Description: A Hilbert lattice has the exchange property. Part of Definition 7.8 of MaedaMaeda p. 32. (Contributed by NM, 15-Nov-2011) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | hlexch3.b | |- B = ( Base ` K ) |
|
| hlexch3.l | |- .<_ = ( le ` K ) |
||
| hlexch3.j | |- .\/ = ( join ` K ) |
||
| hlexch3.m | |- ./\ = ( meet ` K ) |
||
| hlexch3.z | |- .0. = ( 0. ` K ) |
||
| hlexch3.a | |- A = ( Atoms ` K ) |
||
| Assertion | hlexch4N | |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P ./\ X ) = .0. ) -> ( P .<_ ( X .\/ Q ) <-> ( X .\/ P ) = ( X .\/ Q ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlexch3.b | |- B = ( Base ` K ) |
|
| 2 | hlexch3.l | |- .<_ = ( le ` K ) |
|
| 3 | hlexch3.j | |- .\/ = ( join ` K ) |
|
| 4 | hlexch3.m | |- ./\ = ( meet ` K ) |
|
| 5 | hlexch3.z | |- .0. = ( 0. ` K ) |
|
| 6 | hlexch3.a | |- A = ( Atoms ` K ) |
|
| 7 | hlcvl | |- ( K e. HL -> K e. CvLat ) |
|
| 8 | 1 2 3 4 5 6 | cvlexch4N | |- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P ./\ X ) = .0. ) -> ( P .<_ ( X .\/ Q ) <-> ( X .\/ P ) = ( X .\/ Q ) ) ) |
| 9 | 7 8 | syl3an1 | |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P ./\ X ) = .0. ) -> ( P .<_ ( X .\/ Q ) <-> ( X .\/ P ) = ( X .\/ Q ) ) ) |