mapdh8aa

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 12-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 ) )
	mapdh8aa.f	\|- ( ph -> F e. D )
	mapdh8aa.mn	\|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
	mapdh8aa.eg	\|- ( ph -> ( I ` <. X , F , Y >. ) = G )
	mapdh8aa.ee	\|- ( ph -> ( I ` <. X , F , Z >. ) = E )
	mapdh8aa.x	\|- ( ph -> X e. ( V \ { .0. } ) )
	mapdh8aa.y	\|- ( ph -> Y e. ( V \ { .0. } ) )
	mapdh8aa.z	\|- ( ph -> Z e. ( V \ { .0. } ) )
	mapdh8aa.zt	\|- ( ph -> ( N ` { Z } ) =/= ( N ` { T } ) )
	mapdh8aa.t	\|- ( ph -> T e. ( V \ { .0. } ) )
	mapdh8aa.yn	\|- ( ph -> -. Y e. ( N ` { Z , T } ) )
	mapdh8aa.xn	\|- ( ph -> -. X e. ( N ` { Y , Z } ) )
Assertion	mapdh8aa	\|- ( ph -> ( I ` <. Y , G , T >. ) = ( I ` <. Z , E , 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		mapdh8aa.f	\|- ( ph -> F e. D )
16		mapdh8aa.mn	\|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
17		mapdh8aa.eg	\|- ( ph -> ( I ` <. X , F , Y >. ) = G )
18		mapdh8aa.ee	\|- ( ph -> ( I ` <. X , F , Z >. ) = E )
19		mapdh8aa.x	\|- ( ph -> X e. ( V \ { .0. } ) )
20		mapdh8aa.y	\|- ( ph -> Y e. ( V \ { .0. } ) )
21		mapdh8aa.z	\|- ( ph -> Z e. ( V \ { .0. } ) )
22		mapdh8aa.zt	\|- ( ph -> ( N ` { Z } ) =/= ( N ` { T } ) )
23		mapdh8aa.t	\|- ( ph -> T e. ( V \ { .0. } ) )
24		mapdh8aa.yn	\|- ( ph -> -. Y e. ( N ` { Z , T } ) )
25		mapdh8aa.xn	\|- ( ph -> -. X e. ( N ` { Y , Z } ) )
26	20	eldifad	\|- ( ph -> Y e. V )
27	1 2 14	dvhlvec	\|- ( ph -> U e. LVec )
28	19	eldifad	\|- ( ph -> X e. V )
29	21	eldifad	\|- ( ph -> Z e. V )
30	3 6 27 28 26 29 25	lspindpi	\|- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
31	30	simpld	\|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
32	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 26 31	mapdhcl	\|- ( ph -> ( I ` <. X , F , Y >. ) e. D )
33	17 32	eqeltrrd	\|- ( ph -> G e. D )
34	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 20 33 31	mapdheq	\|- ( ph -> ( ( I ` <. X , F , Y >. ) = G <-> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) )
35	17 34	mpbid	\|- ( ph -> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) )
36	35	simpld	\|- ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) )
37	23	eldifad	\|- ( ph -> T e. V )
38	3 6 27 26 29 37 24	lspindpi	\|- ( ph -> ( ( N ` { Y } ) =/= ( N ` { Z } ) /\ ( N ` { Y } ) =/= ( N ` { T } ) ) )
39	38	simpld	\|- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
40	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 39 25 19 20 21	mapdh75d	\|- ( ph -> ( I ` <. Y , G , Z >. ) = E )
41	1 2 3 4 5 6 7 8 9 10 11 12 13 14 33 36 40 20 21 22 23 24	mapdh8a	\|- ( ph -> ( I ` <. Z , E , T >. ) = ( I ` <. Y , G , T >. ) )
42	41	eqcomd	\|- ( ph -> ( I ` <. Y , G , T >. ) = ( I ` <. Z , E , T >. ) )

Metamath Proof Explorer

Theorem mapdh8aa

Proof