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Theorem hdmapfnN 32630
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 6553 . . 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 2436 . . 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 5573 . 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 2436 . . . 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 2436 . . . 4  |-  ( LSpan `  U )  =  (
LSpan `  U )
9 eqid 2436 . . . 4  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
10 eqid 2436 . . . 4  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
11 eqid 2436 . . . 4  |-  ( (HVMap `  K ) `  W
)  =  ( (HVMap `  K ) `  W
)
12 eqid 2436 . . . 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 32628 . . 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 5536 . 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 225 1  |-  ( ph  ->  S  Fn  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    u. cun 3318   {csn 3814   <.cop 3817   <.cotp 3818    e. cmpt 4266    _I cid 4493    |` cres 4880    Fn wfn 5449   ` cfv 5454   iota_crio 6542   Basecbs 13469   LSpanclspn 16047   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   DVecHcdvh 31876  LCDualclcd 32384  HVMapchvm 32554  HDMap1chdma1 32590  HDMapchdma 32591
This theorem is referenced by:  hdmaprnlem11N  32661  hdmaprnlem17N  32664  hdmaprnN  32665  hdmapf1oN  32666  hgmaprnlem4N  32700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-ot 3824  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-riota 6549  df-hdmap 32593
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