Step |
Hyp |
Ref |
Expression |
1 |
|
dochkrsm.h |
|- H = ( LHyp ` K ) |
2 |
|
dochkrsm.i |
|- I = ( ( DIsoH ` K ) ` W ) |
3 |
|
dochkrsm.o |
|- ._|_ = ( ( ocH ` K ) ` W ) |
4 |
|
dochkrsm.u |
|- U = ( ( DVecH ` K ) ` W ) |
5 |
|
dochkrsm.p |
|- .(+) = ( LSSum ` U ) |
6 |
|
dochkrsm.f |
|- F = ( LFnl ` U ) |
7 |
|
dochkrsm.l |
|- L = ( LKer ` U ) |
8 |
|
dochkrsm.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
9 |
|
dochkrsm.x |
|- ( ph -> X e. ran I ) |
10 |
|
dochkrsm.g |
|- ( ph -> G e. F ) |
11 |
|
eqid |
|- ( LSAtoms ` U ) = ( LSAtoms ` U ) |
12 |
8
|
adantr |
|- ( ( ph /\ ( ._|_ ` ( L ` G ) ) e. ( LSAtoms ` U ) ) -> ( K e. HL /\ W e. H ) ) |
13 |
9
|
adantr |
|- ( ( ph /\ ( ._|_ ` ( L ` G ) ) e. ( LSAtoms ` U ) ) -> X e. ran I ) |
14 |
|
simpr |
|- ( ( ph /\ ( ._|_ ` ( L ` G ) ) e. ( LSAtoms ` U ) ) -> ( ._|_ ` ( L ` G ) ) e. ( LSAtoms ` U ) ) |
15 |
1 2 4 5 11 12 13 14
|
dihsmatrn |
|- ( ( ph /\ ( ._|_ ` ( L ` G ) ) e. ( LSAtoms ` U ) ) -> ( X .(+) ( ._|_ ` ( L ` G ) ) ) e. ran I ) |
16 |
|
oveq2 |
|- ( ( ._|_ ` ( L ` G ) ) = { ( 0g ` U ) } -> ( X .(+) ( ._|_ ` ( L ` G ) ) ) = ( X .(+) { ( 0g ` U ) } ) ) |
17 |
1 4 8
|
dvhlmod |
|- ( ph -> U e. LMod ) |
18 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
19 |
1 4 2 18
|
dihrnlss |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> X e. ( LSubSp ` U ) ) |
20 |
8 9 19
|
syl2anc |
|- ( ph -> X e. ( LSubSp ` U ) ) |
21 |
18
|
lsssubg |
|- ( ( U e. LMod /\ X e. ( LSubSp ` U ) ) -> X e. ( SubGrp ` U ) ) |
22 |
17 20 21
|
syl2anc |
|- ( ph -> X e. ( SubGrp ` U ) ) |
23 |
|
eqid |
|- ( 0g ` U ) = ( 0g ` U ) |
24 |
23 5
|
lsm01 |
|- ( X e. ( SubGrp ` U ) -> ( X .(+) { ( 0g ` U ) } ) = X ) |
25 |
22 24
|
syl |
|- ( ph -> ( X .(+) { ( 0g ` U ) } ) = X ) |
26 |
16 25
|
sylan9eqr |
|- ( ( ph /\ ( ._|_ ` ( L ` G ) ) = { ( 0g ` U ) } ) -> ( X .(+) ( ._|_ ` ( L ` G ) ) ) = X ) |
27 |
9
|
adantr |
|- ( ( ph /\ ( ._|_ ` ( L ` G ) ) = { ( 0g ` U ) } ) -> X e. ran I ) |
28 |
26 27
|
eqeltrd |
|- ( ( ph /\ ( ._|_ ` ( L ` G ) ) = { ( 0g ` U ) } ) -> ( X .(+) ( ._|_ ` ( L ` G ) ) ) e. ran I ) |
29 |
1 3 4 23 11 6 7 8 10
|
dochsat0 |
|- ( ph -> ( ( ._|_ ` ( L ` G ) ) e. ( LSAtoms ` U ) \/ ( ._|_ ` ( L ` G ) ) = { ( 0g ` U ) } ) ) |
30 |
15 28 29
|
mpjaodan |
|- ( ph -> ( X .(+) ( ._|_ ` ( L ` G ) ) ) e. ran I ) |