ( DVecH `  K ) `  w
) )  =  .x.  )
3534oveqd 6098 . . . . . . . . . 10  |-  ( w  =  W  ->  (
j ( .s `  ( ( DVecH `  K
) `  w )
) x )  =  ( j  .x.  x
) )
3630, 31, 35oveq123d 6102 . . . . . . . . 9  |-  ( w  =  W  ->  (
t ( +g  `  (
( DVecH `  K ) `  w ) ) ( j ( .s `  ( ( DVecH `  K
) `  w )
) x ) )  =  ( t  .+  ( j  .x.  x
) ) )
3736eqeq2d 2447 . . . . . . . 8  |-  ( w  =  W  ->  (
v  =  ( t ( +g  `  (
( DVecH `  K ) `  w ) ) ( j ( .s `  ( ( DVecH `  K
) `  w )
) x ) )  <-> 
v  =  ( t 
.+  ( j  .x.  x ) ) ) )
3827, 37rexeqbidv 2917 . . . . . . 7  |-  ( w  =  W  ->  ( E. t  e.  (
( ( ocH `  K
) `  w ) `  { x } ) v  =  ( t ( +g  `  (
( DVecH `  K ) `  w ) ) ( j ( .s `  ( ( DVecH `  K
) `  w )
) x ) )  <->  E. t  e.  ( O `  { x } ) v  =  ( t  .+  (
j  .x.  x )
) ) )
3923, 38riotaeqbidv 6552 . . . . . 6  |-  ( w  =  W  ->  ( iota_ j  e.  ( Base `  (Scalar `  ( ( DVecH `  K ) `  w ) ) ) E. t  e.  ( ( ( ocH `  K
) `  w ) `  { x } ) v  =  ( t ( +g  `  (
( DVecH `  K ) `  w ) ) ( j ( .s `  ( ( DVecH `  K
) `  w )
) x ) ) )  =  ( iota_ j  e.  R E. t  e.  ( O `  {
x } ) v  =  ( t  .+  ( j  .x.  x
) ) ) )
4012, 39mpteq12dv 4287 . . . . 5  |-  ( w  =  W  ->  (
v  e.  ( Base `  ( ( DVecH `  K
) `  w )
)  |->  ( iota_ j  e.  ( Base `  (Scalar `  ( ( DVecH `  K
) `  w )
) ) E. t  e.  ( ( ( ocH `  K ) `  w
) `  { x } ) v  =  ( t ( +g  `  ( ( DVecH `  K
) `  w )
) ( j ( .s `  ( (
DVecH `  K ) `  w ) ) x ) ) ) )  =  ( v  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  {
x } ) v  =  ( t  .+  ( j  .x.  x
) ) ) ) )
4117, 40mpteq12dv 4287 . . . 4  |-  ( w  =  W  ->  (
x  e.  ( (
Base `  ( ( DVecH `  K ) `  w ) )  \  { ( 0g `  ( ( DVecH `  K
) `  w )
) } )  |->  ( v  e.  ( Base `  ( ( DVecH `  K
) `  w )
)  |->  ( iota_ j  e.  ( Base `  (Scalar `  ( ( DVecH `  K
) `  w )
) ) E. t  e.  ( ( ( ocH `  K ) `  w
) `  { x } ) v  =  ( t ( +g  `  ( ( DVecH `  K
) `  w )
) ( j ( .s `  ( (
DVecH `  K ) `  w ) ) x ) ) ) ) )  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  (
iota_ j  e.  R E. t  e.  ( O `  { x } ) v  =  ( t  .+  (
j  .x.  x )
) ) ) ) )
42 eqid 2436 . . . 4  |-  ( w  e.  H  |->  ( x  e.  ( ( Base `  ( ( DVecH `  K
) `  w )
)  \  { ( 0g `  ( ( DVecH `  K ) `  w
) ) } ) 
|->  ( v  e.  (
Base `  ( ( DVecH `  K ) `  w ) )  |->  (
iota_ j  e.  ( Base `  (Scalar `  (
( DVecH `  K ) `  w ) ) ) E. t  e.  ( ( ( ocH `  K
) `  w ) `  { x } ) v  =  ( t ( +g  `  (
( DVecH `  K ) `  w ) ) ( j ( .s `  ( ( DVecH `  K
) `  w )
) x ) ) ) ) ) )  =  ( w  e.  H  |->  ( x  e.  ( ( Base `  (
( DVecH `  K ) `  w ) )  \  { ( 0g `  ( ( DVecH `  K
) `  w )
) } )  |->  ( v  e.  ( Base `  ( ( DVecH `  K
) `  w )
)  |->  ( iota_ j  e.  ( Base `  (Scalar `  ( ( DVecH `  K
) `  w )
) ) E. t  e.  ( ( ( ocH `  K ) `  w
) `  { x } ) v  =  ( t ( +g  `  ( ( DVecH `  K
) `  w )
) ( j ( .s `  ( (
DVecH `  K ) `  w ) ) x ) ) ) ) ) )
43 fvex 5742 . . . . . . 7  |-  ( Base `  U )  e.  _V
4411, 43eqeltri 2506 . . . . . 6  |-  V  e. 
_V
45 difexg 4351 . . . . . 6  |-  ( V  e.  _V  ->  ( V  \  {  .0.  }
)  e.  _V )
4644, 45ax-mp 8 . . . . 5  |-  ( V 
\  {  .0.  }
)  e.  _V
4746mptex 5966 . . . 4  |-  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  (
iota_ j  e.  R E. t  e.  ( O `  { x } ) v  =  ( t  .+  (
j  .x.  x )
) ) ) )  e.  _V
4841, 42, 47fvmpt 5806 . . 3  |-  ( W  e.  H  ->  (
( w  e.  H  |->  ( x  e.  ( ( Base `  (
( DVecH `  K ) `  w ) )  \  { ( 0g `  ( ( DVecH `  K
) `  w )
) } )  |->  ( v  e.  ( Base `  ( ( DVecH `  K
) `  w )
)  |->  ( iota_ j  e.  ( Base `  (Scalar `  ( ( DVecH `  K
) `  w )
) ) E. t  e.  ( ( ( ocH `  K ) `  w
) `  { x } ) v  =  ( t ( +g  `  ( ( DVecH `  K
) `  w )
) ( j ( .s `  ( (
DVecH `  K ) `  w ) ) x ) ) ) ) ) ) `  W
)  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  (
iota_ j  e.  R E. t  e.  ( O `  { x } ) v  =  ( t  .+  (
j  .x.  x )
) ) ) ) )
496, 48sylan9eq 2488 . 2  |-  ( ( K  e.  A  /\  W  e.  H )  ->  M  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  (
iota_ j  e.  R E. t  e.  ( O `  { x } ) v  =  ( t  .+  (
j  .x.  x )
) ) ) ) )
501, 49syl 16 1  |-  ( ph  ->  M  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  (
iota_ j  e.  R E. t  e.  ( O `  { x } ) v  =  ( t  .+  (
j  .x.  x )
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   _Vcvv 2956    \ cdif 3317   {csn 3814    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   iota_crio 6542   Basecbs 13469   +g cplusg 13529  Scalarcsca 13532   .scvsca 13533   0gc0g 13723   LHypclh 30781   DVecHcdvh 31876   ocHcoch 32145  HVMapchvm 32554
This theorem is referenced by:  hvmapval  32558  hvmap1o  32561  hvmaplkr  32566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-riota 6549  df-hvmap 32555
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