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Theorem hdmap1val 31989
Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 31914.) TODO: change  I  =  ( x  e.  _V  |->... to  ( ph  ->  ( I `  <. X ,  F ,  Y  >  )  =... in e.g. mapdh8 31979 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 3650 . . . 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 4719 . . . . . 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 4719 . . . . 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 2367 . . 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 31988 . 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 6136 . . . . 5  |-  ( Y  e.  V  ->  ( 2nd `  <. X ,  F ,  Y >. )  =  Y )
2620, 25syl 15 . . . 4  |-  ( ph  ->  ( 2nd `  <. X ,  F ,  Y >. )  =  Y )
2726eqeq1d 2291 . . 3  |-  ( ph  ->  ( ( 2nd `  <. X ,  F ,  Y >. )  =  .0.  <->  Y  =  .0.  ) )
2826sneqd 3653 . . . . . . . 8  |-  ( ph  ->  { ( 2nd `  <. X ,  F ,  Y >. ) }  =  { Y } )
2928fveq2d 5529 . . . . . . 7  |-  ( ph  ->  ( N `  {
( 2nd `  <. X ,  F ,  Y >. ) } )  =  ( N `  { Y } ) )
3029fveq2d 5529 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) )  =  ( M `
 ( N `  { Y } ) ) )
3130eqeq1d 2291 . . . . 5  |-  ( ph  ->  ( ( M `  ( N `  { ( 2nd `  <. X ,  F ,  Y >. ) } ) )  =  ( J `  {
h } )  <->  ( M `  ( N `  { Y } ) )  =  ( J `  {
h } ) ) )
32 ot1stg 6134 . . . . . . . . . . 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 5876 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) )  =  ( X 
.-  Y ) )
3534sneqd 3653 . . . . . . . 8  |-  ( ph  ->  { ( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) }  =  { ( X 
.-  Y ) } )
3635fveq2d 5529 . . . . . . 7  |-  ( ph  ->  ( N `  {
( ( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } )  =  ( N `  {
( X  .-  Y
) } ) )
3736fveq2d 5529 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { (
( 1st `  ( 1st `  <. X ,  F ,  Y >. ) )  .-  ( 2nd `  <. X ,  F ,  Y >. ) ) } ) )  =  ( M `  ( N `  { ( X  .-  Y ) } ) ) )
38 ot2ndg 6135 . . . . . . . . . 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 5873 . . . . . . . 8  |-  ( ph  ->  ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h )  =  ( F R h ) )
4140sneqd 3653 . . . . . . 7  |-  ( ph  ->  { ( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) }  =  {
( F R h ) } )
4241fveq2d 5529 . . . . . 6  |-  ( ph  ->  ( J `  {
( ( 2nd `  ( 1st `  <. X ,  F ,  Y >. ) ) R h ) } )  =  ( J `  { ( F R h ) } ) )
4337, 42eqeq12d 2297 . . . . 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 6306 . . 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 3585 . 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 2315 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 1623    e. wcel 1684   ifcif 3565   {csn 3640   <.cop 3643   <.cotp 3644    X. cxp 4687   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   iota_crio 6297   Basecbs 13148   0gc0g 13400   -gcsg 14365   LSpanclspn 15728   LHypclh 30173   DVecHcdvh 31268  LCDualclcd 31776  mapdcmpd 31814  HDMap1chdma1 31982
This theorem is referenced by:  hdmap1val0  31990  hdmap1val2  31991  hdmap1valc  31994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-hdmap1 31984
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