| Step |
Hyp |
Ref |
Expression |
| 1 |
|
latlem.b |
|- B = ( Base ` K ) |
| 2 |
|
latlem.j |
|- .\/ = ( join ` K ) |
| 3 |
|
latlem.m |
|- ./\ = ( meet ` K ) |
| 4 |
|
simp1 |
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> K e. Lat ) |
| 5 |
|
simp2 |
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> X e. B ) |
| 6 |
|
simp3 |
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> Y e. B ) |
| 7 |
|
opelxpi |
|- ( ( X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) ) |
| 8 |
7
|
3adant1 |
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) ) |
| 9 |
1 2 3
|
islat |
|- ( K e. Lat <-> ( K e. Poset /\ ( dom .\/ = ( B X. B ) /\ dom ./\ = ( B X. B ) ) ) ) |
| 10 |
|
simprl |
|- ( ( K e. Poset /\ ( dom .\/ = ( B X. B ) /\ dom ./\ = ( B X. B ) ) ) -> dom .\/ = ( B X. B ) ) |
| 11 |
9 10
|
sylbi |
|- ( K e. Lat -> dom .\/ = ( B X. B ) ) |
| 12 |
11
|
3ad2ant1 |
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> dom .\/ = ( B X. B ) ) |
| 13 |
8 12
|
eleqtrrd |
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> <. X , Y >. e. dom .\/ ) |
| 14 |
1 2 4 5 6 13
|
joincl |
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X .\/ Y ) e. B ) |
| 15 |
|
simprr |
|- ( ( K e. Poset /\ ( dom .\/ = ( B X. B ) /\ dom ./\ = ( B X. B ) ) ) -> dom ./\ = ( B X. B ) ) |
| 16 |
9 15
|
sylbi |
|- ( K e. Lat -> dom ./\ = ( B X. B ) ) |
| 17 |
16
|
3ad2ant1 |
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> dom ./\ = ( B X. B ) ) |
| 18 |
8 17
|
eleqtrrd |
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> <. X , Y >. e. dom ./\ ) |
| 19 |
1 3 4 5 6 18
|
meetcl |
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) e. B ) |
| 20 |
14 19
|
jca |
|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X .\/ Y ) e. B /\ ( X ./\ Y ) e. B ) ) |