Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmapfval Structured version   Unicode version

Theorem hdmapfval 35107
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 35106 . . . . 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 5883 . . . 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 2482 . . 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 5881 . . . . . . . . 9  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
87reseq2d 5125 . . . . . . . 8  |-  ( w  =  W  ->  (  _I  |`  ( ( LTrn `  K ) `  w
) )  =  (  _I  |`  ( ( LTrn `  K ) `  W ) ) )
98opeq2d 4197 . . . . . . 7  |-  ( w  =  W  ->  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  w ) ) >.  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
)
10 fveq2 5881 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
11 fveq2 5881 . . . . . . . . . 10  |-  ( w  =  W  ->  (
(HDMap1 `  K ) `  w )  =  ( (HDMap1 `  K ) `  W ) )
12 fveq2 5881 . . . . . . . . . . . . . 14  |-  ( w  =  W  ->  (
(LCDual `  K ) `  w )  =  ( (LCDual `  K ) `  W ) )
1312fveq2d 5885 . . . . . . . . . . . . 13  |-  ( w  =  W  ->  ( Base `  ( (LCDual `  K ) `  w
) )  =  (
Base `  ( (LCDual `  K ) `  W
) ) )
14 fveq2 5881 . . . . . . . . . . . . . . . . . . . . 21  |-  ( w  =  W  ->  (
(HVMap `  K ) `  w )  =  ( (HVMap `  K ) `  W ) )
1514fveq1d 5883 . . . . . . . . . . . . . . . . . . . 20  |-  ( w  =  W  ->  (
( (HVMap `  K
) `  w ) `  e )  =  ( ( (HVMap `  K
) `  W ) `  e ) )
1615oteq2d 4203 . . . . . . . . . . . . . . . . . . 19  |-  ( w  =  W  ->  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >.  =  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. )
1716fveq2d 5885 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  W  ->  (
i `  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >. )  =  ( i `  <. e ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.
) )
1817oteq2d 4203 . . . . . . . . . . . . . . . . 17  |-  ( w  =  W  ->  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e
) ,  z >.
) ,  t >.  =  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. ) ,  t >. )
1918fveq2d 5885 . . . . . . . . . . . . . . . 16  |-  ( w  =  W  ->  (
i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >. ) ,  t >. )  =  ( i `  <. z ,  ( i `
 <. e ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. ) ,  t >.
) )
2019eqeq2d 2443 . . . . . . . . . . . . . . 15  |-  ( w  =  W  ->  (
y  =  ( i `
 <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w
) `  e ) ,  z >. ) ,  t >. )  <->  y  =  ( i `  <. z ,  ( i `
 <. e ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. ) ,  t >.
) ) )
2120imbi2d 317 . . . . . . . . . . . . . 14  |-  ( 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 >. )
) ) )
2221ralbidv 2871 . . . . . . . . . . . . 13  |-  ( 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 >. )
) ) )
2313, 22riotaeqbidv 6270 . . . . . . . . . . . 12  |-  ( 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 >.
) ) ) )
2423mpteq2dv 4513 . . . . . . . . . . 11  |-  ( 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 >.
) ) ) ) )
2524eleq2d 2499 . . . . . . . . . 10  |-  ( 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 >.
) ) ) ) ) )
2611, 25sbceqbid 3312 . . . . . . . . 9  |-  ( 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 >.
) ) ) ) ) )
2726sbcbidv 3360 . . . . . . . 8  |-  ( 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 >.
) ) ) ) ) )
2810, 27sbceqbid 3312 . . . . . . 7  |-  ( 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 >.
) ) ) ) ) )
299, 28sbceqbid 3312 . . . . . 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 >.
