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Theorem hvmapvalvalN 31951
Description: Value of value of map (i.e. functional value) from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.) (New usage is discouraged.)
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.  }
) )
hvmapval.y  |-  ( ph  ->  Y  e.  V )
Assertion
Ref Expression
hvmapvalvalN  |-  ( ph  ->  ( ( M `  X ) `  Y
)  =  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) ) )
Distinct variable groups:    t, j, K    t, W    t, O    R, j    j, W    j, X, t    j, Y, t
Allowed substitution hints:    ph( t, j)    A( t, j)    .+ ( t, j)    R( t)    S( t, j)    .x. ( t, j)    U( t, j)    H( t, j)    M( t, j)    O( j)    V( t, j)    .0. ( t, j)

Proof of Theorem hvmapvalvalN
Dummy variable  y 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
) )
12 hvmapval.x . . . 4  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12hvmapval 31950 . . 3  |-  ( ph  ->  ( M `  X
)  =  ( y  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) y  =  ( t  .+  (
j  .x.  X )
) ) ) )
1413fveq1d 5527 . 2  |-  ( ph  ->  ( ( M `  X ) `  Y
)  =  ( ( y  e.  V  |->  (
iota_ j  e.  R E. t  e.  ( O `  { X } ) y  =  ( t  .+  (
j  .x.  X )
) ) ) `  Y ) )
15 hvmapval.y . . 3  |-  ( ph  ->  Y  e.  V )
16 riotaex 6308 . . 3  |-  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) )  e.  _V
17 eqeq1 2289 . . . . . 6  |-  ( y  =  Y  ->  (
y  =  ( t 
.+  ( j  .x.  X ) )  <->  Y  =  ( t  .+  (
j  .x.  X )
) ) )
1817rexbidv 2564 . . . . 5  |-  ( y  =  Y  ->  ( E. t  e.  ( O `  { X } ) y  =  ( t  .+  (
j  .x.  X )
)  <->  E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) ) )
1918riotabidv 6306 . . . 4  |-  ( y  =  Y  ->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) y  =  ( t 
.+  ( j  .x.  X ) ) )  =  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) ) )
20 eqid 2283 . . . 4  |-  ( y  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) y  =  ( t  .+  (
j  .x.  X )
) ) )  =  ( y  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) y  =  ( t  .+  (
j  .x.  X )
) ) )
2119, 20fvmptg 5600 . . 3  |-  ( ( Y  e.  V  /\  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) )  e.  _V )  ->  ( ( y  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) y  =  ( t  .+  (
j  .x.  X )
) ) ) `  Y )  =  (
iota_ j  e.  R E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) ) )
2215, 16, 21sylancl 643 . 2  |-  ( ph  ->  ( ( y  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) y  =  ( t  .+  (
j  .x.  X )
) ) ) `  Y )  =  (
iota_ j  e.  R E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) ) )
2314, 22eqtrd 2315 1  |-  ( ph  ->  ( ( M `  X ) `  Y
)  =  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) Y  =  ( 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 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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