Theorem mapdh8a

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 5-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 ) )
	mapdh8a.f	|- ( ph -> F e. D )
	mapdh8a.mn	|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
	mapdh8a.a	|- ( ph -> ( I ` <. X , F , Y >. ) = G )
	mapdh8a.x	|- ( ph -> X e. ( V \ { .0. } ) )
	mapdh8a.y	|- ( ph -> Y e. ( V \ { .0. } ) )
	mapdh8a.yz	|- ( ph -> ( N ` { Y } ) =/= ( N ` { T } ) )
	mapdh8a.xt	|- ( ph -> T e. ( V \ { .0. } ) )
	mapdh8a.xn	|- ( ph -> -. X e. ( N ` { Y , T } ) )
Assertion	mapdh8a	|- ( ph -> ( I ` <. Y , G , T >. ) = ( I ` <. X , F , 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		mapdh8a.f	|- ( ph -> F e. D )
16		mapdh8a.mn	|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
17		mapdh8a.a	|- ( ph -> ( I ` <. X , F , Y >. ) = G )
18		mapdh8a.x	|- ( ph -> X e. ( V \ { .0. } ) )
19		mapdh8a.y	|- ( ph -> Y e. ( V \ { .0. } ) )
20		mapdh8a.yz	|- ( ph -> ( N ` { Y } ) =/= ( N ` { T } ) )
21		mapdh8a.xt	|- ( ph -> T e. ( V \ { .0. } ) )
22		mapdh8a.xn	|- ( ph -> -. X e. ( N ` { Y , T } ) )
23		eqidd	|- ( ph -> ( I ` <. X , F , T >. ) = ( I ` <. X , F , T >. ) )
24	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 18 19 21 22 20 17 23	mapdheq4	|- ( ph -> ( I ` <. Y , G , T >. ) = ( I ` <. X , F , T >. ) )