| Step |
Hyp |
Ref |
Expression |
| 1 |
|
pmapmeet.b |
|- B = ( Base ` K ) |
| 2 |
|
pmapmeet.m |
|- ./\ = ( meet ` K ) |
| 3 |
|
pmapmeet.a |
|- A = ( Atoms ` K ) |
| 4 |
|
pmapmeet.p |
|- P = ( pmap ` K ) |
| 5 |
|
eqid |
|- ( glb ` K ) = ( glb ` K ) |
| 6 |
|
simp1 |
|- ( ( K e. HL /\ X e. B /\ Y e. B ) -> K e. HL ) |
| 7 |
|
simp2 |
|- ( ( K e. HL /\ X e. B /\ Y e. B ) -> X e. B ) |
| 8 |
|
simp3 |
|- ( ( K e. HL /\ X e. B /\ Y e. B ) -> Y e. B ) |
| 9 |
5 2 6 7 8
|
meetval |
|- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( ( glb ` K ) ` { X , Y } ) ) |
| 10 |
9
|
fveq2d |
|- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( P ` ( X ./\ Y ) ) = ( P ` ( ( glb ` K ) ` { X , Y } ) ) ) |
| 11 |
|
prssi |
|- ( ( X e. B /\ Y e. B ) -> { X , Y } C_ B ) |
| 12 |
11
|
3adant1 |
|- ( ( K e. HL /\ X e. B /\ Y e. B ) -> { X , Y } C_ B ) |
| 13 |
|
prnzg |
|- ( X e. B -> { X , Y } =/= (/) ) |
| 14 |
13
|
3ad2ant2 |
|- ( ( K e. HL /\ X e. B /\ Y e. B ) -> { X , Y } =/= (/) ) |
| 15 |
1 5 4
|
pmapglb |
|- ( ( K e. HL /\ { X , Y } C_ B /\ { X , Y } =/= (/) ) -> ( P ` ( ( glb ` K ) ` { X , Y } ) ) = |^|_ x e. { X , Y } ( P ` x ) ) |
| 16 |
6 12 14 15
|
syl3anc |
|- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( P ` ( ( glb ` K ) ` { X , Y } ) ) = |^|_ x e. { X , Y } ( P ` x ) ) |
| 17 |
|
fveq2 |
|- ( x = X -> ( P ` x ) = ( P ` X ) ) |
| 18 |
|
fveq2 |
|- ( x = Y -> ( P ` x ) = ( P ` Y ) ) |
| 19 |
17 18
|
iinxprg |
|- ( ( X e. B /\ Y e. B ) -> |^|_ x e. { X , Y } ( P ` x ) = ( ( P ` X ) i^i ( P ` Y ) ) ) |
| 20 |
19
|
3adant1 |
|- ( ( K e. HL /\ X e. B /\ Y e. B ) -> |^|_ x e. { X , Y } ( P ` x ) = ( ( P ` X ) i^i ( P ` Y ) ) ) |
| 21 |
10 16 20
|
3eqtrd |
|- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( P ` ( X ./\ Y ) ) = ( ( P ` X ) i^i ( P ` Y ) ) ) |