Theorem hvmapfval 37048

Description: Map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)

Hypotheses

Ref	Expression
hvmapval.h	|- H = ( LHyp ` K )
hvmapval.u	|- U = ( ( DVecH ` K ) ` W )
hvmapval.o	|- O = ( ( ocH ` K ) ` W )
hvmapval.v	|- V = ( Base ` U )
hvmapval.p	|- .+ = ( +g ` U )
hvmapval.t	|- .x. = ( .s ` U )
hvmapval.z	|- .0. = ( 0g ` U )
hvmapval.s	|- S = (Scalar ` U )
hvmapval.r	|- R = ( Base ` S )
hvmapval.m	|- M = ( (HVMap ` K ) ` W )
hvmapval.k	|- ( ph -> ( K e. A /\ W e. H ) )

Assertion

Ref	Expression
hvmapfval	|- ( ph -> M = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) ) )

Distinct variable groups:   t, j, v, x, K   t, W   t, O   R, j   x, V   j, W, v, x   x, .0.

Allowed substitution hints:   ph( x, v, t, j)   A( x, v, t, j)   .+ ( x, v, t, j)   R( x, v, t)   S( x, v, t, j)   .x. ( x, v, t, j)   U( x, v, t, j)   H( x, v, t, j)   M( x, v, t, j)   O( x, v, j)   V( v, t, j)   .0. ( v, t, j)

