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Theorem hgmapval 32373
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 32368. (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 32372 . . 3  |-  ( ph  ->  I  =  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) ) )
1312fveq1d 5689 . 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 6512 . . 3  |-  ( iota_ y  e.  B A. v  e.  V  ( M `  ( X  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) )  e.  _V
16 oveq1 6047 . . . . . . . 8  |-  ( x  =  X  ->  (
x  .x.  v )  =  ( X  .x.  v ) )
1716fveq2d 5691 . . . . . . 7  |-  ( x  =  X  ->  ( M `  ( x  .x.  v ) )  =  ( M `  ( X  .x.  v ) ) )
1817eqeq1d 2412 . . . . . 6  |-  ( x  =  X  ->  (
( M `  (
x  .x.  v )
)  =  ( y 
.xb  ( M `  v ) )  <->  ( M `  ( X  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) )
1918ralbidv 2686 . . . . 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 6510 . . . 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 2404 . . . 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 5763 . . 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 644 . 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 2436 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 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   iota_crio 6501   Basecbs 13424  Scalarcsca 13487   .scvsca 13488   LHypclh 30466   DVecHcdvh 31561  LCDualclcd 32069  HDMapchdma 32276  HGMapchg 32369
This theorem is referenced by:  hgmapcl  32375  hgmapvs  32377
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-riota 6508  df-hgmap 32370
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