mapdh8d

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 6-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 ) )
	mapdh8d.f	\|- ( ph -> F e. D )
	mapdh8d.mn	\|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
	mapdh8b.eg	\|- ( ph -> ( I ` <. X , F , Y >. ) = G )
	mapdh8d.x	\|- ( ph -> X e. ( V \ { .0. } ) )
	mapdh8d.y	\|- ( ph -> Y e. ( V \ { .0. } ) )
	mapdh8d.xt	\|- ( ph -> T e. ( V \ { .0. } ) )
	mapdh8d.yz	\|- ( ph -> ( N ` { Y } ) =/= ( N ` { T } ) )
	mapdh8d.w	\|- ( ph -> w e. ( V \ { .0. } ) )
	mapdh8d.wt	\|- ( ph -> ( N ` { w } ) =/= ( N ` { T } ) )
	mapdh8d.ut	\|- ( ph -> ( N ` { X } ) =/= ( N ` { T } ) )
	mapdh8d.vw	\|- ( ph -> ( N ` { Y } ) =/= ( N ` { w } ) )
	mapdh8d.xn	\|- ( ph -> -. X e. ( N ` { Y , w } ) )
Assertion	mapdh8d	\|- ( 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		mapdh8d.f	\|- ( ph -> F e. D )
16		mapdh8d.mn	\|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
17		mapdh8b.eg	\|- ( ph -> ( I ` <. X , F , Y >. ) = G )
18		mapdh8d.x	\|- ( ph -> X e. ( V \ { .0. } ) )
19		mapdh8d.y	\|- ( ph -> Y e. ( V \ { .0. } ) )
20		mapdh8d.xt	\|- ( ph -> T e. ( V \ { .0. } ) )
21		mapdh8d.yz	\|- ( ph -> ( N ` { Y } ) =/= ( N ` { T } ) )
22		mapdh8d.w	\|- ( ph -> w e. ( V \ { .0. } ) )
23		mapdh8d.wt	\|- ( ph -> ( N ` { w } ) =/= ( N ` { T } ) )
24		mapdh8d.ut	\|- ( ph -> ( N ` { X } ) =/= ( N ` { T } ) )
25		mapdh8d.vw	\|- ( ph -> ( N ` { Y } ) =/= ( N ` { w } ) )
26		mapdh8d.xn	\|- ( ph -> -. X e. ( N ` { Y , w } ) )
27	14	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( K e. HL /\ W e. H ) )
28	19	eldifad	\|- ( ph -> Y e. V )
29	1 2 14	dvhlvec	\|- ( ph -> U e. LVec )
30	18	eldifad	\|- ( ph -> X e. V )
31	22	eldifad	\|- ( ph -> w e. V )
32	3 6 29 30 28 31 26	lspindpi	\|- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { w } ) ) )
33	32	simpld	\|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
34	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 18 28 33	mapdhcl	\|- ( ph -> ( I ` <. X , F , Y >. ) e. D )
35	17 34	eqeltrrd	\|- ( ph -> G e. D )
36	35	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> G e. D )
37	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 18 19 35 33	mapdheq	\|- ( ph -> ( ( I ` <. X , F , Y >. ) = G <-> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) )
38	17 37	mpbid	\|- ( ph -> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) )
39	38	simpld	\|- ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) )
40	39	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) )
41	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 25 22 26	mapdh8a	\|- ( ph -> ( I ` <. Y , G , w >. ) = ( I ` <. X , F , w >. ) )
42	41	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( I ` <. Y , G , w >. ) = ( I ` <. X , F , w >. ) )
43	19	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> Y e. ( V \ { .0. } ) )
44	22	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> w e. ( V \ { .0. } ) )
45	23	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( N ` { w } ) =/= ( N ` { T } ) )
46	20	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> T e. ( V \ { .0. } ) )
47	25	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( N ` { Y } ) =/= ( N ` { w } ) )
48		simpr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> X e. ( N ` { Y , T } ) )
49	26	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> -. X e. ( N ` { Y , w } ) )
50	1 2 3 4 5 6 7 8 9 10 11 12 13 27 36 40 42 43 44 45 46 47 48 49	mapdh8b	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) = ( I ` <. Y , G , T >. ) )
51	15	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> F e. D )
52	16	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
53		eqidd	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( I ` <. X , F , w >. ) = ( I ` <. X , F , w >. ) )
54	18	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> X e. ( V \ { .0. } ) )
55	21	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( N ` { Y } ) =/= ( N ` { T } ) )
56	24	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( N ` { X } ) =/= ( N ` { T } ) )
57	1 2 3 4 5 6 7 8 9 10 11 12 13 27 51 52 53 54 43 46 55 44 45 56 47 48 49	mapdh8c	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) = ( I ` <. X , F , T >. ) )
58	50 57	eqtr3d	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( I ` <. Y , G , T >. ) = ( I ` <. X , F , T >. ) )
59	14	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> ( K e. HL /\ W e. H ) )
60	15	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> F e. D )
61	16	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
62	17	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> ( I ` <. X , F , Y >. ) = G )
63	18	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> X e. ( V \ { .0. } ) )
64	19	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> Y e. ( V \ { .0. } ) )
65	21	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> ( N ` { Y } ) =/= ( N ` { T } ) )
66	20	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> T e. ( V \ { .0. } ) )
67		simpr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> -. X e. ( N ` { Y , T } ) )
68	1 2 3 4 5 6 7 8 9 10 11 12 13 59 60 61 62 63 64 65 66 67	mapdh8a	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> ( I ` <. Y , G , T >. ) = ( I ` <. X , F , T >. ) )
69	58 68	pm2.61dan	\|- ( ph -> ( I ` <. Y , G , T >. ) = ( I ` <. X , F , T >. ) )

Metamath Proof Explorer

Theorem mapdh8d

Proof