) ) ) ) ) )
30 opex 4686 . . . . . . 7  |-  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  e.  _V
31 fvex 5891 . . . . . . 7  |-  ( (
DVecH `  K ) `  W )  e.  _V
32 fvex 5891 . . . . . . 7  |-  ( Base `  u )  e.  _V
33 simp1 1005 . . . . . . . . 9  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  e  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
)
34 hdmapfval.e . . . . . . . . 9  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
3533, 34syl6eqr 2488 . . . . . . . 8  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  e  =  E )
36 simp2 1006 . . . . . . . . 9  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  u  =  ( ( DVecH `  K ) `  W
) )
37 hdmapfval.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
3836, 37syl6eqr 2488 . . . . . . . 8  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  u  =  U )
39 simp3 1007 . . . . . . . . . 10  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  v  =  ( Base `  u
) )
4038fveq2d 5885 . . . . . . . . . 10  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  ( Base `  u )  =  ( Base `  U
) )
4139, 40eqtrd 2470 . . . . . . . . 9  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  v  =  ( Base `  U
) )
42 hdmapfval.v . . . . . . . . 9  |-  V  =  ( Base `  U
)
4341, 42syl6eqr 2488 . . . . . . . 8  |-  ( ( e  =  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  /\  u  =  (
( DVecH `  K ) `  W )  /\  v  =  ( Base `  u
) )  ->  v  =  V )
44 fvex 5891 . . . . . . . . . 10  |-  ( (HDMap1 `  K ) `  W
)  e.  _V
45 id 23 . . . . . . . . . . . 12  |-  ( i  =  ( (HDMap1 `  K ) `  W
)  ->  i  =  ( (HDMap1 `  K ) `  W ) )
46 hdmapfval.i . . . . . . . . . . . 12  |-  I  =  ( (HDMap1 `  K
) `  W )
4745, 46syl6eqr 2488 . . . . . . . . . . 11  |-  ( i  =  ( (HDMap1 `  K ) `  W
)  ->  i  =  I )
48 fveq1 5880 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  I  ->  (
i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. ) ,  t >. )  =  ( I `  <. z ,  ( i `
 <. e ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. ) ,  t >.
) )
49 fveq1 5880 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  I  ->  (
i `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. )  =  ( I `  <. e ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.
) )
5049oteq2d 4203 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  I  ->  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.
) ,  t >.  =  <. z ,  ( I `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. ) ,  t >. )
5150fveq2d 5885 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  I  ->  (
I `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. ) ,  t >. )  =  ( I `  <. z ,  ( I `
 <. e ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. ) ,  t >.
) )
5248, 51eqtrd 2470 . . . . . . . . . . . . . . . . 17  |-  ( i  =  I  ->  (
i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. ) ,  t >. )  =  ( I `  <. z ,  ( I `
 <. e ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. ) ,  t >.
) )
5352eqeq2d 2443 . . . . . . . . . . . . . . . 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 >.
) ) )
5453imbi2d 317 . . . . . . . . . . . . . . 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 >. )
) ) )
5554ralbidv 2871 . . . . . . . . . . . . . 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 >. )
) ) )
5655riotabidv 6269 . . . . . . . . . . . . 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 >.
) ) ) )
5756mpteq2dv 4513 . . . . . . . . . . . 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 >.
) ) ) ) )
5857eleq2d 2499 . . . . . . . . . . 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 >.
) ) ) ) ) )
5947, 58syl 17 . . . . . . . . . 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 >.
) ) ) ) ) )
6044, 59sbcie 3340 . . . . . . . . 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 >.
) ) ) ) )
61 simp3 1007 . . . . . . . . . . 11  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  v  =  V )
62 hdmapfval.d . . . . . . . . . . . . . 14  |-  D  =  ( Base `  C
)
63 hdmapfval.c . . . . . . . . . . . . . . 15  |-  C  =  ( (LCDual `  K
) `  W )
6463fveq2i 5884 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  ( (LCDual `  K ) `  W
) )
6562, 64eqtr2i 2459 . . . . . . . . . . . . 13  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  D
6665a1i 11 . . . . . . . . . . . 12  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( Base `  (
(LCDual `  K ) `  W ) )  =  D )
67 simp2 1006 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  u  =  U )
6867fveq2d 5885 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( LSpan `  u )  =  ( LSpan `  U
) )
69 hdmapfval.n . . . . . . . . . . . . . . . . . . 19  |-  N  =  ( LSpan `  U )
7068, 69syl6eqr 2488 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( LSpan `  u )  =  N )
71 simp1 1005 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  e  =  E )
