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Theorem hdmap1fval 35334
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 35333 . . . . 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 5883 . . . 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 2475 . . 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 5881 . . . . . . 7  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
8 fveq2 5881 . . . . . . . . . 10  |-  ( w  =  W  ->  (
(LCDual `  K ) `  w )  =  ( (LCDual `  K ) `  W ) )
9 fveq2 5881 . . . . . . . . . . . . 13  |-  ( w  =  W  ->  (
(mapd `  K ) `  w )  =  ( (mapd `  K ) `  W ) )
109sbceq1d 3304 . . . . . . . . . . . 12  |-  ( 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 ) } ) ) ) ) ) ) )
1110sbcbidv 3354 . . . . . . . . . . 11  |-  ( 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 ) } ) ) ) ) ) ) )
1211sbcbidv 3354 . . . . . . . . . 10  |-  ( 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 ) } ) ) ) ) ) ) )
138, 12sbceqbid 3306 . . . . . . . . 9  |-  ( 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 ) } ) ) ) ) ) ) )
1413sbcbidv 3354 . . . . . . . 8  |-  ( 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 ) } ) ) ) ) ) ) )
1514sbcbidv 3354 . . . . . . 7  |-  ( 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 ) } ) ) ) ) ) ) )
167, 15sbceqbid 3306 . . . . . 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 ) } ) ) ) ) ) ) )
17 fvex 5891 . . . . . . 7  |-  ( (
DVecH `  K ) `  W )  e.  _V
18 fvex 5891 . . . . . . 7  |-  ( Base `  u )  e.  _V
19 fvex 5891 . . . . . . 7  |-  ( LSpan `  u )  e.  _V
20 hdmap1fval.u . . . . . . . . . . 11  |-  U  =  ( ( DVecH `  K
) `  W )
2120eqeq2i 2440 . . . . . . . . . 10  |-  ( u  =  U  <->  u  =  ( ( DVecH `  K
) `  W )
)
2221biimpri 209 . . . . . . . . 9  |-  ( u  =  ( ( DVecH `  K ) `  W
)  ->  u  =  U )
23223ad2ant1 1026 . . . . . . . 8  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  u  =  U )
24 simp2 1006 . . . . . . . . . 10  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  v  =  ( Base `  u )
)
2522fveq2d 5885 . . . . . . . . . . 11  |-  ( u  =  ( ( DVecH `  K ) `  W
)  ->  ( Base `  u )  =  (
Base `  U )
)
26253ad2ant1 1026 . . . . . . . . . 10  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  ( Base `  u )  =  (
Base `  U )
)
2724, 26eqtrd 2463 . . . . . . . . 9  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  v  =  ( Base `  U )
)
28 hdmap1fval.v . . . . . . . . 9  |-  V  =  ( Base `  U
)
2927, 28syl6eqr 2481 . . . . . . . 8  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  v  =  V )
30 simp3 1007 . . . . . . . . . 10  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  n  =  ( LSpan `  u )
)
3123fveq2d 5885 . . . . . . . . . 10  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  ( LSpan `  u )  =  (
LSpan `  U ) )
3230, 31eqtrd 2463 . . . . . . . . 9  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  n  =  ( LSpan `  U )
)
33 hdmap1fval.n . . . . . . . . 9  |-  N  =  ( LSpan `  U )
3432, 33syl6eqr 2481 . . . . . . . 8  |-  ( ( u  =  ( (
DVecH `  K ) `  W )  /\  v  =  ( Base `  u
)  /\  n  =  ( LSpan `  u )
)  ->  n  =  N )
35 fvex 5891 . . . . . . . . . 10  |-  ( (LCDual `  K ) `  W
)  e.  _V
36 fvex 5891 . . . . . . . . . 10  |-  ( Base `  c )  e.  _V
37 fvex 5891 . . . . . . . . . 10  |-  ( LSpan `  c )  e.  _V
38 id 22 . . . . . . . . . . . . 13  |-  ( c  =  ( (LCDual `  K ) `  W
)  ->  c  =  ( (LCDual `  K ) `  W ) )
39 hdmap1fval.c . . . . . . . . . . . . 13  |-  C  =  ( (LCDual `  K
) `  W )
4038, 39syl6eqr 2481 . . . . . . . . . . . 12  |-  ( c  =  ( (LCDual `  K ) `  W
)  ->  c  =  C )
41403ad2ant1 1026 . . . . . . . . . . 11  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  c  =  C )
