| Step |
Hyp |
Ref |
Expression |
| 1 |
|
polpmap.b |
|- B = ( Base ` K ) |
| 2 |
|
polpmap.o |
|- ._|_ = ( oc ` K ) |
| 3 |
|
polpmap.m |
|- M = ( pmap ` K ) |
| 4 |
|
polpmap.p |
|- P = ( _|_P ` K ) |
| 5 |
|
eqid |
|- ( Atoms ` K ) = ( Atoms ` K ) |
| 6 |
1 5 3
|
pmapssat |
|- ( ( K e. HL /\ X e. B ) -> ( M ` X ) C_ ( Atoms ` K ) ) |
| 7 |
|
eqid |
|- ( lub ` K ) = ( lub ` K ) |
| 8 |
7 2 5 3 4
|
polval2N |
|- ( ( K e. HL /\ ( M ` X ) C_ ( Atoms ` K ) ) -> ( P ` ( M ` X ) ) = ( M ` ( ._|_ ` ( ( lub ` K ) ` ( M ` X ) ) ) ) ) |
| 9 |
6 8
|
syldan |
|- ( ( K e. HL /\ X e. B ) -> ( P ` ( M ` X ) ) = ( M ` ( ._|_ ` ( ( lub ` K ) ` ( M ` X ) ) ) ) ) |
| 10 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
| 11 |
1 10 5 3
|
pmapval |
|- ( ( K e. HL /\ X e. B ) -> ( M ` X ) = { p e. ( Atoms ` K ) | p ( le ` K ) X } ) |
| 12 |
11
|
fveq2d |
|- ( ( K e. HL /\ X e. B ) -> ( ( lub ` K ) ` ( M ` X ) ) = ( ( lub ` K ) ` { p e. ( Atoms ` K ) | p ( le ` K ) X } ) ) |
| 13 |
|
hlomcmat |
|- ( K e. HL -> ( K e. OML /\ K e. CLat /\ K e. AtLat ) ) |
| 14 |
1 10 7 5
|
atlatmstc |
|- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B ) -> ( ( lub ` K ) ` { p e. ( Atoms ` K ) | p ( le ` K ) X } ) = X ) |
| 15 |
13 14
|
sylan |
|- ( ( K e. HL /\ X e. B ) -> ( ( lub ` K ) ` { p e. ( Atoms ` K ) | p ( le ` K ) X } ) = X ) |
| 16 |
12 15
|
eqtrd |
|- ( ( K e. HL /\ X e. B ) -> ( ( lub ` K ) ` ( M ` X ) ) = X ) |
| 17 |
16
|
fveq2d |
|- ( ( K e. HL /\ X e. B ) -> ( ._|_ ` ( ( lub ` K ) ` ( M ` X ) ) ) = ( ._|_ ` X ) ) |
| 18 |
17
|
fveq2d |
|- ( ( K e. HL /\ X e. B ) -> ( M ` ( ._|_ ` ( ( lub ` K ) ` ( M ` X ) ) ) ) = ( M ` ( ._|_ ` X ) ) ) |
| 19 |
9 18
|
eqtrd |
|- ( ( K e. HL /\ X e. B ) -> ( P ` ( M ` X ) ) = ( M ` ( ._|_ ` X ) ) ) |