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Theorem hgmapfval 34873
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 34872 . . . . 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 5805 . . . 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 2453 . . 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 5803 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
8 hgmapfval.u . . . . . . . 8  |-  U  =  ( ( DVecH `  K
) `  W )
97, 8syl6eqr 2459 . . . . . . 7  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
10 fveq2 5803 . . . . . . . . . 10  |-  ( w  =  W  ->  (
(HDMap `  K ) `  w )  =  ( (HDMap `  K ) `  W ) )
11 hgmapfval.m . . . . . . . . . 10  |-  M  =  ( (HDMap `  K
) `  W )
1210, 11syl6eqr 2459 . . . . . . . . 9  |-  ( w  =  W  ->  (
(HDMap `  K ) `  w )  =  M )
13 fveq2 5803 . . . . . . . . . . . . . . . 16  |-  ( w  =  W  ->  (
(LCDual `  K ) `  w )  =  ( (LCDual `  K ) `  W ) )
1413fveq2d 5807 . . . . . . . . . . . . . . 15  |-  ( w  =  W  ->  ( .s `  ( (LCDual `  K ) `  w
) )  =  ( .s `  ( (LCDual `  K ) `  W
) ) )
1514oveqd 6249 . . . . . . . . . . . . . 14  |-  ( w  =  W  ->  (
y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  W ) ) ( m `  v ) ) )
1615eqeq2d 2414 . . . . . . . . . . . . 13  |-  ( 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 ) ) ) )
1716ralbidv 2840 . . . . . . . . . . . 12  |-  ( 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 ) ) ) )
1817riotabidv 6196 . . . . . . . . . . 11  |-  ( 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 ) ) ) )
1918mpteq2dv 4479 . . . . . . . . . 10  |-  ( 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 ) ) ) ) )
2019eleq2d 2470 . . . . . . . . 9  |-  ( 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 ) ) ) ) ) )
2112, 20sbceqbid 3281 . . . . . . . 8  |-  ( 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 ) ) ) ) ) )
2221sbcbidv 3329 . . . . . . 7  |-  ( 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 ) ) ) ) ) )
239, 22sbceqbid 3281 . . . . . 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 ) ) ) ) ) )
24 fvex 5813 . . . . . . . 8  |-  ( (
DVecH `  K ) `  W )  e.  _V
258, 24eqeltri 2484 . . . . . . 7  |-  U  e. 
_V
26 fvex 5813 . . . . . . 7  |-  ( Base `  (Scalar `  u )
)  e.  _V
27 fvex 5813 . . . . . . . 8  |-  ( (HDMap `  K ) `  W
)  e.  _V
2811, 27eqeltri 2484 . . . . . . 7  |-  M  e. 
_V
29 simp2 996 . . . . . . . . . 10  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  b  =  ( Base `  (Scalar `  u ) ) )
30 simp1 995 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  u  =  U )
3130fveq2d 5807 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  (Scalar `  u )  =  (Scalar `  U ) )
32 hgmapfval.r . . . . . . . . . . . 12  |-  R  =  (Scalar `  U )
3331, 32syl6eqr 2459 . . . . . . . . . . 11  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  (Scalar `  u )  =  R )
3433fveq2d 5807 . . . . . . . . . 10  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  ( Base `  (Scalar `  u
) )  =  (
Base `  R )
)
3529, 34eqtrd 2441 . . . . . . . . 9  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  b  =  ( Base `  R
) )
36 hgmapfval.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
3735, 36syl6eqr 2459 . . . . . . . 8  |-  ( ( u  =  U  /\  b  =  ( Base `  (Scalar `  u )
)  /\  m  =  M )  ->  b  =  B )
38 simp2 996 . . . . . . . . . 10  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  b  =  B )
39 simp1 995 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  u  =  U )
4039fveq2d 5807 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( Base `  u
)  =  ( Base `  U ) )
41 hgmapfval.v . . . . . . . . . . . . 13  |-  V  =  ( Base `  U
)
4240, 41syl6eqr 2459 . . . . . . . . . . . 12  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( Base `  u
