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Theorem hvmapval 31950
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 31949 . . 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 5527 . 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 5539 . . . . 5  |-  ( Base `  U )  e.  _V
164, 15eqeltri 2353 . . . 4  |-  V  e. 
_V
1716mptex 5746 . . 3  |-  ( v  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) v  =  ( t  .+  (
j  .x.  X )
) ) )  e. 
_V
18 sneq 3651 . . . . . . . 8  |-  ( x  =  X  ->  { x }  =  { X } )
1918fveq2d 5529 . . . . . . 7  |-  ( x  =  X  ->  ( O `  { x } )  =  ( O `  { X } ) )
20 oveq2 5866 . . . . . . . . 9  |-  ( x  =  X  ->  (
j  .x.  x )  =  ( j  .x.  X ) )
2120oveq2d 5874 . . . . . . . 8  |-  ( x  =  X  ->  (
t  .+  ( j  .x.  x ) )  =  ( t  .+  (
j  .x.  X )
) )
2221eqeq2d 2294 . . . . . . 7  |-  ( x  =  X  ->  (
v  =  ( t 
.+  ( j  .x.  x ) )  <->  v  =  ( t  .+  (
j  .x.  X )
) ) )
2319, 22rexeqbidv 2749 . . . . . 6  |-  ( x  =  X  ->  ( E. t  e.  ( O `  { x } ) v  =  ( t  .+  (
j  .x.  x )
)  <->  E. t  e.  ( O `  { X } ) v  =  ( t  .+  (
j  .x.  X )
) ) )
2423riotabidv 6306 . . . . 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 4107 . . . 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 2283 . . . 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 5600 . . 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 643 . 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 2315 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 358    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    \ cdif 3149   {csn 3640    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   LHypclh 30173   DVecHcdvh 31268   ocHcoch 31537  HVMapchvm 31946
This theorem is referenced by:  hvmapvalvalN  31951  hvmapidN  31952  hdmapevec2  32029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-hvmap 31947
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