hdmap1l6e

Description: Lemmma for hdmap1l6 . Part (6) in Baer p. 47 line 38. (Contributed by NM, 1-May-2015)

	Ref	Expression
Hypotheses	hdmap1l6.h	\|- H = ( LHyp ` K )
	hdmap1l6.u	\|- U = ( ( DVecH ` K ) ` W )
	hdmap1l6.v	\|- V = ( Base ` U )
	hdmap1l6.p	\|- .+ = ( +g ` U )
	hdmap1l6.s	\|- .- = ( -g ` U )
	hdmap1l6c.o	\|- .0. = ( 0g ` U )
	hdmap1l6.n	\|- N = ( LSpan ` U )
	hdmap1l6.c	\|- C = ( ( LCDual ` K ) ` W )
	hdmap1l6.d	\|- D = ( Base ` C )
	hdmap1l6.a	\|- .+b = ( +g ` C )
	hdmap1l6.r	\|- R = ( -g ` C )
	hdmap1l6.q	\|- Q = ( 0g ` C )
	hdmap1l6.l	\|- L = ( LSpan ` C )
	hdmap1l6.m	\|- M = ( ( mapd ` K ) ` W )
	hdmap1l6.i	\|- I = ( ( HDMap1 ` K ) ` W )
	hdmap1l6.k	\|- ( ph -> ( K e. HL /\ W e. H ) )
	hdmap1l6.f	\|- ( ph -> F e. D )
	hdmap1l6cl.x	\|- ( ph -> X e. ( V \ { .0. } ) )
	hdmap1l6.mn	\|- ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
	hdmap1l6d.xn	\|- ( ph -> -. X e. ( N ` { Y , Z } ) )
	hdmap1l6d.yz	\|- ( ph -> ( N ` { Y } ) = ( N ` { Z } ) )
	hdmap1l6d.y	\|- ( ph -> Y e. ( V \ { .0. } ) )
	hdmap1l6d.z	\|- ( ph -> Z e. ( V \ { .0. } ) )
	hdmap1l6d.w	\|- ( ph -> w e. ( V \ { .0. } ) )
	hdmap1l6d.wn	\|- ( ph -> -. w e. ( N ` { X , Y } ) )
Assertion	hdmap1l6e	\|- ( ph -> ( I ` <. X , F , ( ( w .+ Y ) .+ Z ) >. ) = ( ( I ` <. X , F , ( w .+ Y ) >. ) .+b ( I ` <. X , F , Z >. ) ) )

