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Theorem hdmapfval 32642
Description: Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmapval.h  |-  H  =  ( LHyp `  K
)
hdmapfval.e  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
hdmapfval.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmapfval.v  |-  V  =  ( Base `  U
)
hdmapfval.n  |-  N  =  ( LSpan `  U )
hdmapfval.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmapfval.d  |-  D  =  ( Base `  C
)
hdmapfval.j  |-  J  =  ( (HVMap `  K
) `  W )
hdmapfval.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmapfval.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmapfval.k  |-  ( ph  ->  ( K  e.  A  /\  W  e.  H
) )
Assertion
Ref Expression
hdmapfval  |-  ( ph  ->  S  =  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) ) )
Distinct variable groups:    y, t,
z, K    y, D    t, E, y, z    t, I, y, z    t, U, y, z    t, V, y, z    t, W, y, z
Allowed substitution hints:    ph( y, z, t)    A( y, z, t)    C( y, z, t)    D( z, t)    S( y, z, t)    H( y, z, t)    J( y, z, t)    N( y, z, t)

Proof of Theorem hdmapfval
Dummy variables  w  e  a  i  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hdmapfval.k . 2  |-  ( ph  ->  ( K  e.  A  /\  W  e.  H
) )
2 hdmapfval.s . . . 4  |-  S  =  ( (HDMap `  K
) `  W )
3 hdmapval.h . . . . . 6  |-  H  =  ( LHyp `  K
)
43hdmapffval 32641 . . . . 5  |-  ( K  e.  A  ->  (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 >.
) ) ) ) } ) )
54fveq1d 5543 . . . 4  |-  ( K  e.  A  ->  (
(HDMap `  K ) `  W )  =  ( ( 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 >.
) ) ) ) } ) `  W
) )
62, 5syl5eq 2340 . . 3  |-  ( K  e.  A  ->  S  =  ( ( 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 >.
) ) ) ) } ) `  W
) )
7 fveq2 5541 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
87reseq2d 4971 . . . . . . . . 9  |-  ( w  =  W  ->  (  _I  |`  ( ( LTrn `  K ) `  w
) )  =  (  _I  |`  ( ( LTrn `  K ) `  W ) ) )
98opeq2d 3819 . . . . . . . 8  |-  ( w  =  W  ->  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  w ) ) >.  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
)
10 dfsbcq 3006 . . . . . . . 8  |-  ( <.
(  _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 >.
) ) ) ) ) )
119, 10syl 15 . . . . . . 7  |-  ( w  =  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 >.
) ) ) ) ) )
12 fveq2 5541 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
13 dfsbcq 3006 . . . . . . . . . 10  |-  ( ( ( 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 >.
) ) ) ) ) )
1412, 13syl 15 . . . . . . . . 9  |-  ( w  =  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 >.
) ) ) ) ) )
15 fveq2 5541 . . . . . . . . . . . . 13  |-  ( w  =  W  ->  (
(HDMap1 `  K ) `  w )  =  ( (HDMap1 `  K ) `  W ) )
16 dfsbcq 3006 . . . . . . . . . . . . 13  |-  ( ( (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 >.
) ) ) ) ) )
1715, 16syl 15 . . . . . . . . . . . 12  |-  ( w  =  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 >.
) ) ) ) ) )
18 fveq2 5541 . . . . . . . . . . . . . . . . 17  |-  ( w  =  W  ->  (
(LCDual `  K ) `  w )  =  ( (LCDual `  K ) `  W ) )
1918fveq2d 5545 . . . . . . . . . . . . . . . 16  |-  ( w  =  W  ->  ( Base `  ( (LCDual `  K ) `  w
) )  =  (
Base `  ( (LCDual `  K ) `  W
) ) )
20 fveq2 5541 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( w  =  W  ->  (
(HVMap `  K ) `  w )  =  ( (HVMap `  K ) `  W ) )
2120fveq1d 5543 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( w  =  W  ->  (
( (HVMap `  K
) `  w ) `  e )  =  ( ( (HVMap `  K
) `  W ) `  e ) )
22 oteq2 3822 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( (HVMap `  K
) `  w ) `  e )  =  ( ( (HVMap `  K
) `  W ) `  e )  ->  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >.  =  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. )
2321, 22syl 15 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( w  =  W  ->  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >.  =  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. )
2423fveq2d 5545 . . . . . . . . . . . . . . . . . . . . 21  |-  ( w  =  W  ->  (
i `  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >. )  =  ( i `  <. e ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.
) )
25 oteq2 3822 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 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 >.
