mapdh8i

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 11-May-2015)

	Ref	Expression
Hypotheses	mapdh8a.h	\|- H = ( LHyp ` K )
	mapdh8a.u	\|- U = ( ( DVecH ` K ) ` W )
	mapdh8a.v	\|- V = ( Base ` U )
	mapdh8a.s	\|- .- = ( -g ` U )
	mapdh8a.o	\|- .0. = ( 0g ` U )
	mapdh8a.n	\|- N = ( LSpan ` U )
	mapdh8a.c	\|- C = ( ( LCDual ` K ) ` W )
	mapdh8a.d	\|- D = ( Base ` C )
	mapdh8a.r	\|- R = ( -g ` C )
	mapdh8a.q	\|- Q = ( 0g ` C )
	mapdh8a.j	\|- J = ( LSpan ` C )
	mapdh8a.m	\|- M = ( ( mapd ` K ) ` W )
	mapdh8a.i	\|- I = ( 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 ) } ) ) ) ) )
	mapdh8a.k	\|- ( ph -> ( K e. HL /\ W e. H ) )
	mapdh8h.f	\|- ( ph -> F e. D )
	mapdh8h.mn	\|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
	mapdh8i.x	\|- ( ph -> X e. ( V \ { .0. } ) )
	mapdh8i.y	\|- ( ph -> Y e. ( V \ { .0. } ) )
	mapdh8i.z	\|- ( ph -> Z e. ( V \ { .0. } ) )
	mapdh8i.xy	\|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
	mapdh8i.xz	\|- ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) )
	mapdh8i.yt	\|- ( ph -> ( N ` { Y } ) =/= ( N ` { T } ) )
	mapdh8i.zt	\|- ( ph -> ( N ` { Z } ) =/= ( N ` { T } ) )
	mapdh8i.t	\|- ( ph -> T e. ( V \ { .0. } ) )
	mapdh8i.xt	\|- ( ph -> ( N ` { X } ) =/= ( N ` { T } ) )
Assertion	mapdh8i	\|- ( ph -> ( I ` <. Y , ( I ` <. X , F , Y >. ) , T >. ) = ( I ` <. Z , ( I ` <. X , F , Z >. ) , T >. ) )

Proof

Step	Hyp	Ref	Expression
1		mapdh8a.h	\|- H = ( LHyp ` K )
2		mapdh8a.u	\|- U = ( ( DVecH ` K ) ` W )
3		mapdh8a.v	\|- V = ( Base ` U )
4		mapdh8a.s	\|- .- = ( -g ` U )
5		mapdh8a.o	\|- .0. = ( 0g ` U )
6		mapdh8a.n	\|- N = ( LSpan ` U )
7		mapdh8a.c	\|- C = ( ( LCDual ` K ) ` W )
8		mapdh8a.d	\|- D = ( Base ` C )
9		mapdh8a.r	\|- R = ( -g ` C )
10		mapdh8a.q	\|- Q = ( 0g ` C )
11		mapdh8a.j	\|- J = ( LSpan ` C )
12		mapdh8a.m	\|- M = ( ( mapd ` K ) ` W )
13		mapdh8a.i	\|- I = ( 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 ) } ) ) ) ) )
14		mapdh8a.k	\|- ( ph -> ( K e. HL /\ W e. H ) )
15		mapdh8h.f	\|- ( ph -> F e. D )
16		mapdh8h.mn	\|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
17		mapdh8i.x	\|- ( ph -> X e. ( V \ { .0. } ) )
18		mapdh8i.y	\|- ( ph -> Y e. ( V \ { .0. } ) )
19		mapdh8i.z	\|- ( ph -> Z e. ( V \ { .0. } ) )
20		mapdh8i.xy	\|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
21		mapdh8i.xz	\|- ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) )
22		mapdh8i.yt	\|- ( ph -> ( N ` { Y } ) =/= ( N ` { T } ) )
23		mapdh8i.zt	\|- ( ph -> ( N ` { Z } ) =/= ( N ` { T } ) )
24		mapdh8i.t	\|- ( ph -> T e. ( V \ { .0. } ) )
25		mapdh8i.xt	\|- ( ph -> ( N ` { X } ) =/= ( N ` { T } ) )
26		eqidd	\|- ( ph -> ( I ` <. X , F , Y >. ) = ( I ` <. X , F , Y >. ) )
27	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 26 17 18 24 20 25 22	mapdh8g	\|- ( ph -> ( I ` <. Y , ( I ` <. X , F , Y >. ) , T >. ) = ( I ` <. X , F , T >. ) )
28		eqidd	\|- ( ph -> ( I ` <. X , F , Z >. ) = ( I ` <. X , F , Z >. ) )
29	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 28 17 19 24 21 25 23	mapdh8g	\|- ( ph -> ( I ` <. Z , ( I ` <. X , F , Z >. ) , T >. ) = ( I ` <. X , F , T >. ) )
30	27 29	eqtr4d	\|- ( ph -> ( I ` <. Y , ( I ` <. X , F , Y >. ) , T >. ) = ( I ` <. Z , ( I ` <. X , F , Z >. ) , T >. ) )

Metamath Proof Explorer

Theorem mapdh8i

Proof