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Theorem hdmapfnN 35796
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 6160 . . 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 2452 . . 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 5642 . 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 2452 . . . 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 2452 . . . 4  |-  ( LSpan `  U )  =  (
LSpan `  U )
9 eqid 2452 . . . 4  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
10 eqid 2452 . . . 4  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
11 eqid 2452 . . . 4  |-  ( (HVMap `  K ) `  W
)  =  ( (HVMap `  K ) `  W
)
12 eqid 2452 . . . 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 35794 . . 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 5604 . 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 369    = wceq 1370    e. wcel 1758   A.wral 2796    u. cun 3429   {csn 3980   <.cop 3986   <.cotp 3988    |-> cmpt 4453    _I cid 4734    |` cres 4945    Fn wfn 5516   ` cfv 5521   iota_crio 6155   Basecbs 14287   LSpanclspn 17170   HLchlt 33314   LHypclh 33947   LTrncltrn 34064   DVecHcdvh 35042  LCDualclcd 35550  HVMapchvm 35720  HDMap1chdma1 35756  HDMapchdma 35757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-ot 3989  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-hdmap 35759
This theorem is referenced by:  hdmaprnlem11N  35827  hdmaprnlem17N  35830  hdmaprnN  35831  hdmapf1oN  35832  hgmaprnlem4N  35866
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