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Theorem hdmapfnN 32022
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 6308 . . 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 2283 . . 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 5372 . 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 2283 . . . 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 2283 . . . 4  |-  ( LSpan `  U )  =  (
LSpan `  U )
9 eqid 2283 . . . 4  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
10 eqid 2283 . . . 4  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
11 eqid 2283 . . . 4  |-  ( (HVMap `  K ) `  W
)  =  ( (HVMap `  K ) `  W
)
12 eqid 2283 . . . 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 32020 . . 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 5335 . 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 224 1  |-  ( ph  ->  S  Fn  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    u. cun 3150   {csn 3640   <.cop 3643   <.cotp 3644    e. cmpt 4077    _I cid 4304    |` cres 4691    Fn wfn 5250   ` cfv 5255   iota_crio 6297   Basecbs 13148   LSpanclspn 15728   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   DVecHcdvh 31268  LCDualclcd 31776  HVMapchvm 31946  HDMap1chdma1 31982  HDMapchdma 31983
This theorem is referenced by:  hdmaprnlem11N  32053  hdmaprnlem17N  32056  hdmaprnN  32057  hdmapf1oN  32058  hgmaprnlem4N  32092
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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