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Theorem hdmap1fval 36469
Description: Preliminary map from vectors to functionals in the closed kernel dual space. TODO: change span  J to the convention  L for this section. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmap1val.h  |-  H  =  ( LHyp `  K
)
hdmap1fval.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap1fval.v  |-  V  =  ( Base `  U
)
hdmap1fval.s  |-  .-  =  ( -g `  U )
hdmap1fval.o  |-  .0.  =  ( 0g `  U )
hdmap1fval.n  |-  N  =  ( LSpan `  U )
hdmap1fval.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap1fval.d  |-  D  =  ( Base `  C
)
hdmap1fval.r  |-  R  =  ( -g `  C
)
hdmap1fval.q  |-  Q  =  ( 0g `  C
)
hdmap1fval.j  |-  J  =  ( LSpan `  C )
hdmap1fval.m  |-  M  =  ( (mapd `  K
) `  W )
hdmap1fval.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmap1fval.k  |-  ( ph  ->  ( K  e.  A  /\  W  e.  H
) )
Assertion
Ref Expression
hdmap1fval  |-  ( ph  ->  I  =  ( x  e.  ( ( V  X.  D )  X.  V )  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) )
Distinct variable groups:    x, h, C    D, h, x    h, J, x    h, M, x   
h, N, x    U, h, x    h, V, x
Allowed substitution hints:    ph( x, h)    A( x, h)    Q( x, h)    R( x, h)    H( x, h)    I( x, h)    K( x, h)    .- ( x, h)    W( x, h)    .0. ( x, h)

Proof of Theorem hdmap1fval
Dummy variables  w  a  c  d  j  m  n  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hdmap1fval.k . 2  |-  ( ph  ->  ( K  e.  A  /\  W  e.  H
) )
2 hdmap1fval.i . . . 4  |-  I  =  ( (HDMap1 `  K
) `  W )
3 hdmap1val.h . . . . . 6  |-  H  =  ( LHyp `  K
)
43hdmap1ffval 36468 . . . . 5  |-  ( K  e.  A  ->  (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 ) } ) ) ) ) ) } ) )
54fveq1d 5859 . . . 4  |-  ( K  e.  A  ->  (
(HDMap1 `  K ) `  W )  =  ( ( 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 ) } ) ) ) ) ) } ) `  W
) )
62, 5syl5eq 2513 . . 3  |-  ( K  e.  A  ->  I  =  ( ( 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 ) } ) ) ) ) ) } ) `  W
) )
7 fveq2 5857 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
8 dfsbcq 3326 . . . . . . . 8  |-  ( ( ( 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 ) } ) ) ) ) ) ) )
97, 8syl 16 . . . . . . 7  |-  ( w  =  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 ) } ) ) ) ) ) ) )
10 fveq2 5857 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (
(LCDual `  K ) `  w )  =  ( (LCDual `  K ) `  W ) )
11 dfsbcq 3326 . . . . . . . . . . . 12  |-  ( ( (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 . . . . . . . . . . 11  |-  ( w  =  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 ) } ) ) ) ) ) ) )
13 fveq2 5857 . . . . . . . . . . . . . . 15  |-  ( w  =  W  ->  (
(mapd `  K ) `  w )  =  ( (mapd `  K ) `  W ) )
14 dfsbcq 3326 . . . . . . . . . . . . . . 15  |-  ( ( (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 ) } ) ) ) ) ) ) )
1513, 14syl 16 . . . . . . . . . . . . . 14  |-  ( w  =  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 ) } ) ) ) ) ) ) )
1615sbcbidv 3383 . . . . . . . . . . . . 13  |-  ( w  =  W  ->  ( [. ( 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 ) } ) ) ) ) ) ) )
1716sbcbidv 3383 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( [. ( 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 ) } ) ) ) ) ) ) )
1817sbcbidv 3383 . . . . . . . . . . 11  |-  ( w  =  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 ) } ) ) ) ) ) ) )
1912, 18bitrd 253 . . . . . . . . . 10  |-  ( w  =  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 ) } ) ) ) ) ) ) )
2019sbcbidv 3383 . . . . . . . . 9  |-  ( w  =  W  ->  ( [. ( 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 ) } ) ) ) ) ) ) )
2120sbcbidv 3383 . . . . . . . 8  |-  ( w  =  W  ->  ( [. ( 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 ) } ) ) ) ) ) ) )