Proof

Step	Hyp	Ref	Expression
1		hdmap1l6.h	\|- H = ( LHyp ` K )
2		hdmap1l6.u	\|- U = ( ( DVecH ` K ) ` W )
3		hdmap1l6.v	\|- V = ( Base ` U )
4		hdmap1l6.p	\|- .+ = ( +g ` U )
5		hdmap1l6.s	\|- .- = ( -g ` U )
6		hdmap1l6c.o	\|- .0. = ( 0g ` U )
7		hdmap1l6.n	\|- N = ( LSpan ` U )
8		hdmap1l6.c	\|- C = ( ( LCDual ` K ) ` W )
9		hdmap1l6.d	\|- D = ( Base ` C )
10		hdmap1l6.a	\|- .+b = ( +g ` C )
11		hdmap1l6.r	\|- R = ( -g ` C )
12		hdmap1l6.q	\|- Q = ( 0g ` C )
13		hdmap1l6.l	\|- L = ( LSpan ` C )
14		hdmap1l6.m	\|- M = ( ( mapd ` K ) ` W )
15		hdmap1l6.i	\|- I = ( ( HDMap1 ` K ) ` W )
16		hdmap1l6.k	\|- ( ph -> ( K e. HL /\ W e. H ) )
17		hdmap1l6.f	\|- ( ph -> F e. D )
18		hdmap1l6cl.x	\|- ( ph -> X e. ( V \ { .0. } ) )
19		hdmap1l6.mn	\|- ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
20		hdmap1l6d.xn	\|- ( ph -> -. X e. ( N ` { Y , Z } ) )
21		hdmap1l6d.yz	\|- ( ph -> ( N ` { Y } ) = ( N ` { Z } ) )
22		hdmap1l6d.y	\|- ( ph -> Y e. ( V \ { .0. } ) )
23		hdmap1l6d.z	\|- ( ph -> Z e. ( V \ { .0. } ) )
24		hdmap1l6d.w	\|- ( ph -> w e. ( V \ { .0. } ) )
25		hdmap1l6d.wn	\|- ( ph -> -. w e. ( N ` { X , Y } ) )
26	1 2 16	dvhlmod	\|- ( ph -> U e. LMod )
27	24	eldifad	\|- ( ph -> w e. V )
28	22	eldifad	\|- ( ph -> Y e. V )
29	3 4	lmodvacl	\|- ( ( U e. LMod /\ w e. V /\ Y e. V ) -> ( w .+ Y ) e. V )
30	26 27 28 29	syl3anc	\|- ( ph -> ( w .+ Y ) e. V )
31	1 2 16	dvhlvec	\|- ( ph -> U e. LVec )
32	18	eldifad	\|- ( ph -> X e. V )
33	3 7 31 27 32 28 25	lspindpi	\|- ( ph -> ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { Y } ) ) )
34	33	simprd	\|- ( ph -> ( N ` { w } ) =/= ( N ` { Y } ) )
35	3 4 6 7 26 27 28 34	lmodindp1	\|- ( ph -> ( w .+ Y ) =/= .0. )
36		eldifsn	\|- ( ( w .+ Y ) e. ( V \ { .0. } ) <-> ( ( w .+ Y ) e. V /\ ( w .+ Y ) =/= .0. ) )
37	30 35 36	sylanbrc	\|- ( ph -> ( w .+ Y ) e. ( V \ { .0. } ) )
38	23	eldifad	\|- ( ph -> Z e. V )
39	3 7 31 32 28 38 20	lspindpi	\|- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
40	39	simpld	\|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
41	3 4 6 7 31 18 22 23 24 21 40 25	mapdindp3	\|- ( ph -> ( N ` { X } ) =/= ( N ` { ( w .+ Y ) } ) )
42	3 4 6 7 31 18 22 23 24 21 40 25	mapdindp4	\|- ( ph -> -. Z e. ( N ` { X , ( w .+ Y ) } ) )
43	3 6 7 31 18 30 38 41 42	lspindp1	\|- ( ph -> ( ( N ` { Z } ) =/= ( N ` { ( w .+ Y ) } ) /\ -. X e. ( N ` { Z , ( w .+ Y ) } ) ) )
44	43	simprd	\|- ( ph -> -. X e. ( N ` { Z , ( w .+ Y ) } ) )
45		prcom	\|- { ( w .+ Y ) , Z } = { Z , ( w .+ Y ) }
46	45	fveq2i	\|- ( N ` { ( w .+ Y ) , Z } ) = ( N ` { Z , ( w .+ Y ) } )
47	46	eleq2i	\|- ( X e. ( N ` { ( w .+ Y ) , Z } ) <-> X e. ( N ` { Z , ( w .+ Y ) } ) )
48	44 47	sylnibr	\|- ( ph -> -. X e. ( N ` { ( w .+ Y ) , Z } ) )
49	3 7 31 38 32 30 42	lspindpi	\|- ( ph -> ( ( N ` { Z } ) =/= ( N ` { X } ) /\ ( N ` { Z } ) =/= ( N ` { ( w .+ Y ) } ) ) )
50	49	simprd	\|- ( ph -> ( N ` { Z } ) =/= ( N ` { ( w .+ Y ) } ) )
51	50	necomd	\|- ( ph -> ( N ` { ( w .+ Y ) } ) =/= ( N ` { Z } ) )
52		eqidd	\|- ( ph -> ( I ` <. X , F , ( w .+ Y ) >. ) = ( I ` <. X , F , ( w .+ Y ) >. ) )
53		eqidd	\|- ( ph -> ( I ` <. X , F , Z >. ) = ( I ` <. X , F , Z >. ) )
54	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 37 23 48 51 52 53	hdmap1l6a	\|- ( ph -> ( I ` <. X , F , ( ( w .+ Y ) .+ Z ) >. ) = ( ( I ` <. X , F , ( w .+ Y ) >. ) .+b ( I ` <. X , F , Z >. ) ) )

Metamath Proof Explorer

Theorem hdmap1l6e

Proof