Theorem shlej2i

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

	Ref	Expression
Hypotheses	shincl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
	shincl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
	shless.1	$$⊢C\in {\mathbf{S}}_{\mathscr{H}}$$
Assertion	shlej2i	$$⊢A\subseteq B\to C{\vee }_{\mathscr{H}}A\subseteq C{\vee }_{\mathscr{H}}B$$

Proof

Step	Hyp	Ref	Expression
1		shincl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
2		shincl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
3		shless.1	$$⊢C\in {\mathbf{S}}_{\mathscr{H}}$$
4	1 2 3	shlej1i	$$⊢A\subseteq B\to A{\vee }_{\mathscr{H}}C\subseteq B{\vee }_{\mathscr{H}}C$$
5	3 1	shjcomi	$$⊢C{\vee }_{\mathscr{H}}A=A{\vee }_{\mathscr{H}}C$$
6	3 2	shjcomi	$$⊢C{\vee }_{\mathscr{H}}B=B{\vee }_{\mathscr{H}}C$$
7	4 5 6	3sstr4g	$$⊢A\subseteq B\to C{\vee }_{\mathscr{H}}A\subseteq C{\vee }_{\mathscr{H}}B$$