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Theorem hgmapfval 32079
Description: Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
Hypotheses
Ref Expression
hgmapval.h  |-  H  =  ( LHyp `  K
)
hgmapfval.u  |-  U  =  ( ( DVecH `  K
) `  W )
hgmapfval.v  |-  V  =  ( Base `  U
)
hgmapfval.t  |-  .x.  =  ( .s `  U )
hgmapfval.r  |-  R  =  (Scalar `  U )
hgmapfval.b  |-  B  =  ( Base `  R
)
hgmapfval.c  |-  C  =  ( (LCDual `  K
) `  W )
hgmapfval.s  |-  .xb  =  ( .s `  C )
hgmapfval.m  |-  M  =  ( (HDMap `  K
) `  W )
hgmapfval.i  |-  I  =  ( (HGMap `  K
) `  W )
hgmapfval.k  |-  ( ph  ->  ( K  e.  Y  /\  W  e.  H
) )
Assertion
Ref Expression
hgmapfval  |-  ( ph  ->  I  =  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) ) )
Distinct variable groups:    x, v,
y, K    v, B, x, y    v, M, x, y    v, U, x, y    v, V    v, W, x, y
Allowed substitution hints:    ph( x, y, v)    C( x, y, v)    R( x, y, v)    .xb ( x, y, v)    .x. ( x, y, v)    H( x, y, v)    I( x, y, v)    V( x, y)    Y( x, y, v)

Proof of Theorem hgmapfval
Dummy variables  w  a  b  m  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hgmapfval.k . 2  |-  ( ph  ->  ( K  e.  Y  /\  W  e.  H
) )
2 hgmapfval.i . . . 4  |-  I  =  ( (HGMap `  K
) `  W )
3 hgmapval.h . . . . . 6  |-  H  =  ( LHyp `  K
)
43hgmapffval 32078 . . . . 5  |-  ( K  e.  Y  ->  (HGMap `  K )  =  ( w  e.  H  |->  { a  |  [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  (Scalar `  u ) )  / 
b ]. [. ( (HDMap `  K ) `  w
)  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) ) } ) )
54fveq1d 5527 . . . 4  |-  ( K  e.  Y  ->  (
(HGMap `  K ) `  W )  =  ( ( w  e.  H  |->  { a  |  [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  (Scalar `  u
) )  /  b ]. [. ( (HDMap `  K ) `  w
)  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) ) } ) `  W ) )
62, 5syl5eq 2327 . . 3  |-  ( K  e.  Y  ->  I  =  ( ( w  e.  H  |->  { a  |  [. ( (
DVecH `  K ) `  w )  /  u ]. [. ( Base `  (Scalar `  u ) )  / 
b ]. [. ( (HDMap `  K ) `  w
)  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) ) } ) `  W ) )
7 fveq2 5525 . . . . . . . . 9  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
8 hgmapfval.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
97, 8syl6eqr 2333 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
10 dfsbcq 2993 . . . . . . . 8  |-  ( ( ( DVecH `  K ) `  w )  =  U  ->  ( [. (
( DVecH `  K ) `  w )  /  u ]. [. ( Base `  (Scalar `  u ) )  / 
b ]. [. ( (HDMap `  K ) `  w
)  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) )  <->  [. U  /  u ]. [. ( Base `  (Scalar `  u )
)  /  b ]. [. ( (HDMap `  K
) `  w )  /  m ]. a  e.  ( x  e.  b 
|->  ( iota_ y  e.  b A. v  e.  (
Base `  u )
( m `  (
x ( .s `  u ) v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  w ) ) ( m `  v ) ) ) ) ) )
119, 10syl 15 . . . . . . 7  |-  ( w  =  W  ->  ( [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  (Scalar `  u
) )  /  b ]. [. ( (HDMap `  K ) `  w
)  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) )  <->  [. U  /  u ]. [. ( Base `  (Scalar `  u )
)  /  b ]. [. ( (HDMap `  K
) `  w )  /  m ]. a  e.  ( x  e.  b 
|->  ( iota_ y  e.  b A. v  e.  (
Base `  u )
( m `  (
x ( .s `  u ) v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  w ) ) ( m `  v ) ) ) ) ) )
12 fveq2 5525 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (
(HDMap `  K ) `  w )  =  ( (HDMap `  K ) `  W ) )
13 hgmapfval.m . . . . . . . . . . . 12  |-  M  =  ( (HDMap `  K
) `  W )
1412, 13syl6eqr 2333 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
(HDMap `  K ) `  w )  =  M )
15 dfsbcq 2993 . . . . . . . . . . 11  |-  ( ( (HDMap `  K ) `  w )  =  M  ->  ( [. (
(HDMap `  K ) `  w )  /  m ]. a  e.  (
x  e.  b  |->  (
iota_ y  e.  b A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u
) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w
) ) ( m `
 v ) ) ) )  <->  [. M  /  m ]. a  e.  ( x  e.  b  |->  (
iota_ y  e.  b A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u
) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w
) ) ( m `
 v ) ) ) ) ) )
1614, 15syl 15 . . . . . . . . . 10  |-  ( w  =  W  ->  ( [. ( (HDMap `  K
) `  w )  /  m ]. a  e.  ( x  e.  b 
|->  ( iota_ y  e.  b A. v  e.  (
Base `  u )
( m `  (
