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Theorem hdmap1valc 35807
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 35806 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 3451 . . 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 35802 . 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 35806 . . 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 35727 . 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 2498 1  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  ( L `
 <. X ,  F ,  Y >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    \ cdif 3436   ifcif 3902   {csn 3988   <.cotp 3996    |-> cmpt 4461   ` cfv 5529   iota_crio 6163  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689   Basecbs 14295   0gc0g 14500   -gcsg 15535   LSpanclspn 17178   HLchlt 33353   LHypclh 33986   DVecHcdvh 35081  LCDualclcd 35589  mapdcmpd 35627  HDMap1chdma1 35795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-ot 3997  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-1st 6690  df-2nd 6691  df-hdmap1 35797
This theorem is referenced by:  hdmap1cl  35808  hdmap1eq2  35809  hdmap1eq4N  35810  hdmap1eulem  35827  hdmap1eulemOLDN  35828
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