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Theorem hdmap1ffval 31912
Description: Preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 14-May-2015.)
Hypothesis
Ref Expression
hdmap1val.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
hdmap1ffval  |-  ( K  e.  X  ->  (HDMap1 `  K )  =  ( w  e.  H  |->  { a  |  [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( LSpan `  u )  /  n ]. [. (
(LCDual `  K ) `  w )  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  K ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) } ) )
Distinct variable groups:    w, H    a, c, d, j, m, n, u, v, w, K    h, a, x, c, d, j, m, n, u, v, w
Allowed substitution hints:    H( x, v, u, h, j, m, n, a, c, d)    K( x, h)    X( x, w, v, u, h, j, m, n, a, c, d)

Proof of Theorem hdmap1ffval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2908 . 2  |-  ( K  e.  X  ->  K  e.  _V )
2 fveq2 5669 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 hdmap1val.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2438 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5669 . . . . . . . 8  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
65fveq1d 5671 . . . . . . 7  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
7 dfsbcq 3107 . . . . . . 7  |-  ( ( ( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w )  ->  ( [. ( ( DVecH `  k
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( LSpan `  u )  /  n ]. [. ( (LCDual `  k ) `  w
)  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  k ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( LSpan `  u )  /  n ]. [. ( (LCDual `  k ) `  w
)  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  k ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
86, 7syl 16 . . . . . 6  |-  ( k  =  K  ->  ( [. ( ( DVecH `  k
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( LSpan `  u )  /  n ]. [. ( (LCDual `  k ) `  w
)  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  k ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( LSpan `  u )  /  n ]. [. ( (LCDual `  k ) `  w
)  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  k ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
9 fveq2 5669 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (LCDual `  k )  =  (LCDual `  K ) )
109fveq1d 5671 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
(LCDual `  k ) `  w )  =  ( (LCDual `  K ) `  w ) )
11 dfsbcq 3107 . . . . . . . . . . 11  |-  ( ( (LCDual `  k ) `  w )  =  ( (LCDual `  K ) `  w )  ->  ( [. ( (LCDual `  k
) `  w )  /  c ]. [. ( Base `  c )  / 
d ]. [. ( LSpan `  c )  /  j ]. [. ( (mapd `  k ) `  w
)  /  m ]. a  e.  ( x  e.  ( ( v  X.  d )  X.  v
)  |->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  ( iota_ h  e.  d ( ( m `
 ( n `  { ( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( (LCDual `  K
) `  w )  /  c ]. [. ( Base `  c )  / 
d ]. [. ( LSpan `  c )  /  j ]. [. ( (mapd `  k ) `  w
)  /  m ]. a  e.  ( x  e.  ( ( v  X.  d )  X.  v
)  |->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  ( iota_ h  e.  d ( ( m `
 ( n `  { ( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
1210, 11syl 16 . . . . . . . . . 10  |-  ( k  =  K  ->  ( [. ( (LCDual `  k
) `  w )  /  c ]. [. ( Base `  c )  / 
d ]. [. ( LSpan `  c )  /  j ]. [. ( (mapd `  k ) `  w
)  /  m ]. a  e.  ( x  e.  ( ( v  X.  d )  X.  v
)  |->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  ( iota_ h  e.  d ( ( m `
 ( n `  { ( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( (LCDual `  K
) `  w )  /  c ]. [. ( Base `  c )  / 
d ]. [. ( LSpan `  c )  /  j ]. [. ( (mapd `  k ) `  w
)  /  m ]. a  e.  ( x  e.  ( ( v  X.  d )  X.  v
)  |->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  ( iota_ h  e.  d ( ( m `
 ( n `  { ( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
13 fveq2 5669 . . . . . . . . . . . . . . 15  |-  ( k  =  K  ->  (mapd `  k )  =  (mapd `  K ) )
1413fveq1d 5671 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  (
(mapd `  k ) `  w )  =  ( (mapd `  K ) `  w ) )
15 dfsbcq 3107 . . . . . . . . . . . . . 14  |-  ( ( (mapd `  k ) `  w )  =  ( (mapd `  K ) `  w )  ->  ( [. ( (mapd `  k
) `  w )  /  m ]. a  e.  ( x  e.  ( ( v  X.  d
)  X.  v ) 
|->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  (
iota_ h  e.  d
( ( m `  ( n `  {
( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( (mapd `  K
) `  w )  /  m ]. a  e.  ( x  e.  ( ( v  X.  d
)  X.  v ) 
|->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  (
iota_ h  e.  d
( ( m `  ( n `  {
( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
1614, 15syl 16 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( [. ( (mapd `  k
) `  w )  /  m ]. a  e.  ( x  e.  ( ( v  X.  d
)  X.  v ) 
|->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  (
iota_ h  e.  d
( ( m `  ( n `  {
( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( (mapd `  K
) `  w )  /  m ]. a  e.  ( x  e.  ( ( v  X.  d
)  X.  v ) 
|->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  (
iota_ h  e.  d
( ( m `  ( n `  {
( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
1716sbcbidv 3159 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( [. ( LSpan `  c )  /  j ]. [. (
(mapd `  k ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( LSpan `  c )  /  j ]. [. (
(mapd `  K ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
1817sbcbidv 3159 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  k ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  K ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
1918sbcbidv 3159 . . . . . . . . . 10  |-  ( k  =  K  ->  ( [. ( (LCDual `  K
) `  w )  /  c ]. [. ( Base `  c )  / 
d ]. [. ( LSpan `  c )  /  j ]. [. ( (mapd `  k ) `  w
)  /  m ]. a  e.  ( x  e.  ( ( v  X.  d )  X.  v
)  |->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  ( iota_ h  e.  d ( ( m `
 ( n `  { ( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( (LCDual `  K
) `  w )  /  c ]. [. ( Base `  c )  / 