x ( .s `  u ) v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  w ) ) ( m `  v ) ) ) )  <->  [. M  /  m ]. a  e.  ( x  e.  b  |->  (
iota_ y  e.  b A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u
) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w
) ) ( m `
 v ) ) ) ) ) )
17 fveq2 5525 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  W  ->  (
(LCDual `  K ) `  w )  =  ( (LCDual `  K ) `  W ) )
1817fveq2d 5529 . . . . . . . . . . . . . . . . 17  |-  ( w  =  W  ->  ( .s `  ( (LCDual `  K ) `  w
) )  =  ( .s `  ( (LCDual `  K ) `  W
) ) )
1918oveqd 5875 . . . . . . . . . . . . . . . 16  |-  ( w  =  W  ->  (
y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  W ) ) ( m `  v ) ) )
2019eqeq2d 2294 . . . . . . . . . . . . . . 15  |-  ( w  =  W  ->  (
( m `  (
x ( .s `  u ) v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  w ) ) ( m `  v ) )  <->  ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W ) ) ( m `  v ) ) ) )
2120ralbidv 2563 . . . . . . . . . . . . . 14  |-  ( w  =  W  ->  ( A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u
) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w
) ) ( m `
 v ) )  <->  A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u
) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W
) ) ( m `
 v ) ) ) )
2221riotabidv 6306 . . . . . . . . . . . . 13  |-  ( w  =  W  ->  ( iota_ y  e.  b A. v  e.  ( Base `  u ) ( m `
 ( x ( .s `  u ) v ) )  =  ( y ( .s
`  ( (LCDual `  K ) `  w
) ) ( m `
 v ) ) )  =  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W ) ) ( m `  v ) ) ) )
2322mpteq2dv 4107 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (
x  e.  b  |->  (
iota_ y  e.  b A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u
) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w
) ) ( m `
 v ) ) ) )  =  ( x  e.  b  |->  (
iota_ y  e.  b A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u
) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W
) ) ( m `
 v ) ) ) ) )
2423eleq2d 2350 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) )  <->  a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  (
Base `  u )
( m `  (
x ( .s `  u ) v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  W ) ) ( m `  v ) ) ) ) ) )
2524sbcbidv 3045 . . . . . . . . . 10  |-  ( w  =  W  ->  ( [. M  /  m ]. a  e.  (
x  e.  b  |->  (
iota_ y  e.  b A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u
) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w
) ) ( m `
 v ) ) ) )  <->  [. M  /  m ]. a  e.  ( x  e.  b  |->  (
iota_ y  e.  b A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u
) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W
) ) ( m `
 v ) ) ) ) ) )
2616, 25bitrd 244 . . . . . . . . 9  |-  ( w  =  W  ->  ( [. ( (HDMap `  K
) `  w )  /  m ]. a  e.  ( x  e.  b 
|->  ( iota_ y  e.  b A. v  e.  (
Base `  u )
( m `  (
x ( .s `  u ) v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  w ) ) ( m `  v ) ) ) )  <->  [. M  /  m ]. a  e.  ( x  e.  b  |->  (
iota_ y  e.  b A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u
) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W
) ) ( m `
 v ) ) ) ) ) )
2726sbcbidv 3045 . . . . . . . 8  |-  ( w  =  W  ->  ( [. ( Base `  (Scalar `  u ) )  / 
b ]. [. ( (HDMap `  K ) `  w
)  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) )  <->  [. ( Base `  (Scalar `  u )
)  /  b ]. [. M  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W ) ) ( m `  v ) ) ) ) ) )
2827sbcbidv 3045 . . . . . . 7  |-  ( w  =  W  ->  ( [. U  /  u ]. [. ( Base `  (Scalar `  u ) )  / 
b ]. [. ( (HDMap `  K ) `  w
)  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) )  <->  [. U  /  u ]. [. ( Base `  (Scalar `  u )
)  /  b ]. [. M  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W ) ) ( m `  v ) ) ) ) ) )
2911, 28bitrd 244 . . . . . 6  |-  ( w  =  W  ->  ( [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  (Scalar `  u
) )  /  b ]. [. ( (HDMap `  K ) `  w
)  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) )  <->  [. U  /  u ]. [. ( Base `  (Scalar `  u )
)  /  b ]. [. M  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W ) ) ( m `  v ) ) ) ) ) )
30 fvex 5539 . . . . . . . 8  |-  ( (
DVecH `  K ) `  W )  e.  _V
318, 30eqeltri 2353 . . . . . . 7  |-  U  e. 
