Theorem hdmap1vallem 37087

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

Hypotheses

Ref	Expression
hdmap1val.h	|- H = ( LHyp ` K )
hdmap1fval.u	|- U = ( ( DVecH ` K ) ` W )
hdmap1fval.v	|- V = ( Base ` U )
hdmap1fval.s	|- .- = ( -g ` U )
hdmap1fval.o	|- .0. = ( 0g ` U )
hdmap1fval.n	|- N = ( LSpan ` U )
hdmap1fval.c	|- C = ( (LCDual ` K ) ` W )
hdmap1fval.d	|- D = ( Base ` C )
hdmap1fval.r	|- R = ( -g ` C )
hdmap1fval.q	|- Q = ( 0g ` C )
hdmap1fval.j	|- J = ( LSpan ` C )
hdmap1fval.m	|- M = ( (mapd ` K ) ` W )
hdmap1fval.i	|- I = ( (HDMap1 ` K ) ` W )
hdmap1fval.k	|- ( ph -> ( K e. A /\ W e. H ) )
hdmap1val.t	|- ( ph -> T e. ( ( V X. D ) X. V ) )

Assertion

Ref	Expression
hdmap1vallem	|- ( ph -> ( I ` T ) = if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) )

Distinct variable groups:   C, h   D, h   h, J   h, M   h, N   U, h   h, V   T, h

Allowed substitution hints:   ph( h)   A( h)   Q( h)   R( h)   H( h)   I( h)   K( h)   .- ( h)   W( h)   .0. ( h)

Proof of Theorem hdmap1vallem

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmap1val.h	. . . 4 |- H = ( LHyp ` K )
2		hdmap1fval.u	. . . 4 |- U = ( ( DVecH ` K ) ` W )
3		hdmap1fval.v	. . . 4 |- V = ( Base ` U )
4		hdmap1fval.s	. . . 4 |- .- = ( -g ` U )
5		hdmap1fval.o	. . . 4 |- .0. = ( 0g ` U )
6		hdmap1fval.n	. . . 4 |- N = ( LSpan ` U )
7		hdmap1fval.c	. . . 4 |- C = ( (LCDual ` K ) ` W )
8		hdmap1fval.d	. . . 4 |- D = ( Base ` C )
9		hdmap1fval.r	. . . 4 |- R = ( -g ` C )
10		hdmap1fval.q	. . . 4 |- Q = ( 0g ` C )
11		hdmap1fval.j	. . . 4 |- J = ( LSpan ` C )
12		hdmap1fval.m	. . . 4 |- M = ( (mapd ` K ) ` W )
13		hdmap1fval.i	. . . 4 |- I = ( (HDMap1 ` K ) ` W )
14		hdmap1fval.k	. . . 4 |- ( ph -> ( K e. A /\ W e. H ) )
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	hdmap1fval 37086	. . 3 |- ( ph -> I = ( x e. ( ( V X. D ) X. V ) |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) )
16	15	fveq1d 6193	. 2 |- ( ph -> ( I ` T ) = ( ( x e. ( ( V X. D ) X. V ) |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) ` T ) )
17		hdmap1val.t	. . 3 |- ( ph -> T e. ( ( V X. D ) X. V ) )
18		fvex 6201	. . . . 5 |- ( 0g ` C ) e. _V
19	10, 18	eqeltri 2697	. . . 4 |- Q e. _V
20		riotaex 6615	. . . 4 |- ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) e. _V
21	19, 20	ifex 4156	. . 3 |- if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) e. _V
22		fveq2 6191	. . . . . 6 |- ( x = T -> ( 2nd ` x ) = ( 2nd ` T ) )
23	22	eqeq1d 2624	. . . . 5 |- ( x = T -> ( ( 2nd ` x ) = .0. <-> ( 2nd ` T ) = .0. ) )
24	22	sneqd 4189	. . . . . . . . . 10 |- ( x = T -> { ( 2nd ` x ) } = { ( 2nd ` T ) } )
25	24	fveq2d 6195	. . . . . . . . 9 |- ( x = T -> ( N ` { ( 2nd ` x ) } ) = ( N ` { ( 2nd ` T ) } ) )
26	25	fveq2d 6195	. . . . . . . 8 |- ( x = T -> ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( M ` ( N ` { ( 2nd ` T ) } ) ) )
27	26	eqeq1d 2624	. . . . . . 7 |- ( x = T -> ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) <-> ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) ) )
28		fveq2 6191	. . . . . . . . . . . . 13 |- ( x = T -> ( 1st ` x ) = ( 1st ` T ) )
29	28	fveq2d 6195	. . . . . . . . . . . 12 |- ( x = T -> ( 1st ` ( 1st ` x ) ) = ( 1st ` ( 1st ` T ) ) )
30	29, 22	oveq12d 6668	. . . . . . . . . . 11 |- ( x = T -> ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) = ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) )
31	30	sneqd 4189	. . . . . . . . . 10 |- ( x = T -> { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } = { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } )
32	31	fveq2d 6195	. . . . . . . . 9 |- ( x = T -> ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) = ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) )
33	32	fveq2d 6195	. . . . . . . 8 |- ( x = T -> ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) )
34	28	fveq2d 6195	. . . . . . . . . . 11 |- ( x = T -> ( 2nd ` ( 1st ` x ) ) = ( 2nd ` ( 1st ` T ) ) )
35	34	oveq1d 6665	. . . . . . . . . 10 |- ( x = T -> ( ( 2nd ` ( 1st ` x ) ) R h ) = ( ( 2nd ` ( 1st ` T ) ) R h ) )
36	35	sneqd 4189	. . . . . . . . 9 |- ( x = T -> { ( ( 2nd ` ( 1st ` x ) ) R h ) } = { ( ( 2nd ` ( 1st ` T ) ) R h ) } )
37	36	fveq2d 6195	. . . . . . . 8 |- ( x = T -> ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) )
38	33, 37	eqeq12d 2637	. . . . . . 7 |- ( x = T -> ( ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) <-> ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) )
39	27, 38	anbi12d 747	. . . . . 6 |- ( x = T -> ( ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) <-> ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) )
40	39	riotabidv 6613	. . . . 5 |- ( x = T -> ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) = ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) )
41	23, 40	ifbieq2d 4111	. . . 4 |- ( x = T -> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) = if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) )
42		eqid 2622	. . . 4 |- ( x e. ( ( V X. D ) X. V ) |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) = ( x e. ( ( V X. D ) X. V ) |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
43	41, 42	fvmptg 6280	. . 3 |- ( ( T e. ( ( V X. D ) X. V ) /\ if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) e. _V ) -> ( ( x e. ( ( V X. D ) X. V ) |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) ` T ) = if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) )
44	17, 21, 43	sylancl 694	. 2 |- ( ph -> ( ( x e. ( ( V X. D ) X. V ) |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) ` T ) = if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) )
45	16, 44	eqtrd 2656	1 |- ( ph -> ( I ` T ) = if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   {csn 4177   |-> cmpt 4729   X. cxp 5112   ` cfv 5888   iota_crio 6610  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   0gc0g 16100   -gcsg 17424   LSpanclspn 18971   LHypclh 35270   DVecHcdvh 36367  LCDualclcd 36875  mapdcmpd 36913  HDMap1chdma1 37081

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-pw 4160  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-hdmap1 37083

This theorem is referenced by:  hdmap1val  37088