Theorem hdmap1val2 37090

Description: Value of preliminary map from vectors to functionals in the closed kernel dual space, for nonzero Y. (Contributed by NM, 16-May-2015.)

Hypotheses

Ref	Expression
hdmap1val2.h	|- H = ( LHyp ` K )
hdmap1val2.u	|- U = ( ( DVecH ` K ) ` W )
hdmap1val2.v	|- V = ( Base ` U )
hdmap1val2.s	|- .- = ( -g ` U )
hdmap1val2.o	|- .0. = ( 0g ` U )
hdmap1val2.n	|- N = ( LSpan ` U )
hdmap1val2.c	|- C = ( (LCDual ` K ) ` W )
hdmap1val2.d	|- D = ( Base ` C )
hdmap1val2.r	|- R = ( -g ` C )
hdmap1val2.l	|- L = ( LSpan ` C )
hdmap1val2.m	|- M = ( (mapd ` K ) ` W )
hdmap1val2.i	|- I = ( (HDMap1 ` K ) ` W )
hdmap1val2.k	|- ( ph -> ( K e. HL /\ W e. H ) )
hdmap1val2.x	|- ( ph -> X e. V )
hdmap1val2.f	|- ( ph -> F e. D )
hdmap1val2.y	|- ( ph -> Y e. ( V \ { .0. } ) )

Assertion

Ref	Expression
hdmap1val2	|- ( ph -> ( I ` <. X , F , Y >. ) = ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( L ` { ( F R h ) } ) ) ) )

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

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

Proof of Theorem hdmap1val2

Step	Hyp	Ref	Expression
1		hdmap1val2.h	. . 3 |- H = ( LHyp ` K )
2		hdmap1val2.u	. . 3 |- U = ( ( DVecH ` K ) ` W )
3		hdmap1val2.v	. . 3 |- V = ( Base ` U )
4		hdmap1val2.s	. . 3 |- .- = ( -g ` U )
5		hdmap1val2.o	. . 3 |- .0. = ( 0g ` U )
6		hdmap1val2.n	. . 3 |- N = ( LSpan ` U )
7		hdmap1val2.c	. . 3 |- C = ( (LCDual ` K ) ` W )
8		hdmap1val2.d	. . 3 |- D = ( Base ` C )
9		hdmap1val2.r	. . 3 |- R = ( -g ` C )
10		eqid 2622	. . 3 |- ( 0g ` C ) = ( 0g ` C )
11		hdmap1val2.l	. . 3 |- L = ( LSpan ` C )
12		hdmap1val2.m	. . 3 |- M = ( (mapd ` K ) ` W )
13		hdmap1val2.i	. . 3 |- I = ( (HDMap1 ` K ) ` W )
14		hdmap1val2.k	. . 3 |- ( ph -> ( K e. HL /\ W e. H ) )
15		hdmap1val2.x	. . 3 |- ( ph -> X e. V )
16		hdmap1val2.f	. . 3 |- ( ph -> F e. D )
17		hdmap1val2.y	. . . 4 |- ( ph -> Y e. ( V \ { .0. } ) )
18	17	eldifad 3586	. . 3 |- ( ph -> Y e. V )
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18	hdmap1val 37088	. 2 |- ( ph -> ( I ` <. X , F , Y >. ) = if ( Y = .0. , ( 0g ` C ) , ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( L ` { ( F R h ) } ) ) ) ) )
20		eldifsni 4320	. . . 4 |- ( Y e. ( V \ { .0. } ) -> Y =/= .0. )
21	20	neneqd 2799	. . 3 |- ( Y e. ( V \ { .0. } ) -> -. Y = .0. )
22		iffalse 4095	. . 3 |- ( -. Y = .0. -> if ( Y = .0. , ( 0g ` C ) , ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( L ` { ( F R h ) } ) ) ) ) = ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( L ` { ( F R h ) } ) ) ) )
23	17, 21, 22	3syl 18	. 2 |- ( ph -> if ( Y = .0. , ( 0g ` C ) , ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( L ` { ( F R h ) } ) ) ) ) = ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( L ` { ( F R h ) } ) ) ) )
24	19, 23	eqtrd 2656	1 |- ( ph -> ( I ` <. X , F , Y >. ) = ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( L ` { ( F R h ) } ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   ifcif 4086   {csn 4177   <.cotp 4185   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   0gc0g 16100   -gcsg 17424   LSpanclspn 18971   HLchlt 34637   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:  hdmap1eq  37091