Step |
Hyp |
Ref |
Expression |
1 |
|
atmod.b |
|- B = ( Base ` K ) |
2 |
|
atmod.l |
|- .<_ = ( le ` K ) |
3 |
|
atmod.j |
|- .\/ = ( join ` K ) |
4 |
|
atmod.m |
|- ./\ = ( meet ` K ) |
5 |
|
atmod.a |
|- A = ( Atoms ` K ) |
6 |
|
simpl |
|- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) ) -> K e. HL ) |
7 |
|
simpr2 |
|- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) ) -> X e. B ) |
8 |
|
simpr1 |
|- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) ) -> P e. A ) |
9 |
|
eqid |
|- ( pmap ` K ) = ( pmap ` K ) |
10 |
|
eqid |
|- ( +P ` K ) = ( +P ` K ) |
11 |
1 3 5 9 10
|
pmapjat2 |
|- ( ( K e. HL /\ X e. B /\ P e. A ) -> ( ( pmap ` K ) ` ( P .\/ X ) ) = ( ( ( pmap ` K ) ` P ) ( +P ` K ) ( ( pmap ` K ) ` X ) ) ) |
12 |
6 7 8 11
|
syl3anc |
|- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) ) -> ( ( pmap ` K ) ` ( P .\/ X ) ) = ( ( ( pmap ` K ) ` P ) ( +P ` K ) ( ( pmap ` K ) ` X ) ) ) |
13 |
1 5
|
atbase |
|- ( P e. A -> P e. B ) |
14 |
1 2 3 4 9 10
|
hlmod1i |
|- ( ( K e. HL /\ ( P e. B /\ X e. B /\ Y e. B ) ) -> ( ( P .<_ Y /\ ( ( pmap ` K ) ` ( P .\/ X ) ) = ( ( ( pmap ` K ) ` P ) ( +P ` K ) ( ( pmap ` K ) ` X ) ) ) -> ( ( P .\/ X ) ./\ Y ) = ( P .\/ ( X ./\ Y ) ) ) ) |
15 |
13 14
|
syl3anr1 |
|- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) ) -> ( ( P .<_ Y /\ ( ( pmap ` K ) ` ( P .\/ X ) ) = ( ( ( pmap ` K ) ` P ) ( +P ` K ) ( ( pmap ` K ) ` X ) ) ) -> ( ( P .\/ X ) ./\ Y ) = ( P .\/ ( X ./\ Y ) ) ) ) |
16 |
12 15
|
mpan2d |
|- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) ) -> ( P .<_ Y -> ( ( P .\/ X ) ./\ Y ) = ( P .\/ ( X ./\ Y ) ) ) ) |
17 |
16
|
3impia |
|- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ Y ) -> ( ( P .\/ X ) ./\ Y ) = ( P .\/ ( X ./\ Y ) ) ) |
18 |
17
|
eqcomd |
|- ( ( K e. HL /\ ( P e. A /\ X e. B /\ Y e. B ) /\ P .<_ Y ) -> ( P .\/ ( X ./\ Y ) ) = ( ( P .\/ X ) ./\ Y ) ) |