Theorem lecmii

Description: Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of Beran p. 40. (Contributed by NM, 7-Aug-2004) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3		lecmi.1	$$⊢A\subseteq B$$
4	1 2	lecmi	$$⊢A\subseteq B\to A{𝐶}_{\mathscr{H}}B$$
5	3 4	ax-mp	$$⊢A{𝐶}_{\mathscr{H}}B$$