Theorem chlub

Description: Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 12-Jun-2004) (New usage is discouraged.)

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

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		shlub	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\left(A\subseteq C\wedge B\subseteq C\right)\leftrightarrow A{\vee }_{\mathscr{H}}B\subseteq C\right)$$
4	2 3	syl3an2	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\left(A\subseteq C\wedge B\subseteq C\right)\leftrightarrow A{\vee }_{\mathscr{H}}B\subseteq C\right)$$
5	1 4	syl3an1	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\left(A\subseteq C\wedge B\subseteq C\right)\leftrightarrow A{\vee }_{\mathscr{H}}B\subseteq C\right)$$