Step |
Hyp |
Ref |
Expression |
1 |
|
cvlsupr5.a |
|- A = ( Atoms ` K ) |
2 |
|
cvlsupr5.j |
|- .\/ = ( join ` K ) |
3 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
4 |
1 3 2
|
cvlsupr2 |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= Q ) -> ( ( P .\/ R ) = ( Q .\/ R ) <-> ( R =/= P /\ R =/= Q /\ R ( le ` K ) ( P .\/ Q ) ) ) ) |
5 |
|
simp1 |
|- ( ( R =/= P /\ R =/= Q /\ R ( le ` K ) ( P .\/ Q ) ) -> R =/= P ) |
6 |
4 5
|
syl6bi |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= Q ) -> ( ( P .\/ R ) = ( Q .\/ R ) -> R =/= P ) ) |
7 |
6
|
3exp |
|- ( K e. CvLat -> ( ( P e. A /\ Q e. A /\ R e. A ) -> ( P =/= Q -> ( ( P .\/ R ) = ( Q .\/ R ) -> R =/= P ) ) ) ) |
8 |
7
|
imp4a |
|- ( K e. CvLat -> ( ( P e. A /\ Q e. A /\ R e. A ) -> ( ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) -> R =/= P ) ) ) |
9 |
8
|
3imp |
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> R =/= P ) |