_V
32 fvex 5539 . . . . . . 7  |-  ( Base `  (Scalar `  u )
)  e.  _V
33 fvex 5539 . . . . . . . 8  |-  ( (HDMap `  K ) `  W
)  e.  _V
3413, 33eqeltri 2353 . . . . . . 7  |-  M  e. 
_V
35 simp2 956 . . . . . . . . . 10  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  b  =  ( Base `  (Scalar `  u ) ) )
36 simp1 955 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  u  =  U )
3736fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  (Scalar `  u )  =  (Scalar `  U ) )
38 hgmapfval.r . . . . . . . . . . . 12  |-  R  =  (Scalar `  U )
3937, 38syl6eqr 2333 . . . . . . . . . . 11  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  (Scalar `  u )  =  R )
4039fveq2d 5529 . . . . . . . . . 10  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  ( Base `  (Scalar `  u
) )  =  (
Base `  R )
)
4135, 40eqtrd 2315 . . . . . . . . 9  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  b  =  ( Base `  R
) )
42 hgmapfval.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
4341, 42syl6eqr 2333 . . . . . . . 8  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  b  =  B )
44 simp2 956 . . . . . . . . . 10  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  b  =  B )
45 simp1 955 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  u  =  U )
4645fveq2d 5529 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( Base `  u
)  =  ( Base `  U ) )
47 hgmapfval.v . . . . . . . . . . . . 13  |-  V  =  ( Base `  U
)
4846, 47syl6eqr 2333 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( Base `  u
)  =  V )
49 simp3 957 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  m  =  M )
5045fveq2d 5529 . . . . . . . . . . . . . . . 16  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( .s `  u
)  =  ( .s
`  U ) )
51 hgmapfval.t . . . . . . . . . . . . . . . 16  |-  .x.  =  ( .s `  U )
5250, 51syl6eqr 2333 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( .s `  u
)  =  .x.  )
5352oveqd 5875 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( x ( .s
`  u ) v )  =  ( x 
.x.  v ) )
5449, 53fveq12d 5531 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( m `  (
x ( .s `  u ) v ) )  =  ( M `
 ( x  .x.  v ) ) )
55 eqidd 2284 . . . . . . . . . . . . . . . . 17  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( (LCDual `  K
) `  W )  =  ( (LCDual `  K ) `  W
) )
56 hgmapfval.c . . . . . . . . . . . . . . . . 17  |-  C  =  ( (LCDual `  K
) `  W )
5755, 56syl6eqr 2333 . . . . . . . . . . . . . . . 16  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( (LCDual `  K
) `  W )  =  C )
5857fveq2d 5529 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( .s `  (
(LCDual `  K ) `  W ) )  =  ( .s `  C
) )
59 hgmapfval.s . . . . . . . . . . . . . . 15  |-  .xb  =  ( .s `  C )
6058, 59syl6eqr 2333 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( .s `  (
(LCDual `  K ) `  W ) )  = 
.xb  )
61 eqidd 2284 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  y  =  y )
6249fveq1d 5527 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( m `  v
)  =  ( M `
 v ) )
6360, 61, 62oveq123d 5879 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( y ( .s
`  ( (LCDual `  K ) `  W
) ) ( m `
 v ) )  =  ( y  .xb  ( M `  v ) ) )
6454, 63eqeq12d 2297 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W ) ) ( m `  v ) )  <->  ( M `  ( x  .x.  v ) )  =  ( y 
.xb  ( M `  v ) ) ) )
6548, 64raleqbidv 2748 . . . . . . . . . . 11  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W ) ) ( m `  v ) )  <->  A. v  e.  V  ( M `  ( x 
.x.  v ) )  =  ( y  .xb  ( M `  v ) ) ) )
6644, 65riotaeqbidv 6307 . . . . . . . . . 10  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( iota_ y  e.  b A. v  e.  (
Base `  u )
( m `  (
x ( .s `  u ) v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  W ) ) ( m `  v ) ) )  =  (
