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Theorem hdmap1vallem 36595
Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmap1val.h  |-  H  =  ( LHyp `  K
)
hdmap1fval.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap1fval.v  |-  V  =  ( Base `  U
)
hdmap1fval.s  |-  .-  =  ( -g `  U )
hdmap1fval.o  |-  .0.  =  ( 0g `  U )
hdmap1fval.n  |-  N  =  ( LSpan `  U )
hdmap1fval.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap1fval.d  |-  D  =  ( Base `  C
)
hdmap1fval.r  |-  R  =  ( -g `  C
)
hdmap1fval.q  |-  Q  =  ( 0g `  C
)
hdmap1fval.j  |-  J  =  ( LSpan `  C )
hdmap1fval.m  |-  M  =  ( (mapd `  K
) `  W )
hdmap1fval.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmap1fval.k  |-  ( ph  ->  ( K  e.  A  /\  W  e.  H
) )
hdmap1val.t  |-  ( ph  ->  T  e.  ( ( V  X.  D )  X.  V ) )
Assertion
Ref Expression
hdmap1vallem  |-  ( ph  ->  ( I `  T
)  =  if ( ( 2nd `  T
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  T ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  T ) ) 
.-  ( 2nd `  T
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  T ) ) R h ) } ) ) ) ) )
Distinct variable groups:    C, h    D, h    h, J    h, M    h, N    U, h    h, V    T, h
Allowed substitution hints:    ph( h)    A( h)    Q( h)    R( h)    H( h)    I( h)    K( h)    .- ( h)    W( h)    .0. ( h)

Proof of Theorem hdmap1vallem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hdmap1val.h . . . 4  |-  H  =  ( LHyp `  K
)
2 hdmap1fval.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmap1fval.v . . . 4  |-  V  =  ( Base `  U
)
4 hdmap1fval.s . . . 4  |-  .-  =  ( -g `  U )
5 hdmap1fval.o . . . 4  |-  .0.  =  ( 0g `  U )
6 hdmap1fval.n . . . 4  |-  N  =  ( LSpan `  U )
7 hdmap1fval.c . . . 4  |-  C  =  ( (LCDual `  K
) `  W )
8 hdmap1fval.d . . . 4  |-  D  =  ( Base `  C
)
9 hdmap1fval.r . . . 4  |-  R  =  ( -g `  C
)
10 hdmap1fval.q . . . 4  |-  Q  =  ( 0g `  C
)
11 hdmap1fval.j . . . 4  |-  J  =  ( LSpan `  C )
12 hdmap1fval.m . . . 4  |-  M  =  ( (mapd `  K
) `  W )
13 hdmap1fval.i . . . 4  |-  I  =  ( (HDMap1 `  K
) `  W )
14 hdmap1fval.k . . . 4  |-  ( ph  ->  ( K  e.  A  /\  W  e.  H
) )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14hdmap1fval 36594 . . 3  |-  ( ph  ->  I  =  ( x  e.  ( ( V  X.  D )  X.  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 ) } ) ) ) ) ) )
1615fveq1d 5866 . 2  |-  ( ph  ->  ( I `  T
)  =  ( ( x  e.  ( ( V  X.  D )  X.  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 ) } ) ) ) ) ) `
 T ) )
17 hdmap1val.t . . 3  |-  ( ph  ->  T  e.  ( ( V  X.  D )  X.  V ) )
18 fvex 5874 . . . . 5  |-  ( 0g
`  C )  e. 
_V
1910, 18eqeltri 2551 . . . 4  |-  Q  e. 
_V
20 riotaex 6247 . . . 4  |-  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  T ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  T ) ) 
.-  ( 2nd `  T
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  T ) ) R h ) } ) ) )  e.  _V
2119, 20ifex 4008 . . 3  |-  if ( ( 2nd `  T
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  T ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  T ) ) 
.-  ( 2nd `  T
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  T ) ) R h ) } ) ) ) )  e. 
