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 |
|
mapdh9a.x |
|- ( ph -> X e. ( V \ { .0. } ) ) |
18 |
|
mapdh9a.t |
|- ( ph -> T e. V ) |
19 |
14
|
3ad2ant1 |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
20 |
15
|
3ad2ant1 |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> F e. D ) |
21 |
16
|
3ad2ant1 |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) ) |
22 |
17
|
3ad2ant1 |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> X e. ( V \ { .0. } ) ) |
23 |
|
simp3ll |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> z e. ( V \ { .0. } ) ) |
24 |
|
simp3rl |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> w e. ( V \ { .0. } ) ) |
25 |
|
simplrl |
|- ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { z } ) =/= ( N ` { X } ) ) |
26 |
25
|
3ad2ant3 |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { z } ) =/= ( N ` { X } ) ) |
27 |
26
|
necomd |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { X } ) =/= ( N ` { z } ) ) |
28 |
|
simprrl |
|- ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { w } ) =/= ( N ` { X } ) ) |
29 |
28
|
3ad2ant3 |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { w } ) =/= ( N ` { X } ) ) |
30 |
29
|
necomd |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { X } ) =/= ( N ` { w } ) ) |
31 |
|
simplrr |
|- ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { z } ) =/= ( N ` { T } ) ) |
32 |
31
|
3ad2ant3 |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { z } ) =/= ( N ` { T } ) ) |
33 |
|
simprrr |
|- ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { w } ) =/= ( N ` { T } ) ) |
34 |
33
|
3ad2ant3 |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { w } ) =/= ( N ` { T } ) ) |
35 |
18
|
3ad2ant1 |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> T e. V ) |
36 |
1 2 3 4 5 6 7 8 9 10 11 12 13 19 20 21 22 23 24 27 30 32 34 35
|
mapdh8 |
|- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) |
37 |
36
|
3exp |
|- ( ph -> ( ( z e. V /\ w e. V ) -> ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) ) ) |
38 |
37
|
ralrimivv |
|- ( ph -> A. z e. V A. w e. V ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) ) |
39 |
17
|
eldifad |
|- ( ph -> X e. V ) |
40 |
1 2 3 6 14 39 18
|
dvh3dim |
|- ( ph -> E. z e. V -. z e. ( N ` { X , T } ) ) |
41 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
42 |
1 2 14
|
dvhlmod |
|- ( ph -> U e. LMod ) |
43 |
42
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> U e. LMod ) |
44 |
3 41 6 42 39 18
|
lspprcl |
|- ( ph -> ( N ` { X , T } ) e. ( LSubSp ` U ) ) |
45 |
44
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( N ` { X , T } ) e. ( LSubSp ` U ) ) |
46 |
|
simplr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> z e. V ) |
47 |
|
simpr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> -. z e. ( N ` { X , T } ) ) |
48 |
5 41 43 45 46 47
|
lssneln0 |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> z e. ( V \ { .0. } ) ) |
49 |
1 2 14
|
dvhlvec |
|- ( ph -> U e. LVec ) |
50 |
49
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> U e. LVec ) |
51 |
39
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> X e. V ) |
52 |
18
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> T e. V ) |
53 |
3 6 50 46 51 52 47
|
lspindpi |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) |
54 |
48 53
|
jca |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) |
55 |
54
|
ex |
|- ( ( ph /\ z e. V ) -> ( -. z e. ( N ` { X , T } ) -> ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) ) |
56 |
55
|
reximdva |
|- ( ph -> ( E. z e. V -. z e. ( N ` { X , T } ) -> E. z e. V ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) ) |
57 |
40 56
|
mpd |
|- ( ph -> E. z e. V ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) |
58 |
14
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
59 |
15
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> F e. D ) |
60 |
16
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) ) |
61 |
17
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> X e. ( V \ { .0. } ) ) |
62 |
|
simplr |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> z e. V ) |
63 |
|
