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Theorem hvmapfval 36849
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 36848 . . . . 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 5873 . . . 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 2520 . . 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 5871 . . . . . . . . 9  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
8 hvmapval.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
97, 8syl6eqr 2526 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
109fveq2d 5875 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  (
Base `  U )
)
11 hvmapval.v . . . . . . 7  |-  V  =  ( Base `  U
)
1210, 11syl6eqr 2526 . . . . . 6  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  V )
139fveq2d 5875 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  ( ( DVecH `  K ) `  w
) )  =  ( 0g `  U ) )
14 hvmapval.z . . . . . . . 8  |-  .0.  =  ( 0g `  U )
1513, 14syl6eqr 2526 . . . . . . 7  |-  ( w  =  W  ->  ( 0g `  ( ( DVecH `  K ) `  w
) )  =  .0.  )
1615sneqd 4044 . . . . . 6  |-  ( w  =  W  ->  { ( 0g `  ( (
DVecH `  K ) `  w ) ) }  =  {  .0.  }
)
1712, 16difeq12d 3628 . . . . 5  |-  ( w  =  W  ->  (
( Base `  ( ( DVecH `  K ) `  w ) )  \  { ( 0g `  ( ( DVecH `  K
) `  w )
) } )  =  ( V  \  {  .0.  } ) )
189fveq2d 5875 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  ( ( DVecH `  K
) `  w )
)  =  (Scalar `  U ) )
19 hvmapval.s . . . . . . . . . 10  |-  S  =  (Scalar `  U )
2018, 19syl6eqr 2526 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  ( ( DVecH `  K
) `  w )
)  =  S )
2120fveq2d 5875 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  (Scalar `  (
( DVecH `  K ) `  w ) ) )  =  ( Base `  S
) )
22 hvmapval.r . . . . . . . 8  |-  R  =  ( Base `  S
)
2321, 22syl6eqr 2526 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  (Scalar `  (
( DVecH `  K ) `  w ) ) )  =  R )
24 fveq2 5871 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( ocH `  K
) `  w )  =  ( ( ocH `  K ) `  W
) )
25 hvmapval.o . . . . . . . . . 10  |-  O  =  ( ( ocH `  K
) `  W )
2624, 25syl6eqr 2526 . . . . . . . . 9  |-  ( w  =  W  ->  (
( ocH `  K
) `  w )  =  O )
2726fveq1d 5873 . . . . . . . 8  |-  ( w  =  W  ->  (
( ( ocH `  K
) `  w ) `  { x } )  =  ( O `  { x } ) )
289fveq2d 5875 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( +g  `  ( ( DVecH `  K ) `  w
) )  =  ( +g  `  U ) )
29 hvmapval.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  U )
3028, 29syl6eqr 2526 . . . . . . . . . 10  |-  ( w  =  W  ->  ( +g  `  ( ( DVecH `  K ) `  w
) )  =  .+  )
31 eqidd 2468 . . . . . . . . . 10  |-  ( w  =  W  ->  t  =  t )
329fveq2d 5875 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .s `  ( ( DVecH `  K ) `  w
) )  =  ( .s `  U ) )
33 hvmapval.t . . . . . . . . . . . 12  |-  .x.  =  ( .s `  U )
3432, 33syl6eqr 2526 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .s `  ( ( DVecH `  K ) `  w
) )  =  .x.  )
3534oveqd 6311 . . . . . . . . . 10  |-  ( w  =  W  ->  (
j ( .s `  ( ( DVecH `  K
) `  w )
) x )  =  ( j  .x.  x
) )
3630, 31, 35oveq123d 6315 . . . . . . . . 9  |-  ( w  =  W  ->  (
t ( +g  `  (
( DVecH `  K ) `  w ) ) ( j ( .s `  ( ( DVecH `  K
) `  w )
) x ) )  =  ( t  .+  ( j  .x.  x
) ) )
3736eqeq2d 2481 . . . . . . . 8  |-  ( w  =  W  ->  (
v  =  ( t ( +g  `  (
( DVecH `  K ) `  w ) ) ( j ( .s `  ( ( DVecH `  K
) `  w )
) x ) )  <-> 
v  =  ( t 
.+  ( j  .x.  x ) ) ) )
3827, 37rexeqbidv 3078 . . . . . . 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 6258 . . . . . 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 4530 . . . . 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 4530 . . . 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 2467 . . . 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 5881 . . . . . . 7  |-  ( Base `  U )  e.  _V
4411, 43eqeltri 2551 . . . . . 6  |-  V  e. 
_V
45 difexg 4600 . . . . . 6  |-  ( V  e.  _V  ->  ( V  \  {  .0.  }
)  e.  _V )
4644, 45ax-mp 5 . . . . 5  |-  ( V 
\  {  .0.  }
)  e.  _V
4746mptex 6141 . . . 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 5956 . . 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 2528 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818   _Vcvv 3118    \ cdif 3478   {csn 4032    |-> cmpt 4510   ` cfv 5593   iota_crio 6254  (class class class)co 6294   Basecbs 14502   +g cplusg 14567  Scalarcsca 14570   .scvsca 14571   0gc0g 14707   LHypclh 35073   DVecHcdvh 36168   ocHcoch 36437  HVMapchvm 36846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-hvmap 36847
This theorem is referenced by:  hvmapval  36850  hvmap1o  36853  hvmaplkr  36858
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