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Theorem hdmapval 31288
Description: Value of map from vectors to functionals in the closed kernel dual space. This is the function sigma on line 27 above part 9 in [Baer] p. 48. We select a convenient fixed reference vector  E to be  <. 0 ,  1 >. (corresponding to vector u on p. 48 line 7) whose span is the lattice isomorphism map of the fiducial atom  P  =  ( ( oc `  K
) `  W ) (see dvheveccl 30569). 
( J `  E
) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 31226 shows in Baer's notation (Fu)* = Gu'). Baer's independent vectors v and w on line 7 correspond to our  z that the  A. z  e.  V ranges over. The middle term  ( I `  <. E ,  ( J `
 E ) ,  z >. ) provides isolation to allow  E and  T to assume the same value without conflict. Closure is shown by hdmapcl 31290. If a separate auxiliary vector is known, hdmapval2 31292 provides a version without quantification. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmapval.h  |-  H  =  ( LHyp `  K
)
hdmapfval.e  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
hdmapfval.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmapfval.v  |-  V  =  ( Base `  U
)
hdmapfval.n  |-  N  =  ( LSpan `  U )
hdmapfval.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmapfval.d  |-  D  =  ( Base `  C
)
hdmapfval.j  |-  J  =  ( (HVMap `  K
) `  W )
hdmapfval.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmapfval.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmapfval.k  |-  ( ph  ->  ( K  e.  A  /\  W  e.  H
) )
hdmapval.t  |-  ( ph  ->  T  e.  V )
Assertion
Ref Expression
hdmapval  |-  ( ph  ->  ( S `  T
)  =  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  T >. ) ) ) )
Distinct variable groups:    y, z, K    y, D    y, E, z    y, I, z    y, U, z    y, V, z   
y, W, z    y, T, z
Allowed substitution hints:    ph( y, z)    A( y, z)    C( y, z)    D( z)    S( y, z)    H( y, z)    J( y, z)    N( y, z)

Proof of Theorem hdmapval
StepHypRef Expression
1 hdmapval.h . . . 4  |-  H  =  ( LHyp `  K
)
2 hdmapfval.e . . . 4  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
3 hdmapfval.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
4 hdmapfval.v . . . 4  |-  V  =  ( Base `  U
)
5 hdmapfval.n . . . 4  |-  N  =  ( LSpan `  U )
6 hdmapfval.c . . . 4  |-  C  =  ( (LCDual `  K
) `  W )
7 hdmapfval.d . . . 4  |-  D  =  ( Base `  C
)
8 hdmapfval.j . . . 4  |-  J  =  ( (HVMap `  K
) `  W )
9 hdmapfval.i . . . 4  |-  I  =  ( (HDMap1 `  K
) `  W )
10 hdmapfval.s . . . 4  |-  S  =  ( (HDMap `  K
) `  W )
11 hdmapfval.k . . . 4  |-  ( ph  ->  ( K  e.  A  /\  W  e.  H
) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hdmapfval 31287 . . 3  |-  ( ph  ->  S  =  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) ) )
1312fveq1d 5487 . 2  |-  ( ph  ->  ( S `  T
)  =  ( ( t  e.  V  |->  (
iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  {
t } ) )  ->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  t
>. ) ) ) ) `
 T ) )
14 hdmapval.t . . 3  |-  ( ph  ->  T  e.  V )
15 riotaex 6303 . . 3  |-  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  T >. ) ) )  e.  _V
16 sneq 3652 . . . . . . . . . . 11  |-  ( t  =  T  ->  { t }  =  { T } )
1716fveq2d 5489 . . . . . . . . . 10  |-  ( t  =  T  ->  ( N `  { t } )  =  ( N `  { T } ) )
1817uneq2d 3330 . . . . . . . . 9  |-  ( t  =  T  ->  (
( N `  { E } )  u.  ( N `  { t } ) )  =  ( ( N `  { E } )  u.  ( N `  { T } ) ) )
1918eleq2d 2351 . . . . . . . 8  |-  ( t  =  T  ->  (
z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  <->  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) ) ) )
2019notbid 287 . . . . . . 7  |-  ( t  =  T  ->  ( -.  z  e.  (
( N `  { E } )  u.  ( N `  { t } ) )  <->  -.  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) ) ) )
21 oteq3 3808 . . . . . . . . 9  |-  ( t  =  T  ->  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  t
>.  =  <. z ,  ( I `  <. E ,  ( J `  E ) ,  z
>. ) ,  T >. )
2221fveq2d 5489 . . . . . . . 8  |-  ( t  =  T  ->  (
I `  <. z ,  ( I `  <. E ,  ( J `  E ) ,  z
>. ) ,  t >.
)  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  T >. ) )
2322eqeq2d 2295 . . . . . . 7  |-  ( t  =  T  ->  (
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
)  <->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  T >. ) ) )
2420, 23imbi12d 313 . . . . . 6  |-  ( t  =  T  ->  (
( -.  z  e.  ( ( N `  { E } )  u.  ( N `  {
t } ) )  ->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  t
>. ) )  <->  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  T >. ) ) ) )
2524ralbidv 2564 . . . . 5  |-  ( t  =  T  ->  ( A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  {
t } ) )  ->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  t
>. ) )  <->  A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  T >. ) ) ) )
2625riotabidv 6301 . . . 4  |-  ( t  =  T  ->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  (
( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) )  =  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  T >. ) ) ) )
27 eqid 2284 . . . 4  |-  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) )  =  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  {
t } ) )  ->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  t
>. ) ) ) )
2826, 27fvmptg 5561 . . 3  |-  ( ( T  e.  V  /\  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  T >. ) ) )  e.  _V )  ->  ( ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
) ) ) ) `
 T )  =  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  T >. ) ) ) )
2914, 15, 28sylancl 646 . 2  |-  ( ph  ->  ( ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  {
t } ) )  ->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  t
>. ) ) ) ) `
 T )  =  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  T >. ) ) ) )
3013, 29eqtrd 2316 1  |-  ( ph  ->  ( S `  T
)  =  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  T >. ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1628    e. wcel 1688   A.wral 2544   _Vcvv 2789    u. cun 3151   {csn 3641   <.cop 3644   <.cotp 3645    e. cmpt 4078    _I cid 4303    |` cres 4690   ` cfv 5221   iota_crio 6290   Basecbs 13142   LSpanclspn 15722   LHypclh 29440   LTrncltrn 29557   DVecHcdvh 30535  LCDualclcd 31043  HVMapchvm 31213  HDMap1chdma1 31249  HDMapchdma 31250
This theorem is referenced by:  hdmapcl  31290  hdmapval2lem  31291
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-ot 3651  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-iota 6252  df-riota 6299  df-hdmap 31252
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