mapdh8g

Description: Part of Part (8) in Baer p. 48. Eliminate X e. ( N{ Y , T } ) . TODO: break out T =/= .0. in mapdh8e so we can share hypotheses. Also, look at hypothesis sharing for earlier mapdh8* and mapdh75* stuff. (Contributed by NM, 10-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 ) )
	mapdh8e.f	\|- ( ph -> F e. D )
	mapdh8e.mn	\|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
	mapdh8e.eg	\|- ( ph -> ( I ` <. X , F , Y >. ) = G )
	mapdh8e.x	\|- ( ph -> X e. ( V \ { .0. } ) )
	mapdh8e.y	\|- ( ph -> Y e. ( V \ { .0. } ) )
	mapdh8e.t	\|- ( ph -> T e. ( V \ { .0. } ) )
	mapdh8e.xy	\|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
	mapdh8e.xt	\|- ( ph -> ( N ` { X } ) =/= ( N ` { T } ) )
	mapdh8e.yt	\|- ( ph -> ( N ` { Y } ) =/= ( N ` { T } ) )
Assertion	mapdh8g	\|- ( 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		mapdh8e.f	\|- ( ph -> F e. D )
16		mapdh8e.mn	\|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
17		mapdh8e.eg	\|- ( ph -> ( I ` <. X , F , Y >. ) = G )
18		mapdh8e.x	\|- ( ph -> X e. ( V \ { .0. } ) )
19		mapdh8e.y	\|- ( ph -> Y e. ( V \ { .0. } ) )
20		mapdh8e.t	\|- ( ph -> T e. ( V \ { .0. } ) )
21		mapdh8e.xy	\|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
22		mapdh8e.xt	\|- ( ph -> ( N ` { X } ) =/= ( N ` { T } ) )
23		mapdh8e.yt	\|- ( ph -> ( N ` { Y } ) =/= ( N ` { T } ) )
24	14	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( K e. HL /\ W e. H ) )
25	15	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> F e. D )
26	16	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
27	17	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( I ` <. X , F , Y >. ) = G )
28	18	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> X e. ( V \ { .0. } ) )
29	19	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> Y e. ( V \ { .0. } ) )
30	20	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> T e. ( V \ { .0. } ) )
31	21	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( N ` { X } ) =/= ( N ` { Y } ) )
32	22	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( N ` { X } ) =/= ( N ` { T } ) )
33	23	adantr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( N ` { Y } ) =/= ( N ` { T } ) )
34		simpr	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> X e. ( N ` { Y , T } ) )
35	1 2 3 4 5 6 7 8 9 10 11 12 13 24 25 26 27 28 29 30 31 32 33 34	mapdh8e	\|- ( ( ph /\ X e. ( N ` { Y , T } ) ) -> ( I ` <. Y , G , T >. ) = ( I ` <. X , F , T >. ) )
36	14	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> ( K e. HL /\ W e. H ) )
37	15	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> F e. D )
38	16	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
39	17	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> ( I ` <. X , F , Y >. ) = G )
40	18	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> X e. ( V \ { .0. } ) )
41	19	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> Y e. ( V \ { .0. } ) )
42	23	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> ( N ` { Y } ) =/= ( N ` { T } ) )
43	20	adantr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> T e. ( V \ { .0. } ) )
44		simpr	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> -. X e. ( N ` { Y , T } ) )
45	1 2 3 4 5 6 7 8 9 10 11 12 13 36 37 38 39 40 41 42 43 44	mapdh8a	\|- ( ( ph /\ -. X e. ( N ` { Y , T } ) ) -> ( I ` <. Y , G , T >. ) = ( I ` <. X , F , T >. ) )
46	35 45	pm2.61dan	\|- ( ph -> ( I ` <. Y , G , T >. ) = ( I ` <. X , F , T >. ) )

Metamath Proof Explorer

Theorem mapdh8g

Proof