_V
22 fveq2 5864 . . . . . 6  |-  ( x  =  T  ->  ( 2nd `  x )  =  ( 2nd `  T
) )
2322eqeq1d 2469 . . . . 5  |-  ( x  =  T  ->  (
( 2nd `  x
)  =  .0.  <->  ( 2nd `  T )  =  .0.  ) )
2422sneqd 4039 . . . . . . . . . 10  |-  ( x  =  T  ->  { ( 2nd `  x ) }  =  { ( 2nd `  T ) } )
2524fveq2d 5868 . . . . . . . . 9  |-  ( x  =  T  ->  ( N `  { ( 2nd `  x ) } )  =  ( N `
 { ( 2nd `  T ) } ) )
2625fveq2d 5868 . . . . . . . 8  |-  ( x  =  T  ->  ( M `  ( N `  { ( 2nd `  x
) } ) )  =  ( M `  ( N `  { ( 2nd `  T ) } ) ) )
2726eqeq1d 2469 . . . . . . 7  |-  ( x  =  T  ->  (
( M `  ( N `  { ( 2nd `  x ) } ) )  =  ( J `  { h } )  <->  ( M `  ( N `  {
( 2nd `  T
) } ) )  =  ( J `  { h } ) ) )
28 fveq2 5864 . . . . . . . . . . . . 13  |-  ( x  =  T  ->  ( 1st `  x )  =  ( 1st `  T
) )
2928fveq2d 5868 . . . . . . . . . . . 12  |-  ( x  =  T  ->  ( 1st `  ( 1st `  x
) )  =  ( 1st `  ( 1st `  T ) ) )
3029, 22oveq12d 6300 . . . . . . . . . . 11  |-  ( x  =  T  ->  (
( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) )  =  ( ( 1st `  ( 1st `  T ) ) 
.-  ( 2nd `  T
) ) )
3130sneqd 4039 . . . . . . . . . 10  |-  ( x  =  T  ->  { ( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) }  =  { ( ( 1st `  ( 1st `  T
) )  .-  ( 2nd `  T ) ) } )
3231fveq2d 5868 . . . . . . . . 9  |-  ( x  =  T  ->  ( N `  { (
( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } )  =  ( N `  { ( ( 1st `  ( 1st `  T
) )  .-  ( 2nd `  T ) ) } ) )
3332fveq2d 5868 . . . . . . . 8  |-  ( x  =  T  ->  ( M `  ( N `  { ( ( 1st `  ( 1st `  x
) )  .-  ( 2nd `  x ) ) } ) )  =  ( M `  ( N `  { (
( 1st `  ( 1st `  T ) ) 
.-  ( 2nd `  T
) ) } ) ) )
3428fveq2d 5868 . . . . . . . . . . 11  |-  ( x  =  T  ->  ( 2nd `  ( 1st `  x
) )  =  ( 2nd `  ( 1st `  T ) ) )
3534oveq1d 6297 . . . . . . . . . 10  |-  ( x  =  T  ->  (
( 2nd `  ( 1st `  x ) ) R h )  =  ( ( 2nd `  ( 1st `  T ) ) R h ) )
3635sneqd 4039 . . . . . . . . 9  |-  ( x  =  T  ->  { ( ( 2nd `  ( 1st `  x ) ) R h ) }  =  { ( ( 2nd `  ( 1st `  T ) ) R h ) } )
3736fveq2d 5868 . . . . . . . 8  |-  ( x  =  T  ->  ( J `  { (
( 2nd `  ( 1st `  x ) ) R h ) } )  =  ( J `
 { ( ( 2nd `  ( 1st `  T ) ) R h ) } ) )
3833, 37eqeq12d 2489 . . . . . . 7  |-  ( x  =  T  ->  (
( M `  ( N `  { (
( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } )  <-> 
( M `  ( N `  { (
( 1st `  ( 1st `  T ) ) 
.-  ( 2nd `  T
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  T ) ) R h ) } ) ) )
3927, 38anbi12d 710 . . . . . 6  |-  ( x  =  T  ->  (
( ( M `  ( N `  { ( 2nd `  x ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  x
) ) R h ) } ) )  <-> 
( ( M `  ( N `  { ( 2nd `  T ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  T ) )  .-  ( 2nd `  T ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  T
) ) R h ) } ) ) ) )
4039riotabidv 6245 . . . . 5  |-  ( x  =  T  ->  ( iota_ h  e.  D  ( ( M `  ( N `  { ( 2nd `  x ) } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( ( 1st `  ( 1st `  x
) )  .-  ( 2nd `  x ) ) } ) )  =  ( J `  {
( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) )  =  ( iota_ h  e.  D  ( ( M `  ( N `  { ( 2nd `  T ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  T ) )  .-  ( 2nd `  T ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  T
) ) R h ) } ) ) ) )
4123, 40ifbieq2d 3964 . . . 4  |-  ( x  =  T  ->  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 ) } ) ) ) )  =  if ( ( 2nd `  T )  =  .0. 
,  Q ,  (
iota_ h  e.  D  ( ( M `  ( N `  { ( 2nd `  T ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  T ) )  .-  ( 2nd `  T ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  T
) ) R h ) } ) ) ) ) )
42 eqid 2467 . . . 4  |-  ( x  e.  ( ( V  X.  D )  X.  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 ) } ) ) ) ) )  =  ( x  e.  ( ( V  X.  D )  X.  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 ) } ) ) ) ) )
4341, 42fvmptg 5946 . . 3  |-  ( ( T  e.  ( ( V  X.  D )  X.  V )  /\  if ( ( 2nd `  T
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  T ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  T ) ) 
.-  ( 2nd `  T
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  T ) ) R h ) } ) ) ) )  e. 
_V )  ->  (
( x  e.  ( ( V  X.  D
)  X.  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 ) } ) ) ) ) ) `  T )  =  if ( ( 2nd `  T
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  T ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  T ) ) 
.-  ( 2nd `  T
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  T ) ) R h ) } ) ) ) ) )
4417, 21, 43sylancl 662 . 2  |-  ( ph  ->  ( ( x  e.  ( ( V  X.  D )  X.  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 ) } ) ) ) ) ) `
 T )  =  if ( ( 2nd `  T )  =  .0. 
,  Q ,  (
iota_ h  e.  D  ( ( M `  ( N `  { ( 2nd `  T ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  T ) )  .-  ( 2nd `  T ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  T
) ) R h ) } ) ) ) ) )
4516, 44eqtrd 2508 1  |-  ( ph  ->  ( I `  T
)  =  if ( ( 2nd `  T
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  T ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  T ) ) 
.-  ( 2nd `  T
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  T ) ) R h ) } ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   ifcif 3939   {csn 4027    |-> cmpt 4505    X. cxp 4997   ` cfv 5586   iota_crio 6242  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   Basecbs 14486   0gc0g 14691   -gcsg 15726   LSpanclspn 17400   LHypclh 34780   DVecHcdvh 35875  LCDualclcd 36383  mapdcmpd 36421  HDMap1chdma1 36589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-hdmap1 36591
This theorem is referenced by:  hdmap1val  36596
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