42 simp2 1006 . . . . . . . . . . . 12  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  d  =  (
Base `  c )
)
4341fveq2d 5885 . . . . . . . . . . . . 13  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  ( Base `  c
)  =  ( Base `  C ) )
44 hdmap1fval.d . . . . . . . . . . . . 13  |-  D  =  ( Base `  C
)
4543, 44syl6eqr 2481 . . . . . . . . . . . 12  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  ( Base `  c
)  =  D )
4642, 45eqtrd 2463 . . . . . . . . . . 11  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  d  =  D )
47 simp3 1007 . . . . . . . . . . . 12  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  j  =  (
LSpan `  c ) )
4841fveq2d 5885 . . . . . . . . . . . . 13  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  ( LSpan `  c
)  =  ( LSpan `  C ) )
49 hdmap1fval.j . . . . . . . . . . . . 13  |-  J  =  ( LSpan `  C )
5048, 49syl6eqr 2481 . . . . . . . . . . . 12  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  ( LSpan `  c
)  =  J )
5147, 50eqtrd 2463 . . . . . . . . . . 11  |-  ( ( c  =  ( (LCDual `  K ) `  W
)  /\  d  =  ( Base `  c )  /\  j  =  ( LSpan `  c ) )  ->  j  =  J )
52 fvex 5891 . . . . . . . . . . . . 13  |-  ( (mapd `  K ) `  W
)  e.  _V
53 id 22 . . . . . . . . . . . . . . 15  |-  ( m  =  ( (mapd `  K ) `  W
)  ->  m  =  ( (mapd `  K ) `  W ) )
54 hdmap1fval.m . . . . . . . . . . . . . . 15  |-  M  =  ( (mapd `  K
) `  W )
5553, 54syl6eqr 2481 . . . . . . . . . . . . . 14  |-  ( m  =  ( (mapd `  K ) `  W
)  ->  m  =  M )
56 fveq1 5880 . . . . . . . . . . . . . . . . . . . 20  |-  ( m  =  M  ->  (
m `  ( n `  { ( 2nd `  x
) } ) )  =  ( M `  ( n `  {
( 2nd `  x
) } ) ) )
5756eqeq1d 2424 . . . . . . . . . . . . . . . . . . 19  |-  ( m  =  M  ->  (
( m `  (
n `  { ( 2nd `  x ) } ) )  =  ( j `  { h } )  <->  ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( j `  { h } ) ) )
58 fveq1 5880 . . . . . . . . . . . . . . . . . . . 20  |-  ( m  =  M  ->  (
m `  ( n `  { ( ( 1st `  ( 1st `  x
) ) ( -g `  u ) ( 2nd `  x ) ) } ) )  =  ( M `  ( n `
 { ( ( 1st `  ( 1st `  x ) ) (
-g `  u )
( 2nd `  x
) ) } ) ) )
5958eqeq1d 2424 . . . . . . . . . . . . . . . . . . 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 ) } ) ) )
6057, 59anbi12d 715 . . . . . . . . . . . . . . . . . 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 ) } ) ) ) )
6160riotabidv 6269 . . . . . . . . . . . . . . . . 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 ) } ) ) ) )
6261ifeq2d 3930 . . . . . . . . . . . . . . . 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 ) } ) ) ) ) )
6362mpteq2dv 4511 . . . . . . . . . . . . . . 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 ) } ) ) ) ) ) )
6463eleq2d 2492 . . . . . . . . . . . . . 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 ) } ) ) ) ) ) ) )
6555, 64syl 17 . . . . . . . . . . . . 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 ) } ) ) ) ) ) ) )
6652, 65sbcie 3334 . . . . . . . . . . . 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 ) } ) ) ) ) ) )
67 simp2 1006 . . . . . . . . . . . . . . 15  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  d  =  D )
68 xpeq2 4868 . . . . . . . . . . . . . . . 16  |-  ( d  =  D  ->  (
v  X.  d )  =  ( v  X.  D ) )
6968xpeq1d 4876 . . . . . . . . . . . . . . 15  |-  ( d  =  D  ->  (
( v  X.  d
)  X.  v )  =  ( ( v  X.  D )  X.  v ) )
7067, 69syl 17 . . . . . . . . . . . . . 14  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( ( v  X.  d )  X.  v
)  =  ( ( v  X.  D )  X.  v ) )
71 simp1 1005 . . . . . . . . . . . . . . . . 17  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  c  =  C )
7271fveq2d 5885 . . . . . . . . . . . . . . . 16  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( 0g `  c
)  =  ( 0g
`  C ) )
73 hdmap1fval.q . . . . . . . . . . . . . . . 16  |-  Q  =  ( 0g `  C
)
