Step |
Hyp |
Ref |
Expression |
1 |
|
dihelval2.l |
|- .<_ = ( le ` K ) |
2 |
|
dihelval2.a |
|- A = ( Atoms ` K ) |
3 |
|
dihelval2.h |
|- H = ( LHyp ` K ) |
4 |
|
dihelval2.p |
|- P = ( ( oc ` K ) ` W ) |
5 |
|
dihelval2.t |
|- T = ( ( LTrn ` K ) ` W ) |
6 |
|
dihelval2.e |
|- E = ( ( TEndo ` K ) ` W ) |
7 |
|
dihelval2.i |
|- I = ( ( DIsoH ` K ) ` W ) |
8 |
|
dihelval2.g |
|- G = ( iota_ g e. T ( g ` P ) = Q ) |
9 |
|
dihelval2.f |
|- F e. _V |
10 |
|
dihelval2.s |
|- S e. _V |
11 |
|
eqid |
|- ( ( DIsoC ` K ) ` W ) = ( ( DIsoC ` K ) ` W ) |
12 |
1 2 3 11 7
|
dihvalcqat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) = ( ( ( DIsoC ` K ) ` W ) ` Q ) ) |
13 |
12
|
eleq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> <. F , S >. e. ( ( ( DIsoC ` K ) ` W ) ` Q ) ) ) |
14 |
1 2 3 4 5 6 11 8 9 10
|
dicopelval2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( ( ( DIsoC ` K ) ` W ) ` Q ) <-> ( F = ( S ` G ) /\ S e. E ) ) ) |
15 |
13 14
|
bitrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> ( F = ( S ` G ) /\ S e. E ) ) ) |