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Theorem hdmap1fval 32609
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 32608 . . . . 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 5543 . . . 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 2340 . . 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 5541 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
8 dfsbcq 3006 . . . . . . . 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 15 . . . . . . 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 5541 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (
(LCDual `  K ) `  w )  =  ( (LCDual `  K ) `  W ) )
11 dfsbcq 3006 . . . . . . . . . . . 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 15 . . . . . . . . . . 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 5541 . . . . . . . . . . . . . . 15  |-  ( w  =  W  ->  (
(mapd `  K ) `  w )  =  ( (mapd `  K ) `  W ) )
14 dfsbcq 3006 . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . 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 3058 . . . . . . . . . . . . 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 3058 . . . . . . . . . . . 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 3058 . . . . . . . . . . 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 244 . . . . . . . . . 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 3058 . . . . . . . . 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 3058 . . . . . . . 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 3058 . . . . . . 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 244 . . . . . 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 5555 . . . . . . 7  |-  ( (
DVecH `  K ) `  W )  e.  _V
25 fvex 5555 . . . . . . 7  |-  ( Base `  u )  e.  _V
26 fvex 5555 . . . . . . 7  |-  ( LSpan `  u )  e.  _V
27 hdmap1fval.u . . . . . . . . . . 11  |-  U  =  ( ( DVecH `  K
) `  W )
2827eqeq2i 2306 . . . . . . . . . 10  |-  ( u  =  U  <->  u  =  ( ( DVecH `  K
) `  W )
)
2928biimpri 197 . . . . . . . . 9  |-  ( u  =  ( ( DVecH `  K ) `  W
)  ->  u  =  U )
30293ad2ant1 976 . . . . . . . 8  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  u  =  U )
31 simp2 956 . . . . . . . . . 10  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  v  =  ( Base `  u )
)
3229fveq2d 5545 . . . . . . . . . . 11  |-  ( u  =  ( ( DVecH `  K ) `  W
)  ->  ( Base `  u )  =  (
Base `  U )
)
33323ad2ant1 976 . . . . . . . . . 10  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  ( Base `  u )  =  (
Base `  U )
)
3431, 33eqtrd 2328 . . . . . . . . 9  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  v  =  ( Base `  U )
)
35 hdmap1fval.v . . . . . . . . 9  |-  V  =  ( Base `  U
)
3634, 35syl6eqr 2346 . . . . . . . 8  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  v  =  V )
37 simp3 957 . . . . . . . . . 10  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  n  =  ( LSpan `  u )
)
3830fveq2d 5545 . . . . . . . . . 10  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  ( LSpan `  u )  =  (
LSpan `  U ) )
3937, 38eqtrd 2328 . . . . . . . . 9  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  n  =  ( LSpan `  U )
)
40 hdmap1fval.n . . . . . . . . 9  |-  N  =  ( LSpan `  U )
4139, 40syl6eqr 2346 . . . . . . . 8  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  n  =  N )
42 fvex 5555 . . . . . . . . . 10  |-  ( (LCDual `  K ) `  W
)  e.  _V
43 fvex 5555 . . . . . . . . . 10  |-  ( Base `  c )  e.  _V
44 fvex 5555 . . . . . . . . . 10  |-  ( LSpan `  c )  e.  _V
45 id 19 . . . . . . . . . . . . 13  |-  ( c  =  ( (LCDual `  K ) `  W
)  ->  c  =  ( (LCDual `  K ) `  W ) )
46 hdmap1fval.c . . . . . . . . . . . . 13  |-  C  =  ( (LCDual `  K
) `  W )
4745, 46syl6eqr 2346 . . . . . . . . . . . 12  |-  ( c  =  ( (LCDual `  K ) `  W
)  ->  c  =  C )
48473ad2ant1 976 . . . . . . . . . . 11  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  c  =  C )
49 simp2 956 . . . . . . . . . . . 12  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  d  =  (
Base `  c )
)
5048fveq2d 5545 . . . . . . . . . . . . 13  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  ( Base `  c
)  =  ( Base `  C ) )
51 hdmap1fval.d . . . . . . . . . . . . 13  |-  D  =  ( Base `  C
)
5250, 51syl6eqr 2346 . . . . . . . . . . . 12  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  ( Base `  c
)  =  D )
5349, 52eqtrd 2328 . . . . . . . . . . 11  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  d  =  D )
54 simp3 957 . . . . . . . . . . . 12  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  j  =  (
