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Theorem hgmapffval 35538
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 2986 . 2  |-  ( K  e.  X  ->  K  e.  _V )
2 fveq2 5696 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 hgmapval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2493 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5696 . . . . . . . 8  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
65fveq1d 5698 . . . . . . 7  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
7 dfsbcq 3193 . . . . . . 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 5696 . . . . . . . . . . 11  |-  ( k  =  K  ->  (HDMap `  k )  =  (HDMap `  K ) )
109fveq1d 5698 . . . . . . . . . 10  |-  ( k  =  K  ->  (
(HDMap `  k ) `  w )  =  ( (HDMap `  K ) `  w ) )
11 dfsbcq 3193 . . . . . . . . . 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 5696 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  K  ->  (LCDual `  k )  =  (LCDual `  K ) )
1413fveq1d 5698 . . . . . . . . . . . . . . . . 17  |-  ( k  =  K  ->  (
(LCDual `  k ) `  w )  =  ( (LCDual `  K ) `  w ) )
1514fveq2d 5700 . . . . . . . . . . . . . . . 16  |-  ( k  =  K  ->  ( .s `  ( (LCDual `  k ) `  w
) )  =  ( .s `  ( (LCDual `  K ) `  w
) ) )
1615oveqd 6113 . . . . . . . . . . . . . . 15  |-  ( k  =  K  ->  (
y ( .s `  ( (LCDual `  k ) `  w ) ) ( m `  v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  w ) ) ( m `  v ) ) )
1716eqeq2d 2454 . . . . . . . . . . . . . 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 2740 . . . . . . . . . . . . 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 6059 . . . . . . . . . . . 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 4384 . . . . . . . . . . 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 2510 . . . . . . . . . 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 3250 . . . . . . . . 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 253 . . . . . . . 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 3250 . . . . . . 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 3250 . . . . . 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 253 . . . . 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 2562 . . . 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 4375 . . 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 35537 . . 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 5706 . . . . 5  |-  ( LHyp `  K )  e.  _V
313, 30eqeltri 2513 . . . 4  |-  H  e. 
_V
3231mptex 5953 . . 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 5779 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720   _Vcvv 2977   [.wsbc 3191    e. cmpt 4355   ` cfv 5423   iota_crio 6056  (class class class)co 6096   Basecbs 14179  Scalarcsca 14246   .scvsca 14247   LHypclh 33633   DVecHcdvh 34728  LCDualclcd 35236  HDMapchdma 35443  HGMapchg 35536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-hgmap 35537
This theorem is referenced by:  hgmapfval  35539
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