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Theorem hdmap1valc 31994
Description: Connect the value of the preliminary map from vectors to functionals  I to the hypothesis  L used by earlier theorems. Note: the  X  e.  ( V  \  {  .0.  } ) hypothesis could be the more general  X  e.  V but the former will be easier to use. TODO: use the  I function directly in those theorems, so this theorem becomes unnecessary? TODO: The hdmap1cbv 31993 is probably unnecessary, but it would mean different $d's later on. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmap1valc.h  |-  H  =  ( LHyp `  K
)
hdmap1valc.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap1valc.v  |-  V  =  ( Base `  U
)
hdmap1valc.s  |-  .-  =  ( -g `  U )
hdmap1valc.o  |-  .0.  =  ( 0g `  U )
hdmap1valc.n  |-  N  =  ( LSpan `  U )
hdmap1valc.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap1valc.d  |-  D  =  ( Base `  C
)
hdmap1valc.r  |-  R  =  ( -g `  C
)
hdmap1valc.q  |-  Q  =  ( 0g `  C
)
hdmap1valc.j  |-  J  =  ( LSpan `  C )
hdmap1valc.m  |-  M  =  ( (mapd `  K
) `  W )
hdmap1valc.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmap1valc.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap1valc.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
hdmap1valc.f  |-  ( ph  ->  F  e.  D )
hdmap1valc.y  |-  ( ph  ->  Y  e.  V )
hdmap1valc.l  |-  L  =  ( x  e.  _V  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )
Assertion
Ref Expression
hdmap1valc  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  ( L `
 <. X ,  F ,  Y >. ) )
Distinct variable groups:    x,  .0.    x, h, D    h, J, x    h, M, x    .- , h, x    h, N, x    R, h, x    x, Q
Allowed substitution hints:    ph( x, h)    C( x, h)    Q( h)    U( x, h)    F( x, h)    H( x, h)    I( x, h)    K( x, h)    L( x, h)    V( x, h)    W( x, h)    X( x, h)    Y( x, h)    .0. ( h)

Proof of Theorem hdmap1valc
Dummy variables  w  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hdmap1valc.h . . 3  |-  H  =  ( LHyp `  K
)
2 hdmap1valc.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmap1valc.v . . 3  |-  V  =  ( Base `  U
)
4 hdmap1valc.s . . 3  |-  .-  =  ( -g `  U )
5 hdmap1valc.o . . 3  |-  .0.  =  ( 0g `  U )
6 hdmap1valc.n . . 3  |-  N  =  ( LSpan `  U )
7 hdmap1valc.c . . 3  |-  C  =  ( (LCDual `  K
) `  W )
8 hdmap1valc.d . . 3  |-  D  =  ( Base `  C
)
9 hdmap1valc.r . . 3  |-  R  =  ( -g `  C
)
10 hdmap1valc.q . . 3  |-  Q  =  ( 0g `  C
)
11 hdmap1valc.j . . 3  |-  J  =  ( LSpan `  C )
12 hdmap1valc.m . . 3  |-  M  =  ( (mapd `  K
) `  W )
13 hdmap1valc.i . . 3  |-  I  =  ( (HDMap1 `  K
) `  W )
14 hdmap1valc.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
15 hdmap1valc.x . . . 4  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
16 eldifi 3298 . . . 4  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
1715, 16syl 15 . . 3  |-  ( ph  ->  X  e.  V )
18 hdmap1valc.f . . 3  |-  ( ph  ->  F  e.  D )
19 hdmap1valc.y . . 3  |-  ( ph  ->  Y  e.  V )
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19hdmap1val 31989 . 2  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  if ( Y  =  .0.  ,  Q ,  ( iota_ g  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( J `  { g } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R g ) } ) ) ) ) )
21 hdmap1valc.l . . . 4  |-  L  =  ( x  e.  _V  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )
2221hdmap1cbv 31993 . . 3  |-  L  =  ( w  e.  _V  |->  if ( ( 2nd `  w
)  =  .0.  ,  Q ,  ( iota_ g  e.  D ( ( M `  ( N `
 { ( 2nd `  w ) } ) )  =  ( J `
 { g } )  /\  ( M `
 ( N `  { ( ( 1st `  ( 1st `  w
) )  .-  ( 2nd `  w ) ) } ) )  =  ( J `  {
( ( 2nd `  ( 1st `  w ) ) R g ) } ) ) ) ) )
2310, 22, 17, 18, 19mapdhval 31914 . 2  |-  ( ph  ->  ( L `  <. X ,  F ,  Y >. )  =  if ( Y  =  .0.  ,  Q ,  ( iota_ g  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( J `  { g } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R g ) } ) ) ) ) )
2420, 23eqtr4d 2318 1  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  ( L `
 <. X ,  F ,  Y >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   ifcif 3565   {csn 3640   <.cotp 3644    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   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:  hdmap1cl  31995  hdmap1eq2  31996  hdmap1eq4N  31997  hdmap1eulem  32014  hdmap1eulemOLDN  32015
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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