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
|
lspsnel5a |
|- ( 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
|
lspsnel5 |
|- ( ( 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 >. ) ) |