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Theorem hdmapfnN 36629
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 6247 . . 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 2467 . . 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 5707 . 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 2467 . . . 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 2467 . . . 4  |-  ( LSpan `  U )  =  (
LSpan `  U )
9 eqid 2467 . . . 4  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
10 eqid 2467 . . . 4  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
11 eqid 2467 . . . 4  |-  ( (HVMap `  K ) `  W
)  =  ( (HVMap `  K ) `  W
)
12 eqid 2467 . . . 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 36627 . . 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 5669 . 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 1379    e. wcel 1767   A.wral 2814    u. cun 3474   {csn 4027   <.cop 4033   <.cotp 4035    |-> cmpt 4505    _I cid 4790    |` cres 5001    Fn wfn 5581   ` cfv 5586   iota_crio 6242   Basecbs 14486   LSpanclspn 17400   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   DVecHcdvh 35875  LCDualclcd 36383  HVMapchvm 36553  HDMap1chdma1 36589  HDMapchdma 36590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-ot 4036  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-hdmap 36592
This theorem is referenced by:  hdmaprnlem11N  36660  hdmaprnlem17N  36663  hdmaprnN  36664  hdmapf1oN  36665  hgmaprnlem4N  36699
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