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Theorem hdmap1val 32611
Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 32536.) TODO: change  I  =  ( x  e.  _V  |->... to  ( ph  ->  ( I `  <. X ,  F ,  Y  >  )  =... in e.g. mapdh8 32601 to shorten proofs with no $d on  x. (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.x  |-  ( ph  ->  X  e.  V )
hdmap1val.f  |-  ( ph  ->  F  e.  D )
hdmap1val.y  |-  ( ph  ->  Y  e.  V )
Assertion
Ref Expression
hdmap1val  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  if ( Y  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R h ) } ) ) ) ) )
Distinct variable groups:    C, h    D, h    h, J    h, M    h, N    U, h    h, V    h, F    h, X    h, Y    ph, h
Allowed substitution hints:    A( h)    Q( h)    R( h)    H( h)    I( h)    K( h)    .- ( h)    W( h)    .0. ( h)

Proof of Theorem hdmap1val
StepHypRef Expression
1 hdmap1val.h . . 3  |-  H  =  ( LHyp `  K
)
2 hdmap1fval.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmap1fval.v . . 3  |-  V  =  ( Base `  U
)
4 hdmap1fval.s . . 3  |-  .-  =  ( -g `  U )
5 hdmap1fval.o . . 3  |-  .0.  =  ( 0g `  U )
6 hdmap1fval.n . . 3  |-  N  =  ( LSpan `  U )
7 hdmap1fval.c . . 3  |-  C  =  ( (LCDual `  K
) `  W )
8 hdmap1fval.d . . 3  |-  D  =  ( Base `  C
)
9 hdmap1fval.r . . 3  |-  R  =  ( -g `  C
)
10 hdmap1fval.q . . 3  |-  Q  =  ( 0g `  C
)
11 hdmap1fval.j . . 3  |-  J  =  ( LSpan `  C )
12 hdmap1fval.m . . 3  |-  M  =  ( (mapd `  K
) `  W )
13 hdmap1fval.i . . 3  |-  I  =  ( (HDMap1 `  K
) `  W )
14 hdmap1fval.k . . 3  |-  ( ph  ->  ( K  e.  A  /\  W  e.  H
) )
15 df-ot 3663 . . . 4  |-  <. X ,  F ,  Y >.  = 
<. <. X ,  F >. ,  Y >.
16 hdmap1val.x . . . . . 6  |-  ( ph  ->  X  e.  V )
17 hdmap1val.f . . . . . 6  |-  ( ph  ->  F  e.  D )
18 opelxp 4735 . . . . . 6  |-  ( <. X ,  F >.  e.  ( V  X.  D
)  <->  ( X  e.  V  /\  F  e.  D ) )
1916, 17, 18sylanbrc 645 . . . . 5  |-  ( ph  -> 
<. X ,  F >.  e.  ( V  X.  D
) )
20 hdmap1val.y . . . . 5  |-  ( ph  ->  Y  e.  V )
21 opelxp 4735 . . . . 5  |-  ( <. <. X ,  F >. ,  Y >.  e.  (
( V  X.  D
)  X.  V )  <-> 
( <. X ,  F >.  e.  ( V  X.  D )  /\  Y  e.  V ) )
2219, 20, 21sylanbrc 645 . . . 4  |-  ( ph  -> 
<. <. X ,  F >. ,  Y >.  e.  ( ( V  X.  D
)  X.  V ) )
2315, 22syl5eqel 2380 . . 3  |-  ( ph  -> 
<. X ,  F ,  Y >.  e.  ( ( V  X.  D )  X.  V ) )
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23hdmap1vallem 32610 . 2  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  if ( ( 2nd `  <. X ,  F ,  Y >. )  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { ( 2nd `  <. X ,  F ,  Y >. ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) } ) ) ) ) )
25 ot3rdg 6152 . . . . 5  |-  ( Y  e.  V  ->  ( 2nd `  <. X ,  F ,  Y >. )  =  Y )
2620, 25syl 15 . . . 4  |-  ( ph  ->  ( 2nd `  <. X ,  F ,  Y >. )  =  Y )
2726eqeq1d 2304 . . 3  |-  ( ph  ->  ( ( 2nd `  <. X ,  F ,  Y >. )  =  .0.  <->  Y  =  .0.  ) )
2826sneqd 3666 . . . . . . . 8  |-  ( ph  ->  { ( 2nd `  <. X ,  F ,  Y >. ) }  =  { Y } )
2928fveq2d 5545 . . . . . . 7  |-  ( ph  ->  ( N `  {
( 2nd `  <. X ,  F ,  Y >. ) } )  =  ( N `  { Y } ) )
3029fveq2d 5545 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) )  =  ( M `
 ( N `  { Y } ) ) )
3130eqeq1d 2304 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) )  =  ( J `  {
h } )  <->  ( M `  ( N `  { Y } ) )  =  ( J `  {
h } ) ) )
32 ot1stg 6150 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  F  e.  D  /\  Y  e.  V )  ->  ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  =  X )
3316, 17, 20, 32syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  =  X )
3433, 26oveq12d 5892 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) )  =  ( X 
.-  Y ) )
3534sneqd 3666 . . . . . . . 8  |-  ( ph  ->  { ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) }  =  { ( X 
.-  Y ) } )
3635fveq2d 5545 . . . . . . 7  |-  ( ph  ->  ( N `  {
( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } )  =  ( N `  {
( X  .-  Y
) } ) )
3736fveq2d 5545 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { (
( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } ) )  =  ( M `  ( N `  { ( X  .-  Y ) } ) ) )
38 ot2ndg 6151 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  F  e.  D  /\  Y  e.  V )  ->  ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) )  =  F )
3916, 17, 20, 38syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) )  =  F )
4039oveq1d 5889 . . . . . . . 8  |-  ( ph  ->  ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h )  =  ( F R h ) )
4140sneqd 3666 . . . . . . 7  |-  ( ph  ->  { ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) }  =  {
( F R h ) } )
4241fveq2d 5545 . . . . . 6  |-  ( ph  ->  ( J `  {
( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) } )  =  ( J `  { ( F R h ) } ) )
4337, 42eqeq12d 2310 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) } )  <->  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( J `  { ( F R h ) } ) ) )
4431, 43anbi12d 691 . . . 4  |-  ( ph  ->  ( ( ( M `
 ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) } ) )  <-> 
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( F R h ) } ) ) ) )
4544riotabidv 6322 . . 3  |-  ( ph  ->  ( iota_ h  e.  D
( ( M `  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) } ) ) )  =  ( iota_ h  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R h ) } ) ) ) )
4627, 45ifbieq2d 3598 . 2  |-  ( ph  ->  if ( ( 2nd `  <. X ,  F ,  Y >. )  =  .0. 
,  Q ,  (
iota_ h  e.  D
( ( M `  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) } ) ) ) )  =  if ( Y  =  .0. 
,  Q ,  (
iota_ h  e.  D
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( F R h ) } ) ) ) ) )
4724, 46eqtrd 2328 1  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  if ( Y  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
 { Y }
) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( F R h ) } ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ifcif 3578   {csn 3653   <.cop 3656   <.cotp 3657    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:  hdmap1val0  32612  hdmap1val2  32613  hdmap1valc  32616
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-ot 3663  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-1st 6138  df-2nd 6139  df-riota 6320  df-hdmap1 32606
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