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Theorem hdmapfnN 37121
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 6615 . . 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 2622 . . 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 6022 . 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 2622 . . . 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 2622 . . . 4 |- ( LSpan ` U ) = ( LSpan ` U )
9 eqid 2622 . . . 4 |- ( (LCDual ` K ) ` W ) = ( (LCDual ` K ) ` W )
10 eqid 2622 . . . 4 |- ( Base ` ( (LCDual ` K ) ` W ) ) = ( Base ` ( (LCDual ` K ) ` W ) )
11 eqid 2622 . . . 4 |- ( (HVMap ` K ) ` W ) = ( (HVMap ` K ) ` W )
12 eqid 2622 . . . 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 37119 . . 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 5981 . 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 248 1 |- ( ph -> S Fn V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   u. cun 3572   {csn 4177   <.cop 4183   <.cotp 4185   |-> cmpt 4729   _I cid 5023   |` cres 5116   Fn wfn 5883   ` cfv 5888   iota_crio 6610   Basecbs 15857   LSpanclspn 18971   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   DVecHcdvh 36367  LCDualclcd 36875  HVMapchvm 37045  HDMap1chdma1 37081  HDMapchdma 37082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-hdmap 37084
This theorem is referenced by:  hdmaprnlem11N  37152  hdmaprnlem17N  37155  hdmaprnN  37156  hdmapf1oN  37157  hgmaprnlem4N  37191
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