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Theorem hvmapvalvalN 37904
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 37903 . . 3  |-  ( ph  ->  ( M `  X
)  =  ( y  e.  V  |->  ( iota_ j  e.  R  E. t  e.  ( O `  { X } ) y  =  ( t  .+  (
j  .x.  X )
) ) ) )
1413fveq1d 5850 . 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 6236 . . 3  |-  ( iota_ j  e.  R  E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) )  e.  _V
17 eqeq1 2458 . . . . . 6  |-  ( y  =  Y  ->  (
y  =  ( t 
.+  ( j  .x.  X ) )  <->  Y  =  ( t  .+  (
j  .x.  X )
) ) )
1817rexbidv 2965 . . . . 5  |-  ( y  =  Y  ->  ( E. t  e.  ( O `  { X } ) y  =  ( t  .+  (
j  .x.  X )
)  <->  E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) ) )
1918riotabidv 6234 . . . 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 2454 . . . 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 5929 . . 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 660 . 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 2495 1  |-  ( ph  ->  ( ( M `  X ) `  Y
)  =  ( iota_ j  e.  R  E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805   _Vcvv 3106    \ cdif 3458   {csn 4016    |-> cmpt 4497   ` cfv 5570   iota_crio 6231  (class class class)co 6270   Basecbs 14719   +g cplusg 14787  Scalarcsca 14790   .scvsca 14791   0gc0g 14932   LHypclh 36124   DVecHcdvh 37221   ocHcoch 37490  HVMapchvm 37899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-hvmap 37900
This theorem is referenced by: (None)
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