Theorem lecmi

Description: Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of Beran p. 40. (Contributed by NM, 16-Jan-2005) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	lecmi	$$⊢A\subseteq B\to A{𝐶}_{\mathscr{H}}B$$

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3		ssinss1	$$⊢A\subseteq B\to A\cap \left(\perp \left(A\right){\vee }_{\mathscr{H}}B\right)\subseteq B$$
4	1 2	cmbr4i	$$⊢A{𝐶}_{\mathscr{H}}B\leftrightarrow A\cap \left(\perp \left(A\right){\vee }_{\mathscr{H}}B\right)\subseteq B$$
5	3 4	sylibr	$$⊢A\subseteq B\to A{𝐶}_{\mathscr{H}}B$$