Step |
Hyp |
Ref |
Expression |
1 |
|
polsubcl.a |
|- A = ( Atoms ` K ) |
2 |
|
polsubcl.p |
|- ._|_ = ( _|_P ` K ) |
3 |
|
polsubcl.c |
|- C = ( PSubCl ` K ) |
4 |
|
eqid |
|- ( lub ` K ) = ( lub ` K ) |
5 |
|
eqid |
|- ( oc ` K ) = ( oc ` K ) |
6 |
|
eqid |
|- ( pmap ` K ) = ( pmap ` K ) |
7 |
4 5 1 6 2
|
polval2N |
|- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) ) |
8 |
|
hlop |
|- ( K e. HL -> K e. OP ) |
9 |
8
|
adantr |
|- ( ( K e. HL /\ X C_ A ) -> K e. OP ) |
10 |
|
hlclat |
|- ( K e. HL -> K e. CLat ) |
11 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
12 |
11 1
|
atssbase |
|- A C_ ( Base ` K ) |
13 |
|
sstr |
|- ( ( X C_ A /\ A C_ ( Base ` K ) ) -> X C_ ( Base ` K ) ) |
14 |
12 13
|
mpan2 |
|- ( X C_ A -> X C_ ( Base ` K ) ) |
15 |
11 4
|
clatlubcl |
|- ( ( K e. CLat /\ X C_ ( Base ` K ) ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) ) |
16 |
10 14 15
|
syl2an |
|- ( ( K e. HL /\ X C_ A ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) ) |
17 |
11 5
|
opoccl |
|- ( ( K e. OP /\ ( ( lub ` K ) ` X ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) e. ( Base ` K ) ) |
18 |
9 16 17
|
syl2anc |
|- ( ( K e. HL /\ X C_ A ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) e. ( Base ` K ) ) |
19 |
11 6 3
|
pmapsubclN |
|- ( ( K e. HL /\ ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) e. ( Base ` K ) ) -> ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) e. C ) |
20 |
18 19
|
syldan |
|- ( ( K e. HL /\ X C_ A ) -> ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) e. C ) |
21 |
7 20
|
eqeltrd |
|- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) e. C ) |