mapdh8

Description: Part (8) in Baer p. 48. Given a reference vector X , the value of function I at a vector T is independent of the choice of auxiliary vectors Y and Z . Unlike Baer's, our version does not require X , Y , and Z to be independent, and also is defined for all Y and Z that are not colinear with X or T . We do this to make the definition of Baer's sigma function more straightforward. (This part eliminates T =/= .0. .) (Contributed by NM, 13-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 } ) )
	mapdh8.t	\|- ( ph -> T e. V )
Assertion	mapdh8	\|- ( 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		mapdh8.t	\|- ( ph -> T e. V )
25		fvexd	\|- ( ph -> ( I ` <. X , F , Y >. ) e. _V )
26	10 13 5 18 25	mapdhval0	\|- ( ph -> ( I ` <. Y , ( I ` <. X , F , Y >. ) , .0. >. ) = Q )
27		fvexd	\|- ( ph -> ( I ` <. X , F , Z >. ) e. _V )
28	10 13 5 19 27	mapdhval0	\|- ( ph -> ( I ` <. Z , ( I ` <. X , F , Z >. ) , .0. >. ) = Q )
29	26 28	eqtr4d	\|- ( ph -> ( I ` <. Y , ( I ` <. X , F , Y >. ) , .0. >. ) = ( I ` <. Z , ( I ` <. X , F , Z >. ) , .0. >. ) )
30	29	adantr	\|- ( ( ph /\ T = .0. ) -> ( I ` <. Y , ( I ` <. X , F , Y >. ) , .0. >. ) = ( I ` <. Z , ( I ` <. X , F , Z >. ) , .0. >. ) )
31		oteq3	\|- ( T = .0. -> <. Y , ( I ` <. X , F , Y >. ) , T >. = <. Y , ( I ` <. X , F , Y >. ) , .0. >. )
32	31	fveq2d	\|- ( T = .0. -> ( I ` <. Y , ( I ` <. X , F , Y >. ) , T >. ) = ( I ` <. Y , ( I ` <. X , F , Y >. ) , .0. >. ) )
33	32	adantl	\|- ( ( ph /\ T = .0. ) -> ( I ` <. Y , ( I ` <. X , F , Y >. ) , T >. ) = ( I ` <. Y , ( I ` <. X , F , Y >. ) , .0. >. ) )
34		oteq3	\|- ( T = .0. -> <. Z , ( I ` <. X , F , Z >. ) , T >. = <. Z , ( I ` <. X , F , Z >. ) , .0. >. )
35	34	fveq2d	\|- ( T = .0. -> ( I ` <. Z , ( I ` <. X , F , Z >. ) , T >. ) = ( I ` <. Z , ( I ` <. X , F , Z >. ) , .0. >. ) )
36	35	adantl	\|- ( ( ph /\ T = .0. ) -> ( I ` <. Z , ( I ` <. X , F , Z >. ) , T >. ) = ( I ` <. Z , ( I ` <. X , F , Z >. ) , .0. >. ) )
37	30 33 36	3eqtr4d	\|- ( ( ph /\ T = .0. ) -> ( I ` <. Y , ( I ` <. X , F , Y >. ) , T >. ) = ( I ` <. Z , ( I ` <. X , F , Z >. ) , T >. ) )
38	14	adantr	\|- ( ( ph /\ T =/= .0. ) -> ( K e. HL /\ W e. H ) )
39	15	adantr	\|- ( ( ph /\ T =/= .0. ) -> F e. D )
40	16	adantr	\|- ( ( ph /\ T =/= .0. ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
41	17	adantr	\|- ( ( ph /\ T =/= .0. ) -> X e. ( V \ { .0. } ) )
42	18	adantr	\|- ( ( ph /\ T =/= .0. ) -> Y e. ( V \ { .0. } ) )
43	19	adantr	\|- ( ( ph /\ T =/= .0. ) -> Z e. ( V \ { .0. } ) )
44	20	adantr	\|- ( ( ph /\ T =/= .0. ) -> ( N ` { X } ) =/= ( N ` { Y } ) )
45	21	adantr	\|- ( ( ph /\ T =/= .0. ) -> ( N ` { X } ) =/= ( N ` { Z } ) )
46	22	adantr	\|- ( ( ph /\ T =/= .0. ) -> ( N ` { Y } ) =/= ( N ` { T } ) )
47	23	adantr	\|- ( ( ph /\ T =/= .0. ) -> ( N ` { Z } ) =/= ( N ` { T } ) )
48	24	anim1i	\|- ( ( ph /\ T =/= .0. ) -> ( T e. V /\ T =/= .0. ) )
49		eldifsn	\|- ( T e. ( V \ { .0. } ) <-> ( T e. V /\ T =/= .0. ) )
50	48 49	sylibr	\|- ( ( ph /\ T =/= .0. ) -> T e. ( V \ { .0. } ) )
51	1 2 3 4 5 6 7 8 9 10 11 12 13 38 39 40 41 42 43 44 45 46 47 50	mapdh8j	\|- ( ( ph /\ T =/= .0. ) -> ( I ` <. Y , ( I ` <. X , F , Y >. ) , T >. ) = ( I ` <. Z , ( I ` <. X , F , Z >. ) , T >. ) )
52	37 51	pm2.61dane	\|- ( ph -> ( I ` <. Y , ( I ` <. X , F , Y >. ) , T >. ) = ( I ` <. Z , ( I ` <. X , F , Z >. ) , T >. ) )

Metamath Proof Explorer

Theorem mapdh8

Proof