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Theorem hdmapval 32021
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 31302). 
( J `  E
) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 31959 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 32023. If a separate auxiliary vector is known, hdmapval2 32025 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
Dummy variable  t is distinct from all other variables.
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 32020 . . 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 5527 . 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 6308 . . 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 3651 . . . . . . . . . . 11  |-  ( t  =  T  ->  { t }  =  { T } )
1716fveq2d 5529 . . . . . . . . . 10  |-  ( t  =  T  ->  ( N `  { t } )  =  ( N `  { T } ) )
1817uneq2d 3329 . . . . . . . . 9  |-  ( t  =  T  ->  (
( N `  { E } )  u.  ( N `  { t } ) )  =  ( ( N `  { E } )  u.  ( N `  { T } ) ) )
1918eleq2d 2350 . . . . . . . 8  |-  ( t  =  T  ->  (
z  e.  ( ( N `  { E } )  u.  ( N `  { t } ) )  <->  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) ) ) )
2019notbid 285 . . . . . . 7  |-  ( t  =  T  ->  ( -.  z  e.  (
( N `  { E } )  u.  ( N `  { t } ) )  <->  -.  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) ) ) )
21 oteq3 3807 . . . . . . . . 9  |-  ( t  =  T  ->  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  t
>.  =  <. z ,  ( I `  <. E ,  ( J `  E ) ,  z
>. ) ,  T >. )
2221fveq2d 5529 . . . . . . . 8  |-  ( t  =  T  ->  (
I `  <. z ,  ( I `  <. E ,  ( J `  E ) ,  z
>. ) ,  t >.
)  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  T >. ) )
2322eqeq2d 2294 . . . . . . 7  |-  ( t  =  T  ->  (
y  =  ( I `
 <. z ,  ( I `  <. E , 
( J `  E
) ,  z >.
) ,  t >.
)  <->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `
 E ) ,  z >. ) ,  T >. ) ) )
2420, 23imbi12d 311 . . . . . 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 2563 . . . . 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 6306 . . . 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 2283 . . . 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 5600 . . 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 643 . 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 2315 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 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    u. cun 3150   {csn 3640   <.cop 3643   <.cotp 3644    e. cmpt 4077    _I cid 4304    |` cres 4691   ` cfv 5255   iota_crio 6297   Basecbs 13148   LSpanclspn 15728   LHypclh 30173   LTrncltrn 30290   DVecHcdvh 31268  LCDualclcd 31776  HVMapchvm 31946  HDMap1chdma1 31982  HDMapchdma 31983
This theorem is referenced by:  hdmapcl  32023  hdmapval2lem  32024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 6304  df-hdmap 31985
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