7472, 73syl6eqr 2481 . . . . . . . . . . . . . . 15  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( 0g `  c
)  =  Q )
75 simp3 1007 . . . . . . . . . . . . . . . . . . 19  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  j  =  J )
7675fveq1d 5883 . . . . . . . . . . . . . . . . . 18  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( j `  {
h } )  =  ( J `  {
h } ) )
7776eqeq2d 2436 . . . . . . . . . . . . . . . . 17  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( j `  { h } )  <-> 
( M `  (
n `  { ( 2nd `  x ) } ) )  =  ( J `  { h } ) ) )
7871fveq2d 5885 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( -g `  c
)  =  ( -g `  C ) )
79 hdmap1fval.r . . . . . . . . . . . . . . . . . . . . . 22  |-  R  =  ( -g `  C
)
8078, 79syl6eqr 2481 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( -g `  c
)  =  R )
8180oveqd 6322 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( ( 2nd `  ( 1st `  x ) ) ( -g `  c
) h )  =  ( ( 2nd `  ( 1st `  x ) ) R h ) )
8281sneqd 4010 . . . . . . . . . . . . . . . . . . 19  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  { ( ( 2nd `  ( 1st `  x
) ) ( -g `  c ) h ) }  =  { ( ( 2nd `  ( 1st `  x ) ) R h ) } )
8375, 82fveq12d 5887 . . . . . . . . . . . . . . . . . 18  |-  ( ( c  =  C  /\  d  =  D  /\  j  =  J )  ->  ( j `  {
( ( 2nd `  ( 1st `  x ) ) ( -g `  c
) h ) } )  =  ( J `
 { ( ( 2nd `  ( 1st `  x ) ) R h ) } ) )
8483eqeq2d 2436 . . . . . . . . . . . . . . . . 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 ) } ) ) )
8577, 84anbi12d 715 . . . . . . . . . . . . . . . 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 ) } ) ) ) )
8667, 85riotaeqbidv 6270 . . . . . . . . . . . . . . 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 ) } ) ) ) )
8774, 86ifeq12d 3931 . . . . . . . . . . . . . 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 ) } ) ) ) ) )
8870, 87mpteq12dv 4502 . . . . . . . . . . . . 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 ) } ) ) ) ) ) )
8988eleq2d 2492 . . . . . . . . . . . 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 ) } ) ) ) ) ) ) )
9066, 89syl5bb 260 . . . . . . . . . . 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 ) } ) ) ) ) ) ) )
9141, 46, 51, 90syl3anc 1264 . . . . . . . . . 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 ) } ) ) ) ) ) ) )
9235, 36, 37, 91sbc3ie 3369 . . . . . . . . 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 ) } ) ) ) ) ) )
93 simp2 1006 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  v  =  V )
9493xpeq1d 4876 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( v  X.  D
)  =  ( V  X.  D ) )
9594, 93xpeq12d 4878 . . . . . . . . . . 11  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( v  X.  D )  X.  v
)  =  ( ( V  X.  D )  X.  V ) )
96 simp1 1005 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  u  =  U )
9796fveq2d 5885 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( 0g `  u
)  =  ( 0g
`  U ) )
98 hdmap1fval.o . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  U )
9997, 98syl6eqr 2481 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( 0g `  u
)  =  .0.  )
10099eqeq2d 2436 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( 2nd `  x
)  =  ( 0g
`  u )  <->  ( 2nd `  x )  =  .0.  ) )
101 simp3 1007 . . . . . . . . . . . . . . . . 17  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  n  =  N )
102101fveq1d 5883 . . . . . . . . . . . . . . . 16  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( n `  {
( 2nd `  x
) } )  =  ( N `  {
( 2nd `  x
) } ) )
103102fveq2d 5885 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( M `  (
n `  { ( 2nd `  x ) } ) )  =  ( M `  ( N `
 { ( 2nd `  x ) } ) ) )
104103eqeq1d 2424 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( M `  ( n `  {
( 2nd `  x
