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Theorem hvmapval 32255
Description: Value of 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
) )
hvmapval.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
Assertion
Ref Expression
hvmapval  |-  ( ph  ->  ( M `  X
)  =  ( v  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) v  =  ( t  .+  (
j  .x.  X )
) ) ) )
Distinct variable groups:    t, j,
v, K    t, W    t, O    R, j    j, W, v    v, V    j, X, t, v
Allowed substitution hints:    ph( v, t, j)    A( v, t, j)    .+ ( v, t, j)    R( v, t)    S( v, t, j)    .x. ( v, t, j)    U( v, t, j)    H( v, t, j)    M( v, t, j)    O( v, j)    V( t, j)    .0. ( v, t, j)

Proof of Theorem hvmapval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hvmapval.h . . . 4  |-  H  =  ( LHyp `  K
)
2 hvmapval.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
3 hvmapval.o . . . 4  |-  O  =  ( ( ocH `  K
) `  W )
4 hvmapval.v . . . 4  |-  V  =  ( Base `  U
)
5 hvmapval.p . . . 4  |-  .+  =  ( +g  `  U )
6 hvmapval.t . . . 4  |-  .x.  =  ( .s `  U )
7 hvmapval.z . . . 4  |-  .0.  =  ( 0g `  U )
8 hvmapval.s . . . 4  |-  S  =  (Scalar `  U )
9 hvmapval.r . . . 4  |-  R  =  ( Base `  S
)
10 hvmapval.m . . . 4  |-  M  =  ( (HVMap `  K
) `  W )
11 hvmapval.k . . . 4  |-  ( ph  ->  ( K  e.  A  /\  W  e.  H
) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hvmapfval 32254 . . 3  |-  ( ph  ->  M  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  (
iota_ j  e.  R E. t  e.  ( O `  { x } ) v  =  ( t  .+  (
j  .x.  x )
) ) ) ) )
1312fveq1d 5697 . 2  |-  ( ph  ->  ( M `  X
)  =  ( ( x  e.  ( V 
\  {  .0.  }
)  |->  ( v  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  {
x } ) v  =  ( t  .+  ( j  .x.  x
) ) ) ) ) `  X ) )
14 hvmapval.x . . 3  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
15 fvex 5709 . . . . 5  |-  ( Base `  U )  e.  _V
164, 15eqeltri 2482 . . . 4  |-  V  e. 
_V
1716mptex 5933 . . 3  |-  ( v  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) v  =  ( t  .+  (
j  .x.  X )
) ) )  e. 
_V
18 sneq 3793 . . . . . . . 8  |-  ( x  =  X  ->  { x }  =  { X } )
1918fveq2d 5699 . . . . . . 7  |-  ( x  =  X  ->  ( O `  { x } )  =  ( O `  { X } ) )
20 oveq2 6056 . . . . . . . . 9  |-  ( x  =  X  ->  (
j  .x.  x )  =  ( j  .x.  X ) )
2120oveq2d 6064 . . . . . . . 8  |-  ( x  =  X  ->  (
t  .+  ( j  .x.  x ) )  =  ( t  .+  (
j  .x.  X )
) )
2221eqeq2d 2423 . . . . . . 7  |-  ( x  =  X  ->  (
v  =  ( t 
.+  ( j  .x.  x ) )  <->  v  =  ( t  .+  (
j  .x.  X )
) ) )
2319, 22rexeqbidv 2885 . . . . . 6  |-  ( x  =  X  ->  ( E. t  e.  ( O `  { x } ) v  =  ( t  .+  (
j  .x.  x )
)  <->  E. t  e.  ( O `  { X } ) v  =  ( t  .+  (
j  .x.  X )
) ) )
2423riotabidv 6518 . . . . 5  |-  ( x  =  X  ->  ( iota_ j  e.  R E. t  e.  ( O `  { x } ) v  =  ( t 
.+  ( j  .x.  x ) ) )  =  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) v  =  ( t  .+  (
j  .x.  X )
) ) )
2524mpteq2dv 4264 . . . 4  |-  ( x  =  X  ->  (
v  e.  V  |->  (
iota_ j  e.  R E. t  e.  ( O `  { x } ) v  =  ( t  .+  (
j  .x.  x )
) ) )  =  ( v  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) v  =  ( t  .+  (
j  .x.  X )
) ) ) )
26 eqid 2412 . . . 4  |-  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  (
iota_ j  e.  R E. t  e.  ( O `  { x } ) v  =  ( t  .+  (
j  .x.  x )
) ) ) )  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  {
x } ) v  =  ( t  .+  ( j  .x.  x
) ) ) ) )
2725, 26fvmptg 5771 . . 3  |-  ( ( X  e.  ( V 
\  {  .0.  }
)  /\  ( v  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) v  =  ( t  .+  (
j  .x.  X )
) ) )  e. 
_V )  ->  (
( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  {
x } ) v  =  ( t  .+  ( j  .x.  x
) ) ) ) ) `  X )  =  ( v  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) v  =  ( t  .+  (
j  .x.  X )
) ) ) )
2814, 17, 27sylancl 644 . 2  |-  ( ph  ->  ( ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  {
x } ) v  =  ( t  .+  ( j  .x.  x
) ) ) ) ) `  X )  =  ( v  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) v  =  ( t  .+  (
j  .x.  X )
) ) ) )
2913, 28eqtrd 2444 1  |-  ( ph  ->  ( M `  X
)  =  ( 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 1649    e. wcel 1721   E.wrex 2675   _Vcvv 2924    \ cdif 3285   {csn 3782    e. cmpt 4234   ` cfv 5421  (class class class)co 6048   iota_crio 6509   Basecbs 13432   +g cplusg 13492  Scalarcsca 13495   .scvsca 13496   0gc0g 13686   LHypclh 30478   DVecHcdvh 31573   ocHcoch 31842  HVMapchvm 32251
This theorem is referenced by:  hvmapvalvalN  32256  hvmapidN  32257  hdmapevec2  32334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-riota 6516  df-hvmap 32252
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