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Theorem hdmapffval 32627
Description: Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
Hypothesis
Ref Expression
hdmapval.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
hdmapffval  |-  ( K  e.  X  ->  (HDMap `  K )  =  ( w  e.  H  |->  { a  |  [. <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  w ) ) >.  /  e ]. [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  K ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) } ) )
Distinct variable groups:    w, H    e, a, i, t, u, v, w, y, z, K
Allowed substitution hints:    H( y, z, v, u, t, e, i, a)    X( y, z, w, v, u, t, e, i, a)

Proof of Theorem hdmapffval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2964 . 2  |-  ( K  e.  X  ->  K  e.  _V )
2 fveq2 5728 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 hdmapval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2486 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5728 . . . . . . . . 9  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
65reseq2d 5146 . . . . . . . 8  |-  ( k  =  K  ->  (  _I  |`  ( Base `  k
) )  =  (  _I  |`  ( Base `  K ) ) )
7 fveq2 5728 . . . . . . . . . 10  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
87fveq1d 5730 . . . . . . . . 9  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
98reseq2d 5146 . . . . . . . 8  |-  ( k  =  K  ->  (  _I  |`  ( ( LTrn `  k ) `  w
) )  =  (  _I  |`  ( ( LTrn `  K ) `  w ) ) )
106, 9opeq12d 3992 . . . . . . 7  |-  ( k  =  K  ->  <. (  _I  |`  ( Base `  k
) ) ,  (  _I  |`  ( ( LTrn `  k ) `  w ) ) >.  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  w ) ) >.
)
11 dfsbcq 3163 . . . . . . 7  |-  ( <.
(  _I  |`  ( Base `  k ) ) ,  (  _I  |`  (
( LTrn `  k ) `  w ) ) >.  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  w ) ) >.  ->  ( [. <. (  _I  |`  ( Base `  k
) ) ,  (  _I  |`  ( ( LTrn `  k ) `  w ) ) >.  /  e ]. [. (
( DVecH `  k ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  k
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <->  [. <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  w ) ) >.  /  e ]. [. (
( DVecH `  k ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  k
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
1210, 11syl 16 . . . . . 6  |-  ( k  =  K  ->  ( [. <. (  _I  |`  ( Base `  k ) ) ,  (  _I  |`  (
( LTrn `  k ) `  w ) ) >.  /  e ]. [. (
( DVecH `  k ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  k
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <->  [. <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  w ) ) >.  /  e ]. [. (
( DVecH `  k ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  k
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
13 fveq2 5728 . . . . . . . . . 10  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
1413fveq1d 5730 . . . . . . . . 9  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
15 dfsbcq 3163 . . . . . . . . 9  |-  ( ( ( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w )  ->  ( [. ( ( DVecH `  k
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( (HDMap1 `  k ) `  w
)  /  i ]. a  e.  ( t  e.  v  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  k ) `  w ) ) A. z  e.  v  ( -.  z  e.  (
( ( LSpan `  u
) `  { e } )  u.  (
( LSpan `  u ) `  { t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <->  [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( (HDMap1 `  k ) `  w
)  /  i ]. a  e.  ( t  e.  v  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  k ) `  w ) ) A. z  e.  v  ( -.  z  e.  (
( ( LSpan `  u
) `  { e } )  u.  (
( LSpan `  u ) `  { t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
1614, 15syl 16 . . . . . . . 8  |-  ( k  =  K  ->  ( [. ( ( DVecH `  k
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( (HDMap1 `  k ) `  w
)  /  i ]. a  e.  ( t  e.  v  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  k ) `  w ) ) A. z  e.  v  ( -.  z  e.  (
( ( LSpan `  u
) `  { e } )  u.  (
( LSpan `  u ) `  { t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <->  [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( (HDMap1 `  k ) `  w
)  /  i ]. a  e.  ( t  e.  v  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  k ) `  w ) ) A. z  e.  v  ( -.  z  e.  (
( ( LSpan `  u
) `  { e } )  u.  (
( LSpan `  u ) `  { t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
17 fveq2 5728 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  (HDMap1 `  k )  =  (HDMap1 `  K ) )
1817fveq1d 5730 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
(HDMap1 `  k ) `  w )  =  ( (HDMap1 `  K ) `  w ) )
19 dfsbcq 3163 . . . . . . . . . . . 12  |-  ( ( (HDMap1 `  k ) `  w )  =  ( (HDMap1 `  K ) `  w )  ->  ( [. ( (HDMap1 `  k
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <->  [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
2018, 19syl 16 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( [. ( (HDMap1 `  k
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <->  [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
21 fveq2 5728 . . . . . . . . . . . . . . . . 17  |-  ( k  =  K  ->  (LCDual `  k )  =  (LCDual `  K ) )
