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Theorem hdmapffval 32088
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 2872 . 2  |-  ( K  e.  X  ->  K  e.  _V )
2 fveq2 5608 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 hdmapval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2408 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5608 . . . . . . . . 9  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
65reseq2d 5037 . . . . . . . 8  |-  ( k  =  K  ->  (  _I  |`  ( Base `  k
) )  =  (  _I  |`  ( Base `  K ) ) )
7 fveq2 5608 . . . . . . . . . 10  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
87fveq1d 5610 . . . . . . . . 9  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
98reseq2d 5037 . . . . . . . 8  |-  ( k  =  K  ->  (  _I  |`  ( ( LTrn `  k ) `  w
) )  =  (  _I  |`  ( ( LTrn `  K ) `  w ) ) )
106, 9opeq12d 3885 . . . . . . 7  |-  ( k  =  K  ->  <. (  _I  |`  ( Base `  k
) ) ,  (  _I  |`  ( ( LTrn `  k ) `  w ) ) >.  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  w ) ) >.
)
11 dfsbcq 3069 . . . . . . 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 15 . . . . . 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 5608 . . . . . . . . . 10  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
1413fveq1d 5610 . . . . . . . . 9  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
15 dfsbcq 3069 . . . . . . . . 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 15 . . . . . . . 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 5608 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  (HDMap1 `  k )  =  (HDMap1 `  K ) )
1817fveq1d 5610 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
(HDMap1 `  k ) `  w )  =  ( (HDMap1 `  K ) `  w ) )
19 dfsbcq 3069 . . . . . . . . . . . 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 15 . . . . . . . . . . 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 5608 . . . . . . . . . . . . . . . . 17  |-  ( k  =  K  ->  (LCDual `  k )  =  (LCDual `  K ) )
2221fveq1d 5610 . . . . . . . . . . . . . . . 16  |-  ( k  =  K  ->  (
(LCDual `  k ) `  w )  =  ( (LCDual `  K ) `  w ) )
2322fveq2d 5612 . . . . . . . . . . . . . . 15  |-  ( k  =  K  ->  ( Base `  ( (LCDual `  k ) `  w
) )  =  (
Base `  ( (LCDual `  K ) `  w
) ) )
24 fveq2 5608 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  K  ->  (HVMap `  k )  =  (HVMap `  K ) )
2524fveq1d 5610 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  K  ->  (
(HVMap `  k ) `  w )  =  ( (HVMap `  K ) `  w ) )
2625fveq1d 5610 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  K  ->  (
( (HVMap `  k
) `  w ) `  e )  =  ( ( (HVMap `  K
) `  w ) `  e ) )
27 oteq2 3887 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( (HVMap `  k
) `  w ) `  e )  =  ( ( (HVMap `  K
) `  w ) `  e )  ->  <. e ,  ( ( (HVMap `  k ) `  w
) `  e ) ,  z >.  =  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >. )
2826, 27syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  K  ->  <. e ,  ( ( (HVMap `  k ) `  w
) `  e ) ,  z >.  =  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >. )
2928fveq2d 5612 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  K  ->  (
i `  <. e ,  ( ( (HVMap `  k ) `  w
) `  e ) ,  z >. )  =  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) )
30 oteq2 3887 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i `  <. e ,  ( ( (HVMap `  k ) `  w
) `  e ) ,  z >. )  =  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
)  ->  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w
) `  e ) ,  z >. ) ,  t >.  =  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.
)
3129, 30syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  K  ->  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e
) ,  z >.
) ,  t >.  =  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >. ) ,  t >. )
3231fveq2d 5612 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  K  ->  (
i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w
) `  e ) ,  z >. ) ,  t >. )  =  ( i `  <. z ,  ( i `
 <. e ,  ( ( (HVMap `  K
) `  w ) `  e ) ,  z
>. ) ,  t >.
) )
3332eqeq2d 2369 . . . . . . . . . . . . . . . . 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 >.
) ) )
3433imbi2d 307 . . . . . . . . . . . . . . . 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 >. )
) ) )
3534ralbidv 2639 . . . . . . . . . . . . . . 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 >. )
) ) )
3623, 35riotaeqbidv 6394 . . . . . . . . . . . . . 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 >.
) ) ) )
3736mpteq2dv 4188 . . . . . . . . . . . . 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 >.
) ) ) ) )
3837eleq2d 2425 . . . . . . . . . . . 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 >.
) ) ) ) ) )
3938sbcbidv 3121 . . . . . . . . . . 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 >.
) ) ) ) ) )
4020, 39bitrd 244 . . . . . . . . . 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 >.
) ) ) ) ) )
4140sbcbidv 3121 . . . . . . . . 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 >.
) ) ) ) ) )
4241sbcbidv 3121 . . . . . . . 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 >.
) ) ) ) ) )
4316, 42bitrd 244 . . . . . . 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 >.
) ) ) ) ) )
4443sbcbidv 3121 . . . . . 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 >.
) ) ) ) ) )
4512, 44bitrd 244 . . . . 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 >.
) ) ) ) ) )
4645abbidv 2472 . . . 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 >.
) ) ) ) } )
474, 46mpteq12dv 4179 . . 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 >.
) ) ) ) } ) )
48 df-hdmap 32054 . . 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 >.
) ) ) ) } ) )
49 fvex 5622 . . . . 5  |-  ( LHyp `  K )  e.  _V
503, 49eqeltri 2428 . . . 4  |-  H  e. 
_V
5150mptex 5832 . . 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
5247, 48, 51fvmpt 5685 . 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 >.
) ) ) ) } ) )
531, 52syl 15 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 176    = wceq 1642    e. wcel 1710   {cab 2344   A.wral 2619   _Vcvv 2864   [.wsbc 3067    u. cun 3226   {csn 3716   <.cop 3719   <.cotp 3720    e. cmpt 4158    _I cid 4386    |` cres 4773   ` cfv 5337   iota_crio 6384   Basecbs 13245   LSpanclspn 15827   LHypclh 30242   LTrncltrn 30359   DVecHcdvh 31337  LCDualclcd 31845  HVMapchvm 32015  HDMap1chdma1 32051  HDMapchdma 32052
This theorem is referenced by:  hdmapfval  32089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-ot 3726  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-riota 6391  df-hdmap 32054
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