Theorem chlej2i

Description: Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
	chlub.1	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chlej2i	$$⊢A\subseteq B\to C{\vee }_{\mathscr{H}}A\subseteq C{\vee }_{\mathscr{H}}B$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3		chlub.1	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}$$
4	1	chshii	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
5	2	chshii	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
6	3	chshii	$$⊢C\in {\mathbf{S}}_{\mathscr{H}}$$
7	4 5 6	shlej2i	$$⊢A\subseteq B\to C{\vee }_{\mathscr{H}}A\subseteq C{\vee }_{\mathscr{H}}B$$