2221fveq1d 5730 . . . . . . . . . . . . . . . 16  |-  ( k  =  K  ->  (
(LCDual `  k ) `  w )  =  ( (LCDual `  K ) `  w ) )
2322fveq2d 5732 . . . . . . . . . . . . . . 15  |-  ( k  =  K  ->  ( Base `  ( (LCDual `  k ) `  w
) )  =  (
Base `  ( (LCDual `  K ) `  w
) ) )
24 fveq2 5728 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  K  ->  (HVMap `  k )  =  (HVMap `  K ) )
2524fveq1d 5730 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  K  ->  (
(HVMap `  k ) `  w )  =  ( (HVMap `  K ) `  w ) )
2625fveq1d 5730 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  K  ->  (
( (HVMap `  k
) `  w ) `  e )  =  ( ( (HVMap `  K
) `  w ) `  e ) )
2726oteq2d 3997 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  K  ->  <. e ,  ( ( (HVMap `  k ) `  w
) `  e ) ,  z >.  =  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >. )
2827fveq2d 5732 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  K  ->  (
i `  <. e ,  ( ( (HVMap `  k ) `  w
) `  e ) ,  z >. )  =  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) )
2928oteq2d 3997 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  K  ->  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.  =  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >. ) ,  t >. )
3029fveq2d 5732 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  K  ->  (
i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w
) `  e ) ,  z >. ) ,  t >. )  =  ( i `  <. z ,  ( i `
 <. e ,  ( ( (HVMap `  K
) `  w ) `  e ) ,  z
>. ) ,  t >.
) )
3130eqeq2d 2447 . . . . . . . . . . . . . . . . 17  |-  ( k  =  K  ->  (
y  =  ( i `
 <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w
) `  e ) ,  z >. ) ,  t >. )  <->  y  =  ( i `  <. z ,  ( i `
 <. e ,  ( ( (HVMap `  K
) `  w ) `  e ) ,  z
>. ) ,  t >.
) ) )
3231imbi2d 308 . . . . . . . . . . . . . . . 16  |-  ( k  =  K  ->  (
( -.  z  e.  ( ( ( LSpan `  u ) `  {
e } )  u.  ( ( LSpan `  u
) `  { t } ) )  -> 
y  =  ( i `
 <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w
) `  e ) ,  z >. ) ,  t >. )
)  <->  ( -.  z  e.  ( ( ( LSpan `  u ) `  {
e } )  u.  ( ( LSpan `  u
) `  { t } ) )  -> 
y  =  ( i `
 <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >. ) ,  t >. )
) ) )
3332ralbidv 2725 . . . . . . . . . . . . . . 15  |-  ( k  =  K  ->  ( A. z  e.  v 
( -.  z  e.  ( ( ( LSpan `  u ) `  {
e } )  u.  ( ( LSpan `  u
) `  { t } ) )  -> 
y  =  ( i `
 <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w
) `  e ) ,  z >. ) ,  t >. )
)  <->  A. z  e.  v  ( -.  z  e.  ( ( ( LSpan `  u ) `  {
e } )  u.  ( ( LSpan `  u
) `  { t } ) )  -> 
y  =  ( i `
 <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >. ) ,  t >. )
) ) )
3423, 33riotaeqbidv 6552 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  ( iota_ y  e.  ( Base `  ( (LCDual `  k
) `  w )
) A. z  e.  v  ( -.  z  e.  ( ( ( LSpan `  u ) `  {
e } )  u.  ( ( LSpan `  u
) `  { t } ) )  -> 
y  =  ( i `
 <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w
) `  e ) ,  z >. ) ,  t >. )
) )  =  (
iota_ y  e.  ( Base `  ( (LCDual `  K ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )
3534mpteq2dv 4296 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  (
t  e.  v  |->  (
iota_ y  e.  ( Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  =  ( t  e.  v  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  K ) `  w ) ) A. z  e.  v  ( -.  z  e.  (
( ( LSpan `  u
) `  { e } )  u.  (
( LSpan `  u ) `  { t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) )
3635eleq2d 2503 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
a  e.  ( t  e.  v  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  k ) `  w ) ) A. z  e.  v  ( -.  z  e.  (
( ( LSpan `  u
) `  { e } )  u.  (
( LSpan `  u ) `  { t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <-> 
a  e.  ( t  e.  v  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  K ) `  w ) ) A. z  e.  v  ( -.  z  e.  (
( ( LSpan `  u
) `  { e } )  u.  (
( LSpan `  u ) `  { t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
3736sbcbidv 3215 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <->  [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  K ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
3820, 37bitrd 245 . . . . . . . . . 10  |-  ( k  =  K  ->  ( [. ( (HDMap1 `  k
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <->  [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  K ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
3938sbcbidv 3215 . . . . . . . . 9  |-  ( k  =  K  ->  ( [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  k
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <->  [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  K ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
4039sbcbidv 3215 . . . . . . . 8  |-  ( k  =  K  ->  ( [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( (HDMap1 `  k ) `  w
)  /  i ]. a  e.  ( t  e.  v  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  k ) `  w ) ) A. z  e.  v  ( -.  z  e.  (
( ( LSpan `  u
) `  { e } )  u.  (
