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Theorem hdmap1ffval 32608
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 2809 . 2  |-  ( K  e.  X  ->  K  e.  _V )
2 fveq2 5541 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 hdmap1val.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2346 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5541 . . . . . . . 8  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
65fveq1d 5543 . . . . . . 7  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
7 dfsbcq 3006 . . . . . . 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 15 . . . . . 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 5541 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (LCDual `  k )  =  (LCDual `  K ) )
109fveq1d 5543 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
(LCDual `  k ) `  w )  =  ( (LCDual `  K ) `  w ) )
11 dfsbcq 3006 . . . . . . . . . . 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 15 . . . . . . . . . 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 5541 . . . . . . . . . . . . . . 15  |-  ( k  =  K  ->  (mapd `  k )  =  (mapd `  K ) )
1413fveq1d 5543 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  (
(mapd `  k ) `  w )  =  ( (mapd `  K ) `  w ) )
15 dfsbcq 3006 . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . 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 3058 . . . . . . . . . . . 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 3058 . . . . . . . . . . 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 3058 . . . . . . . . . 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 244 . . . . . . . . 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 3058 . . . . . . . 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 3058 . . . . . . 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 3058 . . . . . 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 244 . . . . 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 2410 . . . 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 4114 . . 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 32606 . . 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 5555 . . . . 5  |-  ( LHyp `  K )  e.  _V
293, 28eqeltri 2366 . . . 4  |-  H  e. 
_V
3029mptex 5762 . . 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 5618 . 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 15 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   [.wsbc 3004   ifcif 3578   {csn 3653    e. cmpt 4093    X. cxp 4703   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   iota_crio 6313   Basecbs 13164   0gc0g 13416   -gcsg 14381   LSpanclspn 15744   LHypclh 30795   DVecHcdvh 31890  LCDualclcd 32398  mapdcmpd 32436  HDMap1chdma1 32604
This theorem is referenced by:  hdmap1fval  32609
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-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-hdmap1 32606
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