Theorem cmcm2

Description: Commutation with orthocomplement. Theorem 2.3(i) of Beran p. 39. (Contributed by NM, 14-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	cmcm2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝐶ℋ 𝐵 ↔ 𝐴 𝐶ℋ ( ⊥ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cmcm3	⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → ( 𝐵 𝐶ℋ 𝐴 ↔ ( ⊥ ‘ 𝐵 ) 𝐶ℋ 𝐴 ) )
2	1	ancoms	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐵 𝐶ℋ 𝐴 ↔ ( ⊥ ‘ 𝐵 ) 𝐶ℋ 𝐴 ) )
3		cmcm	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝐶ℋ 𝐵 ↔ 𝐵 𝐶ℋ 𝐴 ) )
4		choccl	⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ 𝐵 ) ∈ Cℋ )
5		cmcm	⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ⊥ ‘ 𝐵 ) ∈ Cℋ ) → ( 𝐴 𝐶ℋ ( ⊥ ‘ 𝐵 ) ↔ ( ⊥ ‘ 𝐵 ) 𝐶ℋ 𝐴 ) )
6	4 5	sylan2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝐶ℋ ( ⊥ ‘ 𝐵 ) ↔ ( ⊥ ‘ 𝐵 ) 𝐶ℋ 𝐴 ) )
7	2 3 6	3bitr4d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝐶ℋ 𝐵 ↔ 𝐴 𝐶ℋ ( ⊥ ‘ 𝐵 ) ) )