simprrl |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { z } ) =/= ( N ` { X } ) ) |
64 |
63
|
necomd |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { X } ) =/= ( N ` { z } ) ) |
65 |
10 13 1 12 2 3 4 5 6 7 8 9 11 58 59 60 61 62 64
|
mapdhcl |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. X , F , z >. ) e. D ) |
66 |
|
eqidd |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. X , F , z >. ) = ( I ` <. X , F , z >. ) ) |
67 |
|
simprl |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> z e. ( V \ { .0. } ) ) |
68 |
10 13 1 12 2 3 4 5 6 7 8 9 11 58 59 60 61 67 65 64
|
mapdheq |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( ( I ` <. X , F , z >. ) = ( I ` <. X , F , z >. ) <-> ( ( M ` ( N ` { z } ) ) = ( J ` { ( I ` <. X , F , z >. ) } ) /\ ( M ` ( N ` { ( X .- z ) } ) ) = ( J ` { ( F R ( I ` <. X , F , z >. ) ) } ) ) ) ) |
69 |
66 68
|
mpbid |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( ( M ` ( N ` { z } ) ) = ( J ` { ( I ` <. X , F , z >. ) } ) /\ ( M ` ( N ` { ( X .- z ) } ) ) = ( J ` { ( F R ( I ` <. X , F , z >. ) ) } ) ) ) |
70 |
69
|
simpld |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( M ` ( N ` { z } ) ) = ( J ` { ( I ` <. X , F , z >. ) } ) ) |
71 |
18
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> T e. V ) |
72 |
|
simprrr |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { z } ) =/= ( N ` { T } ) ) |
73 |
10 13 1 12 2 3 4 5 6 7 8 9 11 58 65 70 67 71 72
|
mapdhcl |
|- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) |
74 |
73
|
ex |
|- ( ( ph /\ z e. V ) -> ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) ) |
75 |
74
|
ancld |
|- ( ( ph /\ z e. V ) -> ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) ) ) |
76 |
75
|
reximdva |
|- ( ph -> ( E. z e. V ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> E. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) ) ) |
77 |
57 76
|
mpd |
|- ( ph -> E. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) ) |
78 |
|
eleq1w |
|- ( z = w -> ( z e. ( V \ { .0. } ) <-> w e. ( V \ { .0. } ) ) ) |
79 |
|
sneq |
|- ( z = w -> { z } = { w } ) |
80 |
79
|
fveq2d |
|- ( z = w -> ( N ` { z } ) = ( N ` { w } ) ) |
81 |
80
|
neeq1d |
|- ( z = w -> ( ( N ` { z } ) =/= ( N ` { X } ) <-> ( N ` { w } ) =/= ( N ` { X } ) ) ) |
82 |
80
|
neeq1d |
|- ( z = w -> ( ( N ` { z } ) =/= ( N ` { T } ) <-> ( N ` { w } ) =/= ( N ` { T } ) ) ) |
83 |
81 82
|
anbi12d |
|- ( z = w -> ( ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) <-> ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) |
84 |
78 83
|
anbi12d |
|- ( z = w -> ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) <-> ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) |
85 |
|
oteq1 |
|- ( z = w -> <. z , ( I ` <. X , F , z >. ) , T >. = <. w , ( I ` <. X , F , z >. ) , T >. ) |
86 |
|
oteq3 |
|- ( z = w -> <. X , F , z >. = <. X , F , w >. ) |
87 |
86
|
fveq2d |
|- ( z = w -> ( I ` <. X , F , z >. ) = ( I ` <. X , F , w >. ) ) |
88 |
87
|
oteq2d |
|- ( z = w -> <. w , ( I ` <. X , F , z >. ) , T >. = <. w , ( I ` <. X , F , w >. ) , T >. ) |
89 |
85 88
|
eqtrd |
|- ( z = w -> <. z , ( I ` <. X , F , z >. ) , T >. = <. w , ( I ` <. X , F , w >. ) , T >. ) |
90 |
89
|
fveq2d |
|- ( z = w -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) |
91 |
84 90
|
reusv3 |
|- ( E. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) -> ( A. z e. V A. w e. V ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) <-> E. y e. D A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) ) |
92 |
77 91
|
syl |
|- ( ph -> ( A. z e. V A. w e. V ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) <-> E. y e. D A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) ) |
93 |
38 92
|
mpbid |
|- ( ph -> E. y e. D A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) |
94 |
|
ioran |
|- ( -. ( z e. ( N ` { X } ) \/ z e. ( N ` { T } ) ) <-> ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) |
95 |
|
elun |
|- ( z e. ( ( N ` { X } ) u. ( N ` { T } ) ) <-> ( z e. ( N ` { X } ) \/ z e. ( N ` { T } ) ) ) |
96 |
94 95
|
xchnxbir |
|- ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) <-> ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) |
97 |
42
