Theorem hvmapval 37049

Description: Value of 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 ) )
hvmapval.x	|- ( ph -> X e. ( V \ { .0. } ) )

Assertion

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

Distinct variable groups:   t, j, v, K   t, W   t, O   R, j   j, W, v   v, V   j, X, t, v

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

Proof of Theorem hvmapval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hvmapval.h	. . . 4 |- H = ( LHyp ` K )
2		hvmapval.u	. . . 4 |- U = ( ( DVecH ` K ) ` W )
3		hvmapval.o	. . . 4 |- O = ( ( ocH ` K ) ` W )
4		hvmapval.v	. . . 4 |- V = ( Base ` U )
5		hvmapval.p	. . . 4 |- .+ = ( +g ` U )
6		hvmapval.t	. . . 4 |- .x. = ( .s ` U )
7		hvmapval.z	. . . 4 |- .0. = ( 0g ` U )
8		hvmapval.s	. . . 4 |- S = (Scalar ` U )
9		hvmapval.r	. . . 4 |- R = ( Base ` S )
10		hvmapval.m	. . . 4 |- M = ( (HVMap ` K ) ` W )
11		hvmapval.k	. . . 4 |- ( ph -> ( K e. A /\ W e. H ) )
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	hvmapfval 37048	. . 3 |- ( ph -> M = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) ) )
13	12	fveq1d 6193	. 2 |- ( ph -> ( M ` X ) = ( ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) ) ` X ) )
14		hvmapval.x	. . 3 |- ( ph -> X e. ( V \ { .0. } ) )
15		fvex 6201	. . . . 5 |- ( Base ` U ) e. _V
16	4, 15	eqeltri 2697	. . . 4 |- V e. _V
17	16	mptex 6486	. . 3 |- ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { X } ) v = ( t .+ ( j .x. X ) ) ) ) e. _V
18		sneq 4187	. . . . . . . 8 |- ( x = X -> { x } = { X } )
19	18	fveq2d 6195	. . . . . . 7 |- ( x = X -> ( O ` { x } ) = ( O ` { X } ) )
20		oveq2 6658	. . . . . . . . 9 |- ( x = X -> ( j .x. x ) = ( j .x. X ) )
21	20	oveq2d 6666	. . . . . . . 8 |- ( x = X -> ( t .+ ( j .x. x ) ) = ( t .+ ( j .x. X ) ) )
22	21	eqeq2d 2632	. . . . . . 7 |- ( x = X -> ( v = ( t .+ ( j .x. x ) ) <-> v = ( t .+ ( j .x. X ) ) ) )
23	19, 22	rexeqbidv 3153	. . . . . 6 |- ( x = X -> ( E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) <-> E. t e. ( O ` { X } ) v = ( t .+ ( j .x. X ) ) ) )
24	23	riotabidv 6613	. . . . 5 |- ( x = X -> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) = ( iota_ j e. R E. t e. ( O ` { X } ) v = ( t .+ ( j .x. X ) ) ) )
25	24	mpteq2dv 4745	. . . 4 |- ( x = X -> ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) = ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { X } ) v = ( t .+ ( j .x. X ) ) ) ) )
26		eqid 2622	. . . 4 |- ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) ) = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) )
27	25, 26	fvmptg 6280	. . 3 |- ( ( X e. ( V \ { .0. } ) /\ ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { X } ) v = ( t .+ ( j .x. X ) ) ) ) e. _V ) -> ( ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) ) ` X ) = ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { X } ) v = ( t .+ ( j .x. X ) ) ) ) )
28	14, 17, 27	sylancl 694	. 2 |- ( ph -> ( ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { x } ) v = ( t .+ ( j .x. x ) ) ) ) ) ` X ) = ( v e. V |-> ( iota_ j e. R E. t e. ( O ` { X } ) v = ( t .+ ( j .x. X ) ) ) ) )
29	13, 28	eqtrd 2656	1 |- ( ph -> ( M ` X ) = ( 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:  hvmapvalvalN  37050  hvmapidN  37051  hdmapevec2  37128