Theorem cmcm4i

Description: Commutation with orthocomplement. Remark in Kalmbach p. 23. (Contributed by NM, 7-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	cmcm4i	$$⊢A{𝐶}_{\mathscr{H}}B\leftrightarrow \perp \left(A\right){𝐶}_{\mathscr{H}}\perp \left(B\right)$$

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	1 2	cmcm2i	$$⊢A{𝐶}_{\mathscr{H}}B\leftrightarrow A{𝐶}_{\mathscr{H}}\perp \left(B\right)$$
4	2	choccli	$$⊢\perp \left(B\right)\in {\mathbf{C}}_{\mathscr{H}}$$
5	1 4	cmcm3i	$$⊢A{𝐶}_{\mathscr{H}}\perp \left(B\right)\leftrightarrow \perp \left(A\right){𝐶}_{\mathscr{H}}\perp \left(B\right)$$
6	3 5	bitri	$$⊢A{𝐶}_{\mathscr{H}}B\leftrightarrow \perp \left(A\right){𝐶}_{\mathscr{H}}\perp \left(B\right)$$