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Theorem hgmapffval 32700
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 2809 . 2  |-  ( K  e.  X  ->  K  e.  _V )
2 fveq2 5541 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 hgmapval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2346 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5541 . . . . . . . 8  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
65fveq1d 5543 . . . . . . 7  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
7 dfsbcq 3006 . . . . . . 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 15 . . . . . 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 5541 . . . . . . . . . . 11  |-  ( k  =  K  ->  (HDMap `  k )  =  (HDMap `  K ) )
109fveq1d 5543 . . . . . . . . . 10  |-  ( k  =  K  ->  (
(HDMap `  k ) `  w )  =  ( (HDMap `  K ) `  w ) )
11 dfsbcq 3006 . . . . . . . . . 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 15 . . . . . . . . 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 5541 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  K  ->  (LCDual `  k )  =  (LCDual `  K ) )
1413fveq1d 5543 . . . . . . . . . . . . . . . . 17  |-  ( k  =  K  ->  (
(LCDual `  k ) `  w )  =  ( (LCDual `  K ) `  w ) )
1514fveq2d 5545 . . . . . . . . . . . . . . . 16  |-  ( k  =  K  ->  ( .s `  ( (LCDual `  k ) `  w
) )  =  ( .s `  ( (LCDual `  K ) `  w
) ) )
1615oveqd 5891 . . . . . . . . . . . . . . 15  |-  ( k  =  K  ->  (
y ( .s `  ( (LCDual `  k ) `  w ) ) ( m `  v ) )  =  ( y ( .s `  (
(LCDual `  K ) `  w ) ) ( m `  v ) ) )
1716eqeq2d 2307 . . . . . . . . . . . . . 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 2576 . . . . . . . . . . . . 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 6322 . . . . . . . . . . . 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 4123 . . . . . . . . . . 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 2363 . . . . . . . . . 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 3058 . . . . . . . . 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 244 . . . . . . . 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 3058 . . . . . . 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 3058 . . . . . 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 244 . . . . 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 2410 . . . 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 4114 . . 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 32699 . . 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 5555 . . . . 5  |-  ( LHyp `  K )  e.  _V
313, 30eqeltri 2366 . . . 4  |-  H  e. 
_V
3231mptex 5762 . . 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 5618 . 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 15 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 176    = 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:  hgmapfval  32701
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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