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	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	cmcm3i	$$⊢A{𝐶}_{\mathscr{H}}B\leftrightarrow \perp \left(A\right){𝐶}_{\mathscr{H}}B$$

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	2 1	cmcm2i	$$⊢B{𝐶}_{\mathscr{H}}A\leftrightarrow B{𝐶}_{\mathscr{H}}\perp \left(A\right)$$
4	1 2	cmcmi	$$⊢A{𝐶}_{\mathscr{H}}B\leftrightarrow B{𝐶}_{\mathscr{H}}A$$
5	1	choccli	$$⊢\perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}$$
6	5 2	cmcmi	$$⊢\perp \left(A\right){𝐶}_{\mathscr{H}}B\leftrightarrow B{𝐶}_{\mathscr{H}}\perp \left(A\right)$$
7	3 4 6	3bitr4i	$$⊢A{𝐶}_{\mathscr{H}}B\leftrightarrow \perp \left(A\right){𝐶}_{\mathscr{H}}B$$