| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dihglblem.b |
|- B = ( Base ` K ) |
| 2 |
|
dihglblem.l |
|- .<_ = ( le ` K ) |
| 3 |
|
dihglblem.m |
|- ./\ = ( meet ` K ) |
| 4 |
|
dihglblem.g |
|- G = ( glb ` K ) |
| 5 |
|
dihglblem.h |
|- H = ( LHyp ` K ) |
| 6 |
|
dihglblem.t |
|- T = { u e. B | E. v e. S u = ( v ./\ W ) } |
| 7 |
|
dihglblem.i |
|- J = ( ( DIsoB ` K ) ` W ) |
| 8 |
|
dihglblem.ih |
|- I = ( ( DIsoH ` K ) ` W ) |
| 9 |
1 2 3 4 5 6
|
dihglblem2N |
|- ( ( ( K e. HL /\ W e. H ) /\ S C_ B /\ ( G ` S ) .<_ W ) -> ( G ` S ) = ( G ` T ) ) |
| 10 |
9
|
3adant2r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( G ` S ) = ( G ` T ) ) |
| 11 |
10
|
fveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( I ` ( G ` S ) ) = ( I ` ( G ` T ) ) ) |
| 12 |
1 2 3 4 5 6 7 8
|
dihglblem3N |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( I ` ( G ` T ) ) = |^|_ x e. T ( I ` x ) ) |
| 13 |
11 12
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( I ` ( G ` S ) ) = |^|_ x e. T ( I ` x ) ) |