2221sbcbidv 3383 . . . . . . 7  |-  ( w  =  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 ) } ) ) ) ) ) ) )
239, 22bitrd 253 . . . . . 6  |-  ( w  =  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 ) } ) ) ) ) ) ) )
24 fvex 5867 . . . . . . 7  |-  ( (
DVecH `  K ) `  W )  e.  _V
25 fvex 5867 . . . . . . 7  |-  ( Base `  u )  e.  _V
26 fvex 5867 . . . . . . 7  |-  ( LSpan `  u )  e.  _V
27 hdmap1fval.u . . . . . . . . . . 11  |-  U  =  ( ( DVecH `  K
) `  W )
2827eqeq2i 2478 . . . . . . . . . 10  |-  ( u  =  U  <->  u  =  ( ( DVecH `  K
) `  W )
)
2928biimpri 206 . . . . . . . . 9  |-  ( u  =  ( ( DVecH `  K ) `  W
)  ->  u  =  U )
30293ad2ant1 1012 . . . . . . . 8  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  u  =  U )
31 simp2 992 . . . . . . . . . 10  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  v  =  ( Base `  u )
)
3229fveq2d 5861 . . . . . . . . . . 11  |-  ( u  =  ( ( DVecH `  K ) `  W
)  ->  ( Base `  u )  =  (
Base `  U )
)
33323ad2ant1 1012 . . . . . . . . . 10  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  ( Base `  u )  =  (
Base `  U )
)
3431, 33eqtrd 2501 . . . . . . . . 9  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  v  =  ( Base `  U )
)
35 hdmap1fval.v . . . . . . . . 9  |-  V  =  ( Base `  U
)
3634, 35syl6eqr 2519 . . . . . . . 8  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  v  =  V )
37 simp3 993 . . . . . . . . . 10  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  n  =  ( LSpan `  u )
)
3830fveq2d 5861 . . . . . . . . . 10  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  ( LSpan `  u )  =  (
LSpan `  U ) )
3937, 38eqtrd 2501 . . . . . . . . 9  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  n  =  ( LSpan `  U )
)
40 hdmap1fval.n . . . . . . . . 9  |-  N  =  ( LSpan `  U )
4139, 40syl6eqr 2519 . . . . . . . 8  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  n  =  N )
42 fvex 5867 . . . . . . . . . 10  |-  ( (LCDual `  K ) `  W
)  e.  _V
43 fvex 5867 . . . . . . . . . 10  |-  ( Base `  c )  e.  _V
44 fvex 5867 . . . . . . . . . 10  |-  ( LSpan `  c )  e.  _V
45 id 22 . . . . . . . . . . . . 13  |-  ( c  =  ( (LCDual `  K ) `  W
)  ->  c  =  ( (LCDual `  K ) `  W ) )
46 hdmap1fval.c . . . . . . . . . . . . 13  |-  C  =  ( (LCDual `  K
) `  W )
4745, 46syl6eqr 2519 . . . . . . . . . . . 12  |-  ( c  =  ( (LCDual `  K ) `  W
)  ->  c  =  C )
48473ad2ant1 1012 . . . . . . . . . . 11  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  c  =  C )
49 simp2 992 . . . . . . . . . . . 12  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  d  =  (
Base `  c )
)
5048fveq2d 5861 . . . . . . . . . . . . 13  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  ( Base `  c
)  =  ( Base `  C ) )
51 hdmap1fval.d . . . . . . . . . . . . 13  |-  D  =  ( Base `  C
)
5250, 51syl6eqr 2519 . . . . . . . . . . . 12  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  ( Base `  c
)  =  D )
5349, 52eqtrd 2501 . . . . . . . . . . 11  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  d  =  D )
54 simp3 993 . . . . . . . . . . . 12  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  j  =  (
LSpan `  c ) )
5548fveq2d 5861 . . . . . . . . . . . . 13  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  ( LSpan `  c
)  =  ( LSpan `  C ) )
56 hdmap1fval.j . . . . . . . . . . . . 13  |-  J  =  ( LSpan `  C )
5755, 56syl6eqr 2519 . . . . . . . . . . . 12  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  ( LSpan `  c
)  =  J )
5854, 57eqtrd 2501 . . . . . . . . . . 11  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  j  =  J )
59 fvex 5867 . . . . . . . . . . . . 13  |-  ( (mapd `  K ) `  W
)  e.  _V