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> U e. LMod ) |
98 |
3 41 6
|
lspsncl |
|- ( ( U e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` U ) ) |
99 |
42 39 98
|
syl2anc |
|- ( ph -> ( N ` { X } ) e. ( LSubSp ` U ) ) |
100 |
99
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> ( N ` { X } ) e. ( LSubSp ` U ) ) |
101 |
|
simplr |
|- ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> z e. V ) |
102 |
|
simprl |
|- ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> -. z e. ( N ` { X } ) ) |
103 |
5 41 97 100 101 102
|
lssneln0 |
|- ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> z e. ( V \ { .0. } ) ) |
104 |
103
|
ex |
|- ( ( ph /\ z e. V ) -> ( ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) -> z e. ( V \ { .0. } ) ) ) |
105 |
42
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> U e. LMod ) |
106 |
|
simplr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> z e. V ) |
107 |
39
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> X e. V ) |
108 |
|
simpr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> -. z e. ( N ` { X } ) ) |
109 |
3 6 105 106 107 108
|
lspsnne2 |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> ( N ` { z } ) =/= ( N ` { X } ) ) |
110 |
109
|
ex |
|- ( ( ph /\ z e. V ) -> ( -. z e. ( N ` { X } ) -> ( N ` { z } ) =/= ( N ` { X } ) ) ) |
111 |
42
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> U e. LMod ) |
112 |
|
simplr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> z e. V ) |
113 |
18
|
ad2antrr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> T e. V ) |
114 |
|
simpr |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> -. z e. ( N ` { T } ) ) |
115 |
3 6 111 112 113 114
|
lspsnne2 |
|- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> ( N ` { z } ) =/= ( N ` { T } ) ) |
116 |
115
|
ex |
|- ( ( ph /\ z e. V ) -> ( -. z e. ( N ` { T } ) -> ( N ` { z } ) =/= ( N ` { T } ) ) ) |
117 |
110 116
|
anim12d |
|- ( ( ph /\ z e. V ) -> ( ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) -> ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) |
118 |
104 117
|
jcad |
|- ( ( ph /\ z e. V ) -> ( ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) -> ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) ) |
119 |
96 118
|
syl5bi |
|- ( ( ph /\ z e. V ) -> ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) ) |
120 |
119
|
imim1d |
|- ( ( ph /\ z e. V ) -> ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) -> ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) ) |
121 |
120
|
ralimdva |
|- ( ph -> ( A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) -> A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) ) |
122 |
121
|
reximdv |
|- ( ph -> ( E. y e. D A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) -> E. y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) ) |
123 |
93 122
|
mpd |
|- ( ph -> E. y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) |
124 |
3 6 42 39 18
|
lspprid1 |
|- ( ph -> X e. ( N ` { X , T } ) ) |
125 |
41 6 42 44 124
|
lspsnel5a |
|- ( ph -> ( N ` { X } ) C_ ( N ` { X , T } ) ) |
126 |
3 6 42 39 18
|
lspprid2 |
|- ( ph -> T e. ( N ` { X , T } ) ) |
127 |
41 6 42 44 126
|
lspsnel5a |
|- ( ph -> ( N ` { T } ) C_ ( N ` { X , T } ) ) |
128 |
125 127
|
unssd |
|- ( ph -> ( ( N ` { X } ) u. ( N ` { T } ) ) C_ ( N ` { X , T } ) ) |
129 |
128
|
ssneld |
|- ( ph -> ( -. z e. ( N ` { X , T } ) -> -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) ) ) |
130 |
129
|
reximdv |
|- ( ph -> ( E. z e. V -. z e. ( N ` { X , T } ) -> E. z e. V -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) ) ) |
131 |
40 130
|
mpd |
|- ( ph -> E. z e. V -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) ) |
132 |
|
reusv1 |
|- ( E. z e. V -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> ( E! y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) <-> E. y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) ) |
133 |
131 132
|
syl |
|- ( ph -> ( E! y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) <-> E. y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) ) |
134 |
123 133
|
mpbird |
|- ( ph -> E! y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) |