Step |
Hyp |
Ref |
Expression |
1 |
|
hlatjcom.j |
|- .\/ = ( join ` K ) |
2 |
|
hlatjcom.a |
|- A = ( Atoms ` K ) |
3 |
1 2
|
hlatjcom |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) ) |
4 |
3
|
3adant3r3 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .\/ Q ) = ( Q .\/ P ) ) |
5 |
4
|
oveq1d |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( Q .\/ P ) .\/ R ) ) |
6 |
1 2
|
hlatjass |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( P .\/ ( Q .\/ R ) ) ) |
7 |
|
simpl |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. HL ) |
8 |
|
simpr2 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> Q e. A ) |
9 |
|
simpr1 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. A ) |
10 |
|
simpr3 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. A ) |
11 |
1 2
|
hlatjass |
|- ( ( K e. HL /\ ( Q e. A /\ P e. A /\ R e. A ) ) -> ( ( Q .\/ P ) .\/ R ) = ( Q .\/ ( P .\/ R ) ) ) |
12 |
7 8 9 10 11
|
syl13anc |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( Q .\/ P ) .\/ R ) = ( Q .\/ ( P .\/ R ) ) ) |
13 |
5 6 12
|
3eqtr3d |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .\/ ( Q .\/ R ) ) = ( Q .\/ ( P .\/ R ) ) ) |