Theorem cvp

Description: The Hilbert lattice satisfies the covering property of Definition 7.4 of MaedaMaeda p. 31 and its converse. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	cvp	|- ( ( A e. CH /\ B e. HAtoms ) -> ( ( A i^i B ) = 0H <-> A

Proof

Step	Hyp	Ref	Expression
1		atelch	|- ( B e. HAtoms -> B e. CH )
2		chincl	|- ( ( A e. CH /\ B e. CH ) -> ( A i^i B ) e. CH )
3	1 2	sylan2	|- ( ( A e. CH /\ B e. HAtoms ) -> ( A i^i B ) e. CH )
4		atcveq0	|- ( ( ( A i^i B ) e. CH /\ B e. HAtoms ) -> ( ( A i^i B ) ( A i^i B ) = 0H ) )
5	3 4	sylancom	|- ( ( A e. CH /\ B e. HAtoms ) -> ( ( A i^i B ) ( A i^i B ) = 0H ) )
6		cvexch	|- ( ( A e. CH /\ B e. CH ) -> ( ( A i^i B ) A
7	1 6	sylan2	|- ( ( A e. CH /\ B e. HAtoms ) -> ( ( A i^i B ) A
8	5 7	bitr3d	|- ( ( A e. CH /\ B e. HAtoms ) -> ( ( A i^i B ) = 0H <-> A