LSpan `  c ) )
5548fveq2d 5545 . . . . . . . . . . . . 13  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  ( LSpan `  c
)  =  ( LSpan `  C ) )
56 hdmap1fval.j . . . . . . . . . . . . 13  |-  J  =  ( LSpan `  C )
5755, 56syl6eqr 2346 . . . . . . . . . . . 12  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  ( LSpan `  c
)  =  J )
5854, 57eqtrd 2328 . . . . . . . . . . 11  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  j  =  J )
59 fvex 5555 . . . . . . . . . . . . 13  |-  ( (mapd `  K ) `  W
)  e.  _V
60 id 19 . . . . . . . . . . . . . . 15  |-  ( m  =  ( (mapd `  K ) `  W
)  ->  m  =  ( (mapd `  K ) `  W ) )
61 hdmap1fval.m . . . . . . . . . . . . . . 15  |-  M  =  ( (mapd `  K
) `  W )
6260, 61syl6eqr 2346 . . . . . . . . . . . . . 14  |-  ( m  =  ( (mapd `  K ) `  W
)  ->  m  =  M )
63 fveq1 5540 . . . . . . . . . . . . . . . . . . . 20  |-  ( m  =  M  ->  (
m `  ( n `  { ( 2nd `  x
) } ) )  =  ( M `  ( n `  {
( 2nd `  x
) } ) ) )
6463eqeq1d 2304 . . . . . . . . . . . . . . . . . . 19  |-  ( m  =  M  ->  (
( m `  (
n `  { ( 2nd `  x ) } ) )  =  ( j `  { h } )  <->  ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( j `  { h } ) ) )
65 fveq1 5540 . . . . . . . . . . . . . . . . . . . 20  |-  ( m  =  M  ->  (
m `  ( n `  { ( ( 1st `  ( 1st `  x
) ) ( -g `  u ) ( 2nd `  x ) ) } ) )  =  ( M `  ( n `
 { ( ( 1st `  ( 1st `  x ) ) (
-g `  u )
( 2nd `  x
) ) } ) ) )
6665eqeq1d 2304 . . . . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . . . 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 6322 . . . . . . . . . . . . . . . . 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 3593 . . . . . . . . . . . . . . . 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 4123 . . . . . . . . . . . . . . 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 2363 . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . 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 3038 . . . . . . . . . . . 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 956 . . . . . . . . . . . . . . 15  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  d  =  D )
75 xpeq2 4720 . . . . . . . . . . . . . . . 16  |-  ( d  =  D  ->  (
v  X.  d )  =  ( v  X.  D ) )
7675xpeq1d 4728 . . . . . . . . . . . . . . 15  |-  ( d  =  D  ->  (
( v  X.  d
)  X.  v )  =  ( ( v  X.  D )  X.  v ) )
7774, 76syl 15 . . . . . . . . . . . . . 14  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( ( v  X.  d )  X.  v
)  =  ( ( v  X.  D )  X.  v ) )
78 simp1 955 . . . . . . . . . . . . . . . . 17  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  c  =  C )
7978fveq2d 5545 . . . . . . . . . . . . . . . 16  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( 0g `  c
)  =  ( 0g
`  C ) )
80 hdmap1fval.q . . . . . . . . . . . . . . . 16  |-  Q  =  ( 0g `  C
)
8179, 80syl6eqr 2346 . . . . . . . . . . . . . . 15  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( 0g `  c
)  =  Q )
82 simp3 957 . . . . . . . . . . . . . . . . . . 19  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  j  =  J )
8382fveq1d 5543 . . . . . . . . . . . . . . . . . 18  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( j `  {
h } )  =  ( J `  {
h } ) )
8483eqeq2d 2307 . . . . . . . . . . . . . . . . 17  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( j `  { h } )  <-> 
( M `  (
n `  { ( 2nd `  x ) } ) )  =  ( J `  { h } ) ) )
8578fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( -g `  c
)  =  ( -g `  C ) )
86 hdmap1fval.r . . . . . . . . . . . . . . . . . . . . . 22  |-  R  =  ( -g `  C
)
8785, 86syl6eqr 2346 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( -g `  c
)  =  R )
8887oveqd 5891 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( ( 2nd `  ( 1st `  x ) ) ( -g `  c
) h )  =  ( ( 2nd `  ( 1st `  x ) ) R h ) )
8988sneqd 3666 . . . . . . . . . . . . . . . . . . 19  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  { ( ( 2nd `  ( 1st `  x
) ) ( -g `  c ) h ) }  =  { ( ( 2nd `  ( 1st `  x ) ) R h ) } )
9082, 89fveq12d 5547 . . . . . . . . . . . . . . . . . 18  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( j `  {
( ( 2nd `  ( 1st `  x ) ) ( -g `  c
) h ) } )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) )
