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 |
19
|
simprd |
|- ( ph -> ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) |
21 |
20
|
simpld |
|- ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { i } ) ) |
22 |
|
eqid |
|- ( LSAtoms ` U ) = ( LSAtoms ` U ) |
23 |
|
eqid |
|- ( LSAtoms ` C ) = ( LSAtoms ` C ) |
24 |
1 3 12
|
dvhlmod |
|- ( ph -> U e. LMod ) |
25 |
4 7 6 22 24 14
|
lsatlspsn |
|- ( ph -> ( N ` { Y } ) e. ( LSAtoms ` U ) ) |
26 |
1 2 3 22 8 23 12 25
|
mapdat |
|- ( ph -> ( M ` ( N ` { Y } ) ) e. ( LSAtoms ` C ) ) |
27 |
21 26
|
eqeltrrd |
|- ( ph -> ( J ` { i } ) e. ( LSAtoms ` C ) ) |
28 |
|
eqid |
|- ( 0g ` C ) = ( 0g ` C ) |
29 |
1 8 12
|
lcdlmod |
|- ( ph -> C e. LMod ) |
30 |
19
|
simpld |
|- ( ph -> i e. F ) |
31 |
9 11 28 23 29 30
|
lsatspn0 |
|- ( ph -> ( ( J ` { i } ) e. ( LSAtoms ` C ) <-> i =/= ( 0g ` C ) ) ) |
32 |
27 31
|
mpbid |
|- ( ph -> i =/= ( 0g ` C ) ) |