Step |
Hyp |
Ref |
Expression |
1 |
|
hdmapinvlem3.h |
|- H = ( LHyp ` K ) |
2 |
|
hdmapinvlem3.e |
|- E = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. |
3 |
|
hdmapinvlem3.o |
|- O = ( ( ocH ` K ) ` W ) |
4 |
|
hdmapinvlem3.u |
|- U = ( ( DVecH ` K ) ` W ) |
5 |
|
hdmapinvlem3.v |
|- V = ( Base ` U ) |
6 |
|
hdmapinvlem3.p |
|- .+ = ( +g ` U ) |
7 |
|
hdmapinvlem3.m |
|- .- = ( -g ` U ) |
8 |
|
hdmapinvlem3.q |
|- .x. = ( .s ` U ) |
9 |
|
hdmapinvlem3.r |
|- R = ( Scalar ` U ) |
10 |
|
hdmapinvlem3.b |
|- B = ( Base ` R ) |
11 |
|
hdmapinvlem3.t |
|- .X. = ( .r ` R ) |
12 |
|
hdmapinvlem3.z |
|- .0. = ( 0g ` R ) |
13 |
|
hdmapinvlem3.s |
|- S = ( ( HDMap ` K ) ` W ) |
14 |
|
hdmapinvlem3.g |
|- G = ( ( HGMap ` K ) ` W ) |
15 |
|
hdmapinvlem3.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
16 |
|
hdmapinvlem3.c |
|- ( ph -> C e. ( O ` { E } ) ) |
17 |
|
hdmapinvlem3.d |
|- ( ph -> D e. ( O ` { E } ) ) |
18 |
|
hdmapinvlem3.i |
|- ( ph -> I e. B ) |
19 |
|
hdmapinvlem3.j |
|- ( ph -> J e. B ) |
20 |
|
hdmapinvlem3.ij |
|- ( ph -> ( I .X. ( G ` J ) ) = ( ( S ` D ) ` C ) ) |
21 |
|
eqid |
|- ( ( LCDual ` K ) ` W ) = ( ( LCDual ` K ) ` W ) |
22 |
|
eqid |
|- ( -g ` ( ( LCDual ` K ) ` W ) ) = ( -g ` ( ( LCDual ` K ) ` W ) ) |
23 |
1 4 15
|
dvhlmod |
|- ( ph -> U e. LMod ) |
24 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
25 |
|
eqid |
|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W ) |
26 |
|
eqid |
|- ( 0g ` U ) = ( 0g ` U ) |
27 |
1 24 25 4 5 26 2 15
|
dvheveccl |
|- ( ph -> E e. ( V \ { ( 0g ` U ) } ) ) |
28 |
27
|
eldifad |
|- ( ph -> E e. V ) |
29 |
5 9 8 10
|
lmodvscl |
|- ( ( U e. LMod /\ J e. B /\ E e. V ) -> ( J .x. E ) e. V ) |
30 |
23 19 28 29
|
syl3anc |
|- ( ph -> ( J .x. E ) e. V ) |
31 |
28
|
snssd |
|- ( ph -> { E } C_ V ) |
32 |
1 4 5 3
|
dochssv |
|- ( ( ( K e. HL /\ W e. H ) /\ { E } C_ V ) -> ( O ` { E } ) C_ V ) |
33 |
15 31 32
|
syl2anc |
|- ( ph -> ( O ` { E } ) C_ V ) |
34 |
33 17
|
sseldd |
|- ( ph -> D e. V ) |
35 |
1 4 5 7 21 22 13 15 30 34
|
hdmapsub |
|- ( ph -> ( S ` ( ( J .x. E ) .- D ) ) = ( ( S ` ( J .x. E ) ) ( -g ` ( ( LCDual ` K ) ` W ) ) ( S ` D ) ) ) |
36 |
35
|
fveq1d |
|- ( ph -> ( ( S ` ( ( J .x. E ) .- D ) ) ` ( ( I .x. E ) .+ C ) ) = ( ( ( S ` ( J .x. E ) ) ( -g ` ( ( LCDual ` K ) ` W ) ) ( S ` D ) ) ` ( ( I .x. E ) .+ C ) ) ) |
37 |
|
eqid |
|- ( -g ` R ) = ( -g ` R ) |
38 |
|
eqid |
|- ( Base ` ( ( LCDual ` K ) ` W ) ) = ( Base ` ( ( LCDual ` K ) ` W ) ) |
39 |
1 4 5 21 38 13 15 30
|
hdmapcl |
|- ( ph -> ( S ` ( J .x. E ) ) e. ( Base ` ( ( LCDual ` K ) ` W ) ) ) |
40 |
1 4 5 21 38 13 15 34
|
hdmapcl |
