Step |
Hyp |
Ref |
Expression |
1 |
|
mapdpg.h |
|- H = ( LHyp ` K ) |
2 |
|
mapdpg.m |
|- M = ( ( mapd ` K ) ` W ) |
3 |
|
mapdpg.u |
|- U = ( ( DVecH ` K ) ` W ) |
4 |
|
mapdpg.v |
|- V = ( Base ` U ) |
5 |
|
mapdpg.s |
|- .- = ( -g ` U ) |
6 |
|
mapdpg.z |
|- .0. = ( 0g ` U ) |
7 |
|
mapdpg.n |
|- N = ( LSpan ` U ) |
8 |
|
mapdpg.c |
|- C = ( ( LCDual ` K ) ` W ) |
9 |
|
mapdpg.f |
|- F = ( Base ` C ) |
10 |
|
mapdpg.r |
|- R = ( -g ` C ) |
11 |
|
mapdpg.j |
|- J = ( LSpan ` C ) |
12 |
|
mapdpg.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
13 |
|
mapdpg.x |
|- ( ph -> X e. ( V \ { .0. } ) ) |
14 |
|
mapdpg.y |
|- ( ph -> Y e. ( V \ { .0. } ) ) |
15 |
|
mapdpg.g |
|- ( ph -> G e. F ) |
16 |
|
mapdpg.ne |
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
17 |
|
mapdpg.e |
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) ) |
18 |
|
mapdpgem25.h1 |
|- ( ph -> ( h e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) ) ) |
19 |
|
mapdpgem25.i1 |
|- ( ph -> ( i e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) |
20 |
|
mapdpglem26.a |
|- A = ( Scalar ` U ) |
21 |
|
mapdpglem26.b |
|- B = ( Base ` A ) |
22 |
|
mapdpglem26.t |
|- .x. = ( .s ` C ) |
23 |
|
mapdpglem26.o |
|- O = ( 0g ` A ) |
24 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
|
mapdpglem25 |
|- ( ph -> ( ( J ` { h } ) = ( J ` { i } ) /\ ( J ` { ( G R h ) } ) = ( J ` { ( G R i ) } ) ) ) |
25 |
24
|
simpld |
|- ( ph -> ( J ` { h } ) = ( J ` { i } ) ) |
26 |
|
eqid |
|- ( Scalar ` C ) = ( Scalar ` C ) |
27 |
|
eqid |
|- ( Base ` ( Scalar ` C ) ) = ( Base ` ( Scalar ` C ) ) |
28 |
|
eqid |
|- ( 0g ` ( Scalar ` C ) ) = ( 0g ` ( Scalar ` C ) ) |
29 |
1 8 12
|
lcdlvec |
|- ( ph -> C e. LVec ) |
30 |
18
|
simpld |
|- ( ph -> h e. F ) |
31 |
19
|
simpld |
|- ( ph -> i e. F ) |
32 |
9 26 27 28 22 11 29 30 31
|
lspsneq |
|- ( ph -> ( ( J ` { h } ) = ( J ` { i } ) <-> E. u e. ( ( Base ` ( Scalar ` C ) ) \ { ( 0g ` ( Scalar ` C ) ) } ) h = ( u .x. i ) ) ) |
33 |
1 3 20 21 8 26 27 12
|
lcdsbase |
|- ( ph -> ( Base ` ( Scalar ` C ) ) = B ) |
34 |
1 3 20 23 8 26 28 12
|
lcd0 |
|- ( ph -> ( 0g ` ( Scalar ` C ) ) = O ) |
35 |
34
|
sneqd |
|- ( ph -> { ( 0g ` ( Scalar ` C ) ) } = { O } ) |
36 |
33 35
|
difeq12d |
|- ( ph -> ( ( Base ` ( Scalar ` C ) ) \ { ( 0g ` ( Scalar ` C ) ) } ) = ( B \ { O } ) ) |
37 |
36
|
rexeqdv |
|- ( ph -> ( E. u e. ( ( Base ` ( Scalar ` C ) ) \ { ( 0g ` ( Scalar ` C ) ) } ) h = ( u .x. i ) <-> E. u e. ( B \ { O } ) h = ( u .x. i ) ) ) |
38 |
32 37
|
bitrd |
|- ( ph -> ( ( J ` { h } ) = ( J ` { i } ) <-> E. u e. ( B \ { O } ) h = ( u .x. i ) ) ) |
39 |
25 38
|
mpbid |
|- ( ph -> E. u e. ( B \ { O } ) h = ( u .x. i ) ) |