)  =  V )
43 simp3 997 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  m  =  M )
4439fveq2d 5807 . . . . . . . . . . . . . . . 16  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( .s `  u
)  =  ( .s
`  U ) )
45 hgmapfval.t . . . . . . . . . . . . . . . 16  |-  .x.  =  ( .s `  U )
4644, 45syl6eqr 2459 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( .s `  u
)  =  .x.  )
4746oveqd 6249 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( x ( .s
`  u ) v )  =  ( x 
.x.  v ) )
4843, 47fveq12d 5809 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( m `  (
x ( .s `  u ) v ) )  =  ( M `
 ( x  .x.  v ) ) )
49 eqidd 2401 . . . . . . . . . . . . . . . . 17  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( (LCDual `  K
) `  W )  =  ( (LCDual `  K ) `  W
) )
50 hgmapfval.c . . . . . . . . . . . . . . . . 17  |-  C  =  ( (LCDual `  K
) `  W )
5149, 50syl6eqr 2459 . . . . . . . . . . . . . . . 16  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( (LCDual `  K
) `  W )  =  C )
5251fveq2d 5807 . . . . . . . . . . . . . . 15  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( .s `  (
(LCDual `  K ) `  W ) )  =  ( .s `  C
) )
53 hgmapfval.s . . . . . . . . . . . . . . 15  |-  .xb  =  ( .s `  C )
5452, 53syl6eqr 2459 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( .s `  (
(LCDual `  K ) `  W ) )  = 
.xb  )
55 eqidd 2401 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  y  =  y )
5643fveq1d 5805 . . . . . . . . . . . . . 14  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( m `  v
)  =  ( M `
 v ) )
5754, 55, 56oveq123d 6253 . . . . . . . . . . . . 13  |-  ( ( u  =  U  /\  b  =  B  /\  m  =  M )  ->  ( y ( .s
`  ( (LCDual `  K ) `  W
) ) ( m `
 v ) )  =  ( y  .xb  ( M `  v ) ) )
5848, 57eqeq12d 2422 . . . . . . . . . . . 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 ) ) ) )
5942, 58raleqbidv 3015 . . . . . . . . . . 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 ) ) ) )
6038, 59riotaeqbidv 6197 . . . . . . . . . 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 ) ) ) )
6138, 60mpteq12dv 4470 . . . . . . . . 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 ) ) ) ) )
6261eleq2d 2470 . . . . . . . 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 ) ) ) ) ) )
6337, 62syld3an2 1275 . . . . . . 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 ) ) ) ) ) )
6425, 26, 28, 63sbc3ie 3344 . . . . . 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 ) ) ) ) )
6523, 64syl6bb 261 . . . . 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 ) ) ) ) ) )
6665abbi1dv 2538 . . . 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 ) ) ) ) )
67 eqid 2400 . . . 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 ) ) ) ) } )
68 fvex 5813 . . . . . 6  |-  ( Base `  R )  e.  _V
6936, 68eqeltri 2484 . . . . 5  |-  B  e. 
_V
7069mptex 6078 . . . 4  |-  ( x  e.  B  |->  ( iota_ y  e.  B  A. v  e.  V  ( M `  ( x  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) )  e.  _V
7166, 67, 70fvmpt 5886 . . 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 ) ) ) ) )
726, 71sylan9eq 2461 . 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 ) ) ) ) )
731, 72syl 17 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   {cab 2385   A.wral 2751   _Vcvv 3056   [.wsbc 3274    |-> cmpt 4450   ` cfv 5523   iota_crio 6193  (class class class)co 6232   Basecbs 14731  Scalarcsca 14802   .scvsca 14803   LHypclh 32965   DVecHcdvh 34062  LCDualclcd 34570  HDMapchdma 34777  HGMapchg 34870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-hgmap 34871
This theorem is referenced by:  hgmapval  34874  hgmapfnN  34875
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