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 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ( 𝐴 ∩ 𝐵 ) = 0ℋ ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		atelch	⊢ ( 𝐵 ∈ HAtoms → 𝐵 ∈ Cℋ )
2		chincl	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∩ 𝐵 ) ∈ Cℋ )
3	1 2	sylan2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ∩ 𝐵 ) ∈ Cℋ )
4		atcveq0	⊢ ( ( ( 𝐴 ∩ 𝐵 ) ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) )
5	3 4	sylancom	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) )
6		cvexch	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )
7	1 6	sylan2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ( 𝐴 ∩ 𝐵 ) ⋖ℋ 𝐵 ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )
8	5 7	bitr3d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ( 𝐴 ∩ 𝐵 ) = 0ℋ ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )