Description: A Hilbert lattice has the covering property. Proposition 1(ii) in Kalmbach p. 140 (and its converse). ( chcv1 analog.) (Contributed by NM, 17-Nov-2011)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cvr1.b | |- B = ( Base ` K ) |
|
| cvr1.l | |- .<_ = ( le ` K ) |
||
| cvr1.j | |- .\/ = ( join ` K ) |
||
| cvr1.c | |- C = ( |
||
| cvr1.a | |- A = ( Atoms ` K ) |
||
| Assertion | cvr1 | |- ( ( K e. HL /\ X e. B /\ P e. A ) -> ( -. P .<_ X <-> X C ( X .\/ P ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cvr1.b | |- B = ( Base ` K ) |
|
| 2 | cvr1.l | |- .<_ = ( le ` K ) |
|
| 3 | cvr1.j | |- .\/ = ( join ` K ) |
|
| 4 | cvr1.c | |- C = ( |
|
| 5 | cvr1.a | |- A = ( Atoms ` K ) |
|
| 6 | hlomcmcv | |- ( K e. HL -> ( K e. OML /\ K e. CLat /\ K e. CvLat ) ) |
|
| 7 | 1 2 3 4 5 | cvlcvr1 | |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ X e. B /\ P e. A ) -> ( -. P .<_ X <-> X C ( X .\/ P ) ) ) |
| 8 | 6 7 | syl3an1 | |- ( ( K e. HL /\ X e. B /\ P e. A ) -> ( -. P .<_ X <-> X C ( X .\/ P ) ) ) |