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Theorem hdmap1vallem 32610
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 32609 . . 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 5543 . 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 5555 . . . . 5  |-  ( 0g
`  C )  e. 
_V
1910, 18eqeltri 2366 . . . 4  |-  Q  e. 
_V
20 riotaex 6324 . . . 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 3636 . . 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 5541 . . . . . 6  |-  ( x  =  T  ->  ( 2nd `  x )  =  ( 2nd `  T
) )
2322eqeq1d 2304 . . . . 5  |-  ( x  =  T  ->  (
( 2nd `  x
)  =  .0.  <->  ( 2nd `  T )  =  .0.  ) )
2422sneqd 3666 . . . . . . . . . 10  |-  ( x  =  T  ->  { ( 2nd `  x ) }  =  { ( 2nd `  T ) } )
2524fveq2d 5545 . . . . . . . . 9  |-  ( x  =  T  ->  ( N `  { ( 2nd `  x ) } )  =  ( N `
 { ( 2nd `  T ) } ) )
2625fveq2d 5545 . . . . . . . 8  |-  ( x  =  T  ->  ( M `  ( N `  { ( 2nd `  x
) } ) )  =  ( M `  ( N `  { ( 2nd `  T ) } ) ) )
2726eqeq1d 2304 . . . . . . 7  |-  ( x  =  T  ->  (
( M `  ( N `  { ( 2nd `  x ) } ) )  =  ( J `  { h } )  <->  ( M `  ( N `  {
( 2nd `  T
) } ) )  =  ( J `  { h } ) ) )
28 fveq2 5541 . . . . . . . . . . . . 13  |-  ( x  =  T  ->  ( 1st `  x )  =  ( 1st `  T
) )
2928fveq2d 5545 . . . . . . . . . . . 12  |-  ( x  =  T  ->  ( 1st `  ( 1st `  x
) )  =  ( 1st `  ( 1st `  T ) ) )
3029, 22oveq12d 5892 . . . . . . . . . . 11  |-  ( x  =  T  ->  (
( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) )  =  ( ( 1st `  ( 1st `  T ) ) 
.-  ( 2nd `  T
) ) )
3130sneqd 3666 . . . . . . . . . 10  |-  ( x  =  T  ->  { ( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) }  =  { ( ( 1st `  ( 1st `  T
) )  .-  ( 2nd `  T ) ) } )
3231fveq2d 5545 . . . . . . . . 9  |-  ( x  =  T  ->  ( N `  { (
( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } )  =  ( N `  { ( ( 1st `  ( 1st `  T
) )  .-  ( 2nd `  T ) ) } ) )
3332fveq2d 5545 . . . . . . . 8  |-  ( x  =  T  ->  ( M `  ( N `  { ( ( 1st `  ( 1st `  x
) )  .-  ( 2nd `  x ) ) } ) )  =  ( M `  ( N `  { (
( 1st `  ( 1st `  T ) ) 
.-  ( 2nd `  T
) ) } ) ) )
3428fveq2d 5545 . . . . . . . . . . 11  |-  ( x  =  T  ->  ( 2nd `  ( 1st `  x
) )  =  ( 2nd `  ( 1st `  T ) ) )
3534oveq1d 5889 . . . . . . . . . 10  |-  ( x  =  T  ->  (
( 2nd `  ( 1st `  x ) ) R h )  =  ( ( 2nd `  ( 1st `  T ) ) R h ) )
3635sneqd 3666 . . . . . . . . 9  |-  ( x  =  T  ->  { ( ( 2nd `  ( 1st `  x ) ) R h ) }  =  { ( ( 2nd `  ( 1st `  T ) ) R h ) } )
3736fveq2d 5545 . . . . . . . 8  |-  ( x  =  T  ->  ( J `  { (
( 2nd `  ( 1st `  x ) ) R h ) } )  =  ( J `
 { ( ( 2nd `  ( 1st `  T ) ) R h ) } ) )
3833, 37eqeq12d 2310 . . . . . . 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 691 . . . . . 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 6322 . . . . 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 3598 . . . 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 2296 . . . 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 5616 . . 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 643 . 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 2328 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   ifcif 3578   {csn 3653    e. cmpt 4093    X. cxp 4703   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   iota_crio 6313   Basecbs 13164   0gc0g 13416   -gcsg 14381   LSpanclspn 15744   LHypclh 30795   DVecHcdvh 31890  LCDualclcd 32398  mapdcmpd 32436  HDMap1chdma1 32604
This theorem is referenced by:  hdmap1val  32611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-hdmap1 32606
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