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Theorem hdmap1val2 37980
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 2396 . . 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 3418 . . 3  |-  ( ph  ->  Y  e.  V )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18hdmap1val 37978 . 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 4087 . . . 4  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  =/=  .0.  )
2120neneqd 2598 . . 3  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  -.  Y  =  .0.  )
22 iffalse 3883 . . 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 20 . 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 2437 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836    \ cdif 3403   ifcif 3874   {csn 3961   <.cotp 3969   ` cfv 5513   iota_crio 6179  (class class class)co 6218   Basecbs 14657   0gc0g 14870   -gcsg 16195   LSpanclspn 17753   HLchlt 35527   LHypclh 36160   DVecHcdvh 37257  LCDualclcd 37765  mapdcmpd 37803  HDMap1chdma1 37971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-ot 3970  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-1st 6721  df-2nd 6722  df-hdmap1 37973
This theorem is referenced by:  hdmap1eq  37981
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