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Theorem hdmap1valc 34788
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 34787 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.  }
) )
1615eldifad 3423 . . 3  |-  ( ph  ->  X  e.  V )
17 hdmap1valc.f . . 3  |-  ( ph  ->  F  e.  D )
18 hdmap1valc.y . . 3  |-  ( ph  ->  Y  e.  V )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18hdmap1val 34783 . 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 ) } ) ) ) ) )
20 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 ) } ) ) ) ) )
2120hdmap1cbv 34787 . . 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 ) } ) ) ) ) )
2210, 21, 16, 17, 18mapdhval 34708 . 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 ) } ) ) ) ) )
2319, 22eqtr4d 2444 1  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  ( L `
 <. X ,  F ,  Y >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840   _Vcvv 3056    \ cdif 3408   ifcif 3882   {csn 3969   <.cotp 3977    |-> cmpt 4450   ` cfv 5523   iota_crio 6193  (class class class)co 6232   1stc1st 6734   2ndc2nd 6735   Basecbs 14731   0gc0g 14944   -gcsg 16269   LSpanclspn 17827   HLchlt 32332   LHypclh 32965   DVecHcdvh 34062  LCDualclcd 34570  mapdcmpd 34608  HDMap1chdma1 34776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-ot 3978  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-1st 6736  df-2nd 6737  df-hdmap1 34778
This theorem is referenced by:  hdmap1cl  34789  hdmap1eq2  34790  hdmap1eq4N  34791  hdmap1eulem  34808  hdmap1eulemOLDN  34809
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