Theorem cmj1i

Description: A Hilbert lattice element commutes with its join. (Contributed by NM, 7-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	cmj1i	$$⊢A{𝐶}_{\mathscr{H}}\left(A{\vee }_{\mathscr{H}}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		pjoml2.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjoml2.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	1 2	chjcli	$$⊢A{\vee }_{\mathscr{H}}B\in {\mathbf{C}}_{\mathscr{H}}$$
4	1 2	chub1i	$$⊢A\subseteq A{\vee }_{\mathscr{H}}B$$
5	1 3 4	lecmii	$$⊢A{𝐶}_{\mathscr{H}}\left(A{\vee }_{\mathscr{H}}B\right)$$