Step |
Hyp |
Ref |
Expression |
1 |
|
dochoccl.h |
|- H = ( LHyp ` K ) |
2 |
|
dochoccl.i |
|- I = ( ( DIsoH ` K ) ` W ) |
3 |
|
dochoccl.u |
|- U = ( ( DVecH ` K ) ` W ) |
4 |
|
dochoccl.v |
|- V = ( Base ` U ) |
5 |
|
dochoccl.o |
|- ._|_ = ( ( ocH ` K ) ` W ) |
6 |
|
dochoccl.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
7 |
|
dochoccl.g |
|- ( ph -> X C_ V ) |
8 |
1 2 5
|
dochoc |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ._|_ ` ( ._|_ ` X ) ) = X ) |
9 |
6 8
|
sylan |
|- ( ( ph /\ X e. ran I ) -> ( ._|_ ` ( ._|_ ` X ) ) = X ) |
10 |
|
simpr |
|- ( ( ph /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` ( ._|_ ` X ) ) = X ) |
11 |
1 3 4 5
|
dochssv |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) C_ V ) |
12 |
6 7 11
|
syl2anc |
|- ( ph -> ( ._|_ ` X ) C_ V ) |
13 |
1 2 3 4 5
|
dochcl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ._|_ ` X ) C_ V ) -> ( ._|_ ` ( ._|_ ` X ) ) e. ran I ) |
14 |
6 12 13
|
syl2anc |
|- ( ph -> ( ._|_ ` ( ._|_ ` X ) ) e. ran I ) |
15 |
14
|
adantr |
|- ( ( ph /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` ( ._|_ ` X ) ) e. ran I ) |
16 |
10 15
|
eqeltrrd |
|- ( ( ph /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> X e. ran I ) |
17 |
9 16
|
impbida |
|- ( ph -> ( X e. ran I <-> ( ._|_ ` ( ._|_ ` X ) ) = X ) ) |