Step |
Hyp |
Ref |
Expression |
1 |
|
dochexmidlem1.h |
|- H = ( LHyp ` K ) |
2 |
|
dochexmidlem1.o |
|- ._|_ = ( ( ocH ` K ) ` W ) |
3 |
|
dochexmidlem1.u |
|- U = ( ( DVecH ` K ) ` W ) |
4 |
|
dochexmidlem1.v |
|- V = ( Base ` U ) |
5 |
|
dochexmidlem1.s |
|- S = ( LSubSp ` U ) |
6 |
|
dochexmidlem1.n |
|- N = ( LSpan ` U ) |
7 |
|
dochexmidlem1.p |
|- .(+) = ( LSSum ` U ) |
8 |
|
dochexmidlem1.a |
|- A = ( LSAtoms ` U ) |
9 |
|
dochexmidlem1.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
10 |
|
dochexmidlem1.x |
|- ( ph -> X e. S ) |
11 |
|
dochexmidlem6.pp |
|- ( ph -> p e. A ) |
12 |
|
dochexmidlem6.z |
|- .0. = ( 0g ` U ) |
13 |
|
dochexmidlem6.m |
|- M = ( X .(+) p ) |
14 |
|
dochexmidlem6.xn |
|- ( ph -> X =/= { .0. } ) |
15 |
|
dochexmidlem6.c |
|- ( ph -> ( ._|_ ` ( ._|_ ` X ) ) = X ) |
16 |
|
dochexmidlem6.pl |
|- ( ph -> -. p C_ ( X .(+) ( ._|_ ` X ) ) ) |
17 |
1 3 9
|
dvhlmod |
|- ( ph -> U e. LMod ) |
18 |
5
|
lsssssubg |
|- ( U e. LMod -> S C_ ( SubGrp ` U ) ) |
19 |
17 18
|
syl |
|- ( ph -> S C_ ( SubGrp ` U ) ) |
20 |
19 10
|
sseldd |
|- ( ph -> X e. ( SubGrp ` U ) ) |
21 |
5 8 17 11
|
lsatlssel |
|- ( ph -> p e. S ) |
22 |
19 21
|
sseldd |
|- ( ph -> p e. ( SubGrp ` U ) ) |
23 |
7
|
lsmub2 |
|- ( ( X e. ( SubGrp ` U ) /\ p e. ( SubGrp ` U ) ) -> p C_ ( X .(+) p ) ) |
24 |
20 22 23
|
syl2anc |
|- ( ph -> p C_ ( X .(+) p ) ) |
25 |
24 13
|
sseqtrrdi |
|- ( ph -> p C_ M ) |
26 |
4 5
|
lssss |
|- ( X e. S -> X C_ V ) |
27 |
10 26
|
syl |
|- ( ph -> X C_ V ) |
28 |
1 3 4 5 2
|
dochlss |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) e. S ) |
29 |
9 27 28
|
syl2anc |
|- ( ph -> ( ._|_ ` X ) e. S ) |
30 |
19 29
|
sseldd |
|- ( ph -> ( ._|_ ` X ) e. ( SubGrp ` U ) ) |
31 |
7
|
lsmub1 |
|- ( ( X e. ( SubGrp ` U ) /\ ( ._|_ ` X ) e. ( SubGrp ` U ) ) -> X C_ ( X .(+) ( ._|_ ` X ) ) ) |
32 |
20 30 31
|
syl2anc |
|- ( ph -> X C_ ( X .(+) ( ._|_ ` X ) ) ) |
33 |
|
sstr2 |
|- ( p C_ X -> ( X C_ ( X .(+) ( ._|_ ` X ) ) -> p C_ ( X .(+) ( ._|_ ` X ) ) ) ) |
34 |
32 33
|
syl5com |
|- ( ph -> ( p C_ X -> p C_ ( X .(+) ( ._|_ ` X ) ) ) ) |
35 |
16 34
|
mtod |
|- ( ph -> -. p C_ X ) |
36 |
|
sseq2 |
|- ( M = X -> ( p C_ M <-> p C_ X ) ) |
37 |
36
|
biimpcd |
|- ( p C_ M -> ( M = X -> p C_ X ) ) |
38 |
37
|
necon3bd |
|- ( p C_ M -> ( -. p C_ X -> M =/= X ) ) |
39 |
25 35 38
|
sylc |
|- ( ph -> M =/= X ) |