|- ( ph -> ( S ` D ) e. ( Base ` ( ( LCDual ` K ) ` W ) ) ) |
41 |
5 9 8 10
|
lmodvscl |
|- ( ( U e. LMod /\ I e. B /\ E e. V ) -> ( I .x. E ) e. V ) |
42 |
23 18 28 41
|
syl3anc |
|- ( ph -> ( I .x. E ) e. V ) |
43 |
33 16
|
sseldd |
|- ( ph -> C e. V ) |
44 |
5 6
|
lmodvacl |
|- ( ( U e. LMod /\ ( I .x. E ) e. V /\ C e. V ) -> ( ( I .x. E ) .+ C ) e. V ) |
45 |
23 42 43 44
|
syl3anc |
|- ( ph -> ( ( I .x. E ) .+ C ) e. V ) |
46 |
1 4 5 9 37 21 38 22 15 39 40 45
|
lcdvsubval |
|- ( ph -> ( ( ( S ` ( J .x. E ) ) ( -g ` ( ( LCDual ` K ) ` W ) ) ( S ` D ) ) ` ( ( I .x. E ) .+ C ) ) = ( ( ( S ` ( J .x. E ) ) ` ( ( I .x. E ) .+ C ) ) ( -g ` R ) ( ( S ` D ) ` ( ( I .x. E ) .+ C ) ) ) ) |
47 |
|
eqid |
|- ( +g ` R ) = ( +g ` R ) |
48 |
1 4 5 6 9 47 13 15 42 43 30
|
hdmaplna1 |
|- ( ph -> ( ( S ` ( J .x. E ) ) ` ( ( I .x. E ) .+ C ) ) = ( ( ( S ` ( J .x. E ) ) ` ( I .x. E ) ) ( +g ` R ) ( ( S ` ( J .x. E ) ) ` C ) ) ) |
49 |
1 4 5 8 9 10 11 13 14 15 42 28 19
|
hdmapglnm2 |
|- ( ph -> ( ( S ` ( J .x. E ) ) ` ( I .x. E ) ) = ( ( ( S ` E ) ` ( I .x. E ) ) .X. ( G ` J ) ) ) |
50 |
1 4 5 8 9 10 11 13 15 28 28 18
|
hdmaplnm1 |
|- ( ph -> ( ( S ` E ) ` ( I .x. E ) ) = ( I .X. ( ( S ` E ) ` E ) ) ) |
51 |
|
eqid |
|- ( ( HVMap ` K ) ` W ) = ( ( HVMap ` K ) ` W ) |
52 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
53 |
1 2 51 13 15 4 9 52
|
hdmapevec2 |
|- ( ph -> ( ( S ` E ) ` E ) = ( 1r ` R ) ) |
54 |
53
|
oveq2d |
|- ( ph -> ( I .X. ( ( S ` E ) ` E ) ) = ( I .X. ( 1r ` R ) ) ) |
55 |
9
|
lmodring |
|- ( U e. LMod -> R e. Ring ) |
56 |
23 55
|
syl |
|- ( ph -> R e. Ring ) |
57 |
10 11 52
|
ringridm |
|- ( ( R e. Ring /\ I e. B ) -> ( I .X. ( 1r ` R ) ) = I ) |
58 |
56 18 57
|
syl2anc |
|- ( ph -> ( I .X. ( 1r ` R ) ) = I ) |
59 |
50 54 58
|
3eqtrd |
|- ( ph -> ( ( S ` E ) ` ( I .x. E ) ) = I ) |
60 |
59
|
oveq1d |
|- ( ph -> ( ( ( S ` E ) ` ( I .x. E ) ) .X. ( G ` J ) ) = ( I .X. ( G ` J ) ) ) |
61 |
49 60
|
eqtrd |
|- ( ph -> ( ( S ` ( J .x. E ) ) ` ( I .x. E ) ) = ( I .X. ( G ` J ) ) ) |
62 |
1 4 5 8 9 10 11 13 14 15 43 28 19
|
hdmapglnm2 |
|- ( ph -> ( ( S ` ( J .x. E ) ) ` C ) = ( ( ( S ` E ) ` C ) .X. ( G ` J ) ) ) |
63 |
1 2 3 4 5 9 10 11 12 13 15 16
|
hdmapinvlem1 |
|- ( ph -> ( ( S ` E ) ` C ) = .0. ) |
64 |
63
|
oveq1d |
|- ( ph -> ( ( ( S ` E ) ` C ) .X. ( G ` J ) ) = ( .0. .X. ( G ` J ) ) ) |
65 |
1 4 9 10 14 15 19
|
hgmapcl |
|- ( ph -> ( G ` J ) e. B ) |
66 |
10 11 12
|
ringlz |
|- ( ( R e. Ring /\ ( G ` J ) e. B ) -> ( .0. .X. ( G ` J ) ) = .0. ) |
67 |
56 65 66
|
syl2anc |
|- ( ph -> ( .0. .X. ( G ` J ) ) = .0. ) |
68 |
62 64 67
|
3eqtrd |
|- ( ph -> ( ( S ` ( J .x. E ) ) ` C ) = .0. ) |
69 |
61 68
|
