Step |
Hyp |
Ref |
Expression |
1 |
|
1psubcl.a |
|- A = ( Atoms ` K ) |
2 |
|
1psubcl.c |
|- C = ( PSubCl ` K ) |
3 |
|
ssidd |
|- ( K e. HL -> A C_ A ) |
4 |
|
eqid |
|- ( _|_P ` K ) = ( _|_P ` K ) |
5 |
1 4
|
pol1N |
|- ( K e. HL -> ( ( _|_P ` K ) ` A ) = (/) ) |
6 |
5
|
fveq2d |
|- ( K e. HL -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` A ) ) = ( ( _|_P ` K ) ` (/) ) ) |
7 |
1 4
|
pol0N |
|- ( K e. HL -> ( ( _|_P ` K ) ` (/) ) = A ) |
8 |
6 7
|
eqtrd |
|- ( K e. HL -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` A ) ) = A ) |
9 |
1 4 2
|
ispsubclN |
|- ( K e. HL -> ( A e. C <-> ( A C_ A /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` A ) ) = A ) ) ) |
10 |
3 8 9
|
mpbir2and |
|- ( K e. HL -> A e. C ) |