Theorem cmcm3i

Description: Commutation with orthocomplement. Remark in Kalmbach p. 23. (Contributed by NM, 4-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	⊢ 𝐴 ∈ Cℋ
	pjoml2.2	⊢ 𝐵 ∈ Cℋ
Assertion	cmcm3i	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝐶ℋ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	⊢ 𝐴 ∈ Cℋ
2		pjoml2.2	⊢ 𝐵 ∈ Cℋ
3	2 1	cmcm2i	⊢ ( 𝐵 𝐶ℋ 𝐴 ↔ 𝐵 𝐶ℋ ( ⊥ ‘ 𝐴 ) )
4	1 2	cmcmi	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ 𝐵 𝐶ℋ 𝐴 )
5	1	choccli	⊢ ( ⊥ ‘ 𝐴 ) ∈ Cℋ
6	5 2	cmcmi	⊢ ( ( ⊥ ‘ 𝐴 ) 𝐶ℋ 𝐵 ↔ 𝐵 𝐶ℋ ( ⊥ ‘ 𝐴 ) )
7	3 4 6	3bitr4i	⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝐶ℋ 𝐵 )