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Theorem hdmap1valc 31812
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 31811 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 3332 . . . 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 31807 . 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 31811 . . 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 31732 . 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 2351 1  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  ( L `
 <. X ,  F ,  Y >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822    \ cdif 3183   ifcif 3599   {csn 3674   <.cotp 3678    e. cmpt 4114   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163   iota_crio 6339   Basecbs 13195   0gc0g 13449   -gcsg 14414   LSpanclspn 15777   HLchlt 29358   LHypclh 29991   DVecHcdvh 31086  LCDualclcd 31594  mapdcmpd 31632  HDMap1chdma1 31800
This theorem is referenced by:  hdmap1cl  31813  hdmap1eq2  31814  hdmap1eq4N  31815  hdmap1eulem  31832  hdmap1eulemOLDN  31833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-ot 3684  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-1st 6164  df-2nd 6165  df-riota 6346  df-hdmap1 31802
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