Theorem dmdsym

Description: Dual M-symmetry of the Hilbert lattice. (Contributed by NM, 25-Jul-2007) (New usage is discouraged.)

	Ref	Expression
Assertion	dmdsym	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ* 𝐵 ↔ 𝐵 𝑀ℋ* 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		choccl	⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )
2		choccl	⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ 𝐵 ) ∈ Cℋ )
3		mdsym	⊢ ( ( ( ⊥ ‘ 𝐴 ) ∈ Cℋ ∧ ( ⊥ ‘ 𝐵 ) ∈ Cℋ ) → ( ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) ↔ ( ⊥ ‘ 𝐵 ) 𝑀ℋ ( ⊥ ‘ 𝐴 ) ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) ↔ ( ⊥ ‘ 𝐵 ) 𝑀ℋ ( ⊥ ‘ 𝐴 ) ) )
5		dmdmd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ* 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) ) )
6		dmdmd	⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → ( 𝐵 𝑀ℋ* 𝐴 ↔ ( ⊥ ‘ 𝐵 ) 𝑀ℋ ( ⊥ ‘ 𝐴 ) ) )
7	6	ancoms	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐵 𝑀ℋ* 𝐴 ↔ ( ⊥ ‘ 𝐵 ) 𝑀ℋ ( ⊥ ‘ 𝐴 ) ) )
8	4 5 7	3bitr4d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ* 𝐵 ↔ 𝐵 𝑀ℋ* 𝐴 ) )