Step |
Hyp |
Ref |
Expression |
1 |
|
dochfln0.h |
|- H = ( LHyp ` K ) |
2 |
|
dochfln0.o |
|- ._|_ = ( ( ocH ` K ) ` W ) |
3 |
|
dochfln0.u |
|- U = ( ( DVecH ` K ) ` W ) |
4 |
|
dochfln0.v |
|- V = ( Base ` U ) |
5 |
|
dochfln0.r |
|- R = ( Scalar ` U ) |
6 |
|
dochfln0.n |
|- N = ( 0g ` R ) |
7 |
|
dochfln0.z |
|- .0. = ( 0g ` U ) |
8 |
|
dochfln0.f |
|- F = ( LFnl ` U ) |
9 |
|
dochfln0.l |
|- L = ( LKer ` U ) |
10 |
|
dochfln0.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
11 |
|
dochfln0.g |
|- ( ph -> G e. F ) |
12 |
|
dochfln0.x |
|- ( ph -> X e. ( ( ._|_ ` ( L ` G ) ) \ { .0. } ) ) |
13 |
1 3 10
|
dvhlmod |
|- ( ph -> U e. LMod ) |
14 |
4 8 9 13 11
|
lkrssv |
|- ( ph -> ( L ` G ) C_ V ) |
15 |
1 3 4 2
|
dochssv |
|- ( ( ( K e. HL /\ W e. H ) /\ ( L ` G ) C_ V ) -> ( ._|_ ` ( L ` G ) ) C_ V ) |
16 |
10 14 15
|
syl2anc |
|- ( ph -> ( ._|_ ` ( L ` G ) ) C_ V ) |
17 |
16
|
ssdifd |
|- ( ph -> ( ( ._|_ ` ( L ` G ) ) \ { .0. } ) C_ ( V \ { .0. } ) ) |
18 |
17 12
|
sseldd |
|- ( ph -> X e. ( V \ { .0. } ) ) |
19 |
1 2 3 4 7 10 18
|
dochnel |
|- ( ph -> -. X e. ( ._|_ ` { X } ) ) |
20 |
12
|
eldifad |
|- ( ph -> X e. ( ._|_ ` ( L ` G ) ) ) |
21 |
16 20
|
sseldd |
|- ( ph -> X e. V ) |
22 |
21
|
biantrurd |
|- ( ph -> ( ( G ` X ) = N <-> ( X e. V /\ ( G ` X ) = N ) ) ) |
23 |
4 5 6 8 9
|
ellkr |
|- ( ( U e. LMod /\ G e. F ) -> ( X e. ( L ` G ) <-> ( X e. V /\ ( G ` X ) = N ) ) ) |
24 |
13 11 23
|
syl2anc |
|- ( ph -> ( X e. ( L ` G ) <-> ( X e. V /\ ( G ` X ) = N ) ) ) |
25 |
1 2 3 4 7 8 9 10 11 12
|
dochsnkr |
|- ( ph -> ( L ` G ) = ( ._|_ ` { X } ) ) |
26 |
25
|
eleq2d |
|- ( ph -> ( X e. ( L ` G ) <-> X e. ( ._|_ ` { X } ) ) ) |
27 |
22 24 26
|
3bitr2rd |
|- ( ph -> ( X e. ( ._|_ ` { X } ) <-> ( G ` X ) = N ) ) |
28 |
27
|
necon3bbid |
|- ( ph -> ( -. X e. ( ._|_ ` { X } ) <-> ( G ` X ) =/= N ) ) |
29 |
19 28
|
mpbid |
|- ( ph -> ( G ` X ) =/= N ) |