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Theorem hgmapval 31210
Description: Value of map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. Function sigma of scalar f in part 14 of [Baer] p. 50 line 4. TODO: variable names are inherited from older version. Maybe make more consistent with hdmap14lem15 31205. (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
) )
hgmapval.x  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
hgmapval  |-  ( ph  ->  ( I `  X
)  =  ( iota_ y  e.  B A. v  e.  V  ( M `  ( X  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) )
Distinct variable groups:    y, v, K    v, B, y    v, M, y    v, U, y   
v, V    v, W, y    v, X, y
Allowed substitution hints:    ph( y, v)    C( y, v)    R( y, v)    .xb ( y, v)    .x. ( y,
v)    H( y, v)    I(
y, v)    V( y)    Y( y, v)

Proof of Theorem hgmapval
StepHypRef Expression
1 hgmapval.h . . . 4  |-  H  =  ( LHyp `  K
)
2 hgmapfval.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
3 hgmapfval.v . . . 4  |-  V  =  ( Base `  U
)
4 hgmapfval.t . . . 4  |-  .x.  =  ( .s `  U )
5 hgmapfval.r . . . 4  |-  R  =  (Scalar `  U )
6 hgmapfval.b . . . 4  |-  B  =  ( Base `  R
)
7 hgmapfval.c . . . 4  |-  C  =  ( (LCDual `  K
) `  W )
8 hgmapfval.s . . . 4  |-  .xb  =  ( .s `  C )
9 hgmapfval.m . . . 4  |-  M  =  ( (HDMap `  K
) `  W )
10 hgmapfval.i . . . 4  |-  I  =  ( (HGMap `  K
) `  W )
11 hgmapfval.k . . . 4  |-  ( ph  ->  ( K  e.  Y  /\  W  e.  H
) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hgmapfval 31209 . . 3  |-  ( ph  ->  I  =  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) ) )
1312fveq1d 5425 . 2  |-  ( ph  ->  ( I `  X
)  =  ( ( x  e.  B  |->  (
iota_ y  e.  B A. v  e.  V  ( M `  ( x 
.x.  v ) )  =  ( y  .xb  ( M `  v ) ) ) ) `  X ) )
14 hgmapval.x . . 3  |-  ( ph  ->  X  e.  B )
15 riotaex 6241 . . 3  |-  ( iota_ y  e.  B A. v  e.  V  ( M `  ( X  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) )  e.  _V
16 oveq1 5764 . . . . . . . 8  |-  ( x  =  X  ->  (
x  .x.  v )  =  ( X  .x.  v ) )
1716fveq2d 5427 . . . . . . 7  |-  ( x  =  X  ->  ( M `  ( x  .x.  v ) )  =  ( M `  ( X  .x.  v ) ) )
1817eqeq1d 2264 . . . . . 6  |-  ( x  =  X  ->  (
( M `  (
x  .x.  v )
)  =  ( y 
.xb  ( M `  v ) )  <->  ( M `  ( X  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) )
1918ralbidv 2534 . . . . 5  |-  ( x  =  X  ->  ( A. v  e.  V  ( M `  ( x 
.x.  v ) )  =  ( y  .xb  ( M `  v ) )  <->  A. v  e.  V  ( M `  ( X 
.x.  v ) )  =  ( y  .xb  ( M `  v ) ) ) )
2019riotabidv 6239 . . . 4  |-  ( x  =  X  ->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v ) )  =  ( y  .xb  ( M `  v )
) )  =  (
iota_ y  e.  B A. v  e.  V  ( M `  ( X 
.x.  v ) )  =  ( y  .xb  ( M `  v ) ) ) )
21 eqid 2256 . . . 4  |-  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) )  =  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) )
2220, 21fvmptg 5499 . . 3  |-  ( ( X  e.  B  /\  ( iota_ y  e.  B A. v  e.  V  ( M `  ( X 
.x.  v ) )  =  ( y  .xb  ( M `  v ) ) )  e.  _V )  ->  ( ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) ) `  X )  =  ( iota_ y  e.  B A. v  e.  V  ( M `  ( X  .x.  v ) )  =  ( y 
.xb  ( M `  v ) ) ) )
2314, 15, 22sylancl 646 . 2  |-  ( ph  ->  ( ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v ) )  =  ( y 
.xb  ( M `  v ) ) ) ) `  X )  =  ( iota_ y  e.  B A. v  e.  V  ( M `  ( X  .x.  v ) )  =  ( y 
.xb  ( M `  v ) ) ) )
2413, 23eqtrd 2288 1  |-  ( ph  ->  ( I `  X
)  =  ( 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   _Vcvv 2740    e. cmpt 4017   ` cfv 4638  (class class class)co 5757   iota_crio 6228   Basecbs 13075  Scalarcsca 13138   .scvsca 13139   LHypclh 29303   DVecHcdvh 30398  LCDualclcd 30906  HDMapchdma 31113  HGMapchg 31206
This theorem is referenced by:  hgmapcl  31212  hgmapvs  31214
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-iota 6190  df-riota 6237  df-hgmap 31207
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