Step |
Hyp |
Ref |
Expression |
1 |
|
dochocsn.h |
|- H = ( LHyp ` K ) |
2 |
|
dochocsn.u |
|- U = ( ( DVecH ` K ) ` W ) |
3 |
|
dochocsn.o |
|- ._|_ = ( ( ocH ` K ) ` W ) |
4 |
|
dochocsn.v |
|- V = ( Base ` U ) |
5 |
|
dochocsn.n |
|- N = ( LSpan ` U ) |
6 |
|
dochocsn.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
7 |
|
dochocsn.x |
|- ( ph -> X e. V ) |
8 |
7
|
snssd |
|- ( ph -> { X } C_ V ) |
9 |
1 2 3 4 5 6 8
|
dochocsp |
|- ( ph -> ( ._|_ ` ( N ` { X } ) ) = ( ._|_ ` { X } ) ) |
10 |
9
|
fveq2d |
|- ( ph -> ( ._|_ ` ( ._|_ ` ( N ` { X } ) ) ) = ( ._|_ ` ( ._|_ ` { X } ) ) ) |
11 |
|
eqid |
|- ( ( DIsoH ` K ) ` W ) = ( ( DIsoH ` K ) ` W ) |
12 |
1 2 4 5 11
|
dihlsprn |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. V ) -> ( N ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) ) |
13 |
6 7 12
|
syl2anc |
|- ( ph -> ( N ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) ) |
14 |
1 11 3
|
dochoc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( N ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) ) -> ( ._|_ ` ( ._|_ ` ( N ` { X } ) ) ) = ( N ` { X } ) ) |
15 |
6 13 14
|
syl2anc |
|- ( ph -> ( ._|_ ` ( ._|_ ` ( N ` { X } ) ) ) = ( N ` { X } ) ) |
16 |
10 15
|
eqtr3d |
|- ( ph -> ( ._|_ ` ( ._|_ ` { X } ) ) = ( N ` { X } ) ) |