iota_ y  e.  B A. v  e.  V  ( M `  ( x 
.x.  v ) )  =  ( y  .xb  ( M `  v ) ) ) )
6744, 66mpteq12dv 4098 . . . . . . . . 9  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( x  e.  b 
|->  ( iota_ y  e.  b A. v  e.  (
Base `  u )
( m `  (
x ( .s `  u ) v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  W ) ) ( m `  v ) ) ) )  =  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x 
.x.  v ) )  =  ( y  .xb  ( M `  v ) ) ) ) )
6867eleq2d 2350 . . . . . . . 8  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( a  e.  ( x  e.  b  |->  (
iota_ y  e.  b A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u
) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W
) ) ( m `
 v ) ) ) )  <->  a  e.  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x 
.x.  v ) )  =  ( y  .xb  ( M `  v ) ) ) ) ) )
6943, 68syld3an2 1229 . . . . . . 7  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  (
a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W ) ) ( m `  v ) ) ) )  <->  a  e.  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x 
.x.  v ) )  =  ( y  .xb  ( M `  v ) ) ) ) ) )
7031, 32, 34, 69sbc3ie 3060 . . . . . 6  |-  ( [. U  /  u ]. [. ( Base `  (Scalar `  u
) )  /  b ]. [. M  /  m ]. a  e.  (
x  e.  b  |->  (
iota_ y  e.  b A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u
) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  W
) ) ( m `
 v ) ) ) )  <->  a  e.  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x 
.x.  v ) )  =  ( y  .xb  ( M `  v ) ) ) ) )
7129, 70syl6bb 252 . . . . 5  |-  ( w  =  W  ->  ( [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  (Scalar `  u
) )  /  b ]. [. ( (HDMap `  K ) `  w
)  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) )  <->  a  e.  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x 
.x.  v ) )  =  ( y  .xb  ( M `  v ) ) ) ) ) )
7271abbi1dv 2399 . . . 4  |-  ( w  =  W  ->  { a  |  [. ( (
DVecH `  K ) `  w )  /  u ]. [. ( Base `  (Scalar `  u ) )  / 
b ]. [. ( (HDMap `  K ) `  w
)  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) ) }  =  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v ) )  =  ( y 
.xb  ( M `  v ) ) ) ) )
73 eqid 2283 . . . 4  |-  ( w  e.  H  |->  { a  |  [. ( (
DVecH `  K ) `  w )  /  u ]. [. ( Base `  (Scalar `  u ) )  / 
b ]. [. ( (HDMap `  K ) `  w
)  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) ) } )  =  ( w  e.  H  |->  { a  |  [. ( (
DVecH `  K ) `  w )  /  u ]. [. ( Base `  (Scalar `  u ) )  / 
b ]. [. ( (HDMap `  K ) `  w
)  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) ) } )
74 fvex 5539 . . . . . 6  |-  ( Base `  R )  e.  _V
7542, 74eqeltri 2353 . . . . 5  |-  B  e. 
_V
7675mptex 5746 . . . 4  |-  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) )  e.  _V
7772, 73, 76fvmpt 5602 . . 3  |-  ( W  e.  H  ->  (
( w  e.  H  |->  { a  |  [. ( ( DVecH `  K
) `  w )  /  u ]. [. ( Base `  (Scalar `  u
) )  /  b ]. [. ( (HDMap `  K ) `  w
)  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b A. v  e.  ( Base `  u
) ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) ) } ) `  W )  =  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v ) )  =  ( y 
.xb  ( M `  v ) ) ) ) )
786, 77sylan9eq 2335 . 2  |-  ( ( K  e.  Y  /\  W  e.  H )  ->  I  =  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) ) )
791, 78syl 15 1  |-  ( ph  ->  I  =  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   _Vcvv 2788   [.wsbc 2991    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   LHypclh 30173   DVecHcdvh 31268  LCDualclcd 31776  HDMapchdma 31983  HGMapchg 32076
This theorem is referenced by:  hgmapval  32080  hgmapfnN  32081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-hgmap 32077
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