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Theorem hdmapfnN 34832
Description: Functionality of map from vectors to functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
hdmapfn.h  |-  H  =  ( LHyp `  K
)
hdmapfn.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmapfn.v  |-  V  =  ( Base `  U
)
hdmapfn.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmapfn.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
Assertion
Ref Expression
hdmapfnN  |-  ( ph  ->  S  Fn  V )

Proof of Theorem hdmapfnN
Dummy variables  y 
t  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6243 . . 3  |-  ( iota_ y  e.  ( Base `  (
(LCDual `  K ) `  W ) ) A. z  e.  V  ( -.  z  e.  (
( ( LSpan `  U
) `  { <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. } )  u.  (
( LSpan `  U ) `  { t } ) )  ->  y  =  ( ( (HDMap1 `  K ) `  W
) `  <. z ,  ( ( (HDMap1 `  K ) `  W
) `  <. <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. ,  ( ( (HVMap `  K ) `  W
) `  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.
) ,  z >.
) ,  t >.
) ) )  e. 
_V
2 eqid 2402 . . 3  |-  ( t  e.  V  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  K ) `  W ) ) A. z  e.  V  ( -.  z  e.  (
( ( LSpan `  U
) `  { <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. } )  u.  (
( LSpan `  U ) `  { t } ) )  ->  y  =  ( ( (HDMap1 `  K ) `  W
) `  <. z ,  ( ( (HDMap1 `  K ) `  W
) `  <. <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. ,  ( ( (HVMap `  K ) `  W
) `  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.
) ,  z >.
) ,  t >.
) ) ) )  =  ( t  e.  V  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  K ) `  W ) ) A. z  e.  V  ( -.  z  e.  (
( ( LSpan `  U
) `  { <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. } )  u.  (
( LSpan `  U ) `  { t } ) )  ->  y  =  ( ( (HDMap1 `  K ) `  W
) `  <. z ,  ( ( (HDMap1 `  K ) `  W
) `  <. <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. ,  ( ( (HVMap `  K ) `  W
) `  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.
) ,  z >.
) ,  t >.
) ) ) )
31, 2fnmpti 5691 . 2  |-  ( t  e.  V  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  K ) `  W ) ) A. z  e.  V  ( -.  z  e.  (
( ( LSpan `  U
) `  { <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. } )  u.  (
( LSpan `  U ) `  { t } ) )  ->  y  =  ( ( (HDMap1 `  K ) `  W
) `  <. z ,  ( ( (HDMap1 `  K ) `  W
) `  <. <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. ,  ( ( (HVMap `  K ) `  W
) `  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.
) ,  z >.
) ,  t >.
) ) ) )  Fn  V
4 hdmapfn.h . . . 4  |-  H  =  ( LHyp `  K
)
5 eqid 2402 . . . 4  |-  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
6 hdmapfn.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
7 hdmapfn.v . . . 4  |-  V  =  ( Base `  U
)
8 eqid 2402 . . . 4  |-  ( LSpan `  U )  =  (
LSpan `  U )
9 eqid 2402 . . . 4  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
10 eqid 2402 . . . 4  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
11 eqid 2402 . . . 4  |-  ( (HVMap `  K ) `  W
)  =  ( (HVMap `  K ) `  W
)
12 eqid 2402 . . . 4  |-  ( (HDMap1 `  K ) `  W
)  =  ( (HDMap1 `  K ) `  W
)
13 hdmapfn.s . . . 4  |-  S  =  ( (HDMap `  K
) `  W )
14 hdmapfn.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
154, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14hdmapfval 34830 . . 3  |-  ( ph  ->  S  =  ( t  e.  V  |->  ( iota_ y  e.  ( Base `  (
(LCDual `  K ) `  W ) ) A. z  e.  V  ( -.  z  e.  (
( ( LSpan `  U
) `  { <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. } )  u.  (
( LSpan `  U ) `  { t } ) )  ->  y  =  ( ( (HDMap1 `  K ) `  W
) `  <. z ,  ( ( (HDMap1 `  K ) `  W
) `  <. <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. ,  ( ( (HVMap `  K ) `  W
) `  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.
) ,  z >.
) ,  t >.
) ) ) ) )
1615fneq1d 5651 . 2  |-  ( ph  ->  ( S  Fn  V  <->  ( t  e.  V  |->  (
iota_ y  e.  ( Base `  ( (LCDual `  K ) `  W
) ) A. z  e.  V  ( -.  z  e.  ( (
( LSpan `  U ) `  { <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >. } )  u.  (
( LSpan `  U ) `  { t } ) )  ->  y  =  ( ( (HDMap1 `  K ) `  W
) `  <. z ,  ( ( (HDMap1 `  K ) `  W
) `  <. <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >. ,  ( ( (HVMap `  K ) `  W
) `  <. (  _I  |`  ( Base `  K
) ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) ) >.
) ,  z >.
) ,  t >.
) ) ) )  Fn  V ) )
173, 16mpbiri 233 1  |-  ( ph  ->  S  Fn  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753    u. cun 3411   {csn 3971   <.cop 3977   <.cotp 3979    |-> cmpt 4452    _I cid 4732    |` cres 4824    Fn wfn 5563   ` cfv 5568   iota_crio 6238   Basecbs 14839   LSpanclspn 17935   HLchlt 32348   LHypclh 32981   LTrncltrn 33098   DVecHcdvh 34078  LCDualclcd 34586  HVMapchvm 34756  HDMap1chdma1 34792  HDMapchdma 34793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-ot 3980  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-hdmap 34795
This theorem is referenced by:  hdmaprnlem11N  34863  hdmaprnlem17N  34866  hdmaprnN  34867  hdmapf1oN  34868  hgmaprnlem4N  34902
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