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Theorem hvmapfval 35409
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 35408 . . . . 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 5698 . . . 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 2487 . . 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 5696 . . . . . . . . 9  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
8 hvmapval.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
97, 8syl6eqr 2493 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
109fveq2d 5700 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  (
Base `  U )
)
11 hvmapval.v . . . . . . 7  |-  V  =  ( Base `  U
)
1210, 11syl6eqr 2493 . . . . . 6  |-  ( w  =  W  ->  ( Base `  ( ( DVecH `  K ) `  w
) )  =  V )
139fveq2d 5700 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  ( ( DVecH `  K ) `  w
) )  =  ( 0g `  U ) )
14 hvmapval.z . . . . . . . 8  |-  .0.  =  ( 0g `  U )
1513, 14syl6eqr 2493 . . . . . . 7  |-  ( w  =  W  ->  ( 0g `  ( ( DVecH `  K ) `  w
) )  =  .0.  )
1615sneqd 3894 . . . . . 6  |-  ( w  =  W  ->  { ( 0g `  ( (
DVecH `  K ) `  w ) ) }  =  {  .0.  }
)
1712, 16difeq12d 3480 . . . . 5  |-  ( w  =  W  ->  (
( Base `  ( ( DVecH `  K ) `  w ) )  \  { ( 0g `  ( ( DVecH `  K
) `  w )
) } )  =  ( V  \  {  .0.  } ) )
189fveq2d 5700 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  ( ( DVecH `  K
) `  w )
)  =  (Scalar `  U ) )
19 hvmapval.s . . . . . . . . . 10  |-  S  =  (Scalar `  U )
2018, 19syl6eqr 2493 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  ( ( DVecH `  K
) `  w )
)  =  S )
2120fveq2d 5700 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  (Scalar `  (
( DVecH `  K ) `  w ) ) )  =  ( Base `  S
) )
22 hvmapval.r . . . . . . . 8  |-  R  =  ( Base `  S
)
2321, 22syl6eqr 2493 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  (Scalar `  (
( DVecH `  K ) `  w ) ) )  =  R )
24 fveq2 5696 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( ocH `  K
) `  w )  =  ( ( ocH `  K ) `  W
) )
25 hvmapval.o . . . . . . . . . 10  |-  O  =  ( ( ocH `  K
) `  W )
2624, 25syl6eqr 2493 . . . . . . . . 9  |-  ( w  =  W  ->  (
( ocH `  K
) `  w )  =  O )
2726fveq1d 5698 . . . . . . . 8  |-  ( w  =  W  ->  (
( ( ocH `  K
) `  w ) `  { x } )  =  ( O `  { x } ) )
289fveq2d 5700 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( +g  `  ( ( DVecH `  K ) `  w
) )  =  ( +g  `  U ) )
29 hvmapval.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  U )
3028, 29syl6eqr 2493 . . . . . . . . . 10  |-  ( w  =  W  ->  ( +g  `  ( ( DVecH `  K ) `  w
) )  =  .+  )
31 eqidd 2444 . . . . . . . . . 10  |-  ( w  =  W  ->  t  =  t )
329fveq2d 5700 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .s `  ( ( DVecH `  K ) `  w
) )  =  ( .s `  U ) )
33 hvmapval.t . . . . . . . . . . . 12  |-  .x.  =  ( .s `  U )
3432, 33syl6eqr 2493 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .s `  ( ( DVecH `  K ) `  w
) )  =  .x.  )
3534oveqd 6113 . . . . . . . . . 10  |-  ( w  =  W  ->  (
j ( .s `  ( ( DVecH `  K
) `  w )
) x )  =  ( j  .x.  x
) )
3630, 31, 35oveq123d 6117 . . . . . . . . 9  |-  ( w  =  W  ->  (
t ( +g  `  (
( DVecH `  K ) `  w ) ) ( j ( .s `  ( ( DVecH `  K
) `  w )
) x ) )  =  ( t  .+  ( j  .x.  x
) ) )
3736eqeq2d 2454 . . . . . . . 8  |-  ( w  =  W  ->  (
v  =  ( t ( +g  `  (
( DVecH `  K ) `  w ) ) ( j ( .s `  ( ( DVecH `  K
) `  w )
) x ) )  <-> 
v  =  ( t 
.+  ( j  .x.  x ) ) ) )
3827, 37rexeqbidv 2937 . . . . . . 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 6060 . . . . . 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 4375 . . . . 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 4375 . . . 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 2443 . . . 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 5706 . . . . . . 7  |-  ( Base `  U )  e.  _V
4411, 43eqeltri 2513 . . . . . 6  |-  V  e. 
_V
45 difexg 4445 . . . . . 6  |-  ( V  e.  _V  ->  ( V  \  {  .0.  }
)  e.  _V )
4644, 45ax-mp 5 . . . . 5  |-  ( V 
\  {  .0.  }
)  e.  _V
4746mptex 5953 . . . 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 5779 . . 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 2495 . 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 1369    e. wcel 1756   E.wrex 2721   _Vcvv 2977    \ cdif 3330   {csn 3882    e. cmpt 4355   ` cfv 5423   iota_crio 6056  (class class class)co 6096   Basecbs 14179   +g cplusg 14243  Scalarcsca 14246   .scvsca 14247   0gc0g 14383   LHypclh 33633   DVecHcdvh 34728   ocHcoch 34997  HVMapchvm 35406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-hvmap 35407
This theorem is referenced by:  hvmapval  35410  hvmap1o  35413  hvmaplkr  35418
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