Theorem cmcm2ii

Description: Commutation with orthocomplement. Theorem 2.3(i) of Beran p. 39. (Contributed by NM, 7-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
	cmcmi.1	$$⊢A{𝐶}_{\mathscr{H}}B$$
Assertion	cmcm2ii	$$⊢A{𝐶}_{\mathscr{H}}\perp \left(B\right)$$

Proof

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