Theorem hdmapval 37120

Description: Value of map from vectors to functionals in the closed kernel dual space. This is the function sigma on line 27 above part 9 in [Baer] p. 48. We select a convenient fixed reference vector E to be <. 0 , 1 >. (corresponding to vector u on p. 48 line 7) whose span is the lattice isomorphism map of the fiducial atom P = ( ( oc ` K ) ` W ) (see dvheveccl 36401). ( J ` E ) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 37058 shows in Baer's notation (Fu)* = Gu'). Baer's independent vectors v and w on line 7 correspond to our z that the A. z e. V ranges over. The middle term ( I ` <. E , ( J ` E ) , z >. ) provides isolation to allow E and T to assume the same value without conflict. Closure is shown by hdmapcl 37122. If a separate auxiliary vector is known, hdmapval2 37124 provides a version without quantification. (Contributed by NM, 15-May-2015.)

Hypotheses

Ref	Expression
hdmapval.h	|- H = ( LHyp ` K )
hdmapfval.e	|- E = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >.
hdmapfval.u	|- U = ( ( DVecH ` K ) ` W )
hdmapfval.v	|- V = ( Base ` U )
hdmapfval.n	|- N = ( LSpan ` U )
hdmapfval.c	|- C = ( (LCDual ` K ) ` W )
hdmapfval.d	|- D = ( Base ` C )
hdmapfval.j	|- J = ( (HVMap ` K ) ` W )
hdmapfval.i	|- I = ( (HDMap1 ` K ) ` W )
hdmapfval.s	|- S = ( (HDMap ` K ) ` W )
hdmapfval.k	|- ( ph -> ( K e. A /\ W e. H ) )
hdmapval.t	|- ( ph -> T e. V )

Assertion

Ref	Expression
hdmapval	|- ( ph -> ( S ` T ) = ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , T >. ) ) ) )

Distinct variable groups:   y, z, K   y, D   y, E, z   y, I, z   y, U, z   y, V, z   y, W, z   y, T, z

Allowed substitution hints:   ph( y, z)   A( y, z)   C( y, z)   D( z)   S( y, z)   H( y, z)   J( y, z)   N( y, z)

Proof of Theorem hdmapval

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmapval.h	. . . 4 |- H = ( LHyp ` K )
2		hdmapfval.e	. . . 4 |- E = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >.
3		hdmapfval.u	. . . 4 |- U = ( ( DVecH ` K ) ` W )
4		hdmapfval.v	. . . 4 |- V = ( Base ` U )
5		hdmapfval.n	. . . 4 |- N = ( LSpan ` U )
6		hdmapfval.c	. . . 4 |- C = ( (LCDual ` K ) ` W )
7		hdmapfval.d	. . . 4 |- D = ( Base ` C )
8		hdmapfval.j	. . . 4 |- J = ( (HVMap ` K ) ` W )
9		hdmapfval.i	. . . 4 |- I = ( (HDMap1 ` K ) ` W )
10		hdmapfval.s	. . . 4 |- S = ( (HDMap ` K ) ` W )
11		hdmapfval.k	. . . 4 |- ( ph -> ( K e. A /\ W e. H ) )
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	hdmapfval 37119	. . 3 |- ( ph -> S = ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) )
13	12	fveq1d 6193	. 2 |- ( ph -> ( S ` T ) = ( ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) ` T ) )
14		hdmapval.t	. . 3 |- ( ph -> T e. V )
15		riotaex 6615	. . 3 |- ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , T >. ) ) ) e. _V
16		sneq 4187	. . . . . . . . . . 11 |- ( t = T -> { t } = { T } )
17	16	fveq2d 6195	. . . . . . . . . 10 |- ( t = T -> ( N ` { t } ) = ( N ` { T } ) )
18	17	uneq2d 3767	. . . . . . . . 9 |- ( t = T -> ( ( N ` { E } ) u. ( N ` { t } ) ) = ( ( N ` { E } ) u. ( N ` { T } ) ) )
19	18	eleq2d 2687	. . . . . . . 8 |- ( t = T -> ( z e. ( ( N ` { E } ) u. ( N ` { t } ) ) <-> z e. ( ( N ` { E } ) u. ( N ` { T } ) ) ) )
20	19	notbid 308	. . . . . . 7 |- ( t = T -> ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) <-> -. z e. ( ( N ` { E } ) u. ( N ` { T } ) ) ) )
21		oteq3 4413	. . . . . . . . 9 |- ( t = T -> <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. = <. z , ( I ` <. E , ( J ` E ) , z >. ) , T >. )
22	21	fveq2d 6195	. . . . . . . 8 |- ( t = T -> ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , T >. ) )
23	22	eqeq2d 2632	. . . . . . 7 |- ( t = T -> ( y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) <-> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , T >. ) ) )
24	20, 23	imbi12d 334	. . . . . 6 |- ( t = T -> ( ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) <-> ( -. z e. ( ( N ` { E } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , T >. ) ) ) )
25	24	ralbidv 2986	. . . . 5 |- ( t = T -> ( A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) <-> A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , T >. ) ) ) )
26	25	riotabidv 6613	. . . 4 |- ( t = T -> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) = ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , T >. ) ) ) )
27		eqid 2622	. . . 4 |- ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) = ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) )
28	26, 27	fvmptg 6280	. . 3 |- ( ( T e. V /\ ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , T >. ) ) ) e. _V ) -> ( ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) ` T ) = ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , T >. ) ) ) )
29	14, 15, 28	sylancl 694	. 2 |- ( ph -> ( ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) ` T ) = ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , T >. ) ) ) )
30	13, 29	eqtrd 2656	1 |- ( ph -> ( S ` T ) = ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , T >. ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   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:  hdmapcl  37122  hdmapval2lem  37123