Theorem chlej12i

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	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}$$
	chlej12.4	$$⊢D\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chlej12i	$$⊢\left(A\subseteq B\wedge C\subseteq D\right)\to A{\vee }_{\mathscr{H}}C\subseteq B{\vee }_{\mathscr{H}}D$$

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		chlej12.4	$$⊢D\in {\mathbf{C}}_{\mathscr{H}}$$
5	1 2 3	chlej1i	$$⊢A\subseteq B\to A{\vee }_{\mathscr{H}}C\subseteq B{\vee }_{\mathscr{H}}C$$
6	3 4 2	chlej2i	$$⊢C\subseteq D\to B{\vee }_{\mathscr{H}}C\subseteq B{\vee }_{\mathscr{H}}D$$
7	5 6	sylan9ss	$$⊢\left(A\subseteq B\wedge C\subseteq D\right)\to A{\vee }_{\mathscr{H}}C\subseteq B{\vee }_{\mathscr{H}}D$$