60 id 22 . . . . . . . . . . . . . . 15  |-  ( m  =  ( (mapd `  K ) `  W
)  ->  m  =  ( (mapd `  K ) `  W ) )
61 hdmap1fval.m . . . . . . . . . . . . . . 15  |-  M  =  ( (mapd `  K
) `  W )
6260, 61syl6eqr 2519 . . . . . . . . . . . . . 14  |-  ( m  =  ( (mapd `  K ) `  W
)  ->  m  =  M )
63 fveq1 5856 . . . . . . . . . . . . . . . . . . . 20  |-  ( m  =  M  ->  (
m `  ( n `  { ( 2nd `  x
) } ) )  =  ( M `  ( n `  {
( 2nd `  x
) } ) ) )
6463eqeq1d 2462 . . . . . . . . . . . . . . . . . . 19  |-  ( m  =  M  ->  (
( m `  (
n `  { ( 2nd `  x ) } ) )  =  ( j `  { h } )  <->  ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( j `  { h } ) ) )
65 fveq1 5856 . . . . . . . . . . . . . . . . . . . 20  |-  ( m  =  M  ->  (
m `  ( n `  { ( ( 1st `  ( 1st `  x
) ) ( -g `  u ) ( 2nd `  x ) ) } ) )  =  ( M `  ( n `
 { ( ( 1st `  ( 1st `  x ) ) (
-g `  u )
( 2nd `  x
) ) } ) ) )
6665eqeq1d 2462 . . . . . . . . . . . . . . . . . . 19  |-  ( m  =  M  ->  (
( m `  (
n `  { (
( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } )  <-> 
( M `  (
n `  { (
( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) )
6764, 66anbi12d 710 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  M  ->  (
( ( m `  ( n `  {
( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( m `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) )  <->  ( ( M `
 ( n `  { ( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) ) ) )
6867riotabidv 6238 . . . . . . . . . . . . . . . . 17  |-  ( m  =  M  ->  ( 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 ) } ) ) )  =  ( 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 ) } ) ) ) )
6968ifeq2d 3951 . . . . . . . . . . . . . . . 16  |-  ( m  =  M  ->  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 ) } ) ) ) )  =  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 ) } ) ) ) ) )
7069mpteq2dv 4527 . . . . . . . . . . . . . . 15  |-  ( m  =  M  ->  (
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 ) } ) ) ) ) )  =  ( 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 ) } ) ) ) ) ) )
7170eleq2d 2530 . . . . . . . . . . . . . 14  |-  ( m  =  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  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 ) } ) ) ) ) ) ) )
7262, 71syl 16 . . . . . . . . . . . . 13  |-  ( m  =  ( (mapd `  K ) `  W
)  ->  ( 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  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 ) } ) ) ) ) ) ) )
7359, 72sbcie 3359 . . . . . . . . . . . 12  |-  ( [. ( (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  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 ) } ) ) ) ) ) )
74 simp2 992 . . . . . . . . . . . . . . 15  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  d  =  D )
75 xpeq2 5007 . . . . . . . . . . . . . . . 16  |-  ( d  =  D  ->  (
v  X.  d )  =  ( v  X.  D ) )
7675xpeq1d 5015 . . . . . . . . . . . . . . 15  |-  ( d  =  D  ->  (
( v  X.  d
)  X.  v )  =  ( ( v  X.  D )  X.  v ) )
7774, 76syl 16 . . . . . . . . . . . . . 14  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( ( v  X.  d )  X.  v
)  =  ( ( v  X.  D )  X.  v ) )
78 simp1 991 . . . . . . . . . . . . . . . . 17  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  c  =  C )
7978fveq2d 5861 . . . . . . . . . . . . . . . 16  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( 0g `  c
)  =  ( 0g
`  C ) )
80 hdmap1fval.q . . . . . . . . . . . . . . . 16  |-  Q  =  ( 0g `  C
)
8179, 80syl6eqr 2519 . . . . . . . . . . . . . . 15  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( 0g `  c
)  =  Q )
82 simp3 993 . . . . . . . . . . . . . . . . . . 19  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  j  =  J )
8382fveq1d 5859 . . . . . . . . . . . . . . . . . 18  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( j `  {