)
2624, 25syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( w  =  W  ->  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.  =  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. ) ,  t >. )
2726fveq2d 5545 . . . . . . . . . . . . . . . . . . 19  |-  ( w  =  W  ->  (
i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >. ) ,  t >. )  =  ( i `  <. z ,  ( i `
 <. e ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. ) ,  t >.
) )
2827eqeq2d 2307 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  W  ->  (
y  =  ( i `
 <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >. ) ,  t >. )  <->  y  =  ( i `  <. z ,  ( i `
 <. e ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. ) ,  t >.
) ) )
2928imbi2d 307 . . . . . . . . . . . . . . . . 17  |-  ( w  =  W  ->  (
( -.  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 >. )
) ) )
3029ralbidv 2576 . . . . . . . . . . . . . . . 16  |-  ( w  =  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. z  e.  v  ( -.  z  e.  ( ( ( LSpan `  u ) `  {
e } )  u.  ( ( LSpan `  u
) `  { t } ) )  -> 
y  =  ( i `
 <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. ) ,  t >. )
) ) )
3119, 30riotaeqbidv 6323 . . . . . . . . . . . . . . 15  |-  ( w  =  W  ->  ( 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 >.
) ) ) )
3231mpteq2dv 4123 . . . . . . . . . . . . . 14  |-  ( w  =  W  ->  (
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 >.
) ) ) ) )
3332eleq2d 2363 . . . . . . . . . . . . 13  |-  ( w  =  W  ->  (
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 >.
) ) ) ) ) )
3433sbcbidv 3058 . . . . . . . . . . . 12  |-  ( w  =  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 >.
) ) ) ) ) )
3517, 34bitrd 244 . . . . . . . . . . 11  |-  ( w  =  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 >.
) ) ) ) ) )
3635sbcbidv 3058 . . . . . . . . . 10  |-  ( w  =  W  ->  ( [. ( 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 >.
) ) ) ) ) )
3736sbcbidv 3058 . . . . . . . . 9  |-  ( w  =  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 >.
) ) ) ) ) )
3814, 37bitrd 244 . . . . . . . 8  |-  ( w  =  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 >.
) ) ) ) ) )
3938sbcbidv 3058 . . . . . . 7  |-  ( w  =  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 >.
) ) ) ) ) )
4011, 39bitrd 244 . . . . . 6  |-  ( w  =  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 >.
) ) ) ) ) )
41 opex 4253 . . . . . . 7  |-  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  e.  _V
42 fvex 5555 . . . . . . 7  |-  ( (
DVecH `  K ) `  W )  e.  _V
43 fvex 5555 . . . . . . 7  |-  ( Base `  u )  e.  _V
44 simp1 955 . . . . . . . . 9  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  e  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
)
45 hdmapfval.e . . . . . . . . 9  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
4644, 45syl6eqr 2346 . . . . . . . 8  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  e  =  E )
47 simp2 956 . . . . . . . . 9  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  u  =  ( ( DVecH `  K ) `  W
) )
48 hdmapfval.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
4947, 48syl6eqr 2346 . . . . . . . 8  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  u  =  U )
50 simp3 957 . . . . . . . . . 10  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  v  =  ( Base `  u
) )
5149fveq2d 5545 . . . . . . . . . 10  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  ( Base `  u )  =  ( Base `  U
) )
5250, 51eqtrd 2328 . . . . . . . . 9  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  v  =  ( Base `  U
) )
53 hdmapfval.v . . . . . . . . 9  |-  V  =  ( Base `  U
)
5452, 53syl6eqr 2346 . . . . . . . 8  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  v  =  V )
55 fvex 5555 . . . . . . . . . 10  |-  ( (HDMap1 `  K ) `  W
)  e.  _V
56 id 19 . . . . . . . . . . . 12  |-  ( i  =  ( (HDMap1 `  K ) `  W
)  ->  i  =  ( (HDMap1 `  K ) `  W ) )
57 hdmapfval.i . . . . . . . . . . . 12  |-  I  =  ( (HDMap1 `  K
) `  W )
5856, 57syl6eqr 2346 . . . . . . . . . . 11  |-  ( i  =  ( (HDMap1 `  K ) `  W
)  ->  i  =  I )
59 fveq1 5540 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  I  ->  (
i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. ) ,  t >. )  =  ( I `  <. z ,  ( i `
 <. e ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. ) ,  t >.
) )
60 fveq1 5540 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  I  ->  (
i `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. )  =  ( I `  <. e ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.
) )
61 oteq2 3822 . . . . . . . . . . . . . . . . . . . 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 >.
)
6260, 61syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  I  ->  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.