) } ) )  =  ( J `  { h } )  <-> 
( M `  ( N `  { ( 2nd `  x ) } ) )  =  ( J `  { h } ) ) )
10596fveq2d 5885 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( -g `  u
)  =  ( -g `  U ) )
106 hdmap1fval.s . . . . . . . . . . . . . . . . . . . 20  |-  .-  =  ( -g `  U )
107105, 106syl6eqr 2481 . . . . . . . . . . . . . . . . . . 19  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( -g `  u
)  =  .-  )
108107oveqd 6322 . . . . . . . . . . . . . . . . . 18  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) )  =  ( ( 1st `  ( 1st `  x ) ) 
.-  ( 2nd `  x
) ) )
109108sneqd 4010 . . . . . . . . . . . . . . . . 17  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  { ( ( 1st `  ( 1st `  x
) ) ( -g `  u ) ( 2nd `  x ) ) }  =  { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) } )
110101, 109fveq12d 5887 . . . . . . . . . . . . . . . 16  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( n `  {
( ( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } )  =  ( N `  { ( ( 1st `  ( 1st `  x
) )  .-  ( 2nd `  x ) ) } ) )
111110fveq2d 5885 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  v  =  V  /\  n  =  N )  ->  ( M `  (
n `  { (
( 1st `  ( 1st `  x ) ) ( -g `  u
) ( 2nd `  x
) ) } ) )  =  ( M `
 ( N `  { ( ( 1st `  ( 1st `  x
) )  .-  ( 2nd `  x ) ) } ) ) )
112111eqeq1d 2424 . . . . . . . . . . . . . 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 ) } ) ) )
113104, 112anbi12d 715 . . . . . . . . . . . . 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 ) } ) ) ) )
114113riotabidv 6269 . . . . . . . . . . . 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 ) } ) ) ) )
115100, 114ifbieq2d 3936 . . . . . . . . . . 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 ) } ) ) ) ) )
11695, 115mpteq12dv 4502 . . . . . . . . . 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 ) } ) ) ) ) ) )
117116eleq2d 2492 . . . . . . . . 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 ) } ) ) ) ) ) ) )
11892, 117syl5bb 260 . . . . . . . 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 ) } ) ) ) ) ) ) )
11923, 29, 34, 118syl3anc 1264 . . . . . . 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 ) } ) ) ) ) ) ) )
12017, 18, 19, 119sbc3ie 3369 . . . . . 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 ) } ) ) ) ) ) )
12116, 120syl6bb 264 . . . . 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 ) } ) ) ) ) ) ) )
122121abbi1dv 2555 . . . 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 ) } ) ) ) ) ) )
123 eqid 2422 . . . 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 ) } ) ) ) ) ) } )
124 fvex 5891 . . . . . . . 8  |-  ( Base `  U )  e.  _V
12528, 124eqeltri 2503 . . . . . . 7  |-  V  e. 
_V
126 fvex 5891 . . . . . . . 8  |-  ( Base `  C )  e.  _V
12744, 126eqeltri 2503 . . . . . . 7  |-  D  e. 
_V
128125, 127xpex 6609 . . . . . 6  |-  ( V  X.  D )  e. 
_V
129128, 125xpex 6609 . . . . 5  |-  ( ( V  X.  D )  X.  V )  e. 
_V
130129mptex 6151 . . . 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
131122, 123, 130fvmpt 5964 . . 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 ) } ) ) ) ) ) )
1326, 131sylan9eq 2483 . 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 ) } ) ) ) ) ) )
1331, 132syl 17 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   {cab 2407   _Vcvv 3080   [.wsbc 3299   ifcif 3911   {csn 3998    |-> cmpt 4482    X. cxp 4851   ` cfv 5601   iota_crio 6266  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   Basecbs 15120   0gc0g 15337   -gcsg 16670   LSpanclspn 18193   LHypclh 33518   DVecHcdvh 34615  LCDualclcd 35123  mapdcmpd 35161  HDMap1chdma1 35329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-hdmap1 35331
This theorem is referenced by:  hdmap1vallem  35335
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