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Theorem hgmapval 32750
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 32745. (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
Dummy variable  x is distinct from all other variables.
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 32749 . . 3  |-  ( ph  ->  I  =  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) ) )
1312fveq1d 5732 . 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 6555 . . 3  |-  ( iota_ y  e.  B A. v  e.  V  ( M `  ( X  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) )  e.  _V
16 oveq1 6090 . . . . . . . 8  |-  ( x  =  X  ->  (
x  .x.  v )  =  ( X  .x.  v ) )
1716fveq2d 5734 . . . . . . 7  |-  ( x  =  X  ->  ( M `  ( x  .x.  v ) )  =  ( M `  ( X  .x.  v ) ) )
1817eqeq1d 2446 . . . . . 6  |-  ( x  =  X  ->  (
( M `  (
x  .x.  v )
)  =  ( y 
.xb  ( M `  v ) )  <->  ( M `  ( X  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) )
1918ralbidv 2727 . . . . 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 6553 . . . 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 2438 . . . 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 5806 . . 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 645 . 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 2470 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 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    e. cmpt 4268   ` cfv 5456  (class class class)co 6083   iota_crio 6544   Basecbs 13471  Scalarcsca 13534   .scvsca 13535   LHypclh 30843   DVecHcdvh 31938  LCDualclcd 32446  HDMapchdma 32653  HGMapchg 32746
This theorem is referenced by:  hgmapcl  32752  hgmapvs  32754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-riota 6551  df-hgmap 32747
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