h } )  =  ( J `  {
h } ) )
8483eqeq2d 2474 . . . . . . . . . . . . . . . . 17  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( j `  { h } )  <-> 
( M `  (
n `  { ( 2nd `  x ) } ) )  =  ( J `  { h } ) ) )
8578fveq2d 5861 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( -g `  c
)  =  ( -g `  C ) )
86 hdmap1fval.r . . . . . . . . . . . . . . . . . . . . . 22  |-  R  =  ( -g `  C
)
8785, 86syl6eqr 2519 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( -g `  c
)  =  R )
8887oveqd 6292 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( ( 2nd `  ( 1st `  x ) ) ( -g `  c
) h )  =  ( ( 2nd `  ( 1st `  x ) ) R h ) )
8988sneqd 4032 . . . . . . . . . . . . . . . . . . 19  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  { ( ( 2nd `  ( 1st `  x
) ) ( -g `  c ) h ) }  =  { ( ( 2nd `  ( 1st `  x ) ) R h ) } )
9082, 89fveq12d 5863 . . . . . . . . . . . . . . . . . 18  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( j `  {
( ( 2nd `  ( 1st `  x ) ) ( -g `  c
) h ) } )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) )
9190eqeq2d 2474 . . . . . . . . . . . . . . . . 17  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } )  <-> 
( M `  (
n `  { (
( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) )
9284, 91anbi12d 710 . . . . . . . . . . . . . . . 16  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( ( ( M `
 ( n `  { ( 2nd `  x
) } ) )  =  ( j `  { h } )  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( j `
 { ( ( 2nd `  ( 1st `  x ) ) (
-g `  c )
h ) } ) )  <->  ( ( M `
 ( n `  { ( 2nd `  x
) } ) )  =  ( J `  { h } )  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) )
9374, 92riotaeqbidv 6239 . . . . . . . . . . . . . . 15  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( 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 ) } ) ) )  =  (
iota_ h  e.  D  ( ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( J `  { h } )  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) )
9481, 93ifeq12d 3952 . . . . . . . . . . . . . 14  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  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 ) } ) ) ) )  =  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  Q ,  (
iota_ h  e.  D  ( ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( J `  { h } )  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )
9577, 94mpteq12dv 4518 . . . . . . . . . . . . 13  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( 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 ) } ) ) ) ) )  =  ( x  e.  ( ( v  X.  D )  X.  v
)  |->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  Q ,  ( iota_ h  e.  D  ( ( M `
 ( n `  { ( 2nd `  x
) } ) )  =  ( J `  { h } )  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) )
9695eleq2d 2530 . . . . . . . . . . . 12  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( 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  e.  ( x  e.  ( ( v  X.  D )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( n `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) ) )
9773, 96syl5bb 257 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  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  e.  ( x  e.  ( ( v  X.  D )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( n `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) ) )
9848, 53, 58, 97syl3anc 1223 . . . . . . . . . 10  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  ( [. (
(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  e.  ( x  e.  ( ( v  X.  D )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( n `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) ) )
