Theorem ch1dle

Description: A 1-dimensional subspace is less than or equal to any member of CH containing its generating vector. (Contributed by NM, 30-May-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	ch1dle	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in A\right)\to \perp \left(\perp \left(\left\{B\right\}\right)\right)\subseteq A$$

Proof

Step	Hyp	Ref	Expression
1		chsh	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to A\in {\mathbf{S}}_{\mathscr{H}}$$
2		sh1dle	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in A\right)\to \perp \left(\perp \left(\left\{B\right\}\right)\right)\subseteq A$$
3	1 2	sylan	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in A\right)\to \perp \left(\perp \left(\left\{B\right\}\right)\right)\subseteq A$$