Theorem cmcmi

Description: Commutation is symmetric. Theorem 2(v) of Kalmbach p. 22. (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	cmcmi	$$⊢A{𝐶}_{\mathscr{H}}B\leftrightarrow B{𝐶}_{\mathscr{H}}A$$

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	cmcmlem	$$⊢A{𝐶}_{\mathscr{H}}B\to B{𝐶}_{\mathscr{H}}A$$
4	2 1	cmcmlem	$$⊢B{𝐶}_{\mathscr{H}}A\to A{𝐶}_{\mathscr{H}}B$$
5	3 4	impbii	$$⊢A{𝐶}_{\mathscr{H}}B\leftrightarrow B{𝐶}_{\mathscr{H}}A$$