Theorem hdmap1valc 37093

Description: Connect the value of the preliminary map from vectors to functionals I to the hypothesis L used by earlier theorems. Note: the X e. ( V \ { .0. } ) hypothesis could be the more general X e. V but the former will be easier to use. TODO: use the I function directly in those theorems, so this theorem becomes unnecessary? TODO: The hdmap1cbv 37092 is probably unnecessary, but it would mean different $d's later on. (Contributed by NM, 15-May-2015.)

Hypotheses

Ref	Expression
hdmap1valc.h	|- H = ( LHyp ` K )
hdmap1valc.u	|- U = ( ( DVecH ` K ) ` W )
hdmap1valc.v	|- V = ( Base ` U )
hdmap1valc.s	|- .- = ( -g ` U )
hdmap1valc.o	|- .0. = ( 0g ` U )
hdmap1valc.n	|- N = ( LSpan ` U )
hdmap1valc.c	|- C = ( (LCDual ` K ) ` W )
hdmap1valc.d	|- D = ( Base ` C )
hdmap1valc.r	|- R = ( -g ` C )
hdmap1valc.q	|- Q = ( 0g ` C )
hdmap1valc.j	|- J = ( LSpan ` C )
hdmap1valc.m	|- M = ( (mapd ` K ) ` W )
hdmap1valc.i	|- I = ( (HDMap1 ` K ) ` W )
hdmap1valc.k	|- ( ph -> ( K e. HL /\ W e. H ) )
hdmap1valc.x	|- ( ph -> X e. ( V \ { .0. } ) )
hdmap1valc.f	|- ( ph -> F e. D )
hdmap1valc.y	|- ( ph -> Y e. V )
hdmap1valc.l	|- L = ( x e. _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 ) } ) ) ) ) )

Assertion

Ref	Expression
hdmap1valc	|- ( ph -> ( I ` <. X , F , Y >. ) = ( L ` <. X , F , Y >. ) )

Distinct variable groups:   x, .0.   x, h, D   h, J, x   h, M, x   .- , h, x   h, N, x   R, h, x   x, Q

Allowed substitution hints:   ph( x, h)   C( x, h)   Q( h)   U( x, h)   F( x, h)   H( x, h)   I( x, h)   K( x, h)   L( x, h)   V( x, h)   W( x, h)   X( x, h)   Y( x, h)   .0. ( h)

Proof of Theorem hdmap1valc

Dummy variables w g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmap1valc.h	. . 3 |- H = ( LHyp ` K )
2		hdmap1valc.u	. . 3 |- U = ( ( DVecH ` K ) ` W )
3		hdmap1valc.v	. . 3 |- V = ( Base ` U )
4		hdmap1valc.s	. . 3 |- .- = ( -g ` U )
5		hdmap1valc.o	. . 3 |- .0. = ( 0g ` U )
6		hdmap1valc.n	. . 3 |- N = ( LSpan ` U )
7		hdmap1valc.c	. . 3 |- C = ( (LCDual ` K ) ` W )
8		hdmap1valc.d	. . 3 |- D = ( Base ` C )
9		hdmap1valc.r	. . 3 |- R = ( -g ` C )
10		hdmap1valc.q	. . 3 |- Q = ( 0g ` C )
11		hdmap1valc.j	. . 3 |- J = ( LSpan ` C )
12		hdmap1valc.m	. . 3 |- M = ( (mapd ` K ) ` W )
13		hdmap1valc.i	. . 3 |- I = ( (HDMap1 ` K ) ` W )
14		hdmap1valc.k	. . 3 |- ( ph -> ( K e. HL /\ W e. H ) )
15		hdmap1valc.x	. . . 4 |- ( ph -> X e. ( V \ { .0. } ) )
16	15	eldifad 3586	. . 3 |- ( ph -> X e. V )
17		hdmap1valc.f	. . 3 |- ( ph -> F e. D )
18		hdmap1valc.y	. . 3 |- ( ph -> Y e. V )
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18	hdmap1val 37088	. 2 |- ( ph -> ( I ` <. X , F , Y >. ) = if ( Y = .0. , Q , ( iota_ g e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { g } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R g ) } ) ) ) ) )
20		hdmap1valc.l	. . . 4 |- L = ( x e. _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 ) } ) ) ) ) )
21	20	hdmap1cbv 37092	. . 3 |- L = ( w e. _V |-> if ( ( 2nd ` w ) = .0. , Q , ( iota_ g e. D ( ( M ` ( N ` { ( 2nd ` w ) } ) ) = ( J ` { g } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` w ) ) .- ( 2nd ` w ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` w ) ) R g ) } ) ) ) ) )
22	10, 21, 16, 17, 18	mapdhval 37013	. 2 |- ( ph -> ( L ` <. X , F , Y >. ) = if ( Y = .0. , Q , ( iota_ g e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { g } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R g ) } ) ) ) ) )
23	19, 22	eqtr4d 2659	1 |- ( ph -> ( I ` <. X , F , Y >. ) = ( L ` <. X , F , Y >. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   ifcif 4086   {csn 4177   <.cotp 4185   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   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:  hdmap1cl  37094  hdmap1eq2  37095  hdmap1eq4N  37096  hdmap1eulem  37113  hdmap1eulemOLDN  37114