) ,  t >.  =  <. z ,  ( I `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. ) ,  t >. )
6362fveq2d 5545 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  I  ->  (
I `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. ) ,  t >. )  =  ( I `  <. z ,  ( I `
 <. e ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. ) ,  t >.
) )
6459, 63eqtrd 2328 . . . . . . . . . . . . . . . . 17  |-  ( i  =  I  ->  (
i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. ) ,  t >. )  =  ( I `  <. z ,  ( I `
 <. e ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. ) ,  t >.
) )
6564eqeq2d 2307 . . . . . . . . . . . . . . . 16  |-  ( i  =  I  ->  (
y  =  ( i `
 <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. ) ,  t >. )  <->  y  =  ( I `  <. z ,  ( I `
 <. e ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. ) ,  t >.
) ) )
6665imbi2d 307 . . . . . . . . . . . . . . 15  |-  ( i  =  I  ->  (
( -.  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 >. )
) ) )
6766ralbidv 2576 . . . . . . . . . . . . . 14  |-  ( i  =  I  ->  ( 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 >. )
) ) )
6867riotabidv 6322 . . . . . . . . . . . . 13  |-  ( i  =  I  ->  ( 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 >.
) ) ) )
6968mpteq2dv 4123 . . . . . . . . . . . 12  |-  ( i  =  I  ->  (
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 >.
) ) ) ) )
7069eleq2d 2363 . . . . . . . . . . 11  |-  ( i  =  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  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 >.
) ) ) ) ) )
7158, 70syl 15 . . . . . . . . . 10  |-  ( i  =  ( (HDMap1 `  K ) `  W
)  ->  ( 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 >.
) ) ) ) ) )
7255, 71sbcie 3038 . . . . . . . . 9  |-  ( [. ( (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  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 >.
) ) ) ) )
73 simp3 957 . . . . . . . . . . 11  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  v  =  V )
74 hdmapfval.d . . . . . . . . . . . . . 14  |-  D  =  ( Base `  C
)
75 hdmapfval.c . . . . . . . . . . . . . . 15  |-  C  =  ( (LCDual `  K
) `  W )
7675fveq2i 5544 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  ( (LCDual `  K ) `  W
) )
7774, 76eqtr2i 2317 . . . . . . . . . . . . 13  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  D
7877a1i 10 . . . . . . . . . . . 12  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( Base `  (
(LCDual `  K ) `  W ) )  =  D )
79 simp2 956 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  u  =  U )
8079fveq2d 5545 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( LSpan `  u )  =  ( LSpan `  U
) )
81 hdmapfval.n . . . . . . . . . . . . . . . . . . 19  |-  N  =  ( LSpan `  U )
8280, 81syl6eqr 2346 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( LSpan `  u )  =  N )
83 simp1 955 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  e  =  E )
8483sneqd 3666 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  { e }  =  { E } )
8582, 84fveq12d 5547 . . . . . . . . . . . . . . . . 17  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( ( LSpan `  u
) `  { e } )  =  ( N `  { E } ) )
8682fveq1d 5543 . . . . . . . . . . . . . . . . 17  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( ( LSpan `  u
) `  { t } )  =  ( N `  { t } ) )
8785, 86uneq12d 3343 . . . . . . . . . . . . . . . 16  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( ( ( LSpan `  u ) `  {
e } )  u.  ( ( LSpan `  u
) `  { t } ) )  =  ( ( N `  { E } )  u.  ( N `  {
t } ) ) )
8887eleq2d 2363 . . . . . . . . . . . . . . 15  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( z  e.  ( ( ( LSpan `  u
) `  { e } )  u.  (
( LSpan `  u ) `  { t } ) )  <->  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) ) ) )
8988notbid 285 . . . . . . . . . . . . . 14  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( -.  z  e.  ( ( ( LSpan `  u ) `  {
e } )  u.  ( ( LSpan `  u
) `  { t } ) )  <->  -.  z  e.  ( ( N `  { E } )  u.  ( N `  {
t } ) ) ) )
90 oteq1 3821 . . . . . . . . . . . . . . . . . . . 20  |-  ( e  =  E  ->  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >.  =  <. E ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. )
9183, 90syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  -> 
<. e ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.  =  <. E ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. )
9283fveq2d 5545 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( ( (HVMap `  K ) `  W
) `  e )  =  ( ( (HVMap `  K ) `  W
) `  E )
)
93 hdmapfval.j . . . . . . . . . . . . . . . . . . . . . 22  |-  J  =  ( (HVMap `  K
) `  W )
9493fveq1i 5542 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J `
 E )  =  ( ( (HVMap `  K ) `  W
) `  E )
9592, 94syl6eqr 2346 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( ( (HVMap `  K ) `  W
) `  e )  =  ( J `  E ) )
96 oteq2 3822 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( (HVMap `  K
) `  W ) `  e )  =  ( J `  E )  ->  <. E ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>.  =  <. E , 
( J `  E
) ,  z >.
