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Theorem hdmap1val2 32599
Description: Value of preliminary map from vectors to functionals in the closed kernel dual space, for nonzero  Y. (Contributed by NM, 16-May-2015.)
Hypotheses
Ref Expression
hdmap1val2.h  |-  H  =  ( LHyp `  K
)
hdmap1val2.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap1val2.v  |-  V  =  ( Base `  U
)
hdmap1val2.s  |-  .-  =  ( -g `  U )
hdmap1val2.o  |-  .0.  =  ( 0g `  U )
hdmap1val2.n  |-  N  =  ( LSpan `  U )
hdmap1val2.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap1val2.d  |-  D  =  ( Base `  C
)
hdmap1val2.r  |-  R  =  ( -g `  C
)
hdmap1val2.l  |-  L  =  ( LSpan `  C )
hdmap1val2.m  |-  M  =  ( (mapd `  K
) `  W )
hdmap1val2.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmap1val2.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap1val2.x  |-  ( ph  ->  X  e.  V )
hdmap1val2.f  |-  ( ph  ->  F  e.  D )
hdmap1val2.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
Assertion
Ref Expression
hdmap1val2  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  ( iota_ h  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( L `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R h ) } ) ) ) )
Distinct variable groups:    C, h    D, h    h, F    h, L    h, M    h, N    U, h    h, V    h, X    h, Y    ph, h
Allowed substitution hints:    R( h)    H( h)    I( h)    K( h)    .- ( h)    W( h)    .0. ( h)

Proof of Theorem hdmap1val2
StepHypRef Expression
1 hdmap1val2.h . . 3  |-  H  =  ( LHyp `  K
)
2 hdmap1val2.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmap1val2.v . . 3  |-  V  =  ( Base `  U
)
4 hdmap1val2.s . . 3  |-  .-  =  ( -g `  U )
5 hdmap1val2.o . . 3  |-  .0.  =  ( 0g `  U )
6 hdmap1val2.n . . 3  |-  N  =  ( LSpan `  U )
7 hdmap1val2.c . . 3  |-  C  =  ( (LCDual `  K
) `  W )
8 hdmap1val2.d . . 3  |-  D  =  ( Base `  C
)
9 hdmap1val2.r . . 3  |-  R  =  ( -g `  C
)
10 eqid 2436 . . 3  |-  ( 0g
`  C )  =  ( 0g `  C
)
11 hdmap1val2.l . . 3  |-  L  =  ( LSpan `  C )
12 hdmap1val2.m . . 3  |-  M  =  ( (mapd `  K
) `  W )
13 hdmap1val2.i . . 3  |-  I  =  ( (HDMap1 `  K
) `  W )
14 hdmap1val2.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
15 hdmap1val2.x . . 3  |-  ( ph  ->  X  e.  V )
16 hdmap1val2.f . . 3  |-  ( ph  ->  F  e.  D )
17 hdmap1val2.y . . . 4  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
1817eldifad 3332 . . 3  |-  ( ph  ->  Y  e.  V )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18hdmap1val 32597 . 2  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  if ( Y  =  .0.  , 
( 0g `  C
) ,  ( iota_ h  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( L `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R h ) } ) ) ) ) )
20 eldifsni 3928 . . . 4  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  =/=  .0.  )
2120neneqd 2617 . . 3  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  -.  Y  =  .0.  )
22 iffalse 3746 . . 3  |-  ( -.  Y  =  .0.  ->  if ( Y  =  .0. 
,  ( 0g `  C ) ,  (
iota_ h  e.  D
( ( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( L `  { ( F R h ) } ) ) ) )  =  ( iota_ h  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( L `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R h ) } ) ) ) )
2317, 21, 223syl 19 . 2  |-  ( ph  ->  if ( Y  =  .0.  ,  ( 0g
`  C ) ,  ( iota_ h  e.  D
( ( M `  ( N `  { Y } ) )  =  ( L `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( L `  { ( F R h ) } ) ) ) )  =  ( iota_ h  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( L `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R h ) } ) ) ) )
2419, 23eqtrd 2468 1  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  ( iota_ h  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( L `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( L `
 { ( F R h ) } ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    \ cdif 3317   ifcif 3739   {csn 3814   <.cotp 3818   ` cfv 5454  (class class class)co 6081   iota_crio 6542   Basecbs 13469   0gc0g 13723   -gcsg 14688   LSpanclspn 16047   HLchlt 30148   LHypclh 30781   DVecHcdvh 31876  LCDualclcd 32384  mapdcmpd 32422  HDMap1chdma1 32590
This theorem is referenced by:  hdmap1eq  32600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-ot 3824  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-riota 6549  df-hdmap1 32592
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