Step |
Hyp |
Ref |
Expression |
1 |
|
mapdpglem.h |
|- H = ( LHyp ` K ) |
2 |
|
mapdpglem.m |
|- M = ( ( mapd ` K ) ` W ) |
3 |
|
mapdpglem.u |
|- U = ( ( DVecH ` K ) ` W ) |
4 |
|
mapdpglem.v |
|- V = ( Base ` U ) |
5 |
|
mapdpglem.s |
|- .- = ( -g ` U ) |
6 |
|
mapdpglem.n |
|- N = ( LSpan ` U ) |
7 |
|
mapdpglem.c |
|- C = ( ( LCDual ` K ) ` W ) |
8 |
|
mapdpglem.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
9 |
|
mapdpglem.x |
|- ( ph -> X e. V ) |
10 |
|
mapdpglem.y |
|- ( ph -> Y e. V ) |
11 |
|
mapdpglem1.p |
|- .(+) = ( LSSum ` C ) |
12 |
|
mapdpglem2.j |
|- J = ( LSpan ` C ) |
13 |
|
mapdpglem3.f |
|- F = ( Base ` C ) |
14 |
|
mapdpglem3.te |
|- ( ph -> t e. ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) ) |
15 |
|
mapdpglem3.a |
|- A = ( Scalar ` U ) |
16 |
|
mapdpglem3.b |
|- B = ( Base ` A ) |
17 |
|
mapdpglem3.t |
|- .x. = ( .s ` C ) |
18 |
|
mapdpglem3.r |
|- R = ( -g ` C ) |
19 |
|
mapdpglem3.g |
|- ( ph -> G e. F ) |
20 |
|
mapdpglem3.e |
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) ) |
21 |
|
mapdpglem4.q |
|- Q = ( 0g ` U ) |
22 |
|
mapdpglem.ne |
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
23 |
|
mapdpglem4.jt |
|- ( ph -> ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) ) |
24 |
|
mapdpglem4.z |
|- .0. = ( 0g ` A ) |
25 |
|
mapdpglem4.g4 |
|- ( ph -> g e. B ) |
26 |
|
mapdpglem4.z4 |
|- ( ph -> z e. ( M ` ( N ` { Y } ) ) ) |
27 |
|
mapdpglem4.t4 |
|- ( ph -> t = ( ( g .x. G ) R z ) ) |
28 |
|
mapdpglem4.xn |
|- ( ph -> X =/= Q ) |
29 |
|
mapdpglem12.yn |
|- ( ph -> Y =/= Q ) |
30 |
|
mapdpglem17.ep |
|- E = ( ( ( invr ` A ) ` g ) .x. z ) |
31 |
1 7 8
|
lcdlvec |
|- ( ph -> C e. LVec ) |
32 |
1 3 8
|
dvhlvec |
|- ( ph -> U e. LVec ) |
33 |
15
|
lvecdrng |
|- ( U e. LVec -> A e. DivRing ) |
34 |
32 33
|
syl |
|- ( ph -> A e. DivRing ) |
35 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
|
mapdpglem11 |
|- ( ph -> g =/= .0. ) |
36 |
|
eqid |
|- ( invr ` A ) = ( invr ` A ) |
37 |
16 24 36
|
drnginvrcl |
|- ( ( A e. DivRing /\ g e. B /\ g =/= .0. ) -> ( ( invr ` A ) ` g ) e. B ) |
38 |
34 25 35 37
|
syl3anc |
|- ( ph -> ( ( invr ` A ) ` g ) e. B ) |
39 |
|
eqid |
|- ( Scalar ` C ) = ( Scalar ` C ) |
40 |
|
eqid |
|- ( Base ` ( Scalar ` C ) ) = ( Base ` ( Scalar ` C ) ) |
41 |
1 3 15 16 7 39 40 8
|
lcdsbase |
|- ( ph -> ( Base ` ( Scalar ` C ) ) = B ) |
42 |
38 41
|
eleqtrrd |
|- ( ph -> ( ( invr ` A ) ` g ) e. ( Base ` ( Scalar ` C ) ) ) |
43 |
16 24 36
|
drnginvrn0 |
|- ( ( A e. DivRing /\ g e. B /\ g =/= .0. ) -> ( ( invr ` A ) ` g ) =/= .0. ) |
44 |
34 25 35 43
|
syl3anc |
|- ( ph -> ( ( invr ` A ) ` g ) =/= .0. ) |
45 |
|
eqid |
|- ( 0g ` ( Scalar ` C ) ) = ( 0g ` ( Scalar ` C ) ) |
46 |
1 3 15 24 7 39 45 8
|
lcd0 |
|- ( ph -> ( 0g ` ( Scalar ` C ) ) = .0. ) |
47 |
44 46
|
neeqtrrd |
|- ( ph -> ( ( invr ` A ) ` g ) =/= ( 0g ` ( Scalar ` C ) ) ) |
48 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|
mapdpglem2a |
|- ( ph -> t e. F ) |
49 |
13 39 17 40 45 12
|
lspsnvs |
|- ( ( C e. LVec /\ ( ( ( invr ` A ) ` g ) e. ( Base ` ( Scalar ` C ) ) /\ ( ( invr ` A ) ` g ) =/= ( 0g ` ( Scalar ` C ) ) ) /\ t e. F ) -> ( J ` { ( ( ( invr ` A ) ` g ) .x. t ) } ) = ( J ` { t } ) ) |
50 |
31 42 47 48 49
|
syl121anc |
|- ( ph -> ( J ` { ( ( ( invr ` A ) ` g ) .x. t ) } ) = ( J ` { t } ) ) |
51 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
|
mapdpglem21 |
|- ( ph -> ( ( ( invr ` A ) ` g ) .x. t ) = ( G R E ) ) |
52 |
51
|
sneqd |
|- ( ph -> { ( ( ( invr ` A ) ` g ) .x. t ) } = { ( G R E ) } ) |
53 |
52
|
fveq2d |
|- ( ph -> ( J ` { ( ( ( invr ` A ) ` g ) .x. t ) } ) = ( J ` { ( G R E ) } ) ) |
54 |
23 50 53
|
3eqtr2d |
|- ( ph -> ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R E ) } ) ) |