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Theorem hdmap1val2 31991
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 2283 . . 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.  }
) )
18 eldifi 3298 . . . 4  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
1917, 18syl 15 . . 3  |-  ( ph  ->  Y  e.  V )
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19hdmap1val 31989 . 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 ) } ) ) ) ) )
21 eldifsni 3750 . . . 4  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  =/=  .0.  )
2221neneqd 2462 . . 3  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  -.  Y  =  .0.  )
23 iffalse 3572 . . 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 ) } ) ) ) )
2417, 22, 233syl 18 . 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 ) } ) ) ) )
2520, 24eqtrd 2315 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 358    = wceq 1623    e. wcel 1684    \ cdif 3149   ifcif 3565   {csn 3640   <.cotp 3644   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   0gc0g 13400   -gcsg 14365   LSpanclspn 15728   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268  LCDualclcd 31776  mapdcmpd 31814  HDMap1chdma1 31982
This theorem is referenced by:  hdmap1eq  31992
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650  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-1st 6122  df-2nd 6123  df-riota 6304  df-hdmap1 31984
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