Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hvmapvalvalN Structured version   Unicode version

Theorem hvmapvalvalN 36433
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 36432 . . 3  |-  ( ph  ->  ( M `  X
)  =  ( y  e.  V  |->  ( iota_ j  e.  R  E. t  e.  ( O `  { X } ) y  =  ( t  .+  (
j  .x.  X )
) ) ) )
1413fveq1d 5859 . 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 6240 . . 3  |-  ( iota_ j  e.  R  E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) )  e.  _V
17 eqeq1 2464 . . . . . 6  |-  ( y  =  Y  ->  (
y  =  ( t 
.+  ( j  .x.  X ) )  <->  Y  =  ( t  .+  (
j  .x.  X )
) ) )
1817rexbidv 2966 . . . . 5  |-  ( y  =  Y  ->  ( E. t  e.  ( O `  { X } ) y  =  ( t  .+  (
j  .x.  X )
)  <->  E. t  e.  ( O `  { X } ) Y  =  ( t  .+  (
j  .x.  X )
) ) )
1918riotabidv 6238 . . . 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 2460 . . . 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 5939 . . 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 662 . 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 2501 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 369    = wceq 1374    e. wcel 1762   E.wrex 2808   _Vcvv 3106    \ cdif 3466   {csn 4020    |-> cmpt 4498   ` cfv 5579   iota_crio 6235  (class class class)co 6275   Basecbs 14479   +g cplusg 14544  Scalarcsca 14547   .scvsca 14548   0gc0g 14684   LHypclh 34655   DVecHcdvh 35750   ocHcoch 36019  HVMapchvm 36428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-hvmap 36429
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator