Theorem hdmapffval 37118

Description: Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)

Hypothesis

Ref	Expression
hdmapval.h	|- H = ( LHyp ` K )

Assertion

Ref

Expression

hdmapffval

|- ( K e. X -> (HDMap ` K ) = ( w e. H |-> { a | [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } ) )

Distinct variable groups:   w, H   e, a, i, t, u, v, w, y, z, K

Allowed substitution hints:   H( y, z, v, u, t, e, i, a)   X( y, z, w, v, u, t, e, i, a)

Proof of Theorem hdmapffval

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. X -> K e. _V )
2		fveq2 6191	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		hdmapval.h	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2674	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( Base ` k ) = ( Base ` K ) )
6	5	reseq2d 5396	. . . . . . 7 |- ( k = K -> ( _I |` ( Base ` k ) ) = ( _I |` ( Base ` K ) ) )
7		fveq2 6191	. . . . . . . . 9 |- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
8	7	fveq1d 6193	. . . . . . . 8 |- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
9	8	reseq2d 5396	. . . . . . 7 |- ( k = K -> ( _I |` ( ( LTrn ` k ) ` w ) ) = ( _I |` ( ( LTrn ` K ) ` w ) ) )
10	6, 9	opeq12d 4410	. . . . . 6 |- ( k = K -> <. ( _I |` ( Base ` k ) ) , ( _I |` ( ( LTrn ` k ) ` w ) ) >. = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. )
11		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( DVecH ` k ) = ( DVecH ` K ) )
12	11	fveq1d 6193	. . . . . . 7 |- ( k = K -> ( ( DVecH ` k ) ` w ) = ( ( DVecH ` K ) ` w ) )
13		fveq2 6191	. . . . . . . . . 10 |- ( k = K -> (HDMap1 ` k ) = (HDMap1 ` K ) )
14	13	fveq1d 6193	. . . . . . . . 9 |- ( k = K -> ( (HDMap1 ` k ) ` w ) = ( (HDMap1 ` K ) ` w ) )
15		fveq2 6191	. . . . . . . . . . . . . 14 |- ( k = K -> (LCDual ` k ) = (LCDual ` K ) )
16	15	fveq1d 6193	. . . . . . . . . . . . 13 |- ( k = K -> ( (LCDual ` k ) ` w ) = ( (LCDual ` K ) ` w ) )
17	16	fveq2d 6195	. . . . . . . . . . . 12 |- ( k = K -> ( Base ` ( (LCDual ` k ) ` w ) ) = ( Base ` ( (LCDual ` K ) ` w ) ) )
18		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = K -> (HVMap ` k ) = (HVMap ` K ) )
19	18	fveq1d 6193	. . . . . . . . . . . . . . . . . . . 20 |- ( k = K -> ( (HVMap ` k ) ` w ) = ( (HVMap ` K ) ` w ) )
20	19	fveq1d 6193	. . . . . . . . . . . . . . . . . . 19 |- ( k = K -> ( ( (HVMap ` k ) ` w ) ` e ) = ( ( (HVMap ` K ) ` w ) ` e ) )
21	20	oteq2d 4415	. . . . . . . . . . . . . . . . . 18 |- ( k = K -> <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. = <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. )
22	21	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( k = K -> ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) = ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) )
23	22	oteq2d 4415	. . . . . . . . . . . . . . . 16 |- ( k = K -> <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. = <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. )
24	23	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( k = K -> ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) )
25	24	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( k = K -> ( y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) <-> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) )
26	25	imbi2d 330	. . . . . . . . . . . . 13 |- ( k = K -> ( ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) ) <-> ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) )
27	26	ralbidv 2986	. . . . . . . . . . . 12 |- ( k = K -> ( A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) ) <-> A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) )
28	17, 27	riotaeqbidv 6614	. . . . . . . . . . 11 |- ( k = K -> ( iota_ y e. ( Base ` ( (LCDual ` k ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) ) ) = ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) )
29	28	mpteq2dv 4745	. . . . . . . . . 10 |- ( k = K -> ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` k ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) = ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) )
30	29	eleq2d 2687	. . . . . . . . 9 |- ( k = K -> ( a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` k ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) <-> a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) ) )
31	14, 30	sbceqbid 3442	. . . . . . . 8 |- ( k = K -> ( [. ( (HDMap1 ` k ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` k ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) <-> [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) ) )
32	31	sbcbidv 3490	. . . . . . 7 |- ( k = K -> ( [. ( Base ` u ) / v ]. [. ( (HDMap1 ` k ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` k ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) <-> [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) ) )
33	12, 32	sbceqbid 3442	. . . . . 6 |- ( k = K -> ( [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` k ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` k ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) <-> [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) ) )
34	10, 33	sbceqbid 3442	. . . . 5 |- ( k = K -> ( [. <. ( _I |` ( Base ` k ) ) , ( _I |` ( ( LTrn ` k ) ` w ) ) >. / e ]. [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` k ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` k ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) <-> [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) ) )
35	34	abbidv 2741	. . . 4 |- ( k = K -> { a | [. <. ( _I |` ( Base ` k ) ) , ( _I |` ( ( LTrn ` k ) ` w ) ) >. / e ]. [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` k ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` k ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } = { a | [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } )
36	4, 35	mpteq12dv 4733	. . 3 |- ( k = K -> ( w e. ( LHyp ` k ) |-> { a | [. <. ( _I |` ( Base ` k ) ) , ( _I |` ( ( LTrn ` k ) ` w ) ) >. / e ]. [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` k ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` k ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } ) = ( w e. H |-> { a | [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } ) )
37		df-hdmap 37084	. . 3 |- HDMap = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { a | [. <. ( _I |` ( Base ` k ) ) , ( _I |` ( ( LTrn ` k ) ` w ) ) >. / e ]. [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` k ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` k ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` k ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } ) )
38		fvex 6201	. . . . 5 |- ( LHyp ` K ) e. _V
39	3, 38	eqeltri 2697	. . . 4 |- H e. _V
40	39	mptex 6486	. . 3 |- ( w e. H |-> { a | [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } ) e. _V
41	36, 37, 40	fvmpt 6282	. 2 |- ( K e. _V -> (HDMap ` K ) = ( w e. H |-> { a | [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } ) )
42	1, 41	syl 17	1 |- ( K e. X -> (HDMap ` K ) = ( w e. H |-> { a | [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   [.wsbc 3435   u. cun 3572   {csn 4177   <.cop 4183   <.cotp 4185   |-> cmpt 4729   _I cid 5023   |` cres 5116   ` cfv 5888   iota_crio 6610   Basecbs 15857   LSpanclspn 18971   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:  hdmapfval  37119