Metamath Proof Explorer

Theorem cvr2N

Description: Less-than and covers equivalence in a Hilbert lattice. ( chcv2 analog.) (Contributed by NM, 7-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cvr2.b 
|- B = ( Base ` K )
cvr2.s 
|- .< = ( lt ` K )
cvr2.j 
|- .\/ = ( join ` K )
cvr2.c 
|- C = (
cvr2.a 
|- A = ( Atoms ` K )
Assertion cvr2N 
|- ( ( K e. HL /\ X e. B /\ P e. A ) -> ( X .< ( X .\/ P ) <-> X C ( X .\/ P ) ) )

Proof

Step Hyp Ref Expression
1 cvr2.b 
 |-  B = ( Base ` K )
2 cvr2.s 
 |-  .< = ( lt ` K )
3 cvr2.j 
 |-  .\/ = ( join ` K )
4 cvr2.c 
 |-  C = (
5 cvr2.a 
 |-  A = ( Atoms ` K )
6 hllat 
 |-  ( K e. HL -> K e. Lat )
7 6 3ad2ant1 
 |-  ( ( K e. HL /\ X e. B /\ P e. A ) -> K e. Lat )
8 simp2 
 |-  ( ( K e. HL /\ X e. B /\ P e. A ) -> X e. B )
9 1 5 atbase 
 |-  ( P e. A -> P e. B )
10 9 3ad2ant3 
 |-  ( ( K e. HL /\ X e. B /\ P e. A ) -> P e. B )
11 eqid 
 |-  ( le ` K ) = ( le ` K )
12 1 11 2 3 latnle 
 |-  ( ( K e. Lat /\ X e. B /\ P e. B ) -> ( -. P ( le ` K ) X <-> X .< ( X .\/ P ) ) )
13 7 8 10 12 syl3anc 
 |-  ( ( K e. HL /\ X e. B /\ P e. A ) -> ( -. P ( le ` K ) X <-> X .< ( X .\/ P ) ) )
14 1 11 3 4 5 cvr1 
 |-  ( ( K e. HL /\ X e. B /\ P e. A ) -> ( -. P ( le ` K ) X <-> X C ( X .\/ P ) ) )
15 13 14 bitr3d 
 |-  ( ( K e. HL /\ X e. B /\ P e. A ) -> ( X .< ( X .\/ P ) <-> X C ( X .\/ P ) ) )