Metamath Proof Explorer

Theorem mapdh8e

Description: Part of Part (8) in Baer p. 48. Eliminate w . (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 } ) )
mapdh8e.e 
|- ( ph -> X e. ( N ` { Y , T } ) )
Assertion mapdh8e 
|- ( 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 mapdh8e.e 
 |-  ( ph -> X e. ( N ` { Y , T } ) )
25 18 eldifad 
 |-  ( ph -> X e. V )
26 19 eldifad 
 |-  ( ph -> Y e. V )
27 1 2 3 6 14 25 26 dvh3dim 
 |-  ( ph -> E. w e. V -. w e. ( N ` { X , Y } ) )
28 14 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( K e. HL /\ W e. H ) )
29 15 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> F e. D )
30 16 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
31 17 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( I ` <. X , F , Y >. ) = G )
32 18 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> X e. ( V \ { .0. } ) )
33 19 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> Y e. ( V \ { .0. } ) )
34 20 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> T e. ( V \ { .0. } ) )
35 23 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( N ` { Y } ) =/= ( N ` { T } ) )
36 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
37 1 2 14 dvhlmod 
 |-  ( ph -> U e. LMod )
38 37 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> U e. LMod )
39 3 36 6 37 25 26 lspprcl 
 |-  ( ph -> ( N ` { X , Y } ) e. ( LSubSp ` U ) )
40 39 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( N ` { X , Y } ) e. ( LSubSp ` U ) )
41 simp2 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> w e. V )
42 simp3 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> -. w e. ( N ` { X , Y } ) )
43 5 36 38 40 41 42 lssneln0 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> w e. ( V \ { .0. } ) )
44 1 2 14 dvhlvec 
 |-  ( ph -> U e. LVec )
45 20 eldifad 
 |-  ( ph -> T e. V )
46 prcom 
 |-  { Y , T } = { T , Y }
47 46 fveq2i 
 |-  ( N ` { Y , T } ) = ( N ` { T , Y } )
48 24 47 eleqtrdi 
 |-  ( ph -> X e. ( N ` { T , Y } ) )
49 3 5 6 44 18 45 26 21 48 lspexch 
 |-  ( ph -> T e. ( N ` { X , Y } ) )
50 36 6 37 39 49 ellspsn5 
 |-  ( ph -> ( N ` { T } ) C_ ( N ` { X , Y } ) )
51 50 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( N ` { T } ) C_ ( N ` { X , Y } ) )
52 37 adantr 
 |-  ( ( ph /\ w e. V ) -> U e. LMod )
53 39 adantr 
 |-  ( ( ph /\ w e. V ) -> ( N ` { X , Y } ) e. ( LSubSp ` U ) )
54 simpr 
 |-  ( ( ph /\ w e. V ) -> w e. V )
55 3 36 6 52 53 54 ellspsn5b 
 |-  ( ( ph /\ w e. V ) -> ( w e. ( N ` { X , Y } ) <-> ( N ` { w } ) C_ ( N ` { X , Y } ) ) )
56 55 biimprd 
 |-  ( ( ph /\ w e. V ) -> ( ( N ` { w } ) C_ ( N ` { X , Y } ) -> w e. ( N ` { X , Y } ) ) )
57 56 con3d 
 |-  ( ( ph /\ w e. V ) -> ( -. w e. ( N ` { X , Y } ) -> -. ( N ` { w } ) C_ ( N ` { X , Y } ) ) )
58 57 3impia 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> -. ( N ` { w } ) C_ ( N ` { X , Y } ) )
59 nssne2 
 |-  ( ( ( N ` { T } ) C_ ( N ` { X , Y } ) /\ -. ( N ` { w } ) C_ ( N ` { X , Y } ) ) -> ( N ` { T } ) =/= ( N ` { w } ) )
60 51 58 59 syl2anc 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( N ` { T } ) =/= ( N ` { w } ) )
61 60 necomd 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( N ` { w } ) =/= ( N ` { T } ) )
62 22 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( N ` { X } ) =/= ( N ` { T } ) )
63 44 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> U e. LVec )
64 25 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> X e. V )
65 26 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> Y e. V )
66 3 6 63 41 64 65 42 lspindpi 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { Y } ) ) )
67 66 simprd 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( N ` { w } ) =/= ( N ` { Y } ) )
68 67 necomd 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( N ` { Y } ) =/= ( N ` { w } ) )
69 21 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( N ` { X } ) =/= ( N ` { Y } ) )
70 3 5 6 63 32 65 41 69 42 lspindp2l 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( ( N ` { Y } ) =/= ( N ` { w } ) /\ -. X e. ( N ` { Y , w } ) ) )
71 70 simprd 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> -. X e. ( N ` { Y , w } ) )
72 1 2 3 4 5 6 7 8 9 10 11 12 13 28 29 30 31 32 33 34 35 43 61 62 68 71 mapdh8d 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( I ` <. Y , G , T >. ) = ( I ` <. X , F , T >. ) )
73 72 rexlimdv3a 
 |-  ( ph -> ( E. w e. V -. w e. ( N ` { X , Y } ) -> ( I ` <. Y , G , T >. ) = ( I ` <. X , F , T >. ) ) )
74 27 73 mpd 
 |-  ( ph -> ( I ` <. Y , G , T >. ) = ( I ` <. X , F , T >. ) )