Theorem cmbri

Description: Binary relation expressing the commutes relation. Definition of commutes in Kalmbach p. 20. (Contributed by NM, 6-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	cmbri	$$⊢A{𝐶}_{\mathscr{H}}B\leftrightarrow A=\left(A\cap B\right){\vee }_{\mathscr{H}}\left(A\cap \perp \left(B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3		cmbr	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝐶}_{\mathscr{H}}B\leftrightarrow A=\left(A\cap B\right){\vee }_{\mathscr{H}}\left(A\cap \perp \left(B\right)\right)\right)$$
4	1 2 3	mp2an	$$⊢A{𝐶}_{\mathscr{H}}B\leftrightarrow A=\left(A\cap B\right){\vee }_{\mathscr{H}}\left(A\cap \perp \left(B\right)\right)$$