( LSpan `  u ) `  { t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <->  [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( (HDMap1 `  K ) `  w
)  /  i ]. a  e.  ( t  e.  v  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  K ) `  w ) ) A. z  e.  v  ( -.  z  e.  (
( ( LSpan `  u
) `  { e } )  u.  (
( LSpan `  u ) `  { t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
4116, 40bitrd 245 . . . . . . 7  |-  ( k  =  K  ->  ( [. ( ( DVecH `  k
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( (HDMap1 `  k ) `  w
)  /  i ]. a  e.  ( t  e.  v  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  k ) `  w ) ) A. z  e.  v  ( -.  z  e.  (
( ( LSpan `  u
) `  { e } )  u.  (
( LSpan `  u ) `  { t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <->  [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( (HDMap1 `  K ) `  w
)  /  i ]. a  e.  ( t  e.  v  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  K ) `  w ) ) A. z  e.  v  ( -.  z  e.  (
( ( LSpan `  u
) `  { e } )  u.  (
( LSpan `  u ) `  { t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
4241sbcbidv 3215 . . . . . 6  |-  ( k  =  K  ->  ( [. <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  w ) ) >.  /  e ]. [. (
( DVecH `  k ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  k
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <->  [. <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  w ) ) >.  /  e ]. [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  K ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
4312, 42bitrd 245 . . . . 5  |-  ( k  =  K  ->  ( [. <. (  _I  |`  ( Base `  k ) ) ,  (  _I  |`  (
( LTrn `  k ) `  w ) ) >.  /  e ]. [. (
( DVecH `  k ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  k
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) )  <->  [. <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  w ) ) >.  /  e ]. [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  K ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) ) )
4443abbidv 2550 . . . 4  |-  ( k  =  K  ->  { a  |  [. <. (  _I  |`  ( Base `  k
) ) ,  (  _I  |`  ( ( LTrn `  k ) `  w ) ) >.  /  e ]. [. (
( DVecH `  k ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  k
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) }  =  { a  |  [. <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  w ) ) >.  /  e ]. [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  K ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) } )
454, 44mpteq12dv 4287 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  { a  |  [. <. (  _I  |`  ( Base `  k
) ) ,  (  _I  |`  ( ( LTrn `  k ) `  w ) ) >.  /  e ]. [. (
( DVecH `  k ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  k
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) } )  =  ( w  e.  H  |->  { a  |  [. <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  w ) ) >.  /  e ]. [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  K ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) } ) )
46 df-hdmap 32593 . . 3  |- HDMap  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { a  |  [. <. (  _I  |`  ( Base `  k
) ) ,  (  _I  |`  ( ( LTrn `  k ) `  w ) ) >.  /  e ]. [. (
( DVecH `  k ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  k
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  k ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) } ) )
47 fvex 5742 . . . . 5  |-  ( LHyp `  K )  e.  _V
483, 47eqeltri 2506 . . . 4  |-  H  e. 
_V
4948mptex 5966 . . 3  |-  ( w  e.  H  |->  { a  |  [. <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  w ) ) >.  /  e ]. [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  K ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) } )  e.  _V
5045, 46, 49fvmpt 5806 . 2  |-  ( K  e.  _V  ->  (HDMap `  K )  =  ( w  e.  H  |->  { a  |  [. <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  w ) ) >.  /  e ]. [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  K ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) } ) )
511, 50syl 16 1  |-  ( K  e.  X  ->  (HDMap `  K )  =  ( w  e.  H  |->  { a  |  [. <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  w ) ) >.  /  e ]. [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( (HDMap1 `  K
) `  w )  /  i ]. a  e.  ( t  e.  v 
|->  ( iota_ y  e.  (
Base `  ( (LCDual `  K ) `  w
) ) A. z  e.  v  ( -.  z  e.  ( (
( LSpan `  u ) `  { e } )  u.  ( ( LSpan `  u ) `  {
t } ) )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
) ) ) ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705   _Vcvv 2956   [.wsbc 3161    u. cun 3318   {csn 3814   <.cop 3817   <.cotp 3818    e. cmpt 4266    _I cid 4493    |` cres 4880   ` cfv 5454   iota_crio 6542   Basecbs 13469   LSpanclspn 16047   LHypclh 30781   LTrncltrn 30898   DVecHcdvh 31876  LCDualclcd 32384  HVMapchvm 32554  HDMap1chdma1 32590  HDMapchdma 32591
This theorem is referenced by:  hdmapfval  32628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-ot 3824  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-riota 6549  df-hdmap 32593
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