9942, 43, 44, 98sbc3ie 3402 . . . . . . . . 9  |-  ( [. ( (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  e.  ( x  e.  ( ( v  X.  D )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( n `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) )
100 simp2 992 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  v  =  V )
101100xpeq1d 5015 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( v  X.  D
)  =  ( V  X.  D ) )
102101, 100xpeq12d 5017 . . . . . . . . . . 11  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( v  X.  D )  X.  v
)  =  ( ( V  X.  D )  X.  V ) )
103 simp1 991 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  u  =  U )
104103fveq2d 5861 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( 0g `  u
)  =  ( 0g
`  U ) )
105 hdmap1fval.o . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  U )
106104, 105syl6eqr 2519 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( 0g `  u
)  =  .0.  )
107106eqeq2d 2474 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( 2nd `  x
)  =  ( 0g
`  u )  <->  ( 2nd `  x )  =  .0.  ) )
108 simp3 993 . . . . . . . . . . . . . . . . 17  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  n  =  N )
109108fveq1d 5859 . . . . . . . . . . . . . . . 16  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( n `  {
( 2nd `  x
) } )  =  ( N `  {
( 2nd `  x
) } ) )
110109fveq2d 5861 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( M `  (
n `  { ( 2nd `  x ) } ) )  =  ( M `  ( N `
 { ( 2nd `  x ) } ) ) )
111110eqeq1d 2462 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( J `  { h } )  <-> 
( M `  ( N `  { ( 2nd `  x ) } ) )  =  ( J `  { h } ) ) )
112103fveq2d 5861 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( -g `  u
)  =  ( -g `  U ) )
113 hdmap1fval.s . . . . . . . . . . . . . . . . . . . 20  |-  .-  =  ( -g `  U )
114112, 113syl6eqr 2519 . . . . . . . . . . . . . . . . . . 19  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( -g `  u
)  =  .-  )
115114oveqd 6292 . . . . . . . . . . . . . . . . . 18  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) )  =  ( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) )
116115sneqd 4032 . . . . . . . . . . . . . . . . 17  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  { ( ( 1st `  ( 1st `  x
) ) ( -g `  u ) ( 2nd `  x ) ) }  =  { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) } )
117108, 116fveq12d 5863 . . . . . . . . . . . . . . . 16  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } )  =  ( N `  { ( ( 1st `  ( 1st `  x
) )  .-  ( 2nd `  x ) ) } ) )
118117fveq2d 5861 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( M `  (
n `  { (
( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( M `
 ( N `  { ( ( 1st `  ( 1st `  x
) )  .-  ( 2nd `  x ) ) } ) ) )
119118eqeq1d 2462 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } )  <-> 
( M `  ( N `  { (
( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) )
120111, 119anbi12d 710 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( ( M `
 ( n `  { ( 2nd `  x
) } ) )  =  ( J `  { h } )  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) )  <->  ( ( M `
 ( N `  { ( 2nd `  x
) } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) )
121120riotabidv 6238 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( iota_ h  e.  D  ( ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( J `  { h } )  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) )  =  (
iota_ h  e.  D  ( ( M `  ( N `  { ( 2nd `  x ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  x
) ) R h ) } ) ) ) )
122107, 121ifbieq2d 3957 . . . . . . . . . . 11  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  Q ,  (
iota_ h  e.  D  ( ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( J `  { h } )  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) )  =  if ( ( 2nd `  x )  =  .0. 
