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 8 16
|
lcdlmod |
|- ( ph -> C e. LMod ) |
27 |
1 2 16
|
dvhlvec |
|- ( ph -> U e. LVec ) |
28 |
24
|
eldifad |
|- ( ph -> w e. V ) |
29 |
18
|
eldifad |
|- ( ph -> X e. V ) |
30 |
22
|
eldifad |
|- ( ph -> Y e. V ) |
31 |
3 7 27 28 29 30 25
|
lspindpi |
|- ( ph -> ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { Y } ) ) ) |
32 |
31
|
simpld |
|- ( ph -> ( N ` { w } ) =/= ( N ` { X } ) ) |
33 |
32
|
necomd |
|- ( ph -> ( N ` { X } ) =/= ( N ` { w } ) ) |
34 |
1 2 3 6 7 8 9 13 14 15 16 17 19 33 18 28
|
hdmap1cl |
|- ( ph -> ( I ` <. X , F , w >. ) e. D ) |
35 |
9 10 12
|
lmod0vrid |
|- ( ( C e. LMod /\ ( I ` <. X , F , w >. ) e. D ) -> ( ( I ` <. X , F , w >. ) .+b Q ) = ( I ` <. X , F , w >. ) ) |
36 |
26 34 35
|
syl2anc |
|- ( ph -> ( ( I ` <. X , F , w >. ) .+b Q ) = ( I ` <. X , F , w >. ) ) |
37 |
36
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( ( I ` <. X , F , w >. ) .+b Q ) = ( I ` <. X , F , w >. ) ) |
38 |
|
oteq3 |
|- ( ( Y .+ Z ) = .0. -> <. X , F , ( Y .+ Z ) >. = <. X , F , .0. >. ) |
39 |
38
|
fveq2d |
|- ( ( Y .+ Z ) = .0. -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( I ` <. X , F , .0. >. ) ) |
40 |
1 2 3 6 8 9 12 15 16 17 29
|
hdmap1val0 |
|- ( ph -> ( I ` <. X , F , .0. >. ) = Q ) |
41 |
39 40
|
sylan9eqr |
|- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = Q ) |
42 |
41
|
oveq2d |
|- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) = ( ( I ` <. X , F , w >. ) .+b Q ) ) |
43 |
|
oveq2 |
|- ( ( Y .+ Z ) = .0. -> ( w .+ ( Y .+ Z ) ) = ( w .+ .0. ) ) |
44 |
1 2 16
|
dvhlmod |
|- ( ph -> U e. LMod ) |
45 |
3 4 6
|
lmod0vrid |
|- ( ( U e. LMod /\ w e. V ) -> ( w .+ .0. ) = w ) |
46 |
44 28 45
|
syl2anc |
|- ( ph -> ( w .+ .0. ) = w ) |
47 |
43 46
|
sylan9eqr |
|- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( w .+ ( Y .+ Z ) ) = w ) |
48 |
47
|
oteq3d |
|- ( ( ph /\ ( Y .+ Z ) = .0. ) -> <. X , F , ( w .+ ( Y .+ Z ) ) >. = <. X , F , w >. ) |
49 |
48
|
fveq2d |
|- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( I ` <. X , F , w >. ) ) |
50 |
37 42 49
|
3eqtr4rd |
|- ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) ) |
51 |
16
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( K e. HL /\ W e. H ) ) |
52 |
17
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> F e. D ) |
53 |
18
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> X e. ( V \ { .0. } ) ) |
54 |
19
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) ) |
55 |
24
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> w e. ( V \ { .0. } ) ) |
56 |
23
|
eldifad |
|- ( ph -> Z e. V ) |
57 |
3 4
|
lmodvacl |
|- ( ( U e. LMod /\ Y e. V /\ Z e. V ) -> ( Y .+ Z ) e. V ) |
58 |
44 30 56 57
|
syl3anc |
|- ( ph -> ( Y .+ Z ) e. V ) |
59 |
58
|
anim1i |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( ( Y .+ Z ) e. V /\ ( Y .+ Z ) =/= .0. ) ) |
60 |
|
eldifsn |
|- ( ( Y .+ Z ) e. ( V \ { .0. } ) <-> ( ( Y .+ Z ) e. V /\ ( Y .+ Z ) =/= .0. ) ) |
61 |
59 60
|
sylibr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( Y .+ Z ) e. ( V \ { .0. } ) ) |
62 |
3 7 27 29 30 56 20
|
lspindpi |
|- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) ) |
63 |
62
|
simpld |
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
64 |
3 4 6 7 27 18 22 23 24 21 63 25
|
mapdindp1 |
|- ( ph -> ( N ` { X } ) =/= ( N ` { ( Y .+ Z ) } ) ) |
65 |
3 4 6 7 27 18 22 23 24 21 63 25
|
mapdindp2 |
|- ( ph -> -. w e. ( N ` { X , ( Y .+ Z ) } ) ) |
66 |
3 6 7 27 18 58 28 64 65
|
lspindp1 |
|- ( ph -> ( ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) /\ -. X e. ( N ` { w , ( Y .+ Z ) } ) ) ) |
67 |
66
|
simprd |
|- ( ph -> -. X e. ( N ` { w , ( Y .+ Z ) } ) ) |
68 |
67
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> -. X e. ( N ` { w , ( Y .+ Z ) } ) ) |
69 |
31
|
simprd |
|- ( ph -> ( N ` { w } ) =/= ( N ` { Y } ) ) |
70 |
3 6 7 27 24 30 69
|
lspsnne1 |
|- ( ph -> -. w e. ( N ` { Y } ) ) |
71 |
|
eqid |
|- ( LSSum ` U ) = ( LSSum ` U ) |
72 |
3 7 71 44 30 56
|
lsmpr |
|- ( ph -> ( N ` { Y , Z } ) = ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) ) |
73 |
21
|
oveq2d |
|- ( ph -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) ) |
74 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
75 |
3 74 7
|
lspsncl |
|- ( ( U e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` U ) ) |
76 |
44 30 75
|
syl2anc |
|- ( ph -> ( N ` { Y } ) e. ( LSubSp ` U ) ) |
77 |
74
|
lsssubg |
|- ( ( U e. LMod /\ ( N ` { Y } ) e. ( LSubSp ` U ) ) -> ( N ` { Y } ) e. ( SubGrp ` U ) ) |
78 |
44 76 77
|
syl2anc |
|- ( ph -> ( N ` { Y } ) e. ( SubGrp ` U ) ) |
79 |
71
|
lsmidm |
|- ( ( N ` { Y } ) e. ( SubGrp ` U ) -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( N ` { Y } ) ) |
80 |
78 79
|
syl |
|- ( ph -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( N ` { Y } ) ) |
81 |
72 73 80
|
3eqtr2d |
|- ( ph -> ( N ` { Y , Z } ) = ( N ` { Y } ) ) |
82 |
70 81
|
neleqtrrd |
|- ( ph -> -. w e. ( N ` { Y , Z } ) ) |
83 |
3 4 7 44 30 56 28 82
|
lspindp4 |
|- ( ph -> -. w e. ( N ` { Y , ( Y .+ Z ) } ) ) |
84 |
3 7 27 28 30 58 83
|
lspindpi |
|- ( ph -> ( ( N ` { w } ) =/= ( N ` { Y } ) /\ ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) ) ) |
85 |
84
|
simprd |
|- ( ph -> ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) ) |
86 |
85
|
adantr |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) ) |
87 |
|
eqidd |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( I ` <. X , F , w >. ) = ( I ` <. X , F , w >. ) ) |
88 |
|
eqidd |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( I ` <. X , F , ( Y .+ Z ) >. ) ) |
89 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 51 52 53 54 55 61 68 86 87 88
|
hdmap1l6a |
|- ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) ) |
90 |
50 89
|
pm2.61dane |
|- ( ph -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) ) |