d ]. [. ( LSpan `  c )  /  j ]. [. ( (mapd `  K ) `  w
)  /  m ]. a  e.  ( x  e.  ( ( v  X.  d )  X.  v
)  |->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  ( iota_ h  e.  d ( ( m `
 ( n `  { ( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
2012, 19bitrd 245 . . . . . . . . 9  |-  ( k  =  K  ->  ( [. ( (LCDual `  k
) `  w )  /  c ]. [. ( Base `  c )  / 
d ]. [. ( LSpan `  c )  /  j ]. [. ( (mapd `  k ) `  w
)  /  m ]. a  e.  ( x  e.  ( ( v  X.  d )  X.  v
)  |->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  ( iota_ h  e.  d ( ( m `
 ( n `  { ( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( (LCDual `  K
) `  w )  /  c ]. [. ( Base `  c )  / 
d ]. [. ( LSpan `  c )  /  j ]. [. ( (mapd `  K ) `  w
)  /  m ]. a  e.  ( x  e.  ( ( v  X.  d )  X.  v
)  |->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  ( iota_ h  e.  d ( ( m `
 ( n `  { ( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
2120sbcbidv 3159 . . . . . . . 8  |-  ( k  =  K  ->  ( [. ( LSpan `  u )  /  n ]. [. (
(LCDual `  k ) `  w )  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  k ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( LSpan `  u )  /  n ]. [. (
(LCDual `  K ) `  w )  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  K ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
2221sbcbidv 3159 . . . . . . 7  |-  ( k  =  K  ->  ( [. ( Base `  u
)  /  v ]. [. ( LSpan `  u )  /  n ]. [. (
(LCDual `  k ) `  w )  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  k ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( Base `  u
)  /  v ]. [. ( LSpan `  u )  /  n ]. [. (
(LCDual `  K ) `  w )  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  K ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
2322sbcbidv 3159 . . . . . 6  |-  ( k  =  K  ->  ( [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( LSpan `  u )  /  n ]. [. ( (LCDual `  k ) `  w
)  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  k ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( LSpan `  u )  /  n ]. [. ( (LCDual `  K ) `  w
)  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  K ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
248, 23bitrd 245 . . . . 5  |-  ( k  =  K  ->  ( [. ( ( DVecH `  k
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( LSpan `  u )  /  n ]. [. ( (LCDual `  k ) `  w
)  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  k ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) )  <->  [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  u )  / 
v ]. [. ( LSpan `  u )  /  n ]. [. ( (LCDual `  K ) `  w
)  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  K ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) ) )
2524abbidv 2502 . . . 4  |-  ( k  =  K  ->  { a  |  [. ( (
DVecH `  k ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( LSpan `  u )  /  n ]. [. (
(LCDual `  k ) `  w )  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  k ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) }  =  { a  |  [. ( (
DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( LSpan `  u )  /  n ]. [. (
(LCDual `  K ) `  w )  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  K ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) } )
264, 25mpteq12dv 4229 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  { a  |  [. ( (
DVecH `  k ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( LSpan `  u )  /  n ]. [. (
(LCDual `  k ) `  w )  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  k ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) } )  =  ( w  e.  H  |->  { a  |  [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( LSpan `  u )  /  n ]. [. (
(LCDual `  K ) `  w )  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  K ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) } ) )
27 df-hdmap1 31910 . . 3  |- HDMap1  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { a  |  [. ( (
DVecH `  k ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( LSpan `  u )  /  n ]. [. (
(LCDual `  k ) `  w )  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  k ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) } ) )
28 fvex 5683 . . . . 5  |-  ( LHyp `  K )  e.  _V
293, 28eqeltri 2458 . . . 4  |-  H  e. 
_V
3029mptex 5906 . . 3  |-  ( w  e.  H  |->  { a  |  [. ( (
DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( LSpan `  u )  /  n ]. [. (
(LCDual `  K ) `  w )  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  K ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) } )  e.  _V
3126, 27, 30fvmpt 5746 . 2  |-  ( K  e.  _V  ->  (HDMap1 `  K )  =  ( w  e.  H  |->  { a  |  [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( LSpan `  u )  /  n ]. [. (
(LCDual `  K ) `  w )  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  K ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) } ) )
321, 31syl 16 1  |-  ( K  e.  X  ->  (HDMap1 `  K )  =  ( w  e.  H  |->  { a  |  [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  u
)  /  v ]. [. ( LSpan `  u )  /  n ]. [. (
(LCDual `  K ) `  w )  /  c ]. [. ( Base `  c
)  /  d ]. [. ( LSpan `  c )  /  j ]. [. (
(mapd `  K ) `  w )  /  m ]. a  e.  (
x  e.  ( ( v  X.  d )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  ( 0g `  c
) ,  ( iota_ h  e.  d ( ( m `  ( n `
 { ( 2nd `  x ) } ) )  =  ( j `
 { h }
)  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) ) ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2374   _Vcvv 2900   [.wsbc 3105   ifcif 3683   {csn 3758    e. cmpt 4208    X. cxp 4817   ` cfv 5395  (class class class)co 6021   1stc1st 6287   2ndc2nd 6288   iota_crio 6479   Basecbs 13397   0gc0g 13651   -gcsg 14616   LSpanclspn 15975   LHypclh 30099   DVecHcdvh 31194  LCDualclcd 31702  mapdcmpd 31740  HDMap1chdma1 31908
This theorem is referenced by:  hdmap1fval  31913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-hdmap1 31910
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