Theorem chlej2

Description: Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chlej2	$$⊢\left(\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\wedge A\subseteq B\right)\to C{\vee }_{\mathscr{H}}A\subseteq C{\vee }_{\mathscr{H}}B$$

Proof

Step	Hyp	Ref	Expression
1		chsh	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to A\in {\mathbf{S}}_{\mathscr{H}}$$
2		chsh	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}\to B\in {\mathbf{S}}_{\mathscr{H}}$$
3		chsh	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}\to C\in {\mathbf{S}}_{\mathscr{H}}$$
4		shlej2	$$⊢\left(\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\wedge C\in {\mathbf{S}}_{\mathscr{H}}\right)\wedge A\subseteq B\right)\to C{\vee }_{\mathscr{H}}A\subseteq C{\vee }_{\mathscr{H}}B$$
5	1 2 3 4	syl3anl	$$⊢\left(\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\wedge A\subseteq B\right)\to C{\vee }_{\mathscr{H}}A\subseteq C{\vee }_{\mathscr{H}}B$$