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Theorem hgmapfval 32701
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 32700 . . . . 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 5543 . . . 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 2340 . . 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 5541 . . . . . . . . 9  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
8 hgmapfval.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
97, 8syl6eqr 2346 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
10 dfsbcq 3006 . . . . . . . 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 5541 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (
(HDMap `  K ) `  w )  =  ( (HDMap `  K ) `  W ) )
13 hgmapfval.m . . . . . . . . . . . 12  |-  M  =  ( (HDMap `  K
) `  W )
1412, 13syl6eqr 2346 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
(HDMap `  K ) `  w )  =  M )
15 dfsbcq 3006 . . . . . . . . . . 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 5541 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  W  ->  (
(LCDual `  K ) `  w )  =  ( (LCDual `  K ) `  W ) )
1817fveq2d 5545 . . . . . . . . . . . . . . . . 17  |-  ( w  =  W  ->  ( .s `  ( (LCDual `  K ) `  w
) )  =  ( .s `  ( (LCDual `  K ) `  W
) ) )
1918oveqd 5891 . . . . . . . . . . . . . . . 16  |-  ( w  =  W  ->  (
y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  W ) ) ( m `  v ) ) )
2019eqeq2d 2307 . . . . . . . . . . . . . . 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 2576 . . . . . . . . . . . . . 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 6322 . . . . . . . . . . . . 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 4123 . . . . . . . . . . . 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 2363 . . . . . . . . . . 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 3058 . . . . . . . . . 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 3058 . . . . . . . 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 3058 . . . . . . 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 5555 . . . . . . . 8  |-  ( (
DVecH `  K ) `  W )  e.  _V
318, 30eqeltri 2366 . . . . . . 7  |-  U  e. 
_V
32 fvex 5555 . . . . . . 7  |-  ( Base `  (Scalar `  u )
)  e.  _V
33 fvex 5555 . . . . . . . 8  |-  ( (HDMap `  K ) `  W
)  e.  _V
3413, 33eqeltri 2366 . . . . . . 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 5545 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  (Scalar `  u )  =  (Scalar `  U ) )
38 hgmapfval.r . . . . . . . . . . . 12  |-  R  =  (Scalar `  U )
3937, 38syl6eqr 2346 . . . . . . . . . . 11  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  (Scalar `  u )  =  R )
4039fveq2d 5545 . . . . . . . . . 10  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  ( Base `  (Scalar `  u
) )  =  (
Base `  R )
)
4135, 40eqtrd 2328 . . . . . . . . 9  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  b  =  ( Base `  R
) )
42 hgmapfval.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
4341, 42syl6eqr 2346 . . . . . . . 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 5545 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( Base `  u
)  =  ( Base `  U ) )
47 hgmapfval.v . . . . . . . . . . . . 13  |-  V  =  ( Base `  U
)
4846, 47syl6eqr 2346 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( Base `  u
)  =  V )
49 simp3 957 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  m  =  M )
5045fveq2d 5545 . . . . . . . . . . . . . . . 16  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( .s `  u
)  =  ( .s
`  U ) )
51 hgmapfval.t . . . . . . . . . . . . . . . 16  |-  .x.  =  ( .s `  U )
5250, 51syl6eqr 2346 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( .s `  u
)  =  .x.  )
5352oveqd 5891 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( x ( .s
`  u ) v )  =  ( x 
.x.  v ) )
5449, 53fveq12d 5547 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( m `  (
x ( .s `  u ) v ) )  =  ( M `
 ( x  .x.  v ) ) )
55 eqidd 2297 . . . . . . . . . . . . . . . . 17  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( (LCDual `  K
) `  W )  =  ( (LCDual `  K ) `  W
) )
56 hgmapfval.c . . . . . . . . . . . . . . . . 17  |-  C  =  ( (LCDual `  K
) `  W )
5755, 56syl6eqr 2346 . . . . . . . . . . . . . . . 16  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( (LCDual `  K
) `  W )  =  C )
5857fveq2d 5545 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( .s `  (
(LCDual `  K ) `  W ) )  =  ( .s `  C
) )
59 hgmapfval.s . . . . . . . . . . . . . . 15  |-  .xb  =  ( .s `  C )
6058, 59syl6eqr 2346 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( .s `  (
(LCDual `  K ) `  W ) )  = 
.xb  )
61 eqidd 2297 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  y  =  y )
6249fveq1d 5543 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( m `  v
)  =  ( M `
 v ) )
6360, 61, 62oveq123d 5895 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( y ( .s
`  ( (LCDual `  K ) `  W
) ) ( m `
 v ) )  =  ( y  .xb  ( M `  v ) ) )
6454, 63eqeq12d 2310 . . . . . . . . . . . 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 2761 . . . . . . . . . . 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 6323 . . . . . . . . . 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 4114 . . . . . . . . 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 2363 . . . . . . . 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 3073 . . . . . 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 2412 . . . 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 2296 . . . 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 5555 . . . . . 6  |-  ( Base `  R )  e.  _V
7542, 74eqeltri 2366 . . . . 5  |-  B  e. 
_V
7675mptex 5762 . . . 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 5618 . . 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 2348 . 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 1632    e. wcel 1696   {cab 2282   A.wral 2556   _Vcvv 2801   [.wsbc 3004    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   LHypclh 30795   DVecHcdvh 31890  LCDualclcd 32398  HDMapchdma 32605  HGMapchg 32698
This theorem is referenced by:  hgmapval  32702  hgmapfnN  32703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-hgmap 32699
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