9190eqeq2d 2307 . . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . 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 3594 . . . . . . . . . . . . . 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 4114 . . . . . . . . . . . . 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 2363 . . . . . . . . . . . 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 248 . . . . . . . . . . 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 1182 . . . . . . . . . 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 3073 . . . . . . . . 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 956 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  v  =  V )
101100xpeq1d 4728 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( v  X.  D
)  =  ( V  X.  D ) )
102101, 100xpeq12d 4730 . . . . . . . . . . 11  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( v  X.  D )  X.  v
)  =  ( ( V  X.  D )  X.  V ) )
103 simp1 955 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  u  =  U )
104103fveq2d 5545 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( 0g `  u
)  =  ( 0g
`  U ) )
105 hdmap1fval.o . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  U )
106104, 105syl6eqr 2346 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( 0g `  u
)  =  .0.  )
107106eqeq2d 2307 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( 2nd `  x
)  =  ( 0g
`  u )  <->  ( 2nd `  x )  =  .0.  ) )
108 simp3 957 . . . . . . . . . . . . . . . . 17  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  n  =  N )
109108fveq1d 5543 . . . . . . . . . . . . . . . 16  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( n `  {
( 2nd `  x
) } )  =  ( N `  {
( 2nd `  x
) } ) )
110109fveq2d 5545 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( M `  (
n `  { ( 2nd `  x ) } ) )  =  ( M `  ( N `
 { ( 2nd `  x ) } ) ) )
111110eqeq1d 2304 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( J `  { h } )  <-> 
( M `  ( N `  { ( 2nd `  x ) } ) )  =  ( J `  { h } ) ) )
112103fveq2d 5545 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( -g `  u
)  =  ( -g `  U ) )
113 hdmap1fval.s . . . . . . . . . . . . . . . . . . . 20  |-  .-  =  ( -g `  U )
114112, 113syl6eqr 2346 . . . . . . . . . . . . . . . . . . 19  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( -g `  u
)  =  .-  )
115114oveqd 5891 . . . . . . . . . . . . . . . . . 18  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) )  =  ( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) )
116115sneqd 3666 . . . . . . . . . . . . . . . . 17  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  { ( ( 1st `  ( 1st `  x
) ) ( -g `  u ) ( 2nd `  x ) ) }  =  { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) } )
117108, 116fveq12d 5547 . . . . . . . . . . . . . . . 16  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } )  =  ( N `  { ( ( 1st `  ( 1st `  x
) )  .-  ( 2nd `  x ) ) } ) )
118117fveq2d 5545 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( M `  (
n `  { (
( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( M `
 ( N `  { ( ( 1st `  ( 1st `  x
) )  .-  ( 2nd `  x ) ) } ) ) )
119118eqeq1d 2304 . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . 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 6322 . . . . . . . . . . . 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 3598 . . . . . . . . . . 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 4114 . . . . . . . . . 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 2363 . . . . . . . . 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 248 . . . . . . . 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 1182 . . . . . . 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 3073 . . . . . 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 252 . . . . 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 2412 . . . 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 2296 . . . 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 5555 . . . . . . . 8  |-  ( Base `  U )  e.  _V
13235, 131eqeltri 2366 . . . . . . 7  |-  V  e. 
_V
133 fvex 5555 . . . . . . . 8  |-  ( Base `  C )  e.  _V
13451, 133eqeltri 2366 . . . . . . 7  |-  D  e. 
_V
135132, 134xpex 4817 . . . . . 6  |-  ( V  X.  D )  e. 
_V
136135, 132xpex 4817 . . . . 5  |-  ( ( V  X.  D )  X.  V )  e. 
_V
137136mptex 5762 . . . 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 5618 . . 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 2348 . 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 15 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = 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:  hdmap1vallem  32610
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-pw 3640  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-ov 5877  df-riota 6320  df-hdmap1 32606
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