Step |
Hyp |
Ref |
Expression |
1 |
|
2polss.a |
|- A = ( Atoms ` K ) |
2 |
|
2polss.p |
|- ._|_ = ( _|_P ` K ) |
3 |
|
hlclat |
|- ( K e. HL -> K e. CLat ) |
4 |
3
|
ad3antrrr |
|- ( ( ( ( K e. HL /\ X C_ A ) /\ p e. A ) /\ p e. X ) -> K e. CLat ) |
5 |
|
simpr |
|- ( ( ( ( K e. HL /\ X C_ A ) /\ p e. A ) /\ p e. X ) -> p e. X ) |
6 |
|
simpllr |
|- ( ( ( ( K e. HL /\ X C_ A ) /\ p e. A ) /\ p e. X ) -> X C_ A ) |
7 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
8 |
7 1
|
atssbase |
|- A C_ ( Base ` K ) |
9 |
6 8
|
sstrdi |
|- ( ( ( ( K e. HL /\ X C_ A ) /\ p e. A ) /\ p e. X ) -> X C_ ( Base ` K ) ) |
10 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
11 |
|
eqid |
|- ( lub ` K ) = ( lub ` K ) |
12 |
7 10 11
|
lubel |
|- ( ( K e. CLat /\ p e. X /\ X C_ ( Base ` K ) ) -> p ( le ` K ) ( ( lub ` K ) ` X ) ) |
13 |
4 5 9 12
|
syl3anc |
|- ( ( ( ( K e. HL /\ X C_ A ) /\ p e. A ) /\ p e. X ) -> p ( le ` K ) ( ( lub ` K ) ` X ) ) |
14 |
13
|
ex |
|- ( ( ( K e. HL /\ X C_ A ) /\ p e. A ) -> ( p e. X -> p ( le ` K ) ( ( lub ` K ) ` X ) ) ) |
15 |
14
|
ss2rabdv |
|- ( ( K e. HL /\ X C_ A ) -> { p e. A | p e. X } C_ { p e. A | p ( le ` K ) ( ( lub ` K ) ` X ) } ) |
16 |
|
sseqin2 |
|- ( X C_ A <-> ( A i^i X ) = X ) |
17 |
16
|
biimpi |
|- ( X C_ A -> ( A i^i X ) = X ) |
18 |
17
|
adantl |
|- ( ( K e. HL /\ X C_ A ) -> ( A i^i X ) = X ) |
19 |
|
dfin5 |
|- ( A i^i X ) = { p e. A | p e. X } |
20 |
18 19
|
eqtr3di |
|- ( ( K e. HL /\ X C_ A ) -> X = { p e. A | p e. X } ) |
21 |
|
eqid |
|- ( pmap ` K ) = ( pmap ` K ) |
22 |
11 1 21 2
|
2polvalN |
|- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` ( ._|_ ` X ) ) = ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) ) |
23 |
|
sstr |
|- ( ( X C_ A /\ A C_ ( Base ` K ) ) -> X C_ ( Base ` K ) ) |
24 |
8 23
|
mpan2 |
|- ( X C_ A -> X C_ ( Base ` K ) ) |
25 |
7 11
|
clatlubcl |
|- ( ( K e. CLat /\ X C_ ( Base ` K ) ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) ) |
26 |
3 24 25
|
syl2an |
|- ( ( K e. HL /\ X C_ A ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) ) |
27 |
7 10 1 21
|
pmapval |
|- ( ( K e. HL /\ ( ( lub ` K ) ` X ) e. ( Base ` K ) ) -> ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) = { p e. A | p ( le ` K ) ( ( lub ` K ) ` X ) } ) |
28 |
26 27
|
syldan |
|- ( ( K e. HL /\ X C_ A ) -> ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) = { p e. A | p ( le ` K ) ( ( lub ` K ) ` X ) } ) |
29 |
22 28
|
eqtrd |
|- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` ( ._|_ ` X ) ) = { p e. A | p ( le ` K ) ( ( lub ` K ) ` X ) } ) |
30 |
15 20 29
|
3sstr4d |
|- ( ( K e. HL /\ X C_ A ) -> X C_ ( ._|_ ` ( ._|_ ` X ) ) ) |