7271sneqd 4014 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  { e }  =  { E } )
7370, 72fveq12d 5887 . . . . . . . . . . . . . . . . 17  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( ( LSpan `  u
) `  { e } )  =  ( N `  { E } ) )
7470fveq1d 5883 . . . . . . . . . . . . . . . . 17  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( ( LSpan `  u
) `  { t } )  =  ( N `  { t } ) )
7573, 74uneq12d 3627 . . . . . . . . . . . . . . . 16  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( ( ( LSpan `  u ) `  {
e } )  u.  ( ( LSpan `  u
) `  { t } ) )  =  ( ( N `  { E } )  u.  ( N `  {
t } ) ) )
7675eleq2d 2499 . . . . . . . . . . . . . . 15  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( z  e.  ( ( ( LSpan `  u
) `  { e } )  u.  (
( LSpan `  u ) `  { t } ) )  <->  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) ) ) )
7776notbid 295 . . . . . . . . . . . . . 14  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( -.  z  e.  ( ( ( LSpan `  u ) `  {
e } )  u.  ( ( LSpan `  u
) `  { t } ) )  <->  -.  z  e.  ( ( N `  { E } )  u.  ( N `  {
t } ) ) ) )
7871oteq1d 4202 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  -> 
<. e ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.  =  <. E ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. )
7971fveq2d 5885 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( ( (HVMap `  K ) `  W
) `  e )  =  ( ( (HVMap `  K ) `  W
) `  E )
)
80 hdmapfval.j . . . . . . . . . . . . . . . . . . . . . 22  |-  J  =  ( (HVMap `  K
) `  W )
8180fveq1i 5882 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J `
 E )  =  ( ( (HVMap `  K ) `  W
) `  E )
8279, 81syl6eqr 2488 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( ( (HVMap `  K ) `  W
) `  e )  =  ( J `  E ) )
8382oteq2d 4203 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  -> 
<. E ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.  =  <. E ,  ( J `  E ) ,  z >. )
8478, 83eqtrd 2470 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  -> 
<. e ,  ( ( (HVMap `  K ) `  W ) `  e
) ,  z >.  =  <. E ,  ( J `  E ) ,  z >. )
8584fveq2d 5885 . . . . . . . . . . . . . . . . 17  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  ->  ( I `  <. e ,  ( ( (HVMap `  K ) `  W
) `  e ) ,  z >. )  =  ( I `  <. E ,  ( J `
 E ) ,  z >. ) )
8685oteq2d 4203 . . . . . . . . . . . . . . . 16  |-  ( ( e  =  E  /\  u  =  U  /\  v  =  V )  -> 
<. z ,  ( I `
 <. e ,  ( ( (HVMap `  K
) `  W ) `  e ) ,  z
>. ) ,  t >.  =  <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
)
8786fveq2d 5885 . . . . . . . . . . . . . . 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 >.
) )
8887eqeq2d 2443 . . . . . . . . . . . . . 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
>. ) ) )
8977, 88imbi12d 321 . . . . . . . . . . . . 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
>. ) ) ) )
9061, 89raleqbidv 3046 . . . . . . . . . . . 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
>. ) ) ) )
9166, 90riotaeqbidv 6270 . . . . . . . . . . 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
>. ) ) ) )
9261, 91mpteq12dv 4504 . . . . . . . . . 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
>. ) ) ) ) )
9392eleq2d 2499 . . . . . . . . 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 >.
) ) ) ) ) )
9460, 93syl5bb 260 . . . . . . . 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 >.
) ) ) ) ) )
9535, 38, 43, 94syl3anc 1264 . . . . . . 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 >.
) ) ) ) ) )
9630, 31, 32, 95sbc3ie 3375 . . . . . 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 >.
) ) ) ) )
9729, 96syl6bb 264 . . . . 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 >.
) ) ) ) ) )
9897abbi1dv 2567 . . . 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 >.
) ) ) ) )
99 eqid 2429 . . . 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 >.
) ) ) ) } )
100 fvex 5891 . . . . . 6  |-  ( Base `  U )  e.  _V
10142, 100eqeltri 2513 . . . . 5  |-  V  e. 
_V
102101mptex 6151 . . . 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
10398, 99, 102fvmpt 5964 . . 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 >.
) ) ) ) )
1046, 103sylan9eq 2490 . 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 >.
) ) ) ) )
1051, 104syl 17 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   {cab 2414   A.wral 2782   _Vcvv 3087   [.wsbc 3305    u. cun 3440   {csn 4002   <.cop 4008   <.cotp 4010    |-> cmpt 4484    _I cid 4764    |` cres 4856   ` cfv 5601   iota_crio 6266   Basecbs 15084   LSpanclspn 18129   LHypclh 33258   LTrncltrn 33375   DVecHcdvh 34355  LCDualclcd 34863  HVMapchvm 35033  HDMap1chdma1 35069  HDMapchdma 35070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-ot 4011  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-hdmap 35072
This theorem is referenced by:  hdmapval  35108  hdmapfnN  35109
  Copyright terms: Public domain W3C validator