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Theorem hgmapffval 32005
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.)
Hypothesis
Ref Expression
hgmapval.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
hgmapffval  |-  ( K  e.  X  ->  (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 ) ) ) ) } ) )
Distinct variable groups:    w, H    a, b, m, u, v, w, x, y, K
Allowed substitution hints:    H( x, y, v, u, m, a, b)    X( x, y, w, v, u, m, a, b)

Proof of Theorem hgmapffval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2909 . 2  |-  ( K  e.  X  ->  K  e.  _V )
2 fveq2 5670 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 hgmapval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2439 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5670 . . . . . . . 8  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
65fveq1d 5672 . . . . . . 7  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
7 dfsbcq 3108 . . . . . . 7  |-  ( ( ( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  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 ) ) ) )  <->  [. ( (
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 ) ) ) ) ) )
86, 7syl 16 . . . . . 6  |-  ( k  =  K  ->  ( [. ( ( 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 ) ) ) )  <->  [. ( (
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 ) ) ) ) ) )
9 fveq2 5670 . . . . . . . . . . 11  |-  ( k  =  K  ->  (HDMap `  k )  =  (HDMap `  K ) )
109fveq1d 5672 . . . . . . . . . 10  |-  ( k  =  K  ->  (
(HDMap `  k ) `  w )  =  ( (HDMap `  K ) `  w ) )
11 dfsbcq 3108 . . . . . . . . . 10  |-  ( ( (HDMap `  k ) `  w )  =  ( (HDMap `  K ) `  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 ) ) ) )  <->  [. ( (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 ) ) ) ) ) )
1210, 11syl 16 . . . . . . . . 9  |-  ( k  =  K  ->  ( [. ( (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 ) ) ) )  <->  [. ( (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 ) ) ) ) ) )
13 fveq2 5670 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  K  ->  (LCDual `  k )  =  (LCDual `  K ) )
1413fveq1d 5672 . . . . . . . . . . . . . . . . 17  |-  ( k  =  K  ->  (
(LCDual `  k ) `  w )  =  ( (LCDual `  K ) `  w ) )
1514fveq2d 5674 . . . . . . . . . . . . . . . 16  |-  ( k  =  K  ->  ( .s `  ( (LCDual `  k ) `  w
) )  =  ( .s `  ( (LCDual `  K ) `  w
) ) )
1615oveqd 6039 . . . . . . . . . . . . . . 15  |-  ( k  =  K  ->  (
y ( .s `  ( (LCDual `  k ) `  w ) ) ( m `  v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  w ) ) ( m `  v ) ) )
1716eqeq2d 2400 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  (
( m `  (
x ( .s `  u ) v ) )  =  ( y ( .s `  (
(LCDual `  k ) `  w ) ) ( m `  v ) )  <->  ( m `  ( x ( .s
`  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `  v ) ) ) )
1817ralbidv 2671 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( 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 ) ) ) )
1918riotabidv 6489 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( 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 ) ) ) )
2019mpteq2dv 4239 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
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 ) ) ) ) )
2120eleq2d 2456 . . . . . . . . . 10  |-  ( k  =  K  ->  (
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 ) ) ) ) ) )
2221sbcbidv 3160 . . . . . . . . 9  |-  ( k  =  K  ->  ( [. ( (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 ) ) ) )  <->  [. ( (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 ) ) ) ) ) )
2312, 22bitrd 245 . . . . . . . 8  |-  ( k  =  K  ->  ( [. ( (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 ) ) ) )  <->  [. ( (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 ) ) ) ) ) )
2423sbcbidv 3160 . . . . . . 7  |-  ( k  =  K  ->  ( [. ( 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 ]. [. ( (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 ) ) ) ) ) )
2524sbcbidv 3160 . . . . . 6  |-  ( k  =  K  ->  ( [. ( ( 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 ) ) ) )  <->  [. ( (
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 ) ) ) ) ) )
268, 25bitrd 245 . . . . 5  |-  ( k  =  K  ->  ( [. ( ( 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 ) ) ) )  <->  [. ( (
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 ) ) ) ) ) )
2726abbidv 2503 . . . 4  |-  ( k  =  K  ->  { 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 ) ) ) ) }  =  { 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 ) ) ) ) } )
284, 27mpteq12dv 4230 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  { 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 ) ) ) ) } ) )
29 df-hgmap 32004 . . 3  |- HGMap  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { 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 ) ) ) ) } ) )
30 fvex 5684 . . . . 5  |-  ( LHyp `  K )  e.  _V
313, 30eqeltri 2459 . . . 4  |-  H  e. 
_V
3231mptex 5907 . . 3  |-  ( 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 ) ) ) ) } )  e.  _V
3328, 29, 32fvmpt 5747 . 2  |-  ( K  e.  _V  ->  (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 ) ) ) ) } ) )
341, 33syl 16 1  |-  ( K  e.  X  ->  (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 ) ) ) ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   {cab 2375   A.wral 2651   _Vcvv 2901   [.wsbc 3106    e. cmpt 4209   ` cfv 5396  (class class class)co 6022   iota_crio 6480   Basecbs 13398  Scalarcsca 13461   .scvsca 13462   LHypclh 30100   DVecHcdvh 31195  LCDualclcd 31703  HDMapchdma 31910  HGMapchg 32003
This theorem is referenced by:  hgmapfval  32006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-riota 6487  df-hgmap 32004
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