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Theorem hgmapval 30769
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 30764. (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 30768 . . 3  |-  ( ph  ->  I  =  ( x  e.  B  |->  ( iota_ y  e.  B A. v  e.  V  ( M `  ( x  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) ) )
1312fveq1d 5379 . 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 6194 . . 3  |-  ( iota_ y  e.  B A. v  e.  V  ( M `  ( X  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) )  e.  _V
16 oveq1 5717 . . . . . . . 8  |-  ( x  =  X  ->  (
x  .x.  v )  =  ( X  .x.  v ) )
1716fveq2d 5381 . . . . . . 7  |-  ( x  =  X  ->  ( M `  ( x  .x.  v ) )  =  ( M `  ( X  .x.  v ) ) )
1817eqeq1d 2261 . . . . . 6  |-  ( x  =  X  ->  (
( M `  (
x  .x.  v )
)  =  ( y 
.xb  ( M `  v ) )  <->  ( M `  ( X  .x.  v
) )  =  ( y  .xb  ( M `  v ) ) ) )
1918ralbidv 2527 . . . . 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 6192 . . . 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 2253 . . . 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 5452 . . 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 2285 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 2509   _Vcvv 2727    e. cmpt 3974   ` cfv 4592  (class class class)co 5710   iota_crio 6181   Basecbs 13022  Scalarcsca 13085   .scvsca 13086   LHypclh 28862   DVecHcdvh 29957  LCDualclcd 30465  HDMapchdma 30672  HGMapchg 30765
This theorem is referenced by:  hgmapcl  30771  hgmapvs  30773
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-iota 6143  df-riota 6190  df-hgmap 30766
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