Theorem hlomcmat

Description: A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012)

	Ref	Expression
Assertion	hlomcmat	⊢ ( 𝐾 ∈ HL → ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) )

Proof

Step	Hyp	Ref	Expression
1		hloml	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OML )
2		hlclat	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CLat )
3		hlatl	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
4	1 2 3	3jca	⊢ ( 𝐾 ∈ HL → ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) )