Proof of Theorem hvmapfval

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hvmapval.k	. 2 |- ( ph -> ( K e. A /\ W e. H ) )
2		hvmapval.m	. . . 4 |- M = ( (HVMap ` K ) ` W )
3		hvmapval.h	. . . . . 6 |- H = ( LHyp ` K )
4	3	hvmapffval 37047	. . . . 5 |- ( K e. A -> (HVMap ` K ) = ( w e. H |-> ( x e. ( ( Base ` ( ( DVecH ` K ) ` w ) ) \ { ( 0g ` ( ( DVecH ` K ) ` w ) ) } ) |-> ( v e. ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( iota_ j e. ( Base ` (Scalar ` ( ( DVecH ` K ) ` w ) ) ) E. t e. ( ( ( ocH ` K ) ` w ) ` { x } ) v = ( t ( +g ` ( ( DVecH ` K ) ` w ) ) ( j ( .s ` ( ( DVecH ` K ) ` w ) ) x ) ) ) ) ) ) )
5	4	fveq1d 6193	. . . 4 |- ( K e. A -> ( (HVMap ` K ) ` W ) = ( ( w e. H |-> ( x e. ( ( Base ` ( ( DVecH ` K ) ` w ) ) \ { ( 0g ` ( ( DVecH ` K ) ` w ) ) } ) |-> ( v e. ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( iota_ j e. ( Base ` (Scalar ` ( ( DVecH ` K ) ` w ) ) ) E. t e. ( ( ( ocH ` K ) ` w ) ` { x } ) v = ( t ( +g ` ( ( DVecH ` K ) ` w ) ) ( j ( .s ` ( ( DVecH ` K ) ` w ) ) x ) ) ) ) ) ) ` W ) )
6	2, 5	syl5eq 2668	. . 3 |- ( K e. A -> M = ( ( w e. H |-> ( x e. ( ( Base ` ( ( DVecH ` K ) ` w ) ) \ { ( 0g ` ( ( DVecH ` K ) ` w ) ) } ) |-> ( v e. ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( iota_ j e. ( Base ` (Scalar ` ( ( DVecH ` K ) ` w ) ) ) E. t e. ( ( ( ocH ` K ) ` w ) ` { x } ) v = ( t ( +g ` ( ( DVecH ` K ) ` w ) ) ( j ( .s ` ( ( DVecH ` K ) ` w ) ) x ) ) ) ) ) ) ` W ) )
7		fveq2 6191	. . . . . . . . 9 |- ( w = W -> ( ( DVecH ` K ) ` w ) = ( ( DVecH ` K ) ` W ) )
8		hvmapval.u	. . . . . . . . 9 |- U = ( ( DVecH ` K ) ` W )
9	7, 8	syl6eqr 2674	. . . . . . . 8 |- ( w = W -> ( ( DVecH ` K ) ` w ) = U )
10	9	fveq2d 6195	. . . . . . 7 |- ( w = W -> ( Base ` ( ( DVecH ` K ) ` w ) ) = ( Base ` U ) )
11		hvmapval.v	. . . . . . 7 |- V = ( Base ` U )
12	10, 11	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( Base ` ( ( DVecH ` K ) ` w ) ) = V )
13	9	fveq2d 6195	. . . . . . . 8 |- ( w = W -> ( 0g ` ( ( DVecH ` K ) ` w ) ) = ( 0g ` U ) )
14		hvmapval.z	. . . . . . . 8 |- .0. = ( 0g ` U )
15	13, 14	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( 0g ` ( ( DVecH ` K ) ` w ) ) = .0. )
16	15	sneqd 4189	. . . . . 6 |- ( w = W -> { ( 0g ` ( ( DVecH ` K ) ` w ) ) } = { .0. } )
17	12, 16	difeq12d 3729	. . . . 5 |- ( w = W -> ( ( Base ` ( ( DVecH ` K ) ` w ) ) \ { ( 0g ` ( ( DVecH ` K ) ` w ) ) } ) = ( V \ { .0. } ) )
18	9	fveq2d 6195	. . . . . . . . . 10 |- ( w = W -> (Scalar ` ( ( DVecH ` K ) ` w ) ) = (Scalar ` U ) )
19		hvmapval.s	. . . . . . . . . 10 |- S = (Scalar ` U )
20	18, 19	syl6eqr 2674	. . . . . . . . 9 |- ( w = W -> (Scalar ` ( ( DVecH ` K ) ` w ) ) = S )
21	20	fveq2d 6195	. . . . . . . 8 |- ( w = W -> ( Base ` (Scalar ` ( ( DVecH ` K ) ` w ) ) ) = ( Base ` S ) )
22		hvmapval.r	. . . . . . . 8 |- R = ( Base ` S )
23	21, 22	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( Base ` (Scalar ` ( ( DVecH ` K ) ` w ) ) ) = R )
24		fveq2 6191	. . . . . . . . . 10 |- ( w = W -> ( ( ocH ` K ) ` w ) = ( ( ocH ` K ) ` W ) )
25		hvmapval.o	. . . . . . . . . 10 |- O = ( ( ocH ` K ) ` W )
26	24, 25	syl6eqr 2674	. . . . . . . . 9 |- ( w = W -> ( ( ocH ` K ) ` w ) = O )
27	26	fveq1d 6193	. . . . . . . 8 |- ( w = W -> ( ( ( ocH ` K ) ` w ) ` { x } ) = ( O ` { x } ) )
28	9	fveq2d 6195	. . . . . . . . . . 11 |- ( w = W -> ( +g ` ( ( DVecH ` K ) ` w ) ) = ( +g ` U ) )
29		hvmapval.p	. . . . . . . . . . 11 |- .+ = ( +g ` U )
30	28, 29	syl6eqr 2674	. . . . . . . . . 10 |- ( w = W -> ( +g ` ( ( DVecH ` K ) ` w ) ) = .+ )
31		eqidd 2623	. . . . . . . . . 10 |- ( w = W -> t = t )
32	9	fveq2d 6195	. . . . . . . . . . . 12 |- ( w = W -> ( .s ` ( ( DVecH ` K ) ` w ) ) = ( .s ` U ) )
33		hvmapval.t	. . . . . . . . . . . 12 |- .x. = ( .s ` U )
34	32, 33	syl6eqr 2674	. . . . . . . . . . 11 |- ( w = W -> ( .s ` ( ( DVecH ` K ) ` w ) ) = .x. )
35	34	oveqd 6667	. . . . . . . . . 10 |- ( w = W -> ( j ( .s ` ( ( DVecH ` K ) ` w ) ) x ) = ( j .x. x ) )
36	30, 31, 35	oveq123d 6671	. . . . . . . . 9 |- ( w = W -> ( t ( +g ` ( ( DVecH ` K ) ` w ) ) ( j ( .s ` ( ( DVecH ` K ) ` w ) ) x ) ) = ( t .+ ( j .x. x ) ) )
37	36	eqeq2d 2632	. . . . . . . 8 |- ( w = W -> ( v = ( t ( +g ` ( ( DVecH ` K ) ` w ) ) ( j ( .s ` ( ( DVecH ` K ) ` w ) ) x ) ) <-> v = ( t .+ ( j .x. x ) ) ) )
38	27, 37	rexeqbidv 3153	. . . . . . 7 |- ( w = W -> ( E. t e. ( ( ( ocH ` K ) ` w ) ` { x } ) v = ( t ( +g ` ( ( DVecH ` K ) ` w ) ) ( j ( .s ` ( ( DVecH ` K ) ` w ) ) x ) ) <-> E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) )
39	23, 38	riotaeqbidv 6614	. . . . . 6 |- ( w = W -> ( iota_ j e. ( Base ` (Scalar ` ( ( DVecH ` K ) ` w ) ) ) E. t e. ( ( ( ocH ` K ) ` w ) ` { x } ) v = ( t ( +g ` ( ( DVecH ` K ) ` w ) ) ( j ( .s ` ( ( DVecH ` K ) ` w ) ) x ) ) ) = ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) )
40	12, 39	mpteq12dv 4733	. . . . 5 |- ( w = W -> ( v e. ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( iota_ j e. ( Base ` (Scalar ` ( ( DVecH ` K ) ` w ) ) ) E. t e. ( ( ( ocH ` K ) ` w ) ` { x } ) v = ( t ( +g ` ( ( DVecH ` K ) ` w ) ) ( j ( .s ` ( ( DVecH ` K ) ` w ) ) x ) ) ) ) = ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) )
41	17, 40	mpteq12dv 4733	. . . 4 |- ( w = W -> ( x e. ( ( Base ` ( ( DVecH ` K ) ` w ) ) \ { ( 0g ` ( ( DVecH ` K ) ` w ) ) } ) |-> ( v e. ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( iota_ j e. ( Base ` (Scalar ` ( ( DVecH ` K ) ` w ) ) ) E. t e. ( ( ( ocH ` K ) ` w ) ` { x } ) v = ( t ( +g ` ( ( DVecH ` K ) ` w ) ) ( j ( .s ` ( ( DVecH ` K ) ` w ) ) x ) ) ) ) ) = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) ) )
42		eqid 2622	. . . 4 |- ( w e. H |-> ( x e. ( ( Base ` ( ( DVecH ` K ) ` w ) ) \ { ( 0g ` ( ( DVecH ` K ) ` w ) ) } ) |-> ( v e. ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( iota_ j e. ( Base ` (Scalar ` ( ( DVecH ` K ) ` w ) ) ) E. t e. ( ( ( ocH ` K ) ` w ) ` { x } ) v = ( t ( +g ` ( ( DVecH ` K ) ` w ) ) ( j ( .s ` ( ( DVecH ` K ) ` w ) ) x ) ) ) ) ) ) = ( w e. H |-> ( x e. ( ( Base ` ( ( DVecH ` K ) ` w ) ) \ { ( 0g ` ( ( DVecH ` K ) ` w ) ) } ) |-> ( v e. ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( iota_ j e. ( Base ` (Scalar ` ( ( DVecH ` K ) ` w ) ) ) E. t e. ( ( ( ocH ` K ) ` w ) ` { x } ) v = ( t ( +g ` ( ( DVecH ` K ) ` w ) ) ( j ( .s ` ( ( DVecH ` K ) ` w ) ) x ) ) ) ) ) )
43		fvex 6201	. . . . . . 7 |- ( Base ` U ) e. _V
44	11, 43	eqeltri 2697	. . . . . 6 |- V e. _V
45		difexg 4808	. . . . . 6 |- ( V e. _V -> ( V \ { .0. } ) e. _V )
46	44, 45	ax-mp 5	. . . . 5 |- ( V \ { .0. } ) e. _V
47	46	mptex 6486	. . . 4 |- ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) ) e. _V
48	41, 42, 47	fvmpt 6282	. . 3 |- ( W e. H -> ( ( w e. H |-> ( x e. ( ( Base ` ( ( DVecH ` K ) ` w ) ) \ { ( 0g ` ( ( DVecH ` K ) ` w ) ) } ) |-> ( v e. ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( iota_ j e. ( Base ` (Scalar ` ( ( DVecH ` K ) ` w ) ) ) E. t e. ( ( ( ocH ` K ) ` w ) ` { x } ) v = ( t ( +g ` ( ( DVecH ` K ) ` w ) ) ( j ( .s ` ( ( DVecH ` K ) ` w ) ) x ) ) ) ) ) ) ` W ) = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) ) )
49	6, 48	sylan9eq 2676	. 2 |- ( ( K e. A /\ W e. H ) -> M = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) ) )
50	1, 49	syl 17	1 |- ( ph -> M = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   \ cdif 3571   {csn 4177   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   LHypclh 35270   DVecHcdvh 36367   ocHcoch 36636  HVMapchvm 37045

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-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-ov 6653  df-hvmap 37046

This theorem is referenced by:  hvmapval  37049  hvmap1o  37052  hvmaplkr  37057