Theorem hdmap1val 37088

Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 37013.) TODO: change I = ( x e. _V |->... to ( ph -> ( I ` <. X , F , Y > ) =... in e.g. mapdh8 37078 to shorten proofs with no $d on x. (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.x	|- ( ph -> X e. V )
hdmap1val.f	|- ( ph -> F e. D )
hdmap1val.y	|- ( ph -> Y e. V )

Assertion

Ref	Expression
hdmap1val	|- ( ph -> ( I ` <. X , F , Y >. ) = if ( Y = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) )

Distinct variable groups:   C, h   D, h   h, J   h, M   h, N   U, h   h, V   h, F   h, X   h, Y   ph, h

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

Proof of Theorem hdmap1val

Step	Hyp	Ref	Expression
1		hdmap1val.h	. . 3 |- H = ( LHyp ` K )
2		hdmap1fval.u	. . 3 |- U = ( ( DVecH ` K ) ` W )
3		hdmap1fval.v	. . 3 |- V = ( Base ` U )
4		hdmap1fval.s	. . 3 |- .- = ( -g ` U )
5		hdmap1fval.o	. . 3 |- .0. = ( 0g ` U )
6		hdmap1fval.n	. . 3 |- N = ( LSpan ` U )
7		hdmap1fval.c	. . 3 |- C = ( (LCDual ` K ) ` W )
8		hdmap1fval.d	. . 3 |- D = ( Base ` C )
9		hdmap1fval.r	. . 3 |- R = ( -g ` C )
10		hdmap1fval.q	. . 3 |- Q = ( 0g ` C )
11		hdmap1fval.j	. . 3 |- J = ( LSpan ` C )
12		hdmap1fval.m	. . 3 |- M = ( (mapd ` K ) ` W )
13		hdmap1fval.i	. . 3 |- I = ( (HDMap1 ` K ) ` W )
14		hdmap1fval.k	. . 3 |- ( ph -> ( K e. A /\ W e. H ) )
15		df-ot 4186	. . . 4 |- <. X , F , Y >. = <. <. X , F >. , Y >.
16		hdmap1val.x	. . . . . 6 |- ( ph -> X e. V )
17		hdmap1val.f	. . . . . 6 |- ( ph -> F e. D )
18		opelxp 5146	. . . . . 6 |- ( <. X , F >. e. ( V X. D ) <-> ( X e. V /\ F e. D ) )
19	16, 17, 18	sylanbrc 698	. . . . 5 |- ( ph -> <. X , F >. e. ( V X. D ) )
20		hdmap1val.y	. . . . 5 |- ( ph -> Y e. V )
21		opelxp 5146	. . . . 5 |- ( <. <. X , F >. , Y >. e. ( ( V X. D ) X. V ) <-> ( <. X , F >. e. ( V X. D ) /\ Y e. V ) )
22	19, 20, 21	sylanbrc 698	. . . 4 |- ( ph -> <. <. X , F >. , Y >. e. ( ( V X. D ) X. V ) )
23	15, 22	syl5eqel 2705	. . 3 |- ( ph -> <. X , F , Y >. e. ( ( V X. D ) X. V ) )
24	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23	hdmap1vallem 37087	. 2 |- ( ph -> ( I ` <. X , F , Y >. ) = if ( ( 2nd ` <. X , F , Y >. ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) ) ) )
25		ot3rdg 7184	. . . . 5 |- ( Y e. V -> ( 2nd ` <. X , F , Y >. ) = Y )
26	20, 25	syl 17	. . . 4 |- ( ph -> ( 2nd ` <. X , F , Y >. ) = Y )
27	26	eqeq1d 2624	. . 3 |- ( ph -> ( ( 2nd ` <. X , F , Y >. ) = .0. <-> Y = .0. ) )
28	26	sneqd 4189	. . . . . . . 8 |- ( ph -> { ( 2nd ` <. X , F , Y >. ) } = { Y } )
29	28	fveq2d 6195	. . . . . . 7 |- ( ph -> ( N ` { ( 2nd ` <. X , F , Y >. ) } ) = ( N ` { Y } ) )
30	29	fveq2d 6195	. . . . . 6 |- ( ph -> ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( M ` ( N ` { Y } ) ) )
31	30	eqeq1d 2624	. . . . 5 |- ( ph -> ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) <-> ( M ` ( N ` { Y } ) ) = ( J ` { h } ) ) )
32		ot1stg 7182	. . . . . . . . . . 11 |- ( ( X e. V /\ F e. D /\ Y e. V ) -> ( 1st ` ( 1st ` <. X , F , Y >. ) ) = X )
33	16, 17, 20, 32	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> ( 1st ` ( 1st ` <. X , F , Y >. ) ) = X )
34	33, 26	oveq12d 6668	. . . . . . . . 9 |- ( ph -> ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) = ( X .- Y ) )
35	34	sneqd 4189	. . . . . . . 8 |- ( ph -> { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } = { ( X .- Y ) } )
36	35	fveq2d 6195	. . . . . . 7 |- ( ph -> ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) = ( N ` { ( X .- Y ) } ) )
37	36	fveq2d 6195	. . . . . 6 |- ( ph -> ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( M ` ( N ` { ( X .- Y ) } ) ) )
38		ot2ndg 7183	. . . . . . . . . 10 |- ( ( X e. V /\ F e. D /\ Y e. V ) -> ( 2nd ` ( 1st ` <. X , F , Y >. ) ) = F )
39	16, 17, 20, 38	syl3anc 1326	. . . . . . . . 9 |- ( ph -> ( 2nd ` ( 1st ` <. X , F , Y >. ) ) = F )
40	39	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) = ( F R h ) )
41	40	sneqd 4189	. . . . . . 7 |- ( ph -> { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } = { ( F R h ) } )
42	41	fveq2d 6195	. . . . . 6 |- ( ph -> ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) = ( J ` { ( F R h ) } ) )
43	37, 42	eqeq12d 2637	. . . . 5 |- ( ph -> ( ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) <-> ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) )
44	31, 43	anbi12d 747	. . . 4 |- ( ph -> ( ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) <-> ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) )
45	44	riotabidv 6613	. . 3 |- ( ph -> ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) ) = ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) )
46	27, 45	ifbieq2d 4111	. 2 |- ( ph -> if ( ( 2nd ` <. X , F , Y >. ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` <. X , F , Y >. ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` <. X , F , Y >. ) ) .- ( 2nd ` <. X , F , Y >. ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` <. X , F , Y >. ) ) R h ) } ) ) ) ) = if ( Y = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) )
47	24, 46	eqtrd 2656	1 |- ( ph -> ( I ` <. X , F , Y >. ) = if ( Y = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   {csn 4177   <.cop 4183   <.cotp 4185   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-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-ov 6653  df-1st 7168  df-2nd 7169  df-hdmap1 37083

This theorem is referenced by:  hdmap1val0  37089  hdmap1val2  37090  hdmap1valc  37093