Theorem shlubi

Description: Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces. (Contributed by NM, 11-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shlub.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
	shlub.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
	shlub.3	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	shlubi	$$⊢\left(A\subseteq C\wedge B\subseteq C\right)\leftrightarrow A{\vee }_{\mathscr{H}}B\subseteq C$$

Proof

Step	Hyp	Ref	Expression
1		shlub.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
2		shlub.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
3		shlub.3	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}$$
4		shlub	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\left(A\subseteq C\wedge B\subseteq C\right)\leftrightarrow A{\vee }_{\mathscr{H}}B\subseteq C\right)$$
5	1 2 3 4	mp3an	$$⊢\left(A\subseteq C\wedge B\subseteq C\right)\leftrightarrow A{\vee }_{\mathscr{H}}B\subseteq C$$