)
9795, 96syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  -> 
<. E ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.  =  <. E ,  ( J `  E ) ,  z >. )
9891, 97eqtrd 2328 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  -> 
<. e ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.  =  <. E ,  ( J `  E ) ,  z >. )
9998fveq2d 5545 . . . . . . . . . . . . . . . . 17  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( I `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. )  =  ( I `  <. E ,  ( J `
 E ) ,  z >. ) )
100 oteq2 3822 . . . . . . . . . . . . . . . . 17  |-  ( ( I `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. )  =  ( I `  <. E ,  ( J `
 E ) ,  z >. )  ->  <. z ,  ( I `  <. e ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.
) ,  t >.  =  <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
)
10199, 100syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  -> 
<. z ,  ( I `
 <. e ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. ) ,  t >.  =  <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
)
102101fveq2d 5545 . . . . . . . . . . . . . . 15  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( I `  <. z ,  ( I `  <. e ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.
) ,  t >.
)  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) )
103102eqeq2d 2307 . . . . . . . . . . . . . 14  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( y  =  ( I `  <. z ,  ( I `  <. e ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.
) ,  t >.
)  <->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  t
>. ) ) )
10489, 103imbi12d 311 . . . . . . . . . . . . 13  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( ( -.  z  e.  ( ( ( LSpan `  u ) `  {
e } )  u.  ( ( LSpan `  u
) `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. ) ,  t >. )
)  <->  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  {
t } ) )  ->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  t
>. ) ) ) )
10573, 104raleqbidv 2761 . . . . . . . . . . . 12  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( 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.  ( ( N `  { E } )  u.  ( N `  {
t } ) )  ->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  t
>. ) ) ) )
10678, 105riotaeqbidv 6323 . . . . . . . . . . 11  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  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 >.
) ) )  =  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  {
t } ) )  ->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  t
>. ) ) ) )
10773, 106mpteq12dv 4114 . . . . . . . . . 10  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( 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.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  {
t } ) )  ->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  t
>. ) ) ) ) )
108107eleq2d 2363 . . . . . . . . 9  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( 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.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) ) ) )
10972, 108syl5bb 248 . . . . . . . 8  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  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  e.  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) ) ) )
11046, 49, 54, 109syl3anc 1182 . . . . . . 7  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  ( [. ( (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  e.  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) ) ) )
11141, 42, 43, 110sbc3ie 3073 . . . . . 6  |-  ( [. <. (  _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  e.  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) ) )
11240, 111syl6bb 252 . . . . 5  |-  ( w  =  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 >.
) ) ) )  <-> 
a  e.  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) ) ) )
113112abbi1dv 2412 . . . 4  |-  ( w  =  W  ->  { 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 >.
) ) ) ) }  =  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) ) )
114 eqid 2296 . . . 4  |-  ( 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 >.
) ) ) ) } )  =  ( 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 >.
) ) ) ) } )
115 fvex 5555 . . . . . 6  |-  ( Base `  U )  e.  _V
11653, 115eqeltri 2366 . . . . 5  |-  V  e. 
_V
117116mptex 5762 . . . 4  |-  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) )  e.  _V
118113, 114, 117fvmpt 5618 . . 3  |-  ( W  e.  H  ->  (
( 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 >.
) ) ) ) } ) `  W
)  =  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) ) )
1196, 118sylan9eq 2348 . 2  |-  ( ( K  e.  A  /\  W  e.  H )  ->  S  =  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) ) )
1201, 119syl 15 1  |-  ( ph  ->  S  =  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   _Vcvv 2801   [.wsbc 3004    u. cun 3163   {csn 3653   <.cop 3656   <.cotp 3657    e. cmpt 4093    _I cid 4320    |` cres 4707   ` cfv 5271   iota_crio 6313   Basecbs 13164   LSpanclspn 15744   LHypclh 30795   LTrncltrn 30912   DVecHcdvh 31890  LCDualclcd 32398  HVMapchvm 32568  HDMap1chdma1 32604  HDMapchdma 32605
This theorem is referenced by:  hdmapval  32643  hdmapfnN  32644
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-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-pr 4230
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-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-riota 6320  df-hdmap 32607
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