Step |
Hyp |
Ref |
Expression |
1 |
|
ltrn2eq.l |
|- .<_ = ( le ` K ) |
2 |
|
ltrn2eq.a |
|- A = ( Atoms ` K ) |
3 |
|
ltrn2eq.h |
|- H = ( LHyp ` K ) |
4 |
|
ltrn2eq.t |
|- T = ( ( LTrn ` K ) ` W ) |
5 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
6 |
5 1 2 3 4
|
ltrnideq |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` P ) = P ) ) |
7 |
6
|
3adant3r3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` P ) = P ) ) |
8 |
5 1 2 3 4
|
ltrnideq |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` Q ) = Q ) ) |
9 |
8
|
3adant3r2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` Q ) = Q ) ) |
10 |
7 9
|
bitr3d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( ( F ` P ) = P <-> ( F ` Q ) = Q ) ) |