Theorem atdmd2

Description: Two Hilbert lattice elements have the dual modular pair property if the second is an atom. (Contributed by NM, 6-Jul-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	atdmd2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → 𝐴 𝑀ℋ* 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		atdmd	⊢ ( ( 𝐵 ∈ HAtoms ∧ 𝐴 ∈ Cℋ ) → 𝐵 𝑀ℋ* 𝐴 )
2	1	ancoms	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → 𝐵 𝑀ℋ* 𝐴 )
3		atelch	⊢ ( 𝐵 ∈ HAtoms → 𝐵 ∈ Cℋ )
4		dmdsym	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ* 𝐵 ↔ 𝐵 𝑀ℋ* 𝐴 ) )
5	3 4	sylan2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 𝑀ℋ* 𝐵 ↔ 𝐵 𝑀ℋ* 𝐴 ) )
6	2 5	mpbird	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → 𝐴 𝑀ℋ* 𝐵 )