oveq12d |
|- ( ph -> ( ( ( S ` ( J .x. E ) ) ` ( I .x. E ) ) ( +g ` R ) ( ( S ` ( J .x. E ) ) ` C ) ) = ( ( I .X. ( G ` J ) ) ( +g ` R ) .0. ) ) |
70 |
|
ringgrp |
|- ( R e. Ring -> R e. Grp ) |
71 |
56 70
|
syl |
|- ( ph -> R e. Grp ) |
72 |
9 10 11
|
lmodmcl |
|- ( ( U e. LMod /\ I e. B /\ ( G ` J ) e. B ) -> ( I .X. ( G ` J ) ) e. B ) |
73 |
23 18 65 72
|
syl3anc |
|- ( ph -> ( I .X. ( G ` J ) ) e. B ) |
74 |
10 47 12
|
grprid |
|- ( ( R e. Grp /\ ( I .X. ( G ` J ) ) e. B ) -> ( ( I .X. ( G ` J ) ) ( +g ` R ) .0. ) = ( I .X. ( G ` J ) ) ) |
75 |
71 73 74
|
syl2anc |
|- ( ph -> ( ( I .X. ( G ` J ) ) ( +g ` R ) .0. ) = ( I .X. ( G ` J ) ) ) |
76 |
48 69 75
|
3eqtrd |
|- ( ph -> ( ( S ` ( J .x. E ) ) ` ( ( I .x. E ) .+ C ) ) = ( I .X. ( G ` J ) ) ) |
77 |
1 4 5 6 9 47 13 15 42 43 34
|
hdmaplna1 |
|- ( ph -> ( ( S ` D ) ` ( ( I .x. E ) .+ C ) ) = ( ( ( S ` D ) ` ( I .x. E ) ) ( +g ` R ) ( ( S ` D ) ` C ) ) ) |
78 |
1 4 5 8 9 10 11 13 15 28 34 18
|
hdmaplnm1 |
|- ( ph -> ( ( S ` D ) ` ( I .x. E ) ) = ( I .X. ( ( S ` D ) ` E ) ) ) |
79 |
1 2 3 4 5 9 10 11 12 13 15 17
|
hdmapinvlem2 |
|- ( ph -> ( ( S ` D ) ` E ) = .0. ) |
80 |
79
|
oveq2d |
|- ( ph -> ( I .X. ( ( S ` D ) ` E ) ) = ( I .X. .0. ) ) |
81 |
10 11 12
|
ringrz |
|- ( ( R e. Ring /\ I e. B ) -> ( I .X. .0. ) = .0. ) |
82 |
56 18 81
|
syl2anc |
|- ( ph -> ( I .X. .0. ) = .0. ) |
83 |
78 80 82
|
3eqtrrd |
|- ( ph -> .0. = ( ( S ` D ) ` ( I .x. E ) ) ) |
84 |
83 20
|
oveq12d |
|- ( ph -> ( .0. ( +g ` R ) ( I .X. ( G ` J ) ) ) = ( ( ( S ` D ) ` ( I .x. E ) ) ( +g ` R ) ( ( S ` D ) ` C ) ) ) |
85 |
10 47 12
|
grplid |
|- ( ( R e. Grp /\ ( I .X. ( G ` J ) ) e. B ) -> ( .0. ( +g ` R ) ( I .X. ( G ` J ) ) ) = ( I .X. ( G ` J ) ) ) |
86 |
71 73 85
|
syl2anc |
|- ( ph -> ( .0. ( +g ` R ) ( I .X. ( G ` J ) ) ) = ( I .X. ( G ` J ) ) ) |
87 |
77 84 86
|
3eqtr2d |
|- ( ph -> ( ( S ` D ) ` ( ( I .x. E ) .+ C ) ) = ( I .X. ( G ` J ) ) ) |
88 |
76 87
|
oveq12d |
|- ( ph -> ( ( ( S ` ( J .x. E ) ) ` ( ( I .x. E ) .+ C ) ) ( -g ` R ) ( ( S ` D ) ` ( ( I .x. E ) .+ C ) ) ) = ( ( I .X. ( G ` J ) ) ( -g ` R ) ( I .X. ( G ` J ) ) ) ) |
89 |
46 88
|
eqtrd |
|- ( ph -> ( ( ( S ` ( J .x. E ) ) ( -g ` ( ( LCDual ` K ) ` W ) ) ( S ` D ) ) ` ( ( I .x. E ) .+ C ) ) = ( ( I .X. ( G ` J ) ) ( -g ` R ) ( I .X. ( G ` J ) ) ) ) |
90 |
10 12 37
|
grpsubid |
|- ( ( R e. Grp /\ ( I .X. ( G ` J ) ) e. B ) -> ( ( I .X. ( G ` J ) ) ( -g ` R ) ( I .X. ( G ` J ) ) ) = .0. ) |
91 |
71 73 90
|
syl2anc |
|- ( ph -> ( ( I .X. ( G ` J ) ) ( -g ` R ) ( I .X. ( G ` J ) ) ) = .0. ) |
92 |
36 89 91
|
3eqtrd |
|- ( ph -> ( ( S ` ( ( J .x. E ) .- D ) ) ` ( ( I .x. E ) .+ C ) ) = .0. ) |