,  Q ,  (
iota_ h  e.  D  ( ( M `  ( N `  { ( 2nd `  x ) } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  x
) ) R h ) } ) ) ) ) )
123102, 122mpteq12dv 4518 . . . . . . . . . 10  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( x  e.  ( ( v  X.  D
)  X.  v ) 
|->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  Q ,  (
iota_ h  e.  D  ( ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( J `  { h } )  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )  =  ( x  e.  ( ( V  X.  D )  X.  V
)  |->  if ( ( 2nd `  x )  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `
 ( N `  { ( 2nd `  x
) } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) )
124123eleq2d 2530 . . . . . . . . 9  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( a  e.  ( x  e.  ( ( v  X.  D )  X.  v )  |->  if ( ( 2nd `  x
)  =  ( 0g
`  u ) ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( n `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )  <-> 
a  e.  ( x  e.  ( ( V  X.  D )  X.  V )  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) ) )
12599, 124syl5bb 257 . . . . . . . 8  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  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  e.  ( x  e.  ( ( V  X.  D )  X.  V )  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) ) )
12630, 36, 41, 125syl3anc 1223 . . . . . . 7  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  ( [. ( (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  e.  ( x  e.  ( ( V  X.  D )  X.  V )  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) ) )
12724, 25, 26, 126sbc3ie 3402 . . . . . 6  |-  ( [. ( ( 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  e.  ( x  e.  ( ( V  X.  D )  X.  V )  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) )
12823, 127syl6bb 261 . . . . 5  |-  ( w  =  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 ) } ) ) ) ) )  <-> 
a  e.  ( x  e.  ( ( V  X.  D )  X.  V )  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) ) )
129128abbi1dv 2598 . . . 4  |-  ( w  =  W  ->  { 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 ) } ) ) ) ) ) }  =  ( x  e.  ( ( V  X.  D )  X.  V )  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) )
130 eqid 2460 . . . 4  |-  ( 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 ) } ) ) ) ) ) } )  =  ( 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 ) } ) ) ) ) ) } )
131 fvex 5867 . . . . . . . 8  |-  ( Base `  U )  e.  _V
13235, 131eqeltri 2544 . . . . . . 7  |-  V  e. 
_V
133 fvex 5867 . . . . . . . 8  |-  ( Base `  C )  e.  _V
13451, 133eqeltri 2544 . . . . . . 7  |-  D  e. 
_V
135132, 134xpex 6704 . . . . . 6  |-  ( V  X.  D )  e. 
_V
136135, 132xpex 6704 . . . . 5  |-  ( ( V  X.  D )  X.  V )  e. 
_V
137136mptex 6122 . . . 4  |-  ( x  e.  ( ( V  X.  D )  X.  V )  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) )  e.  _V
138129, 130, 137fvmpt 5941 . . 3  |-  ( W  e.  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 ) } ) ) ) ) ) } ) `  W
)  =  ( x  e.  ( ( V  X.  D )  X.  V )  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) )
1396, 138sylan9eq 2521 . 2  |-  ( ( K  e.  A  /\  W  e.  H )  ->  I  =  ( x  e.  ( ( V  X.  D )  X.  V )  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) )
1401, 139syl 16 1  |-  ( ph  ->  I  =  ( x  e.  ( ( V  X.  D )  X.  V )  |->  if ( ( 2nd `  x
)  =  .0.  ,  Q ,  ( iota_ h  e.  D  ( ( M `  ( N `
 { ( 2nd `  x ) } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) } ) )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   {cab 2445   _Vcvv 3106   [.wsbc 3324   ifcif 3932   {csn 4020    |-> cmpt 4498    X. cxp 4990   ` cfv 5579   iota_crio 6235  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   Basecbs 14479   0gc0g 14684   -gcsg 15719   LSpanclspn 17393   LHypclh 34655   DVecHcdvh 35750  LCDualclcd 36258  mapdcmpd 36296  HDMap1chdma1 36464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-hdmap1 36466
This theorem is referenced by:  hdmap1vallem  36470
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