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Theorem hvmapvalvalN 32561
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 32560 . . 3  |-  ( ph  ->  ( M `  X
)  =  ( y  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) y  =  ( t  .+  (
j  .x.  X )
) ) ) )
1413fveq1d 5732 . 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 6555 . . 3  |-  ( iota_ j  e.  R E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) )  e.  _V
17 eqeq1 2444 . . . . . 6  |-  ( y  =  Y  ->  (
y  =  ( t 
.+  ( j  .x.  X ) )  <->  Y  =  ( t  .+  (
j  .x.  X )
) ) )
1817rexbidv 2728 . . . . 5  |-  ( y  =  Y  ->  ( E. t  e.  ( O `  { X } ) y  =  ( t  .+  (
j  .x.  X )
)  <->  E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) ) )
1918riotabidv 6553 . . . 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 2438 . . . 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 5806 . . 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 645 . 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 2470 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 360    = wceq 1653    e. wcel 1726   E.wrex 2708   _Vcvv 2958    \ cdif 3319   {csn 3816    e. cmpt 4268   ` cfv 5456  (class class class)co 6083   iota_crio 6544   Basecbs 13471   +g cplusg 13531  Scalarcsca 13534   .scvsca 13535   0gc0g 13725   LHypclh 30783   DVecHcdvh 31878   ocHcoch 32147  HVMapchvm 32556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-riota 6551  df-hvmap 32557
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